Abstract of doctoral thesis in physics: Study of spontaneous symmetry breaking in some nonlinear optical systems

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

The work is accomplished at Vinh University

Supervisors: 1 Prof Dr Dinh Xuan Khoa

2 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

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 double-potential, 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

Chapter 1 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 2 and Chapter 3

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 do 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:

𝑖𝜕𝜓

𝜕𝑧 = −1

2𝜓𝑥𝑥 + 𝜎|𝜓|2𝜓 + 𝑈(𝑥)𝜓 , (1.1) The input pulse power is calculated by the modulus of the slowly varying envelope function:

𝑁 = ∫ |𝜓(𝑥, 𝑧)|+∞ 2

−∞ 𝑑𝑥 = ∫ |𝑢(𝑥)|−∞+∞ 2𝑑𝑥, (1.2) The asymmetry of soliton is characterized by the asymmetry denoted by 𝜈,

exceeds the critical value, this type of symmetry breaking is called supercritical

bifurcation

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

Figure 1.8 The subcritical symmetry-breaking bifurcation in the double-channel

model

𝑁𝑏𝑖𝑓

𝑁𝑏𝑖𝑓

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 2 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:

𝑖𝜕𝜓𝜕𝑧 = −12𝜓𝑥𝑥 + 𝜎|𝜓|2𝜓 + 𝑈(𝑥)𝜓, (2.1) 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

Hì nh 2.1 Normalized double-well potential 𝑈(𝑥) |𝑈(𝑥)|⁄ 𝑚𝑎𝑥 in spatial

coordinates x

The figure 2 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 thế tăng lên chúng ta thấy rằng hàm thế Gauss kép dần tới hàm thế Gauss đơn bắt đầu tại giá trị độ rộng 𝑎 ≈ 1.35 Lưu ý rằng sự phá vỡ đối xứng không xảy ra trong trường hợp một kênh

We consider solitons of system having form 𝜓(𝑥, 𝑧) = 𝑢(𝑥)𝑒𝑖𝜇𝑧 where μ is the propagation constant, 𝑧 is propagation length and 𝑢(𝑥) is the function that satisfy the equation:

−𝜇𝑢 +12𝑢𝑥𝑥− 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)|−∞+∞ 2𝑑𝑥 (2.4) The quantity characteristic for the asymmetry of soliton is defined as the asymmetric ratio:

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:

𝑖𝜕𝜓𝜕𝑧 = −12𝜓𝑥𝑥 − |𝜓|2𝜓 + 𝑈(𝑥)𝜓 (2.6)

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 N bif = 0.925 (or 𝜇𝑏𝑖𝑓 = 0.646) exist,

when N > N bif (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 N bif, 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

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

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

spatial for 𝑁 = 2, 𝑎 = 0.5, respectively

We performed calculations on ten different values of the width 𝑎 of Gaussian function, each of which we obtained the critical points 𝑁𝑏𝑖𝑓 and 𝜇𝑏𝑖𝑓 as figure 2.9 below:

Hì nh 2.9 Figure (a) power at bifurcation points 𝑁𝑏𝑖𝑓 as a function of width 𝑎; figure (b) propagation constant at bifurcation points 𝜇𝑏𝑖𝑓 as a function of width

𝑎

So for a given width of gaussian potential a, if power N of pulse is smaller than

threshold value 𝑁𝑏𝑖𝑓 then the system is in stable symmetrc states, if the N of pulse is

greater than threshold value 𝑁𝑏𝑖𝑓 then the system is in stable asymmetric states Note

that in that value region of N there are also symmetric states but they are unstable

Therefore, we find that in the region (1) there are both stable asymmetric state and unstable symmetric states, whereas in the region (2) there are only stable symmetric states

2.1.3 The system with self-defocusing nonlinearity and double-potential

In this section, we consider self-defocusing nonlinearity case which 𝜎 = +1phương trình (2.1) trở thành:

Figure 2.10 Asymmetry ratio Θ as a function respect of pulse power 𝑁 in defocusing nonlinearity case for the width of double-potential 𝑎 = 1.0

self-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

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

Figure 2.12 (a) propagation in space of soliton corresponds to a =1/3, pulse

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 = −12𝜕𝜕𝑥2𝜙2 + 𝑔(𝑥)|𝜙|2𝜙 − 𝑘𝜓

𝑖𝜕𝜓𝜕𝑧 = −12𝜕𝜕𝑥2𝜓2 + 𝑔(𝑥)|𝜓|2𝜓 − 𝑘𝜙, (2.8) 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 ≡12∫−∞+∞[|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:

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

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 = 1 and propagation constant 𝜇 = 4

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 = 2 Total pulse power of asymmetric state:

(a)