3.3.THE NONLINEAR SCHRODINGER EQUATION73The NSE also shows higher order soliton solutions, that travel at the samespeed, i.e. they posses the same carrier frequency, the so called breathersolutions. Figures 3.5(a) and (b) show the amplitude and spectrum of sucha higher order soliton solution, which has twice the area of thefundamentalsoliton. The simulation starts with a sech-pulse, that has twice the area ofthe fundamental soliton, shown in figure 3.3.Due to the interaction of thetwosolitons.thetemporal shapeandthespectrumexhibitsa complicated butperiodic behaviour.This period is the soliton period z = /4, as mentionedabove.Ascan be seen from Figures 3.5(a) and 3.5(b),the higher ordersoliton dynamics leads to an enormous pulse shortening after half of thesoliton period. This process has been used by Mollenauer, to build his solitonlaser [11]. In the soliton laser, the pulse compression, that occures for ahigher order soliton as shown in Fig. 3.5(a), is exploited for modelocking.Mollenauer pioneered soliton propagation in optical fibers,as proposed byHasegawa and Tappert [3], with the soliton laser, which produced the firstpicosecond pulses at 1.55 μm. A detailed account on the soliton laser is givenbyHaus [12].So far, we have discussed the pure soliton solutions of the NSE. But,what happens if one starts propagation with an input pulse that does notcorrespond to a fundamental or higher order soliton?3.3.4Inverse Scattering TheoryObviously, the NSE has solutions. which are composed of fundamental soli-tons.Thus,the solutions obey a certain superposition principle which isabsolutely surprising for a nonlinear system. Of course, not arbitrary super-positions are possible as in a linear system.The deeper reason for the solutionmanyfold of the NSE can be found by studying its physical and mathemat-ical properties. The mathematical basis for an analytic formulation of thesolutions to the NSE is the inverse scattering theory [13, 14, 4, 15]. It is aspectral tranform method for solving integrable, nonlinear wave equationssimilar to the Fourier transform for the solution of linear wave equations [16]
3.3. THE NONLINEAR SCHRÖDINGER EQUATION 73 The NSE also shows higher order soliton solutions, that travel at the same speed, i.e. they posses the same carrier frequency, the so called breather solutions. Figures 3.5(a) and (b) show the amplitude and spectrum of such a higher order soliton solution, which has twice the area of the fundamental soliton. The simulation starts with a sech-pulse, that has twice the area of the fundamental soliton, shown in figure 3.3. Due to the interaction of the two solitons, the temporal shape and the spectrum exhibits a complicated but periodic behaviour. This period is the soliton period z = π/4, as mentioned above. As can be seen from Figures 3.5(a) and 3.5(b), the higher order soliton dynamics leads to an enormous pulse shortening after half of the soliton period. This process has been used by Mollenauer, to build his soliton laser [11]. In the soliton laser, the pulse compression, that occures for a higher order soliton as shown in Fig. 3.5(a), is exploited for modelocking. Mollenauer pioneered soliton propagation in optical fibers, as proposed by Hasegawa and Tappert [3], with the soliton laser, which produced the first picosecond pulses at 1.55 µm. A detailed account on the soliton laser is given by Haus [12]. So far, we have discussed the pure soliton solutions of the NSE. But, what happens if one starts propagation with an input pulse that does not correspond to a fundamental or higher order soliton? 3.3.4 Inverse Scattering Theory Obviously, the NSE has solutions, which are composed of fundamental solitons. Thus, the solutions obey a certain superposition principle which is absolutely surprising for a nonlinear system. Of course, not arbitrary superpositions are possible as in a linear system. The deeper reason for the solution manyfold of the NSE can be found by studying its physical and mathematical properties. The mathematical basis for an analytic formulation of the solutions to the NSE is the inverse scattering theory [13, 14, 4, 15]. It is a spectral tranform method for solving integrable, nonlinear wave equations, similar to the Fourier transform for the solution of linear wave equations [16]
