Solitons of Waveguide Arrays G14DIS Mathematics 4 th Year Dissertation 2010/11 School of Mathematical Sciences University of Nottingham Katie Salisbury Supervisor: Dr H Susanto Division: Applied Project Code: HS D1 Assessment Type: Investigation May 2011 I have read and understood the School and University guidelines on plagiarism. I confirm that this work is my own apart from the acknowledged references. 1
Abstract This report investigates the existence of discrete solitons in waveguide arrays with linear potential by finding localised solutions to the discrete nonlinear Schrödinger equation with linear potential. We take both a numerical approach, using the Newton-Raphson method, and an asymptotic approach to solving the equation. We consider how each of the parameters effects the behaviour of the light propagation and find that increasing the strength of coupling between the waveguides causes a sudden change in behaviour and a saddle-node bifurcation point to occur, after which localised solutions cease to exist. 2
Contents 1 Introduction 5 2 Background 8 2.1 Waveguide arrays 8 2.2 Solitons 9 3 Discrete Nonlinear Schrödinger equation 11 3.1 Discrete diffraction 11 3.2 Discrete Nonlinear Schrödinger equation with linear potential 13 4 Newton-Raphson Method 15 4.1 Rate of Convergence..16 4.2 Coupling strength.16 4.3 Nonlinear coefficient and linear propagation constant...21 4.4 Threshold condition...22 4.5 Linear potential term....24 4.6 Case I: Exciting one waveguide.27 4.7 Case II: Exciting two waveguides 30 4.8 Case III: Twisted mode..34 5 Asymptotic Expansion 37 5.1 Case I: Exciting one waveguide 37 3
5.2 Case II : Exciting two waveguides...40 5.3 Case III: Twisted mode 42 6 Second numerical solution 47 6.1 Case II: Exciting two waveguides...48 7 Conclusions 51 8 Appendix 52 8.1 Matlab coding...52 8.2 Maple coding 53 8.3 Calculations..55 4
1 Introduction A soliton is a profile of light that remains constant and stable during propagation. This is due to a balance between the dispersion effects and the nonlinear Kerr effect [1]. As a result the travelling pulse of light does not change its shape. The first sighting of solitons as a physical phenomenon was by a man called J. Scott Russel in 1838. He described his discovery, I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation [2]. J. Scott Russel follow the wave along the channel and was amazed to find that it continued its course without changing form or diminishing its speed. Intrigued by what he had seen he built wave tanks in his home so that he could to make practical and theoretical observations in to these waves. He published his findings in the Reports of the Meetings of the British Association for the Advancement of Science in 1844. Initially, there were those who had difficulty accepting his ideas because they seemed at odds with the theories of Isaac Newton and Daniel Bernoulli on hydrodynamics. However by the 1970 s further theoretical treatment and solutions had been published, prompting new research in to this fascinating phenomenon [2]. In 1973, A. Hasegawa of AT&T Bell Labs proposed that solitons could exist in optical fibers [1] and thus could potentially be used as channels for transporting energy. Solitons possess the useful property that they may collide, pass through each other and 5
recover completely their original form after the collision. For this reason, optical solitons have proved to be a very effective and secure way of transmitting large amounts of information over thousands of miles. Soliton-based transmission systems are often used to increase performance of optical telecommunications. A waveguide is an example of an optical fiber and acts as a channel for solitary waves to travel along through total internal reflection. This is the process by which the waveguide keeps the light within its core allowing it to propagate successfully [3]. Solitons that travel along discrete media, such as waveguides are known as discrete solitons. In 1998, the formation of discrete solitons in an array of coupled AlGaAs waveguides was experimentally observed [4]. The experiments demonstrated what had been predicted in theory, that in an array of weakly coupled waveguides, when a low intensity of light is injected in to one or a few neighbouring waveguides it will couple to more and more waveguides as it propagates thereby widening it s spatial distribution. However a high intensity of light changes the refractive index of the input waveguides through the Kerr effect and decouples them from the rest of the array. It was shown that certain light distributions propagated while keeping a fixed profile among a limited number of waveguides; these are discrete spatial solitons [4]. The initial input of light required in order to induce the formation of a solitonic structure is known as the threshold condition. If the initial condition is below this threshold then the light will disperse without the formation of a soliton [5] and we get a trivial solution. Theoretical research is now focusing on the existence of discrete solitons in a lattice such as an array of waveguides. The dynamics of such a system are governed by the discrete nonlinear Schrödinger equation (DNLSE). Current approaches include the search for exact solutions in some limits; effective point particle and variational approaches, perturbation 6
