78CHAPTER3.NONLINEARPULSEPROPAGATION24]. Therefore, it is most important to study what happens to a solitonsolution of the NSE due toperturbing effectslikehigher order dispersionfinite response times of the nonlinearites, gain and the finite gain bandwidthofamplifiers,that compensatefortheinevitablelossinareal system.The investigation of solitons under perturbations is as old as the solitonsitself.Manyauthors treat theperturbing effects in the scattering domain[25, 26].Only recently, a perturbation theory on the basis of the linearizedNSE has been developed,which is much more illustrative then a formulationin the scattering amplitudes. This was first used by Haus [27] and rigorouslyformulated by Kaup [28]. In this section, we will present this approach asfar as it is indispensible for the following.A system, where the most important physical processes are dispersionand self-phase modulation, is described by the NSE complimented with someperturbationtermF0A(z,t)[nIA AP4 + FL.,.(3.14)2In the following, we are interested what happens to a solution of the fullequation (3.14) which is very close to a fundamental soliton, i.e.A(z,t) =a(=) + △A(z,t)| e-jks,(3.15)Here, a(r) is the fundamental soliton according to eq.(3.5)= Ao sech((3.16)and1SAR(3.17)ks-is the phase shift of the soliton per unit length, i.e. the soltion wave vector.A deviation from the ideal soliton can arise either due to the additionaldriving term F on the right side or due to a deviation already present inthe initial condition. We use the form (3.15)as an ansatz to solve the NSEto first order in the perturbation A, i.e. we linearize the NSE around thefundamental solitonandobtainfortheperturbationDAA△A + 2sech2(a) (2A + △A*)0z02+F(A, A*,2)eiks,(3.18)
78 CHAPTER 3. NONLINEAR PULSE PROPAGATION 24]. Therefore, it is most important to study what happens to a soliton solution of the NSE due to perturbing effects like higher order dispersion, finite response times of the nonlinearites, gain and the finite gain bandwidth of amplifiers, that compensate for the inevitable loss in a real system. The investigation of solitons under perturbations is as old as the solitons itself. Many authors treat the perturbing effects in the scattering domain [25, 26]. Only recently, a perturbation theory on the basis of the linearized NSE has been developed, which is much more illustrative then a formulation in the scattering amplitudes. This was first used by Haus [27] and rigorously formulated by Kaup [28]. In this section, we will present this approach as far as it is indispensible for the following. A system, where the most important physical processes are dispersion and self-phase modulation, is described by the NSE complimented with some perturbation term F ∂A(z, t) ∂z = −j ∙ |D2| ∂2A ∂t2 + δ|A| 2 A ¸ + F(A, A∗ , z). (3.14) In the following, we are interested what happens to a solution of the full equation (3.14) which is very close to a fundamental soliton, i.e. A(z, t) = ∙ a( t τ ) + ∆A(z, t) ¸ e−jksz . (3.15) Here, a(x) is the fundamental soliton according to eq.(3.5) a( t τ ) = A0 sech( t τ ), (3.16) and ks = 1 2 δA2 0 (3.17) is the phase shift of the soliton per unit length, i.e. the soltion wave vector. A deviation from the ideal soliton can arise either due to the additional driving term F on the right side or due to a deviation already present in the initial condition. We use the form (3.15) as an ansatz to solve the NSE to first order in the perturbation ∆A, i.e. we linearize the NSE around the fundamental soliton and obtain for the perturbation ∂∆A ∂z = −jks ∙µ ∂2 ∂x2 − 1 ¶ ∆A + 2sech2 (x) (2∆A + ∆A∗ ) ¸ +F(A, A∗ , z)ejksz , (3.18)