74CHAPTER3.NONLINEARPULSEPROPAGATIONF. T.A(z=0,t)A(z=0,w)117A(z=L,t)A(z=L,w)Inv. F. T.Figure 3.6: Fourier transform method for the solution of linear, time invariantpartial differential equations.ScatteringProblemA(z=0,t)ScatteringAmplitudesatz=oDiscrete + Continuous1Spectrum/17A(z=L,t)Scattering Amplitudes at z=LInverse ScatteringSoliton + ContinuumFigure 3.7: Schematic representation of the inverse scattering theory for thesolution of integrable nonlinear partial differential equations.Let's remember briefly, how to solve an initial value problem for a linearpartial differential equation (p.d.e.), like eg.(2.184), that treats the case ofa purely dispersive pulse propagation. The method is sketched in Fig. 3.6.We Fourier tranform the initial pulse into the spectral domain, because, theexponential functions are eigensolutions of the differential operators with
74 CHAPTER 3. NONLINEAR PULSE PROPAGATION Figure 3.6: Fourier transform method for the solution of linear, time invariant partial differential equations. Figure 3.7: Schematic representation of the inverse scattering theory for the solution of integrable nonlinear partial differential equations. Let’s remember briefly, how to solve an initial value problem for a linear partial differential equation (p.d.e.), like eq.(2.184), that treats the case of a purely dispersive pulse propagation. The method is sketched in Fig. 3.6. We Fourier tranform the initial pulse into the spectral domain, because, the exponential functions are eigensolutions of the differential operators with
3.3.THENONLINEAR SCHRODINGEREQUATION75constant coefficients. The right side of (2.184) is only composed of powers ofthe differntial operator,thereforethe exponentialsare eigenfunctions of thecomplete right side. Thus, after Fourier transformation, the p.d.e. becomesa set of ordinary differential equations (o.d.e.), one for each partial wave.The excitation of each wave is given by the spectrum of the initial wave.The eigenvalues of the differential operator, that constitutes the right sideof (2.184), is given by the dispersion relation, k(w), up to the imaginaryunit. The solution of the remaining o.d.e is then a simple exponential of thedispersionrelation.Now,wehavethespectrumof thepropagatedwaveandby inverse Fourier transformation, i.e. we sum over all partial waves, we findthe new temporal shape of the propagated pulse.As in the case of the Fourier transform method for the solution of linearwave equations,the inverse scattering theory is again based on a spectraltransform, (Fig.3.7). However, this transform depends now on the detailsof the wave equation and the initial conditions. This dependence leads toa modified superposition principle.As is shown in [7], one can formulatefor many integrable nonlinear wave equations a related scattering problemlike onedoes in Quantun Theory for the scattering of a particle at apoten-tial well. However, the potential well is now determined by the solution ofthe wave equation. Thus, the initial potential is already given by the ini-tial conditions. The stationary states of the scattering problem, which arethe eigensolutions of the corresponding Hamiltonian, are the analog to themonochromatic complex oscillations, which are the eigenfunctions of the dif-ferential operator. The eigenvalues are the analog to the dispersion relation,and as in the case of the linear p.d.e's, the eigensolutions obey simple linearo.d.e's.A given potential will have a certain number of bound states, that cor-respond to the discrete spectrum and a continuum of scattering states.Thecharacteristic of the continuous eigenvalue spectrumisthe reflection coef-ficient for waves scatterd upon reflection at the potential. Thus, a certainpotential, i.e. a certain initial condition, has a certain discrete spectrum andcontinuum with a corresponding reflection coefficient. From inverse scatter-ing theory for quantum mechanical and electromagnetic scattering problems,we know, that the potenial can be reconstructed from the scattering data,i.e. the reflection coefficient and the data for the discrete spectrum [?]. Thisis true for a very general class of scattering potentials. As one can almostguess now, the discrete eigenstates of the initial conditions will lead to solitonsolutions. We have already studied the dynamics of some of these soliton so-
