68CHAPTER3.NONLINEARPULSEPROPAGATIONnormalized form?A'0A(2,t)+ 2|A1?A'(3.4)ot20zThis is equivalent to set D2 = -1 and d = 2. For the numerical simulationswhich are shown in the next chapters, we simulate the normalized eq.(3.4)and theaxes are in normalized units of position and time3.3.2The Fundamental SolitonWelook fora stationary wavefunction of the NSE (3.3),such that its absolutesquare is a self-consistent potential. A potential of that kind is well knownfrom Quantum Mechanics, the sech?-Potential [8], and therefore the shape ofthe solitary pulse is a sechAs(z, t) = Aosecl(3.5)where is the nonlinear phase shift of the soliton=1sAp29=(3.6)2The soltion phase shift is constant over the pulse with respect to time incontrast to the case of self-phase modulation only, where the phase shift isproportional to the instantaneous power.Thebalance between thenonlineareffects and the linear effects requires that the nonlinear phase shift is equalto the dispersive spreading of the pulse[D2]O(3.7)Since the field amplitude A(z,t) is normalized, such that the absolute squareis the intensity,the soliton energy fluence is given by[A(z,t)’dt = 2ABT.(3.8)From eqs.(3.6)to (3.8),we obtainfor constantpulseenergy fuence, that thewidth of the soliton is proportional to the amount of negative dispersion4|D2](3.9)dw
68 CHAPTER 3. NONLINEAR PULSE PROPAGATION normalized form j ∂A´(z´, t) ∂z´ = ∂2A´ ∂t´ 2 + 2|A´| 2 A´ (3.4) This is equivalent to set D2 = −1 and δ = 2. For the numerical simulations, which are shown in the next chapters, we simulate the normalized eq.(3.4) and the axes are in normalized units of position and time. 3.3.2 The Fundamental Soliton We look for a stationary wave function of the NSE (3.3), such that its absolute square is a self-consistent potential. A potential of that kind is well known from Quantum Mechanics, the sech2-Potential [8], and therefore the shape of the solitary pulse is a sech As(z, t) = A0sech µ t τ ¶ e−jθ, (3.5) where θ is the nonlinear phase shift of the soliton θ = 1 2 δA2 0z (3.6) The soltion phase shift is constant over the pulse with respect to time in contrast to the case of self-phase modulation only, where the phase shift is proportional to the instantaneous power. The balance between the nonlinear effects and the linear effects requires that the nonlinear phase shift is equal to the dispersive spreading of the pulse θ = |D2| τ 2 z. (3.7) Since the field amplitude A(z, t) is normalized, such that the absolute square is the intensity, the soliton energy fluence is given by w = Z ∞ −∞ |As(z, t)| 2 dt = 2A2 0τ. (3.8) From eqs.(3.6) to (3.8), we obtain for constant pulse energy fluence, that the width of the soliton is proportional to the amount of negative dispersion τ = 4|D2| δw . (3.9)
693.3.THENONLINEARSCHRODINGEREQUATIONNote, the pulse area for a fundamental soliton is only determined by thedispersionandtheself-phasemodulationcoefficient[D2]Pulse Area -(3.10)[A(z,t)|dt=πAoT=28Thus, an initial pulse with a different area can not just develope into a puresoliton.1.510.510.6Dis0.8Figure 3.3: Propagation of a fundamental soliton.Fig. 3.3 shows the numerical solution of the NSE for the fundamentalsoliton pulse. The distance, after which the soliton aquires a phase shift of/4,is called the solitonperiod,forreasons,which will become clearinthenext section.Since the dispersion is constant over the frequency, i.e. the NSE hasno higher order dispersion, the center frequency of the soliton can be chosenarbitrarily.However,due to the dispersion, the group velocities of the solitonswith different carrier frequencies will be different.One easily finds by aGallilei tranformation toamovingframe, that the NSEposseses thefollowinggeneral fundamental soliton solutionA.(z,t) = Aosech(r(2,t)e-jo(=,t),(3.11)
3.3. THE NONLINEAR SCHRÖDINGER EQUATION 69 Note, the pulse area for a fundamental soliton is only determined by the dispersion and the self-phase modulation coefficient Pulse Area = Z ∞ −∞ |As(z, t)|dt = πA0τ = π r|D2| 2δ . (3.10) Thus, an initial pulse with a different area can not just develope into a pure soliton. 0 0.2 0.4 0.6 0.8 1 -6 -4 -2 0 2 4 6 0.5 1 1.5 2 Amplitude Distance z Time Figure 3.3: Propagation of a fundamental soliton. Fig. 3.3 shows the numerical solution of the NSE for the fundamental soliton pulse. The distance, after which the soliton aquires a phase shift of π/4, is called the soliton period, for reasons, which will become clear in the next section. Since the dispersion is constant over the frequency, i.e. the NSE has no higher order dispersion, the center frequency of the soliton can be chosen arbitrarily. However, due to the dispersion, the group velocities of the solitons with different carrier frequencies will be different. One easily finds by a Gallilei tranformation to a moving frame, that the NSE posseses the following general fundamental soliton solution As(z, t) = A0sech(x(z, t))e−jθ(z,t) , (3.11)