793.5.SOLITONPERTURBATIONTHEORYwhere r = t/t. Due to the nonlinearity, the field is coupled to its complexconjugate. Thus, eg.(3.18) corresponds actually to two equations, one for theamplitude and one for its complex conjugate. Therefore, we introduce thevectornotationAAAA=(3.19)AAWe further introduce the normalized propagation distance z = ksz and thenormalized time =t/t.The linearized perturbed NSE is then given bya-F(A, A, 2)ei2AA=LAA +(3.20)02ksHere, L is the operator which arises from the linearization of the NSE02(3.21)L=-- 1) + 2 sech (r)(2 + α1)10ar2where i,i = 1, 2, 3 are the Pauli matrices. For a solution of the inhomoge-neous equation (3.20), we need the eigenfunctions and the spectrum of thedifferential operatorL.Wefound in section 3.3.2,thatthefundamental soli-ton has four degrees of freedom, four free parameters. This gives already fourknown eigensolutions and mainsolutions of the linearized NSE, respectively.Theyaredeterminedbythederivativesofthegeneralfundamental solitonsolutions according to eqs.(3.11) to (3.13) with respect to free parameters.These eigenfunctions are1(1 - r tanh a)a(r) (1(3.22)fu(r) =fo(c) = -ja(n)(-1 ),(3.23)(m) = -j ra(d(二)(3.24)f(n) = tanh(n)a(n)()(3.25)and they describe perturbations of the soliton energy,phase, carrier frequencyand timing. One component of each of these vector functions is shown in Fig.3.9
3.5. SOLITON PERTURBATION THEORY 79 where x = t/τ . Due to the nonlinearity, the field is coupled to its complex conjugate. Thus, eq.(3.18) corresponds actually to two equations, one for the amplitude and one for its complex conjugate. Therefore, we introduce the vector notation ∆A = µ ∆A ∆A∗ ¶ . (3.19) We further introduce the normalized propagation distance z0 = ksz and the normalized time x = t/τ . The linearized perturbed NSE is then given by ∂ ∂z0 ∆A = L∆A + 1 ks F(A, A∗ , z)ejz0 (3.20) Here, L is the operator which arises from the linearization of the NSE L = −jσ3 ∙ ( ∂2 ∂x2 − 1) + 2 sech2 (x)(2 + σ1) ¸ , (3.21) where σi, i = 1, 2, 3 are the Pauli matrices. For a solution of the inhomogeneous equation (3.20), we need the eigenfunctions and the spectrum of the differential operator L. We found in section 3.3.2, that the fundamental soliton has four degrees of freedom, four free parameters. This gives already four known eigensolutions and mainsolutions of the linearized NSE, respectively. They are determined by the derivatives of the general fundamental soliton solutions according to eqs.(3.11) to (3.13) with respect to free parameters. These eigenfunctions are fw(x) = 1 w(1 − x tanh x)a(x) µ 1 1 ¶ , (3.22) fθ(x) = −ja(x) µ 1 −1 ¶ , (3.23) fp(x) = −j xτ a(x) µ 1 −1 ¶ , (3.24) ft(x) = 1 τ tanh(x) a(x) µ 1 1 ¶ , (3.25) and they describe perturbations of the soliton energy, phase, carrier frequency and timing. One component of each of these vector functions is shown in Fig. 3.9
80CHAPTER3.NONLINEARPULSEPROPAGATIONeW个11.01.010.80.80.60.60.40.20.40.00.2-0.20.0-0.4X-2-6-4-224024046dctp个个o0.80.60.6i0.40.40.20.2 0.0 0.0-0.2-0.2-0.4-0.4-0.6-0.8-0.6X-2024-4-2024-4Figure 3.9:Perturabations in soliton amplitude (a),phase (b),frequency (c)and timing (d).FigurebyMITOcW.The action of the evolution operator of the linearized NSE on these solitonperturbations is1Lfufo,(3.26)=wLfg=0,(3.27)Lfp = -2r2ft,(3.28)Lft = 0.(3.29)
80 CHAPTER 3. NONLINEAR PULSE PROPAGATION Figure 3.9: Perturabations in soliton amplitude (a), phase (b), frequency (c), and timing (d). The action of the evolution operator of the linearized NSE on these soliton perturbations is Lfw = 1 w fθ, (3.26) Lfθ = 0, (3.27) Lfp = −2τ 2 ft, (3.28) Lft = 0. (3.29) -4 -2 Energy Perturbation, f -0.4 -0.2 0.2 0.0 0.4 0.6 0.8 1.0 0 x x w p t θ 2 4 -4-6 -2 Phase Perturbation, jf -4 -2 Frequency Perturbation, jf Timing Perturbation, f -0.8 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.6 0 x x 2 4 0.0 0.2 0.4 0.6 0.8 1.0 0 2 64 -4 -2 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0 2 4 a b c d Figure by MIT OCW
813.5.SOLITONPERTURBATIONTHEORYEquations (3.26) and (3.28) indicate, that perturbations in energy andcarrier frequency are converted to additional phase and timing fluctuationsof the pulse due to SPM and GVD. This is the base for soliton squeezing inoptical fibers [27].The timing and phase perturbations can increase withoutbounds, because the system is autonomous, the origin for the Gordon-Hauseffect, [29] and there is no phase reference in the system. The full continuousspectrum of the linearized NSE has been studied by Kaup [28] and is givenby(3.30)Lfk = Afk,(3.31)= (K2 +1),(k-jtanhrfi(r) =e-jka(3.32)sechrandLf=fe,(3.33) = -(2 + 1),(3.34)f=oifk.