Chapter 3Nonlinear Pulse PropagationThere are many nonlinear pulse propagation problems worthwhile of beingconsidered in detail, such as pulse propagation through a two-level mediumin the coherent regime, which leads to self-induced transparency and solitonsgoverned by the Sinus-Gordon-Equation. The basic model for the medium isthe two-level atom discussed before with infinitely long relaxation times Ti,2,i.e. assuming that the pulses are much shorter than the dephasing time in themedium. In such a medium pulses exist, where the first half of the pulse fullyinverts the medium and the second half of the pulse extracts the energy fromthe medium. The integral over the Rabi-frequency as defined in Eq.(2.39) isthan a mutiple of 2π. The interested reader is refered to the book of Allenand Eberly [1]. Here, we are interested in the nonlinear dynamics due tothe Kerr-effect which is most important for understanding pulse propagationproblems in optical communications and short pulse generation.3.1 The Optical Kerr-effectIn an isotropic and homogeneous medium, the refractive index can not de-pend on the direction of the electric field. Therefore, to lowest order, therefractive index of such a medium can only depend quadratically on thefield, i.e. on the intensity [22]n = n(w, [A)2) ~ no(w) + n2,L|A/2(3.1)Here, we assume, that the pulse envelope A is normalized such that Aj? isthe intensity of the pulse. This is the optical Kerr effect and n2,L is called63
Chapter 3 Nonlinear Pulse Propagation There are many nonlinear pulse propagation problems worthwhile of being considered in detail, such as pulse propagation through a two-level medium in the coherent regime, which leads to self-induced transparency and solitons governed by the Sinus-Gordon-Equation. The basic model for the medium is the two-level atom discussed before with infinitely long relaxation times T1,2, i.e. assuming that the pulses are much shorter than the dephasing time in the medium. In such a medium pulses exist, where the first half of the pulse fully inverts the medium and the second half of the pulse extracts the energy from the medium. The integral over the Rabi-frequency as defined in Eq.(2.39) is than a mutiple of 2π. The interested reader is refered to the book of Allen and Eberly [1]. Here, we are interested in the nonlinear dynamics due to the Kerr-effect which is most important for understanding pulse propagation problems in optical communications and short pulse generation. 3.1 The Optical Kerr-effect In an isotropic and homogeneous medium, the refractive index can not depend on the direction of the electric field. Therefore, to lowest order, the refractive index of such a medium can only depend quadratically on the field, i.e. on the intensity [22] n = n(ω, |A| 2 ) ≈ n0(ω) + n2,L|A| 2 . (3.1) Here, we assume, that the pulse envelope A is normalized such that |A| 2 is the intensity of the pulse. This is the optical Kerr effect and n2,L is called 63
64CHAPTER3.NONLINEARPULSEPROPAGATIONMaterialn2,,[cm?/W]Refractiveindexn3·10-161.76@850nmSapphire (Al,O3)2.46-10-161.45@1064nmFused Quarz2.9.10-16Glass (LG-760)1.5@1064m6.2·10-16YAG(Y3Al,O12)1.82 @ 1064 nm1.72-1016YLF (LiYF4),ne1.47@1047nm4·10-14Si3.3@1550nmTable3.l:Nonlinear refractive indexcoefficients for different materials.Inthe literature most often the electro-statitic unit svstem is in use. The conversion is n2.L[cm?/W] = 4.19.10-3n2,L[esu]/nothe intensity dependent refractive index coefficient. Note, the nonlinear in-dex depends on the polarization of the field and without going further intodetails, we assume that we treat a linearily polarized electric field. For mosttransparent materialsthe intensity dependent refractiveindexis positive3.2Self-Phase Modulation (SPM)In a purely one dimensional propagation problem, the intensity dependentrefractive index imposes an additional self-phase shift on the pulse envelopeduring propagation, which is proportional to the instantaneous intensity ofthe pulse0A(z,t)(3.2)-jkon2,L|A(z,t)2 