around the linearised case and numerical methods [6]. This equation can be used to model many physical phenomena for example localised modes in molecular systems such as long proteins, polarons in one dimensional ionic crystals, localised modes in electrical lattices and a coupled array of nonlinear waveguides [4]. I will be investigating the existence of discrete solitons in waveguide arrays with linear potential by finding localised solutions to the DNLSE with linear potential. I will solve the equation numerically using the Newton-Raphson method using the computing package MatLab to produce results. I will consider what effect each of the parameters has on the behaviour of the propagating light profile and use the computer package Maple, alongside Matlab, to plot the results allowing for them to be graphically interpreted. I will look at the strength of the coupling between the waveguides required for a solitonic structure to form in particular for these three initial wave formations: Case I: exciting a single waveguide Case II: exciting two waveguides Case III: twisted mode I will investigate the existence of any bifurcation points and for the second case I will find a second solution that collides with the first at a saddle-node bifurcation point. I will also look to solve the DNLSE with linear potential using an asymptotic approach by taking the anti-continuum limit. I will compare the results obtained from using the Newton-Raphson method with those found using an asymptotic expansion. 7
2 Background Before we begin exploring the mathematics behind solitons of waveguide arrays, it is important to introduce the basic physical concepts that we will go on to model. 2.1 Waveguide arrays A waveguide is a physical structure that guides waves; there are many different types of waveguides for different types of waves. I will be focusing on optical (light) waveguides. These are physical structures that guide electromagnetic waves in the optical spectrum. They do this through a process called total internal reflection. This means that the light travels along the waveguide bouncing back and forth off of the boundary. However, the light must enter the waveguide within a certain range of angles known as the acceptance cone in order for it to propagate, or travel along the core of the waveguide [3]. The most common optical waveguides are optical fibers. These are thin, flexible, transparent fibers most commonly made from silica glass. Optical fibers transmit light and signals for long distances with a high signal rate. This means that they are one of the most effective forms of communication for carrying large amounts of data. They are widely used as a medium for telecommunication and networking, often replacing electrical cables. Optical fibers consist of dielectric material, these are electrical insulators that can be polarized by an Figure 1: Optical fibers applied electric field. Typically, the dielectric material in 8
the centre of the fiber has a higher refractive index than the dielectric material surrounding it causing total internal reflection to occur and forcing the light to remain in the core of the fiber [2]. This means that the light is successfully transmitted. A waveguide array, is simply a collection of waveguides running parallel to each other forming a discrete lattice which we will go on to model using the discrete nonlinear Schrödinger equation with linear potential. 2.2 Solitons A soliton is a self-reinforcing solitary wave. A profile of the light travelling through a waveguide array is considered to be a soliton if it remains constant and stable during propagation. This occurs when there is a balance between the nonlinear and linear (dispersive) effects in the medium. If we consider a pulse of light travelling in a waveguide, for example glass. This pulse consists of light profiles of several different frequencies. These different frequencies will travel at different speeds and the shape of the pulse will therefore change over time. However, this is not the case with solitons. If the pulse has just the right shape, the nonlinear effect will exactly cancel the dispersion effect, and the pulse's shape will be the same at any point in time (Figure 2). Optical solitons provide a secure means of carry bits of information over thousands of miles. A particularly useful feature of solitons is that they are able to pass through one another without changing shape which allows for high bandwidths in the optical fibers to transmit large amounts of data. 9