70CHAPTER3.NONLINEARPULSEPROPAGATIONwith(3.12) 2[D2|poz- to),and a nonlinear phase shift = Po(t - to) + [D2l 2+00(3.13)Thus,the energy fluence w or amplitude Ao,the carrier frequency po, thephase So and the origin to, i.e. the timing of the fundamental soliton arenot yet determined. Only the soliton area is fixed. The energy fuence andwidth are determined if one of them is specified, given a certain dispersionand SPM-coefficient.3.3.3Higher Order SolitonsThe NSE has constant dispersion,in our case negative dispersion.Thatmeans the group velocity depends linearly on frequency. We assume, thattwofundamental soltionsarefarapartfromeachother,sothattheydonotinteract.Then this linear superpositon is for all practical purposes anothersolution of theNSE.If we choose the carrier frequency of the soliton, startingat a latertime, higher thanthe one of the soliton in front.thelater solitonwill catch up with the leading soliton due to the negative dispersion and thepulses will collide.Figure 3.4 shows this situation. Obviously, the two pulses recover com-pletely from the collision, i.e. the NSE has true soliton solutions. The solitonshave particle like properties. A solution, composed of several fundamentalsolitons, is called a higher order soliton. If we look closer to figure 3.4, werecognize,that the solitonatrest in thelocal time frame,and which followsthe t = o line without the collision, is somewhat pushed forward due to thecollision.Adetailed analysis of the collision wouldalsoshow,thatthephasesof the solitons have changed [4]. The phase changes due to soliton collisionsare used to built all optical switches [10], using backfolded Mach-Zehnder in-terferometers,whichcanberealized inaself-stabilizedwaybySagnacfiberloops
70 CHAPTER 3. NONLINEAR PULSE PROPAGATION with x = 1 τ (t − 2|D2|p0z − t0), (3.12) and a nonlinear phase shift θ = p0(t − t0) + |D2| µ 1 τ 2 − p2 0 ¶ z + θ0. (3.13) Thus, the energy fluence w or amplitude A0, the carrier frequency p0, the phase θ0 and the origin t0, i.e. the timing of the fundamental soliton are not yet determined. Only the soliton area is fixed. The energy fluence and width are determined if one of them is specified, given a certain dispersion and SPM-coefficient. 3.3.3 Higher Order Solitons The NSE has constant dispersion, in our case negative dispersion. That means the group velocity depends linearly on frequency. We assume, that two fundamental soltions are far apart from each other, so that they do not interact. Then this linear superpositon is for all practical purposes another solution of the NSE. If we choose the carrier frequency of the soliton, starting at a later time, higher than the one of the soliton in front, the later soliton will catch up with the leading soliton due to the negative dispersion and the pulses will collide. Figure 3.4 shows this situation. Obviously, the two pulses recover completely from the collision, i.e. the NSE has true soliton solutions. The solitons have particle like properties. A solution, composed of several fundamental solitons, is called a higher order soliton. If we look closer to figure 3.4, we recognize, that the soliton at rest in the local time frame, and which follows the t = 0 line without the collision, is somewhat pushed forward due to the collision. A detailed analysis of the collision would also show, that the phases of the solitons have changed [4]. The phase changes due to soliton collisions are used to built all optical switches [10], using backfolded Mach-Zehnder interferometers, which can be realized in a self-stabilized way by Sagnac fiber loops
713.3.THENONLINEARSCHRODINGEREQUATION0s10Distance zTime10Figure 3.4: A soliton with high carrier frequency collides with a soliton oflower carrierfrequency.Afterthe collison both pulses recover completely
3.3. THE NONLINEAR SCHRÖDINGER EQUATION 71 0 1 2 3 4 5 -10 -5 0 5 10 0.5 1 1.5 2 Amplitude Distance z Time Figure 3.4: A soliton with high carrier frequency collides with a soliton of lower carrier frequency. After the collison both pulses recover completely
72CHAPTER3.NONLINEARPULSEPROPAGATION3.532.5C2-10.50.5Distance zTime1.5F6050~4030~20~100.50.5Distance z00.5Frequency1.5Figure 3.5: (a) Amplitude and, (b) Spectrum of a higher order soliton com-posed of two fundamental solitons with the same carrier frequency
72 CHAPTER 3. NONLINEAR PULSE PROPAGATION 0 0.5 1 1.5 -6 -4 -2 0 2 4 6 0.5 1 1.5 2 2.5 3 3.5 Amplitude Distance z Time 0 0.5 1 1.5 -1 -0.5 0 0.5 1 10 20 30 40 50 60 Spectrum Distance z Frequency Figure 3.5: (a) Amplitude and, (b) Spectrum of a higher order soliton composed of two fundamental solitons with the same carrier frequency