A(z,t) = -js|A(z,t)A(z,t).0where = kon2.L is the self-phase modulation coefficient. Self-phase modu-lation(SPM)leads onlytoaphase shift inthetimedomain.Therefore,theintensity profile of the pulse does not change only the spectrum of the pulsechanges, as discussed in the class on nonlinear optics. Figure (3.1) showsthe spectrum of a Gaussian pulse subject to SPM during propagation (forS = 2 and normalized units). New frequency components are generated bythe nonlinear process via four wave mixing (FWM). If we look at the phase ofthe pulse during propagation due to self-phase modulation, see Fig. 3.2 (a),we find, that the pulse redistributes its energy, such that the low frequencycontributions are inthefront of thepulse and the highfrequencies in theback of the pulse, similar to the case of positive dispersion
64 CHAPTER 3. NONLINEAR PULSE PROPAGATION Material Refractive index n n2,L[cm2/W] Sapphire (Al2O3) 1.76 @ 850 nm 3·10−16 Fused Quarz 1.45 @ 1064 nm 2.46·10−16 Glass (LG-760) 1.5 @ 1064 nm 2.9·10−16 YAG (Y3Al5O12) 1.82 @ 1064 nm 6.2·10−16 YLF (LiYF4), ne 1.47 @ 1047 nm 1.72·10−16 Si 3.3 @ 1550 nm 4·10−14 Table 3.1: Nonlinear refractive index coefficients for different materials. In the literature most often the electro-statitic unit system is in use. The conversion is n2,L[cm2/W]=4.19 · 10−3n2,L[esu]/n0 the intensity dependent refractive index coefficient. Note, the nonlinear index depends on the polarization of the field and without going further into details, we assume that we treat a linearily polarized electric field. For most transparent materials the intensity dependent refractive index is positive. 3.2 Self-Phase Modulation (SPM) In a purely one dimensional propagation problem, the intensity dependent refractive index imposes an additional self-phase shift on the pulse envelope during propagation, which is proportional to the instantaneous intensity of the pulse ∂A(z, t) ∂z = −jk0n2,L|A(z, t)| 2 A(z, t) = −jδ|A(z, t)| 2 A(z, t). (3.2) where δ = k0n2,L is the self-phase modulation coefficient. Self-phase modulation (SPM) leads only to a phase shift in the time domain. Therefore, the intensity profile of the pulse does not change only the spectrum of the pulse changes, as discussed in the class on nonlinear optics. Figure (3.1) shows the spectrum of a Gaussian pulse subject to SPM during propagation (for δ = 2 and normalized units). New frequency components are generated by the nonlinear process via four wave mixing (FWM). If we look at the phase of the pulse during propagation due to self-phase modulation, see Fig. 3.2 (a), we find, that the pulse redistributes its energy, such that the low frequency contributions are in the front of the pulse and the high frequencies in the back of the pulse, similar to the case of positive dispersion
653.2.SELF-PHASEMODULATION(SPM)10080~3402001.51Distance z0.500.5Frequency-73-1.5Figure 3.1: Spectrum [A(z,w = 2 f)? of a Gaussian pulse subject to self-phase modulation
3.2. SELF-PHASE MODULATION (SPM) 65 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 20 40 60 80 100 Spectrum Distance z Frequency Figure 3.1: Spectrum |Aˆ(z, ω = 2πf)| 2 of a Gaussian pulse subject to selfphase modulation
66CHAPTER3.NONLINEARPULSEPROPAGATION(a)4 IntensityBackFrontTime tPhase(b) Time t(c)InstantaneousFrequencyTime tFigure 3.2: (a) Intensity, (b) phase and (c) instantaneous frequency of aGaussian pulse during propagation through a medium with positive self-phasemodulation
66 CHAPTER 3. NONLINEAR PULSE PROPAGATION (a) Time t Intensity Front Back Time t (b) Phase (c) Instantaneous Frequency Time t Figure 3.2: (a) Intensity, (b) phase and (c) instantaneous frequency of a Gaussian pulse during propagation through a medium with positive selfphase modulation