Figure 2: Soliton Solitons that travel along continuous media are localised solutions of the nonlinear Schrödinger equation (NLSE). Whilst solitons that travel along discrete media, such as waveguide arrays, are known as discrete solitons and are localised solutions of the discrete nonlinear Schrödinger equation (DNLSE). We are going to look in particular at discrete solitons travelling along waveguide arrays with linear potential. Therefore we will be looking to find localised solutions to the DNLSE with linear potential, first numerically using the Newton-Raphson formulae in Section 4 and then by asymptotically expanding it in Section 5. 10
3 Discrete Nonlinear Schr dinger equation The nonlinear Schödinger equation can be used to describe many physical phenomena and represents just one of the types of partial differential equations with solitary solutions. As we discussed in Section 2.1, optical waveguides such as optical fibers are discrete nonlinear media which act as a channel for discrete solitary waves to travel along. Such solitary waves are localised solutions of the discrete nonlinear Schödinger equation, (1) where is the coupling constant, is the constant nonlinear coefficient and represents the light profile of the waveguide. In this model, the distance between the waveguides is. So if is small this represents the waveguides being far apart (weakly coupled) and if is large the waveguides are said to be strongly coupled. The nonlinear term is known as the Kerr effect and balances out the linear (dispersive) effects. We can use this equation to explore the evolution of different propagating light profiles. In Section 4.2-4.3 we will look at what affect the values of and have on the light profile and for which values of a numerical solution ceases to exist. 3.1 Discrete diffraction Suppose we have an array of identical, weakly coupled waveguides. It has been observed experimentally that if a low intensity beam of light is injected in to one, or a few neighbouring waveguides in the centre of the array, the light will spread over the adjacent waveguides as it propagates. By doing so it widens it spatial distribution and this is known as discrete diffraction. Figure 3.a) illustrates this; the image shows the light at the end of the 11
waveguides and you can see that the light is spread fairly evenly across the array of waveguides. However, injecting a sufficiently high intensity input beam, causes the beam to selftrap or in other words to start to amplify itself and it detaches itself from the rest of the array to form a localised state ie. a discrete soliton. Subsequently, many interesting properties of nonlinear lattices and discrete solitons can be observed. The formation of a soliton is illustrated in Figure 3.c) where you can see that the light is most intense in the central waveguides and those immediately adjacent to it. The light gets weaker and weaker towards the edge of the array, eventually fading to darkness in the outer waveguides. In Figure 3.b) there is a medium intensity input beam but not enough power to form a soliton so over time it will eventually look like Figure 3.a). Figure 3 a) Low intensity of light is injected and spreads to adjacent waveguides b) Medium intensity of light is injected, not strong enough to form a soliton c) High intensity of light is injected leading to the formation of a soliton. 12
We will illustrate this behaviour at a numerical level using the Newton-Raphon method, using the DNLS equation with linear potential as a supporting model as well by using an asymptotic approach. 3.2 Discrete Nonlinear Schr dinger equation with linear potential To make the equation more interesting to study, let us add a linear potential term. We will look more closely in Section 4.5 at what effect this term has on the localised solutions. The discrete nonlinear Schr dinger equation with linear potential reads: (2) where is the strength of the linear potential, is the coupling constant and is the constant nonlinear coefficient. We can use this equation to model an array of identical waveguides with linear potential, positioned with equal separations, such that all the coupling constants between them are equal. The equation describes the evolution of, the light profile of the waveguide in the presence of the optical Kerr effect. t c ψ ψ ψ 3 ψ n Figure 4: An array of identical waveguides with equal separations 13
