The studied object is the Kerr nonlinear optical systems and optical resonator tworing systems linear coupling with the presence of linear gain and nonlinear loss.
MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY - NGUYEN DUY CUONG STUDY OF SPONTANEOUS SYMMETRY BREAKING IN SOME NONLINEAR OPTICAL SYSTEMS Specialization: OPTICS Code: 9.44.01.10 ABSTRACT OF DOCTORAL THESIS IN PHYSICS NGHE AN - 2020 i The work is accomplished at Vinh University Supervisors: Prof Dr Dinh Xuan Khoa Prof Dr Marek Trippenbach Reviewer 1: Reviewer 2: Reviewer 3: The thesis was defended before the doctoral admission board of Vinh University at … h… , … , … , 2020 The thesis can be found at: - Information centre - Nguyen Thuc Hao library of Vinh University - Viet Nam National library ii PREFACE Reason to choice the investigation subject Spontaneous symmetry breaking is a common phenomenon in nature as well as in many different physical fields such as: in particle physics, magnetic materials or Bose - Einstein condensate system, etc In optics, the phenomenon of spontaneous symmetry breaking can be understood as a result of the interaction between nonlinear terms and waveguide structures When the nonlinear component is strong, it will cancel the linear coupling between the cores in the parallel waveguide leading to the system of asymmetric states In an optical ring resonance system, spontaneous symmetry breaking is a competition between linear effect and nonlinear effect, for example between linear gain and nonlinear loss, leading to asymmetric states, even in the case of chaos state Spontaneous symmetry breaking of optics has many applications in photonic technology In waveguide system, the effect of optical energy conversion between channels can be used as the basis for the design of all-optical switches, nonlinear amplifiers, stability in wavelength division circuits, logic gates and transmission optical bistability The coupling of two nonlinear fiber optics effectively compresses solitons by controlling the dispersion in the two fibers In the optical ring system, spontaneous symmetry breaking in the system also has many applications in photonic devices For example, the system of optical ring and waveguide bus, due to the interference in the optical circle that some wavelengths are retained, which is used in a wave-select circuit Some ring resonance systems due to symmetry breaking form a state of chaos This state has many applications in optical information such as synchronization, information security or random digital signal "0", "1" Especially, the extremely fast fluctuating dynamics of laser application completely solve the problem of artificial intelligence assumption Because it has many such important applications, spontaneous symmetry breaking has been interested by scientists around the world to study with many different types of optical systems both in theory and experiment In waveguide with the presence of constant Kerr nonlinear, spontaneous symmetry breaking has been studied with various types of linear potentials such as square quadratic doublepotential, H-shaped double-potential, and double-potential separated by delta functions, etc In the case of waveguide has the modulation of the Kerr nonlinearity, the spontaneous symmetry breaking is also considered with various types of modulated nonlinear functions such as delta function form, double-Gauss function, etc For each of the above waveguide systems, there will be different control parameter areas that exist different types of soliton states as well as branch characteristics of different spontaneous symmetry breaking In the optical ring system, in 2017, for the first time, Marek Treppenbach's group proposed a system of two optical resonant rings coupling linearly in the presence of linear gain and non-linear loss The first studied of group is the dynamics of the system with constant coupling, then extended it to single-Gaussian coupling The research results show that in the system, there is spontaneous symmetry breaking which leads to many interesting states promising many applications in technology such as: statinary state, oscillation state, vortex