3.3.THENONLINEAR SCHRODINGEREQUATION673.3The Nonlinear Schrodinger EquationIf both effects, dispersion and self-phase modulation, act simultaneously onthe pulse, the field envelope obeys the equationDA.0A(z,t)+A2A,(3.3)Ot202This equation is called the Nonlinear Schrodinger Equation (NSE) - if weput the imaginary unit on the left hand side -, since it has the form of aSchrodinger Equation.Its called nonlinear,because the potential energyis derived from the square of the wave function itself.As we have seenfrom the discussion in the last sections, positive dispersion and positive self-phase modulation lead to a similar redistribution of the spectral components.This enhances the pulse spreading in time. However, if we have negativedispersion, i.e. a wave packet with high carrier frequency travels faster thana wave packet with a low carrier frequency, then, the high frequency wavepackets generated by self-phase modulation in the front of the pulse havea chance to catch up with the pulse itself due to the negative dispersion.The opposite is the case for the low frequencies. This arrangement resultsin pulses that do not disperse any more, i.e. solitary waves. That negativedispersionisnecessarytocompensatethepositiveKerreffectisalsoobviousfrom the NSE (3.3).Because, for apositiveKerr effect, thepotential energyin the NSE is always negative. There are only bound solutions, i.e. brightsolitary waves, if the kinetic energy term, i.e. the dispersion, has a negativesign, D2 < 0.3.3.1 The Solitons of the NSEIn the following,we study different solutions of the NSE forthe case ofnegative dispersion and positive self-phase modulation.We do not intendto give a full overview over the solution manyfold of the NSE in its fullmathematical depth here, because it is not necessary for the following. Thiscan be found in detail elsewhere [4, 5, 6, 7].Without loss of generality, by normalization of the field amplitude A:2, the propagation distance z= z.+2/D2, and the time t = t. T,the NSE (3.3) with negative dispersion can always be transformed into the
3.3. THE NONLINEAR SCHRÖDINGER EQUATION 67 3.3 The Nonlinear Schrödinger Equation If both effects, dispersion and self-phase modulation, act simultaneously on the pulse, the field envelope obeys the equation j ∂A(z, t) ∂z = −D2 ∂2A ∂t2 + δ|A| 2 A, (3.3) This equation is called the Nonlinear Schrödinger Equation (NSE) - if we put the imaginary unit on the left hand side -, since it has the form of a Schrödinger Equation. Its called nonlinear, because the potential energy is derived from the square of the wave function itself. As we have seen from the discussion in the last sections, positive dispersion and positive selfphase modulation lead to a similar redistribution of the spectral components. This enhances the pulse spreading in time. However, if we have negative dispersion, i.e. a wave packet with high carrier frequency travels faster than a wave packet with a low carrier frequency, then, the high frequency wave packets generated by self-phase modulation in the front of the pulse have a chance to catch up with the pulse itself due to the negative dispersion. The opposite is the case for the low frequencies. This arrangement results in pulses that do not disperse any more, i.e. solitary waves. That negative dispersion is necessary to compensate the positive Kerr effect is also obvious from the NSE (3.3). Because, for a positive Kerr effect, the potential energy in the NSE is always negative. There are only bound solutions, i.e. bright solitary waves, if the kinetic energy term, i.e. the dispersion, has a negative sign, D2 < 0. 3.3.1 The Solitons of the NSE In the following, we study different solutions of the NSE for the case of negative dispersion and positive self-phase modulation. We do not intend to give a full overview over the solution manyfold of the NSE in its full mathematical depth here, because it is not necessary for the following. This can be found in detail elsewhere [4, 5, 6, 7]. Without loss of generality, by normalization of the field amplitude A = A´ τ q2D2 δ , the propagation distance z = z´· τ 2/D2, and the time t = t´· τ , the NSE (3.3) with negative dispersion can always be transformed into the
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