Since we are interested in time independent solutions ie. the static state of this equation let us look for solutions of the form where is independent of time and is the linear propagation constant. Substituting this in to equation (2) gives We can introduce a new time scale and choose such that is real, to give 3 3 (3) We will now look at methods of solving this equation to find discrete solitons in waveguide arrays with linear potential. 14
4 Newton-Raphson Method There are many approaches that can be taken to finding solutions to the discrete nonlinear Schrödinger equation with linear potential, including effective point particle and variational approaches, perturbation around the linearised case and numerical methods. In this section we will look at solving the DNLSE with linear potential numerically, using perturbation methods. We will develop increasingly accurate solutions iteratively using the Newton-Raphson method. The Newton-Raphson formulae for waveguide arrays can be written: where, ( ) and 3 where, so in particular, 3 and 15
3 In order to obtain the localised solutions to equation (4) and hence find the discrete solitons governed by the DNLSE with linear potential equation we need to find for which. We should find that as the number of iterations increases, converges closer to the solution. However, we must make sure, when choosing our initial guess, that it is above the threshold condition or we will end up with a trivial solution. 4.1 Rate of convergence The number of iterations of the Newton-Raphson formulae required for a solution to be found varies depending on the choice of constants however generally the rate of convergence in very fast and a solution is reached in under 10 iterations. For our investigation, for the sake of consistency, in all cases we will iterate 100 times. Creating a program on the numerical computing program Matlab allows us to do these iterations very quickly and saves us doing numerous calculations. 4.2 Coupling strength Let us investigate at how varying the value of (coupling constant) affects the shape of the propagating wave and the intensity of light throughout the waveguides. We also want to find the value of for which localised solution cease to exist. Let us begin by exciting a single waveguide in the centre of the array, 16
{ such that. Let us set the constants, (no linear potential) such that we have waveguide. waveguides and a light beam of intensity is injected in to the (central) Figure 5 is a plot of the intensity of light through the waveguides. The intensity of light in the waveguide is given by, where is the wave function. Here the waveguides are uncoupled. The maximum intensity of light is in the central waveguide and there is complete darkest in the rest of the waveguides. As the value of is increased the intensity of light in the excited waveguide decreases and the light spreads out across the adjacent waveguides. The contrast between the intensity of light in the centre and the outer waveguides decreases and we can see from the Figures 6-8 that the solitons becomes smoother. In Figure 9, for the solution has broken down since is too large. If we continue to increase, the light profiles become very irregular and behave in a seemingly random way. Figure 11 illustrates clearly how the shape of the solitons change as the coupling constant is varied. Figure 6b) is a plot of the wave function rather than the intensity of light throughout the waveguides. This illustrates that the value of the wave function in several of the waveguides is in fact negative. 17
If we change the value of, so increasing or decreasing the number of waveguides, similarly, we find that the stronger the coupling is the smoother the solitons are. So, as we would expect, the closer the waveguides are together, the more light is spread over the adjacent waveguides as it propagates. Figure 5: Figure 6: a) Intensity of light throughout the waveguides b) Wave function throughout the waveguides. 18
Figure 7 Figure 8: Figure 9: 19
Figure 10: Figure 11 20
4.3 Nonlinear coefficient and linear propagation constant Let us set and look at the effect that varying (nonlinear coefficient) and (linear propagation constant) has on the localised solutions. Again, we will initially input a light beam of intensity in to the central waveguide such that. Table 1 gives the maximum light intensity in the central waveguide. We can see that there is a simple linear relationship between the two constants and the light intensity, such that the maximum intensity of light is given by. So increasing weakens the peak light intensity and increasing strenghtens it. Figure 12 illustrates clearly how the peak of the soliton is determined by the relationship between and. If we were to choose a different value of we would see that exactly the same relationship holds. Table 1 Max light intensity 5 5 5 5 5 0 0 1 1 1 0 0 1 2 2 0 0 2 5 2.5 0 0 3 1 0.3333 0 0 16 2 0.1250 21