state, chaotic state Based on these studies, we find that we can extend the study of spontaneous symmetry breaking in the aforementioned optical systems The study of spontaneous symmetry breaking in systems in a complete and systematic manner is essential in the direction of empirical research and the wide range of applications in current technology Given the urgency of the research problem and the reasons mentioned above, we chose the research topic "Study of spontaneous symmetry breaking in some nonlinear optical systems" Purpose - Study the influence of pulse power, propagation constant to spontaneous symmetry breaking in two conserved optical system: the first system is waveguide with the presence of Kerr nonlinear and double-Gaussian linear potential, the second system is two waveguides linear coupling and delta function modulation of Kerr nonlinear - Study the influence of control parameters such as coupling strength, gain parameter, loss parameter, the width of coupling function to spontaneous symmetry breaking and the dynamic process of the optical resonator two-ring systems linear coupling with the presence of linear gain and nonlinear loss Object The studied object is the Kerr nonlinear optical systems and optical resonator tworing systems linear coupling with the presence of linear gain and nonlinear loss Methods: Numerical methods and analytic methods Chapter SOME BASIC CONCEPT IN THEORY OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONs In this Chapter 1, the author has presented the following contents: overview of some concepts in the theory of non-linear partial derivative equations, Schrodinger equation describing some phenomena in different optical systems; next effects of cubic nonlinearity are presented as the Kerr nonlinear effect, the phenomenon of absorbing two-photons; the calculation methods for the Schrodinger equation are studied in detail, including the soliton solution and their stability The method to find soliton solution we applied is the image-time-method The method used to find the final state with the technique evolving under the effect of small disturbances is the Split - Step Fourier method, the linearization of eigenvalues of the perturbation modes method, Vakhitov - Kolokolov method The contents listed above are basic knowledge, calculation methods to study symmetry breaking in some optical systems Next, the author presents some contents related to spontaneous symmetry breaking such as: the nature of symmetrical breaking, the branching characteristics of symmetrical breaking, chaos state, the scenario leads to of chaos states We present these in brief, because they are directly related to the findings of Chapter and Chapter 1.1 The nature of spontaneous symmetry breaking We can rotate it around the symmetry axis at any angle and still keep its shape Now press this string from top to bottom along its axis It is clear that the wire and force are still axial symmetry when the force is small When pressed with strong force, the piece of string is bent in a direction that we not know, but the object under consideration has lost axis symmetry That is SSB If the force strength is a parameter, then the system under consideration loses its original symmetry at some value of the parameter called the critical value Figure 1.6 Symmetry breaking phenomenon of straight steel 1.2 Bifurcation characteristics in conservative nonlinear system We consider a specific example of the simplified nonlinear Schrödinger equation describing the propagation of light pulses in the homogeneous nonlinear optical system with linear double-well potential due to the fact that the refractive index varies with space) as follows: 𝜕𝜓 𝑖 = − 𝜓𝑥𝑥 + 𝜎|𝜓|2 𝜓 + 𝑈(𝑥)𝜓 , (1.1) 𝜕𝑧 The input pulse power is calculated by the modulus of the slowly varying envelope function: +∞ +∞ 𝑁 = ∫−∞ |𝜓(𝑥, 𝑧)|2 𝑑𝑥 = ∫−∞ |𝑢(𝑥)|2 𝑑𝑥, (1.2) The asymmetry of soliton is characterized by the asymmetry denoted by 𝜈, ν= N+ −N− N +∞ = (∫0 |u(x)|2 dx−∫−∞|u(x)|2 dx) +∞ ∫−∞ |u(x)|2 dx (1.3) The emergence of asymmetric stable solitons provided that the pulse power exceeds the critical value, this type of symmetry