3.3. THE NONLINEAR SCHRÖDINGER EQUATION 75 constant coefficients. The right side of (2.184) is only composed of powers of the differntial operator, therefore the exponentials are eigenfunctions of the complete right side. Thus, after Fourier transformation, the p.d.e. becomes a set of ordinary differential equations (o.d.e.), one for each partial wave. The excitation of each wave is given by the spectrum of the initial wave. The eigenvalues of the differential operator, that constitutes the right side of (2.184), is given by the dispersion relation, k(ω), up to the imaginary unit. The solution of the remaining o.d.e is then a simple exponential of the dispersion relation. Now, we have the spectrum of the propagated wave and by inverse Fourier transformation, i.e. we sum over all partial waves, we find the new temporal shape of the propagated pulse. As in the case of the Fourier transform method for the solution of linear wave equations, the inverse scattering theory is again based on a spectral transform, (Fig.3.7). However, this transform depends now on the details of the wave equation and the initial conditions. This dependence leads to a modified superposition principle. As is shown in [7], one can formulate for many integrable nonlinear wave equations a related scattering problem like one does in Quantum Theory for the scattering of a particle at a potential well. However, the potential well is now determined by the solution of the wave equation. Thus, the initial potential is already given by the initial conditions. The stationary states of the scattering problem, which are the eigensolutions of the corresponding Hamiltonian, are the analog to the monochromatic complex oscillations, which are the eigenfunctions of the differential operator. The eigenvalues are the analog to the dispersion relation, and as in the case of the linear p.d.e’s, the eigensolutions obey simple linear o.d.e’s. A given potential will have a certain number of bound states, that correspond to the discrete spectrum and a continuum of scattering states. The characteristic of the continuous eigenvalue spectrum is the reflection coef- ficient for waves scatterd upon reflection at the potential. Thus, a certain potential, i.e. a certain initial condition, has a certain discrete spectrum and continuum with a corresponding reflection coefficient. From inverse scattering theory for quantum mechanical and electromagnetic scattering problems, we know, that the potenial can be reconstructed from the scattering data, i.e. the reflection coefficient and the data for the discrete spectrum [?]. This is true for a very general class of scattering potentials. As one can almost guess now, the discrete eigenstates of the initial conditions will lead to soliton solutions. We have already studied the dynamics of some of these soliton so-
76CHAPTER3.NONLINEARPULSEPROPAGATIONlutions above. The continuous spectrum will lead to a dispersive wave whichis called the continuum.Thus, the most general solution of the NSE, forgiven arbitrary initial conditions, is a superposition of a soliton, maybe ahigher order soliton, and a continuum contribution.The continuum will disperse during propagation, so that only the solitonis recognized after a while. Thus, the continuum becomes an asympthoticallysmall contribution to the solution of the NSE. Therefore.the dvnamics ofthe continuum is completely discribed by the linear dispersion relation of thewave equation.The back transformationfrom the spectral to the timedomain is not assimple as in the case of the Fourier transform for linear p.d.e's. One has tosolve a linear integral equation, the Marchenko equation [17]. Nevertheless,the solution of a nonlinear equation has been reduced to the solution of twolinear problems, which is a tremendous success.2.521.5d1、0.5~00.20.40.62Distancez00.8-2Time-4Figure 3.8:Solution of the NSE for an unchirped and rectangular shapedinitial pulse
76 CHAPTER 3. NONLINEAR PULSE PROPAGATION lutions above. The continuous spectrum will lead to a dispersive wave which is called the continuum. Thus, the most general solution of the NSE, for given arbitrary initial conditions, is a superposition of a soliton, maybe a higher order soliton, and a continuum contribution. The continuum will disperse during propagation, so that only the soliton is recognized after a while. Thus, the continuum becomes an asympthotically small contribution to the solution of the NSE. Therefore, the dynamics of the continuum is completely discribed by the linear dispersion relation of the wave equation. The back transformation from the spectral to the time domain is not as simple as in the case of the Fourier transform for linear p.d.e’s. One has to solve a linear integral equation, the Marchenko equation [17]. Nevertheless, the solution of a nonlinear equation has been reduced to the solution of two linear problems, which is a tremendous success. 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 0.5 1 1.5 2 2.5 A mplitude Distance z Time Figure 3.8: Solution of the NSE for an unchirped and rectangular shaped initial pulse