Λ α Λ α Λ α Λ α Λ α 8 Figure 12 4.4 Threshold condition So far we have we have looked at exciting a single waveguide by initially injecting a light beam of intensity (initial wave function) in to the waveguide in the centre of the array. In term of the DNLS equation with nonlinear potential this corresponds to setting { such that. 22
If we increase the initial wave function, this does not affect the shape of the soliton and the intensity of light throughout the waveguides remains the same. However if we decrease the initial input so it is sufficiently small the light disperses without the formation of a solitonic structure. The point at which a soliton ceases to exist is known as the threshold condition. If the intensity of the initial wave function is below the threshold condition we get a trivial solution. Table 2 (no. of iterations) Threshold condition 2 0 0 1 1 100 0.44721 5 0.44721 10 0.44721 2 0 0.00001 1 1 0.44721 5 0.44721 10 0.44721 2 0 0.0001 1 1 100 0.44721 5 0.44721 10 0.44721 2 0 0.001 1 1 100 0.44721 5 0.47721 10 0.44721 2 0 0.01 1 1 100 0.44716 5 0.44716 10 0.44716 2 0 0.1 1 1 100 0.44266 5 0.44266 10 0.44266 2 0 0.3 1 1 100 0.39967 5 0.39908 10 0.39908 2 0 0.5 1 1 100 0.24935 5 0.16632 10 0.11550 Table 2 gives the threshold condition for over increasing values of for. In general, as gets larger, the threshold condition gets smaller. This 23
means that the closer together the waveguides are, the weaker the initial input of light needs to be for a discrete soliton to be formed or in other words to excite a localised mode. The number of waveguides does not seem to effect the threshold condition for very small values of, however for the threshold condition appears to be slightly higher for lower numbers of waveguides. 4.5 Linear potential term Let us explore what effect adding a linear potential term has on the shape of the propagating wave. Again, we will excite a single waveguide by initially inputting a light beam of intensity in to the central waveguide such that. Table 3 gives the maximum intensity of light in the central waveguide, for. We can see that as increases, the max intensity in the central waveguide decreases. This is illustrated in Figure 13. However at a certain value of, the solution breaks down since the linear terms cannot balance out the nonlinearity effects. For the case the solution breaks down for Table 3 Max light intensity 5 0 0 1 1 1 0.001 0.9940 0.01 0.9400 0.09 0.4600 0.1 0 0.2 0 0.3 0 0.4 0 24
Figure 13 If we increase to, this decreases the max light intensity and we can see from Table 4 that it further decreases as is increased. As is increased, the shape of the solitons become increasingly asymmetrical and irregular, and the solution breaks down for This is illustrated in Figure 14. Table 4 5 0 0.1 1 1 0.9796 0.01 0.9182 0.05 0.6699 0.1 0.4793 0.15 0.2776 0.16 0.2119 0.2 0 0.5 0 1 0 5 0 Max light intensity 25
Figure 14 If we increase to, we can see from Figure 15 that the solitons become very irregular very quickly. The solution breaks down for. Table 5 5 0 0.5 1 1 0.0476 0.01 1.9400 0.1 1.1613 0.3 0.1500 0.35 0.0035 0.4 0 0.5 0 1 0 Max light intensity 26
Figure 15 has the effect of causing asymmetry in the soliton. As is increased the shape of the soliton becomes more irregular and depending on the value of, at a certain point, solutions cease to exist. In general, the larger is, so the closer together the waveguides are, the larger can be before a solitonic structure can no longer be formed. 4.6 Case I: Exciting one waveguide In Section 4.2 we looked at how varying the value of (the coupling constant) affected the light profile for the case We found that increasing caused the soliton to become smoother and the intensity of light in the central waveguide to become weaker. Let us now look at finding localised solutions in waveguides with non zero linear potential term. 27
Let us excite a single waveguide by injecting an initial input of light of intensity in to the central waveguide and set such that the array consists of waveguides. Instead of keeping constant throughout let us run upwards from increasing it by increments of. We will use the solution of the previous increment as the initial guess for the next value of. Figure 16 is a plot of the intensity of light throughout the waveguides. For, there is only light in the central waveguide but as increases the light spreads across the neighboring wave guides and the intensity of light in the central waveguide decreases creating a smoother soliton. This is the same result as we found in Section 4.2, however since the solitons are asymmetrical. If were to be made larger the asymmetry in the solitons would become more exaggerated. Figure 16 28