breaking is called supercritical bifurcation 𝑁𝑏𝑖𝑓 Figure 1.7 The supercritical symmetry-breaking bifurcation in the 1-D model In the second case, the asymmetric stable soliton appears at the value of the input pulse power less than its critical value which is called the subcritical bifurcation 𝑁𝑏𝑖𝑓 Figure 1.8 The subcritical symmetry-breaking bifurcation in the double-channel model 1.3 Chaotic state Chaos state is often referred to as disorder, welter However, it is necessary to distinguish chaos from random For chaos, if we know the present (possibly the first state) then the future (possibly the last state) will determine and if there is a small disturbance in the present (the first state), the future (the final state) will not be determined (as it was) By contrast, if the future (the final state) will not be determined, which is random Chaos has a very important property that is sensitive to its initial condition The "butterfly effect" is an example of this property If we make a small change in the initial state of the nonlinear system, it can result in a large change in the later state Figure 1.9 The trajectory of Lorenz system for ρ = 28, σ = 10, β = 8/3 1.4 Scenarios to chaos Figure 1.10 depicts the three scenarios leading to chaos that is often observed in many dynamical systems when a parameter of the system changes Picture (a) shows doubling of frequencies leading to chaos; picture (b) shows the periodic period leading to chaos; picture (c) show a discontinuous (non-smooth) leading to chaos Figure 1.10 Diagram of three scenarios leading to chaos when a parameter changes In figure 1.10, the symbols "S" mean the steady state, "P1", "P2", respectively, the state of oscillating one frequency, two frequencies, , "C" is a mixed state disorder, "QP" is a state of near circulation, "IM" is a discontinuous state (not smooth) Chapter SPONTANEOUS SYMMETRY BREAKING IN SOME CONSERVED NONLINEAR OPTICAL SYSTEMS In this chapter, we study the influence of pulse power and propagation constant to SSB of two conserved nonlinear optical systems At the same time, we tested the stability of solitons which exist in that systems 2.1 Waveguide system with homogeneous nonlinear and double - potential 2.1.1 Model and equation We study the propagation of light in waveguide with homogeneous Kerr nonlinear optical media and double-well potential The nonlinear Schrödinger equation describes this system, it has the following form: 𝜕𝜓 𝑖 = − 𝜓𝑥𝑥 + 𝜎|𝜓|2 𝜓 + 𝑈(𝑥)𝜓, (2.1) 𝜕𝑧 where 𝜓 = 𝜓(𝑥, 𝑧) slow varying envelop function; 𝜓𝑥𝑥 is the second order partial derivative respect to x of 𝜓(𝑥, 𝑧) ; 𝜎 is nonlinear coefficient ( 𝜎 = −1 , 𝜎 = +1 correspond to self-focusing and self-defocusing); double-well potential has a form of a two-gaussian function: (𝑥+1)2 𝑈(𝑥) = − [𝑒𝑥𝑝 (− ) + 𝑒𝑥𝑝 (− 𝑎 √𝜋 𝑎 where a is the width of the potential well (𝑥−1)2 𝑎2 )], (2.2) Hình 2.1 Normalized double-well potential 𝑈(𝑥)⁄|𝑈(𝑥)|𝑚𝑎𝑥 in spatial coordinates x The figure describes double-gauss potential (it has the formula (2.2)) with different width 𝑎 When the width of potential increase, we see that the double-Gauss tăng lên thấy hàm Gauss kép dần tới hàm Gauss đơn bắt đầu giá trị độ rộng 𝑎 ≈ 1.35 Lưu ý phá vỡ đối xứng không xảy trường hợp kênh We consider solitons of system having form 𝜓(𝑥, 𝑧) = 𝑢(𝑥)𝑒 𝑖𝜇𝑧 where μ is the propagation constant, 𝑧 is propagation length and 𝑢(𝑥) is the function that satisfy the equation: −𝜇𝑢 + 𝑢𝑥𝑥 − 𝑈(𝑥)𝑢 − 𝜎𝑢3 = 0, (2.3) where 𝑢𝑥𝑥 is second order partial derivative respect to x of 𝜓(𝑥, 𝑧) and 𝑢 = 𝑢(𝑥) Pulse power of the system is an invariant quantity: +∞ +∞ 𝑁 = ∫−∞ |𝜓(𝑥, 𝑡)|2 𝑑𝑥 = ∫−∞ |𝑢(𝑥)|2 𝑑𝑥 (2.4) The quantity characteristic for the asymmetry of soliton is defined as the asymmetric ratio: Θ= N+ −N− N +∞ = (∫0 |𝑢(𝑥)|2 𝑑𝑥−∫−∞|𝑢(𝑥)|2 𝑑𝑥) +∞ ∫−∞ |𝑢(𝑥)|2 𝑑𝑥 , (2.5) 2.1.2 The system with self-focusing nonlinearity and double-potential We consider self-focusing nonlinearity case which 𝜎 = −1, the equation (2.1) become to: 𝜕𝜓 𝑖 = − 𝜓𝑥𝑥 − |𝜓|2 𝜓 + 𝑈(𝑥)𝜓 (2.6) 𝜕𝑧 In figure 2.4, the blue lines (solid line) correspond to states of stable solitons, red line (dashed line) are unstable solitons (stability of them will be tested by us