773.4.UNIVERSALITY OFTHENSETo appreciate these properties of the solutions of the NSE, we solve theNSE for a rectangular shaped initial pulse. The result is shown in Fig. 3.8.The scattering problem, that has to be solved for this initial condition,is the same as for a nonrelativistic particle in a rectangular potential box[32]. The depth of the potential is chosen small enough, so that it has onlyone bound state.Thus.we start with a wave composed of a fundamentalsoliton and continuum. It is easy to recognize the continuum contributioni.e. the dispersive wave, that separates from the soliton during propagation.This solution illustrates, that soliton pulse shaping due to the presence ofdispersion and self-phase modulation may have a strong impact on pulsegeneration [18]. When the dispersion and self-phase modulation are properlyadjusted.soliton formation can lead tovery cleanstable.and extremlyshortpulses in a modelocked laser.3.4Universality of the NSEAbove, we derived the NSE in detail for the case of disperison and self-phasemodulation.The input for the NSE is surprisingly low, we only have toadmittthefirstnontrivialdispersiveeffect andthelowestordernonlinealeffect that is possible in an isotropic and homogeneous medium like glass.gas or plasmas. Therefore, the NSE and its properties are important formany other effects like self-focusing [19], Langmuir waves in plasma physics,and waves inproteine molecules [2o].Self-focusing will be treated in moredetail later, because it is the basis for Kerr-Lens Mode Locking.3.5Soliton Perturbation TheoryFrom the previous discussion, we have full knowledge about the possiblesolutions of the NSE that describes a special Hamiltonian system. However,the NSE hardly describes a real physical system such as, for example, a realoptical fiber in all its aspects [21, 22]. Indeed the NSE itself, as we haveseen during the derivation in the previous sections, is only an approximationto the complete wave equation.We approximated the dispersion relationby a parabola at the assumed carrier frequency of the soliton. Also theinstantaneous Kerr effect described by an intensity dependent refractive indexis only an approximation to the real x(3)-nonlinearity of a Kerr-medium [23
3.4. UNIVERSALITY OF THE NSE 77 To appreciate these properties of the solutions of the NSE, we solve the NSE for a rectangular shaped initial pulse. The result is shown in Fig. 3.8. The scattering problem, that has to be solved for this initial condition, is the same as for a nonrelativistic particle in a rectangular potential box [32]. The depth of the potential is chosen small enough, so that it has only one bound state. Thus, we start with a wave composed of a fundamental soliton and continuum. It is easy to recognize the continuum contribution, i.e. the dispersive wave, that separates from the soliton during propagation. This solution illustrates, that soliton pulse shaping due to the presence of dispersion and self-phase modulation may have a strong impact on pulse generation [18]. When the dispersion and self-phase modulation are properly adjusted, soliton formation can lead to very clean, stable, and extremly short pulses in a modelocked laser. 3.4 Universality of the NSE Above, we derived the NSE in detail for the case of disperison and self-phase modulation. The input for the NSE is surprisingly low, we only have to admitt the first nontrivial dispersive effect and the lowest order nonlinear effect that is possible in an isotropic and homogeneous medium like glass, gas or plasmas. Therefore, the NSE and its properties are important for many other effects like self-focusing [19], Langmuir waves in plasma physics, and waves in proteine molecules [20]. Self-focusing will be treated in more detail later, because it is the basis for Kerr-Lens Mode Locking. 3.5 Soliton Perturbation Theory From the previous discussion, we have full knowledge about the possible solutions of the NSE that describes a special Hamiltonian system. However, the NSE hardly describes a real physical system such as, for example, a real optical fiber in all its aspects [21, 22]. Indeed the NSE itself, as we have seen during the derivation in the previous sections, is only an approximation to the complete wave equation. We approximated the dispersion relation by a parabola at the assumed carrier frequency of the soliton. Also the instantaneous Kerr effect described by an intensity dependent refractive index is only an approximation to the real χ(3)-nonlinearity of a Kerr-medium [23