Figure 17 is a plot of the coupling constant against the value of in (central) waveguide. This clearly illustrates how the wave function in the central waveguide decreases as the distance between the waveguides in the array is decreased. The value of forms a smooth curve until after which the value of seems to jump around randomly and all localised solutions cease to exist. At the derivative goes to infinity and this is a saddle node bifurcation point. Figure 17 29
Figure 18 If we overlay the plot of on to Figure 16, we can see that the blue line forms a random pathway and no longer forms a soliton. 4.7 Case II: Exciting two waveguides Suppose we now change the initial input to { such that and set. 30
So we exciting the and waveguides by injecting a light beam of intensity in to both of them. Figure 19 illustrates how the shape of the solitons changes as is increased. There is a dual peak since we have excited two waveguides. The asymmetry in the solitons is caused by the nonlinear potential term. As is increased the solitons becomes smoother as the light spreads across the adjacent waveguides. Figure 19 31
Figure 20 is a plot of against the value of in the waveguide given and Figure 21 is a plot of against the value of in the (central) waveguide. Both plots form a smooth curve and break down at 8 where the derivative goes to infinity. This is a saddle node bifurcation point. Figure 22 is a plot of together with. The difference between and remains almost constant throughout. Figure 20 32
Figure 21 X X Figure 22 33
4.8 Case III: Twisted mode We will now look at a twisted mode by choosing the initial wave function such that the phase difference between the two excited waveguides is. Suppose we inject a light beam of value in to the waveguide and value in to the waveguide. This corresponds to setting { such that. Let us set. Figure 23 is a wave function plot and illustrates how the shape of the solitons alters as the value of across the waveguides changes. As is increased the peak at and trough become less exaggerated and light spreads across the adjacent waveguides. For we can see that the solution has broken down and the profile of light is no longer a soliton. Figure 24 is a plot of against the value of in the waveguide and Figure 25 is a plot of against the value of in the 6 waveguide. Both plots form a smooth curve, has a bifurcation point at and has one at, where the derivatives go off to infinity. These are a saddle node bifurcation points where all localised solutions disappear. Figure 26 is a plot of together with. We can see that the two plots do not have the same bifurcation point, this is because although the profile of light has a single overall bifurcation point, each of the individual waveguides doesn t necessarily have a bifurcation at the same point. 34
Figure 23 Figure 24 35
Figure 25 X X Figure 26 36
5 Asymptotic Expansion Another way of analysing the discrete nonlinear Schrödinger equation with linear potential is to expand it asymptotically. We can then compare the results we obtain with those we found by solving numerically using the Newton-Raphson formulae. The DNLS equation with linear potential reads: (2) where is the linear propagation constant, is the coupling constant, is the constant nonlinear coefficient and represents the light profile of the waveguide. We made the substitution, To obtain equation (3), 3 In the anti-continuum limit (uncoupled, ), 3 (5) or 5.1 Case I: Exciting one waveguide Let us try the asymptotic expansion (6) where, This corresponds to setting, such that we have uncoupled waveguides, and injecting light of intensity in to the (central) waveguide. 37
Substituting equation (6) in to equation (5) gives, ( ) ( ) 3 ( ) Equating terms of similar order, gives ) As in Section 4.6, let us set the constants = 1, and to obtain Figure 27 is a plot of against. In Figure 28 we have plotted against together with against (Figure 17) found in Section 4.6. The points correspond exactly for very low values of however for larger values of, they grow apart. This is because the asymptotic expansion is based on the assumption that is small so the asymptotic expansion is only valid for values of very close to zero. 38
Figure 27 φ X Figure 28 39