next section) Here, we find that the critical value Nbif = 0.925 (or 𝜇𝑏𝑖𝑓 = 0.646) exist, when N > Nbif (or 𝜇 > 𝜇𝑏𝑖𝑓 ) the soliton of the system becomes asymmetric A case of symmetric soliton is represented by point A, the asymmetric solitons is represented by points C and D in Figure 2.4b Note that at the value of N greater than Nbif, there also exist symmetric solitons which is represented by red dashed curves in figure 2.4 One of them is represented by point B, their symmetrical shape is the same as at point A The difference here is that the state of soliton is not stable when it propagates with small perturbations (a) C A (b) B D 𝝁𝒃𝒊𝒇 𝑵𝒃𝒊𝒇 Figure 2.4 Asymmetry ratio as a function of the propagation constant 𝜇 (figure a), and the pulse power N (figure b) Next, we test the stability of solitons by three different methods: the solitons propagated in real spatial with small perturbations by SSF method, linearization of eigenvalues of the perturbation modes method, V-K stability criterion The results are the same, that is what confirms the methods are correct We only show the solitons propagated in real spatial with small perturbations by the SSF method We test for the states show by A, B, C and D points (B) (A) (C, D) (b) (A) z (a) (C, D) (B) (d) (c) z z Figure 2.5 (a) illustrated pulse power 𝑁 respect of propagation constant 𝜇; (b) is propagation of symmetric solitons in real spatial for 𝑁 = 0.5, 𝑎 = 0.5; (c), (d) are propagation of symmetric solitons and asymmetric solitons in real Figure 2.10 Asymmetry ratio Θ as a function respect of pulse power 𝑁 in selfdefocusing nonlinearity case for the width of double-potential 𝑎 = 1.0 The calculation results are simulated on the figures: figure 2.11 illustrates the solitons for 𝑎 = 1/3 and 𝑎 = 1.0, figure 2.12 describes the evolution of soliton in real space The results show the states of the system are highly stable We also calculated with varies value of the widths of Gaussian potential wells and the results showed no symmetry breaking in the self-defocusing nonlinear system, the states are highly stable (a) (b) Figure 2.11 Soliton states in double-well potential correspond to different widths, figure (a) corresponds to a =1/3 and figure (b) corresponds to a =1.0, both two cases the pulse power 𝑁=2 (b) (a) Figure 2.12 (a) propagation in space of soliton corresponds to a =1/3, pulse 10 power N = 2, (b) propagation in space of soliton corresponds to a =1.0, pulse power 𝑁=2 2.2 Two waveguide systems with nonlinear double-well modulation and linear coupling 2.2.1 One dimension equations describe the research system The system is illustrated by a system of nonlinear Schrödinger equations as follows: 𝑖 𝜕𝜙 𝜕𝑧 𝜕𝜓 =− 𝜕2 𝜙 𝜕𝑥 𝜕2 𝜓 + 𝑔(𝑥)|𝜙|2 𝜙 − 𝑘𝜓 , (2.8) { 𝑖 =− + 𝑔(𝑥)|𝜓| 𝜓 − 𝑘𝜙 𝜕𝑧 𝜕𝑥 where 𝜙 and 𝜓 are slow varying envelop function of pulse light in the two waveguides, 𝑥 is horizontal coordinates, 𝑔(𝑥) is the local nonlinear coefficient and 𝑘 coupling strengths, z propagation distance Total pulse power of system has form as follows: +∞ 𝑁 ≡ ∫−∞ [|𝜙(𝑥)|2 + |𝜓(𝑥)|2 ]𝑑𝑥, (2.9) and Hamiltonian of the system: +∞ 𝐻 ≡ ∫−∞ [|𝜙𝑥 |2 + |𝜓𝑥 |2 + 𝑔(𝑥)(|𝜙|4 + |𝜓|4 ) − 2𝑘(𝜙𝜓 ∗ + 𝜙 ∗ 𝜓)]𝑑𝑥 (2.10) here the “*“ is the symbol for the complex conjugate complex We consider space modulation with a delta function has the form: 𝑔(𝑥) = −𝛿(𝑥) (2.11) 2.2.2 Soliton states, bifurcation diagram and stability By analytical methods, we found different solitons Thank to we test the influence of control parameters to SSB of system (a) (b) (c) Figure 2.13 Soliton states: (a) is the symmetric state, (b) is the antisymmetric state and (c) asymmetric state of the system for coupling constant 𝜅 = and propagation constant 𝜇 = The figure 2.13a show that red line and dashed blue line coincide that illustrate symmetric state, fig 2.13b are two curves symmetrically across the horizontal axis so-called atisymmetric states and figure 2.13c are them uncoincide that illustrate the asymmetric state We obtained the total pulse power of system has form as (2.9) of symmetric and antisymmetric states: 𝑁𝑠𝑦𝑚𝑚 = 𝑁𝑎𝑛𝑡𝑖𝑠𝑦𝑚𝑚 = Total pulse power of asymmetric state: 11 𝑁𝑎𝑠𝑦𝑚𝑚 = 3𝜇−√𝜇2 −𝜅2 (2.12) 2√𝜇2 −𝜅2 here limit value 𝑁𝑎𝑠𝑦𝑚𝑚 (𝜇 → ∞) = Total pulse power of symmetric solitons state and asymmetric solitons state dependent on 𝜇, illustrate by figure 2.14a as follows Hamiltonian is determined in equation (2.10) It can also be calculated for all states and has a general expression: 𝐸 = √2𝜇+ |𝐴|2 + √2𝜇− |𝐶 |2 − |𝐴|4 − 𝐴2 (𝐶 ∗ )2 − 4|𝐴|2 |𝐶 |2 − (𝐴∗ )2 𝐶 − 𝜅|𝐴|2 √𝜇+ + 𝜅|𝐶|2 √𝜇− (2.13) Symmetric, antisymmetric and asymmetric correspond to the energy 𝐸𝑠𝑦𝑚𝑚 = −𝑘, 𝐸𝑎𝑛𝑡𝑖𝑠𝑦𝑚𝑚 = 𝑘, and 2(𝜇+𝜅) 2(𝜇−𝜅) 𝐸𝑎𝑠𝑦𝑚 = − 𝜅 [√ +√ 2(𝜇−𝜅) 2(𝜇+𝜅) ] (2.14) Bifurcation point 𝜇𝑏𝑖𝑓 = 𝜅 subsituting into the formula (2.14) lead to 𝐸𝑠𝑦𝑚𝑚 = −𝑘 Figure 2.14 (a) show pulse power and (b) show the energy of symmetric, antisymmetric and asymmetric dependent to propagation constant 𝜇 The asymmetric ratio between two waveguides is defined: +∞ Θ = ∫−∞ [|𝑢2 (𝑥)|−|𝑣 (𝑥)|]𝑑𝑥 +∞ ∫−∞ [|𝑢2 (𝑥)|+|𝑣 (𝑥)|]𝑑𝑥 (2.15) Adding the values 𝑢(𝑥), 𝑣(𝑥) above obtained into (2.15) and integrate we lead to: 3 3 2 2 (2.16) It plotted as a function of total power and propagation constant 𝜇 as in figure 2.15 Note that when the propagation constant 𝜇 → +∞ All asymmetry states are unstable in the model with nonlinear modulation as a delta function form (2.11) 12 Figure 2.15 Asymmetry parameter , is defined by equation (2.15), as a function of the total norm (a) and the chemical potential (b) Solid and dashed lines correspond to stable and unstable states To define the stability of soliton states, we used to V-K stability criterion From figure 2.14a, we clear that slope of the total power curve aspect to propagation constant 𝜇 always negative So the asymmetry soliton states allways unstable In the above figure, solid and dashed line correspond to stable and unstable From figure 2.15 we lead to below: firstly the critical power and critical propagation constant 𝑁𝑏𝑖𝑓 = và𝜇𝑏𝑖𝑓 = 1.25; secondly symmetric solitons and antisymmetric solitons of system exist when the power only by only 𝑁 = 2, the system exist asymmetric soliton when the pulse power < 𝑁 < (or propagation constant 𝜇 > 1.25 ), thirdly asymmetric solitons are unstable Based on the types of bifurcation we find that the bifurcation of the system is subcritical Chapter SPONTANEOUS SYMMETRY BREAKING IN TWO COUPLED RING RESONATOR SIZE ABOUT MICRO-METER 3.1 Model and descriptive equations Two coupled nonlinear Schrodinger equations describe the optical ring resonance system: 𝑖𝜕𝑡 𝜓1 = −𝜕𝑥2 𝜓1 + 𝑖𝛾𝜓1 + (1 − 𝑖𝛤)|𝜓1 |2 𝜓1 + 𝐽(𝑥)𝜓2 (3.1) { 𝑖𝜕𝑡 𝜓2 = −𝜕𝑥2 𝜓2 + 𝑖𝛾𝜓2 + (1 − 𝑖𝛤)|𝜓2 |2 𝜓2 + 𝐽(𝑥)𝜓1 where equations are written in scaled dimensionless units Here 𝜓1 , 𝜓2 are the wavefunction in the first and the second rings Depending on the position between the two rings, the couple function 𝐽(𝑥) can be constant (may also be called constant coupling) is studied by Nguyen Viet Hung and et al, single - Gauss coupling is studied by Aleksandr Ramaniuk and et al, or the couple function is double – Gauss (may also be called double – Gauss coupling) will be studied by us and have publicised The single - Gauss coupling and double – Gauss coupling called by local coupling These coupling functions are described by as following formulas: with constant coupling: 𝐽(𝑥) = 𝑐 = ℎằ𝑛𝑔 𝑠ố, (3.2) with single - Gauss coupling: 𝐽(𝑥) = 𝐽0 √ 𝑥2 𝑒𝑥𝑝 (− 2), 𝜋𝑎 𝑎 13 (3.3) with double - Gauss coupling: 𝐽(𝑥) = 𝐽0 √𝜋𝑎 {𝑒𝑥𝑝 (− 𝜋 2 𝑎2 (𝑥− ) ) + 𝑒𝑥𝑝 (− 𝜋 2 𝑎2 (𝑥+ ) )} (3.4) where 𝐽0 is the coupling strength between two rings, 𝑎 is the width of the coupling function The width is considered to be narrow if 𝑎 ≪ 𝜋, or broad coupling if 𝑎 ≫ 𝜋 Figure 3.1 Schematic view of currents in double rings with linear gain 𝛾 and nonlinear loss Γ In the figure 3.1, 𝛾 and Γ are gain and loss parameter, respectively; 𝑗1 , 𝑗2 , 𝑗⊥ are the density currents in the first ring, the second ring and current density between the two rings The purpose of this chapter is to study SSB in above two rings system To research SSB of this system, we need to use some physical quantity as following defined: The light power in each ring: 2𝜋 𝑁𝑖 (𝑡) = ∫0 |𝜓𝑖 (𝑥, 𝑡)|2 𝑑𝑥, (3.5) with 𝑖 = 1, are indices corresponding to power the first ring and the second ring Total power of two rings is: 2𝜋 𝑁(𝑡) = ∫0 [|𝜓1 (𝑥, 