5.2 Case II: Exciting two waveguides Suppose we look at the asymptotic expansion (7) where,, This corresponds to setting, such that we have waveguides, and exciting two waveguides by injecting light of intensity in to the and (central) waveguide. Substituting equation (7) in to equation (5) and equating terms of similar order gives, 3 (3 ) ( ( 3 ) 3 ) 3 Setting the constants = 1, and we obtain 3 8 8 40
3 Figure 29 is a plot of against. Figure 30 is a plot of against together with against (Figure 20) found in Section 4.7. As we would expect, the points correspond for small values of. Figure 31 is a plot of against. Figure 32 is a plot of against together with against X(6) (Figure 21) found in Section 4.7. Again the plots correspond when is close to zero, but as it becomes larger the asymptotic expansion is no longer valid. Figure 29 41
φ X Figure 30 Figure 31 42
φ X Figure 32 5.3 Case III: Twisted mode Suppose we use the asymptotic expansion (8) where,, This corresponds to a twisted mode, where the phase difference between the light in the and (central) waveguides is. Let us set, such that we have uncoupled waveguides. Substituting equation (8) in to equation (5) and equating terms of similar order gives, 3 (3 ) 43
( ( 3 ) 3 ) 3 Setting the constants = 1, and we obtain 3 8 8 3 Figure 33 is a plot of against. Overlaying the plot of against (Figure 24) found on Section 4.8 gives Figure 34. Figure 35 is a plot of against. Plotting Figure 35 together the plot of against (Figure 25) found in Section 4.8 gives Figure 36. For both Figure 34 and Figure 36, the asymptotic solution matches the numerical solution for low values of but the asymptotic solution becomes invalid for as moves away from zero as we made the assumption that the waveguides were uncoupled. 44
Figure 33 φ X Figure 34 45
Figure 35 φ X Figure 36 46
6 Second numerical solution In Sections 4.6 4.8 we looked at 3 different initial conditions: Case I: Injecting light of intensity in to the (central) waveguide such that for and found there is a bifurcation point at Case II: Injecting light of intensity in to the and (central) waveguides such that for and found there is a bifurcation point at 8 Case III: Injecting light of value in to the and in to the (central) waveguide such that the phase difference between the two excited waveguides is and for and found that there is a bifurcation point at. These are saddle node bifurcation points; local bifurcations in which two fixed points of the dynamical system collide. By definition of a saddle-node, one of these fixed points is stable and the other is unstable. We have found one of the fixed points and now will look at finding the other. 47
For Case I and III, we are not able to find the second solution which meets at the bifurcation point. However for Case II we are able to find a pretty accurate approximation of the second solution. 6.1 Case II: Exciting two waveguides For two excited waveguides, the second solution correspond to initially inputting light of intensity in to the and (central) waveguides and light of intensity in to the and waveguides such that Let us set. Figure 37 is a plot of against. Plotting this solution together with that found in Section 4.7 from inserting light of intensity 1 in to the and (central) waveguides we can see that the two solutions meet at the bifurcation point 8. The fact that the two plots do not form a completely smooth curve, suggests that this solution is not exact but it is as accurate as we are likely to find. Figure 39 is a plot of against and Figure 40 shows this plot together with the values of obtained from our first solution. Again we can see that the two solutions meet at the bifurcation point 8, this time forming a completely smooth curve. 48
Figure 37 Figure 38 49
Figure 39 Figure 47 50
7 Conclusions In this investigation, I have studied the existence of solitons in waveguide arrays with linear potential by solving the discrete nonlinear Schrödinger equation with linear potential. I found that all localised solutions disappear at a saddle node bifurcation point where an unstable and stable solution collide and annihilate each other. I looked at three different initial wave formations. For Case II I looked at exciting two waveguides with light of intensity 1 and found a second solution which collides with the first at a saddle-node bifurcation point. However for the Case I: exciting a single waveguide and Case III: twisted mode my investigations failed to find the second solution corresponding to each of these. These are yet to have been discovered. If I had more time I would carry out a more thorough investigation in to these cases. I found that increasing the strength of the coupling c between the waveguides decreases the light intensity in the excited waveguide resulting in a smoother soliton (given that c is below the bifurcation point). I established that the maximum intensity of light in a single excited waveguide when the array is uncoupled is determined by / α (given that the initial wave function is above the threshold condition) where is the linear propagation constant and α is the constant nonlinear coefficient. I looked at solving the DNLSE with linear potential for each of the three initial wave formations I solved numerically, using a different asymptotic expansion for each. I compared the results with those obtained from solving numerically by plotting the results alongside each other and I found that an asymptotic expansion is only a valid method of solving the DNLSE with linear potential when the waveguides are very weakly coupled. 