𝑡)|2 + |𝜓2 (𝑥, 𝑡)|2 ]𝑑𝑥 (3.6) Fourier transform of total power: ̃ (𝜔) = ℱ(𝑁(𝑡)) 𝑁 (3.7) with ℱ is denote the Fourier transforms, 𝜔 is the frequency in Fourier domain Density currents in each ring: 𝜕𝜓 𝜕𝜓∗ 𝑗𝛼 (𝑥, 𝑡) = (𝜓𝛼∗ 𝛼 + 𝜓𝛼 𝛼 ), with 𝛼 = 1,2 (3.8) 2𝑖 𝜕𝑥 𝜕𝑥 The current density between the two rings: 𝐶 𝑗⊥ (𝑥, 𝑡) = (𝜓1∗ 𝜓2 + 𝜓2∗ 𝜓1 ), (3.9) 2𝑖 and the total current between the two rings: 2𝜋 𝐽⊥ (𝑥, 𝑡) = ∫0 𝑗⊥ (𝑥, 𝑡)𝑑𝑥 (3.10) Topological charge is defined as: 2𝜋 𝜕 𝜅 = ∫0 𝑎𝑟𝑔(𝜓𝑖 )𝑑𝑥 (3.11) 2𝜋 𝜕𝑥 if 𝜅 = then correspond to not vortex and if 𝜅 is an integer then correspond to the vortex 14 Asymmetry ratio in each ring: 𝜋 Θ𝑖 = ∫0 |𝜓𝑖 |2 𝑑𝑥− ∫−𝜋|𝜓𝑖 |2 𝑑𝑥 +𝜋 ∫−𝜋 |𝜓𝑖 |2 𝑑𝑥 (3.12) with 𝑖 = 1, are indices corresponding to first ring and second ring If Θ𝑖 = then the state of the system is evenly symmetric 𝑥 → −𝑥, if Θ𝑖 ≠ then the state of the system is asymmetric or called SSB appear We used two numerical methods to study the dynamics of the system, namely Split-Step-Fourier (SSF) method In the case of no coupling (𝐽0 = 0), the stationary solution can be found analytically as: 𝛾 𝑖𝜅𝑥−𝑖( +𝜅2 )𝑡 Γ 𝜓1,2 (𝑡) = 𝜌1,2 𝑒 , (3.13) where, 𝜌1,2 corresponding to is the modul of the functions in first ring and second ring, 𝜅 is topological charge When the conditions is not vortex (𝜅 = 0) and added into (3.13) then this formula becomes to: 𝛾 𝜓1,2 (𝑥, 𝑡 = 0) = √ (1 ± 𝛽sin(k𝑥)), Γ (3.14) where 𝛽 = 0.01 is a perturbation, k is integer number When two ring couple together, modulation instability lead to appear different states: stationary state, oscillation state and chaos state and phenomena types as SSB, vortex Before moving on to our main research section on the SSB of the system in the case of double Gauss coupling, we will detail the types of states and phenomena that occur in the system 3.2 Several types of states and phenomena appear in the ring resonance system 3.2.1 Stationary state and SSB The stationary state is the state in which the wave function module describes a state that does not change over time In Figure 3.3, Figure (a) shows the total power that does not change over time of the final state Figure (b) is the Fourier transform of the total power, we see only the frequency ω = with the other Fourier transform, it is due to the nature of the stationary state, or other than the steady state, the variable Fourier transformation of total power only obtains a nonzero value at ω = 0, while at other frequencies the Fourier transform is equal to zero Figure (c) is the evolution of a wave function over time, and figure (d) is the module of the two wave functions (a) (b) 15 (c) (d) Figure 3.3 Stationary state for single-Gauss with parameters: 𝛾 = 3, Γ = 1, 𝐽0 = 2, 𝑎 = 3.2.2 Oscillation states The oscillation state is the state in which the wave function module changes periodically Figure 3.9a shows the total power of the light in two rounds changing over time The evolution of a wave function is illustrates in Figure 3.9c To know the frequency number of the oscillation state, we perform a Fourier transform of the total power described in Figure 3.9b Thereby, we find that there are frequencies in the Fourier space that have Fourier transforms of total power are nonzero This is the case of the oscillating state with frequencies Figure 3.9d describe the module of two wave functions at the same time, showing the even x → -x asymmetry of the wave functions (a) (b) (d) (c) Figure 3.9 Oscillation states in the system for constant coupling Fig (a) illustrated total power of two rings, (b) show Fourier transform of total power, (c) describes the evolution of the wavefunction aspect to time and (d) is the module of wavefunctions For parameters Γ = 1, 𝛾 = và𝑐 = 1.25 16 3.2.3 Trạng thái hỗn loạn Chaos state is understood as the distribution of light intensity in uneven, messy and disordered (a) (b) (c) (d) Figure 3.11 Chaos states in the system for constant coupling (here small frames of (a) is the result of the previous work), for parameter Γ = 1, 𝛾 = và𝑐 = 3.3 SSB of the system for double - Gauss coupling 3.3.1 Influence of coupling strength to SSB of system In this section we study the effect of coupling strength on the symmetry breaking of the system with parameter sets: fixed set of amplification parameters γ = 3, loss parameters Γ = and change coupling strength 𝐽0 The different widths of coupling function have been tested and shown the same state transitions when the width of the coupling function is not too large, they have a significant difference when the width hundreds of times bigger than each other Therefore, we will present the symmetry breaking of the system in two cases: narrow coupling width (a = 0.01) and large coupling width (a = 1.0) are two representative cases The results obtained for the region of the parameters that exist for different types of states and with symmetry breaking are summarized in the following figures and diagrams 3.3.1.1 SSB for narrow coupling 17 Diagram 3.1 State changing and SSB due to influence by coupling strength when the width of coupling function 𝑎 = 0.01 Notes: O-SSB unbroken SSB broken S: stationary state O: Oscillation states Chaos: Chaos states Figure 3.16 State changing of the system for 𝜸 = 𝟑, 𝚪 = 𝟏, 𝒂 = 𝟎 𝟎𝟏 depending to coupling strength 𝑱𝟎 ∈ [𝟏 𝟗𝟕, 𝟑 𝟓𝟕] 3.3.1.2 SSB for broad coupling Diagram 3.2 State changing and SSB due to the influence by coupling strength when the width of coupling function 𝑎 = 18 Figure 3.18 State changing of the system for high coupling strength when the width of coupling function 𝑎 = 3.3.2 Influence of gain parameter to SSB of system This title, we consider the influence of gain 𝛾 to SSB of the system in following cases: firs case nonlinear loss Γ = , coupling strength 𝐽0 = 2.85 , the width of coupling function 𝑎 = 0.01 and change gain 𝛾; second case Γ = 1, 𝐽0 = 12.75, 𝑎 = and change 𝛾 3.3.2.1 SSB for narrow coupling Figure 3.19 State changing of the system for, when we fixed Γ = 1, 𝐽0 = 2.85, 𝑎 = 0.01 and the 𝛾 change 19 Diagram 3.3 State changing and SSB due to influence by gain when the width of coupling function 𝑎 = 0.01 3.3.2.2 SSB for broad coupling Figure 3.23 State changing of the system for Γ = 1, 𝐽0 = 12,75, 𝑎 = Diagram 3.4 State changing and SSB due to influence by gain when the width of coupling function 𝑎 = 3.3.3 Influence of loss parameter to SSB of system The same above sections, this title we consider two cases for two types of the width: first case we fixed 𝛾 = 3, 𝐽0 = 2.85, 𝑎 = 0.01 and change loss Γ; second case we consider 𝛾 = 3, 𝐽0 = 12.75, 𝑎 = and change Γ First case Fixed 𝛾 = 3, 𝐽0 = 2.85, 𝑎 = 0.01, change Γ The results are summarized as in figure 3.25 Through the diagram, we see the stationary state of the system exists in the loss parameter ranges Γ ≲ 0.99 and Γ ≳ 20 1.60 corresponding to blue areas on the diagram (denoted by symbol S) In loss parameter ranges 0.99 ≲ Γ ≲ 1.4 then the system exists chaos state corresponds to the area with a wide vertical line perpendicular to the horizontal axis (the area denoted by Chaos) Oscillation states occur in an area of the loss parameter 1.4 ≲ Γ ≲ 1.6 Figure 3.25 State changing of system for 𝛾 = 3, 𝐽0 = 2.85, 𝑎 = 0.01, loss parameter Γ change Diagram 3.5 State changing and SSB due to influence by gain when the width of coupling function 𝑎 = 0.01 Second case Fixed 𝛾 = 3, 𝐽0 = 12.75, 𝑎 = 1, change Γ This case we find that state transitions also change between stationary states, oscillations, and chaos But the not full chaotic state represents the light trails of the chaotic zone discontinuously from frequency 𝜔 = to any frequency Variations between states are not as stable as the case of width 𝑎 = studied above That unstable state change is depicted in the figure 3.26, light trails appear intermittently on a blue background Figure 3.26 State changing of the system for 𝛾 = 3, 𝐽0 = 12.75, 𝑎 = 1, loss parameter Γ change 21 Diagram 3.6 State changing and SSB due to influence by gain when the width of coupling function 𝑎 = SUMMARY In this research, we investigated SSB in some optical systems The obtained results show that For the waveguide system with homogeneous Kerr nonlinear and doubleGaussian linear potential - In the case of self-focusing Kerr nonlinear, the system has spontaneous symmetry breaking We have defined the region parameters of such as pulse power, propagation