51
If we had more time then th e next step to investigate the stability of the solutions I found. 52
8 Appendix 8.1 Matlab coding Figure 5 e = 0 a = 1 d = 1 N=5 for k=1:(2*n+1); for l=1; if k==n+1; X(k,l)=1; else X(k,l)=0; end end end for c = 0:0.01:0; for h=1:100 for k=1:(2*n+1); for l=1; end end if k==1; f(k,l)=-c*x(2) + a*x(1)^3 + (e-d)*x(1); elseif k==2*n+1 f(k,l)=-c*x(2*n) + a*x(2*n+1)^3 + (e*(2*n+1)-d)*x(2*n+1); else f(k,l)=-c*(x(k+1)+ X(k-1)) + a*x(k)^3 + (e*k-d)*x(k); end for k=1:(2*n+1); for l=1:(2*n+1); if k==l; J(k,l)=3*a*X(k)^2+((e*k)-d); elseif k==l-1; J(k,l)=-c; elseif k==l+1; J(k,l)=-c; else J(k,l)=0; 53
end end end f X = X - inv(j)*f end plot(x.^2,'ko-'); title(['c = ', num2str(c)]); %,', max(f) = ',num2str(max(abs(f)))]); getframe; end l = X(N) s = X(N+1) The above coding gives a plot of against after 100 iterations of the Newton-Raphson formulae, where,. The plot shows the intensity of light in each of the waveguides. 8.2 Maple coding Figure 17 >points1:=[[0,0.9695],[0.01,0.9694],[0.02,0.9691],[0.03,0.9685 ],[0.04,0.9678],[0.05,0.9668],[0.06,0.9655],[0.07,0.9641],[0.0 8,0.9624],[0.09,0.9604],[0.1,0.9582],[0.11,0.9558],[0.12,0.953 0],[0.13,0.9500],[0.14,0.9467],[0.15,0.9431],[0.16,0.9391],[0. 17,0.9348],[0.18,0.9302],[0.19,0.9251],[0.2,0.9196],[0.21,0.91 36],[0.22,0.9071],[0.23,0.9001],[0.24,0.8924],[0.25,0.8840],[0.26,0.8748],[0.27,0.8647],[0.28,0.8536],[0.29,0.8412],[0.3,0.8 274],[0.31,0.8117],[0.32,0.7938],[0.33,0.7731],[0.34,0.7486],[ 0.35,0.7176],[0.351,0.7137],[0.352,0.7097],[0.353,0.7052],[0.3 54,0.7002],[0.355,0.6936],[0.356,0.3826]]: > plot(points1, thickness = 2, color = blue, labels = [c,x(6)]); 54
The above Maple coding gives a plot of against up to the bifurcation point with = 1, and. Figure 27 > k(c):= 0.9695-1.0974*(c^2); k( c ) := 0.9695 1.0974 c 2 > plot(k(c), c = 0..0.5, thickness = 2); The above coding gives a plot of against obtained from the asymptotic expansion where, The constants where set as = 1, and Figure 28 > plot([points1, k(c)], c=0..0.5, thickness = 2, color = [blue,red]); The above coding gives a plot of against obtained from using the asymptotic expansion and against with = 1, and. Figure 37 > points3333:=[[0,0.9747],[0.01,0.9747],[0.02,0.9747],[0.03,0.97 48],[0.04,0.9749],[0.05,0.9752],[0.06,0.9755],[0.07,0.9759],[0.08,0.9763],[0.09,0.9769],[0.1,0.9776],[0.11,0.9784],[0.12,0.9 793],[0.13,0.9803],[0.14,0.9814],[0.15,0.9827],[0.16,0.9842],[ 0.17,0.9859],[0.18,0.9877],[0.19,0.9897],[0.2,0.9920],[0.21,0. 9946],[0.22,0.9974],[0.23,1.0006],[0.24,1.0042],[0.25,1.0084], [0.26,1.0131],[0.27,1.0189],[0.279,1.0256],[0.28,1.0266],[0.28 01,1.0267],[0.2802,1.0268],[0.2803,1.0269],[0.2804,1.0270],[0. 55
2805,1.0271],[0.2806,1.0273],[0.2807,1.0274],[0.2808,1.0275],[ 0.2809,1.0276],[0.281,1.0279],[0.2811,1.0300],[0.2812,1.0276]]: > plot(points3333,thickness = 2, color = pink, labels = [c, X(5)]); Figure 38 > plot([points3,points3333], c=0..0.5, thickness = 2, color = [blue,pink], labels = [c,x(5)]); 8.3 Calculations ( ) ( ) 3 ( ) ( ) 3 or if, if, ( ) = 0 if ( ) if ( ) 56
if ( ) if, ( ) ( ) if 3 ( ) ( ) if ( ) ( ) if ( ) ( ) if ( ) ( ) if 8 (8 ) ( ) ( ) (8 ) if 8, 57
Hence, ) The above calculations are the steps involved in calculating for the asymptotic expansion where, given in Section 5.1. 58
References [1] H.A. Haus, Optical Fiber Solitons, Their Properties and Uses (IEEE Explore, 1993) [2] Website http://en.wikipedia.org/wiki/optical_fiber, visited 26 January 2011 [3] Website http://en.wikipedia.org/wiki/total_internal_reflection, visited 26 January 2011 [4] H.S Eisenberg and Y. Silberberg, Discrete Spatial Optical Solitons in Waveguide Arrays Phys. Rev. 81, 3383-3386 (1998) [5] P.G. Kevrekidis, J.A. Espinola-Rocha, Y. Drossinos, A. Stephanov, Dynamical barrier for the formation of solitary waves in discrete lattices (ScienceDirect, 2008). [6] A. Trombettoni and A. Smerzi, Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates Phys. Rev. 86, 2353-2356 (2001) 59
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