constants of the system to exist the types of symmetric, asymmetric solitons as well as the stable and unstable regions of those states - The bifurcation characteristic of symmetry breaking in this case is the supercritical - In the case of self-defocusing Kerr nonlinear systems, the system does not have spontaneous symmetry breaking, the symmetric states of the system are always of high stable For the system with two waveguides including the delta function modulation of Kerr nonlinear and the linear coupling - We have identified regions of parameters such as pulse power, propagation constant that exist for different types - The bifurcation characteristic of the symmetry breaking of this case is subcritical, the asymmetric solitons states are unstable, the symmetric solitons states are always stable For the optical resonator two-ring system linear coupling with the presence of linear gain and nonlinear loss, we have considered the influence of three different control parameters such as coupling strength, gain and loss parameters to spontaneous symmetry breaking and receiver: - The parameter regions of coupling strength, gain parameter, loss parameter to exist different types of states such as stationary state, oscillation state and chaos state - There are two different scenarios leading to chaos state are: the scenario from the stop state suddenly turns into a state of chaos and the scenario from the stop state to the state of discontinuous oscillation that leads to chaos The obtained results are basically vital for experimental research It is also applied in photonic devices such as optical switches, extremely fast switching systems, optical information, and particularly chaos applied in optical information security, code techniques In order to understand more deeply about these phenomena, we can study more detail in the non-linear Schrödinger equation with many other specific physical conditions such as expanding more multi-dimensional model, more complex potential, increasing nonlinear terms or increasing the number of rings in the ring resonance 22 system Also, we can study in other physical fields such as BEC, polariton, etc These are contents that we plan to research in the future The research results were presented in specialized scientific conferences, as well as published in prestigious domestic and foreign journals 23 PUBLICATIONS RELATED TO THESIS [1] Nguyen Duy Cuong, Dinh Xuan Khoa, Cao Long Van, M Trippenbach, Bui Dinh Thuan, and Do Thanh Thuy, Spontaneous Symmetry Breaking of Solitons Trapped in a Double-Gauss Potential, Communications in Physics, Vol 28, No (2018), pp 301-310 [2] Duy Cuong Nguyen; Xuan Khoa Dinh; Xuan The Tai Le; Viet Hung Nguyen; Marek Trippenbach, On the nonlinear dynamics of coupled micro-resonators, Proceedings 11204, 14th Conference on Integrated Optics: Sensors, Sensing Structures, and Methods, (2019), Szcyrk-Gliwice, Poland [3] Nguyen Duy Cuong, Bui Dinh Thuan, Dinh Xuan Khoa, Cao Long Van, Marek Trippenbach, and Do Thanh Thuy, Spontaneous Symmetry Breaking in Coupled Ring Resonators with Linear Gain and Nonlinear Loss, Vinh University Journal of Science 48, 2A (2019), 39-48 [4] Nguyen Duy Cuong, Dinh Xuan Khoa, Cao Long Van, Le Canh Trung, Bui Dinh Thuan, Marek Trippenbach, Two Spot Coupled Ring Resonators, Communications in Physics, Vol 29, No (2019), pp 491-500 [5] Le Xuan The Tai, Nguyen Duy Cuong, Dinh Xuan Khoa, Nguyen Viet Hung, and Marek Trippenbach, Local versus uniform coupling, preparing to submit in Photonics Letters of Poland 24 ... "Study of spontaneous symmetry breaking in some nonlinear optical systems" Purpose - Study the influence of pulse power, propagation constant to spontaneous symmetry breaking in two conserved optical. .. the study of spontaneous symmetry breaking in the aforementioned optical systems The study of spontaneous symmetry breaking in systems in a complete and systematic manner is essential in the direction... presence of linear gain and nonlinear loss Object The studied object is the Kerr nonlinear optical systems and optical resonator tworing systems linear coupling with the presence of linear gain and nonlinear