(3.35)Our definition of the eigenfunctions is slightly different from Kaup [28], be-cause we also define the inner product in the complex space as<u/v>ut(c)v(r)dr(3.36)Adopting this definition, the inner product of a vector with itself in thesubspace where the second component is the complex conjugate of the firstcomponent is the energy of the signal, a physical quantity.The operatorL isnot self-adjointwithrespecttothis innerproduct.Thephysical origin for this mathematical property is, that the linearized systemdoes not conserve energy due to the parametric pumping by the soliton.However, from (3.21) and (3.36), we can easily see that the adjoint operatoris given byL+=-03Lo3,(3.37)and therefore, we obtain for the spectrum of the adjoint operatorL+(t) = (tr(),(3.38)^(+) = -j (2 + 1),(3.39)1r - (+gft,(3.40)
3.5. SOLITON PERTURBATION THEORY 81 Equations (3.26) and (3.28) indicate, that perturbations in energy and carrier frequency are converted to additional phase and timing fluctuations of the pulse due to SPM and GVD. This is the base for soliton squeezing in optical fibers [27]. The timing and phase perturbations can increase without bounds, because the system is autonomous, the origin for the Gordon-Haus effect, [29] and there is no phase reference in the system. The full continuous spectrum of the linearized NSE has been studied by Kaup [28] and is given by Lf k = λkfk, (3.30) λk = j(k2 + 1), (3.31) fk(x) = e−jkx µ (k − jtanhx)2 sech2 x ¶ , (3.32) and L¯f k = λ¯k ¯fk, (3.33) λ¯k = −j(k2 + 1), (3.34) ¯fk = σ1fk. (3.35) Our definition of the eigenfunctions is slightly different from Kaup [28], because we also define the inner product in the complex space as < u|v >= 1 2 Z +∞ −∞ u+(x)v(x)dx. (3.36) Adopting this definition, the inner product of a vector with itself in the subspace where the second component is the complex conjugate of the first component is the energy of the signal, a physical quantity. The operator L is not self-adjoint with respect to this inner product. The physical origin for this mathematical property is, that the linearized system does not conserve energy due to the parametric pumping by the soliton. However, from (3.21) and (3.36), we can easily see that the adjoint operator is given by L+ = −σ3Lσ3, (3.37) and therefore, we obtain for the spectrum of the adjoint operator L+f (+) k = λ(+) k f (+) k , (3.38) λ(+) k = −j (k2 + 1), (3.39) f (+) k = 1 π(k2 + 1)2σ3fk, (3.40)
82CHAPTER3.NONLINEARPULSEPROPAGATIONandL+(+) = (+)(+),(3.41)(+) = j(2 + 1),(3.42)1 - (2+p0d.(3.43)The eigenfunctions to L and its adjoint are mutually orthogonal to eachother,and they are already properly normalized<f(+)f>= (-),<(+)>=8(k-)<(+)f> = <f(+)>= 0.This system, which describes the continuum excitations, is made completeby taking also into account the perturbations of the four degrees of freedomof the soliton (3.22) - (3.25) and their adjointsf(+)() = j2TOsfe(a)=2Ta(r)(3.44) f(+)(g) = -2jTo3fu(r)-2if(1 - r tanh a)a(r)(3.45)w2jT21f(+)(a) tanh ra(a)(3.46)03f(r):wwosf() = 2r22jTf(+)(c)--ra(r)(3.47)wNow, the unity can be decomposed into two projections, one onto the con-tinuum and one onto the perturbation of the soliton variables [28]~dk[f><f(+)}+><(+)](-) =+ [f><f(+)|+[f><f+)(3.48)+ If, >< f(+)/ + [f >< f(+)].Any deviation △A can be decomposed into a contribution that leads to a soli-ton with a shift in the four soliton paramters and a continuum contributionacAA()=w(z)fw+e(2)fe+Ap(z)f,+△t(z)ft+ac(z).(3.49)
82 CHAPTER 3. NONLINEAR PULSE PROPAGATION and L+¯f (+) k = λ¯(+) k ¯f (+) k , (3.41) λ¯(+) k = j(k2 + 1), (3.42) ¯f (+) k = 1 π(k2 + 1)2σ3 ¯fk. (3.43) The eigenfunctions to L and its adjoint are mutually orthogonal to each other, and they are already properly normalized < f (+) k |fk0 > = δ(k − k0 ), < ¯f (+) k | ¯fk0 >= δ(k − k0 ) < ¯f (+) k |fk0 > = < f (+) k | ¯fk0 >= 0. This system, which describes the continuum excitations, is made complete by taking also into account the perturbations of the four degrees of freedom of the soliton (3.22) - (3.25) and their adjoints f (+) w (x) = j2τσ3fθ(x)=2τ a(x) µ 1 1 ¶ , (3.44) f (+) θ (x) = −2jτσ3fw(x) = −2jτ w (1 − x tanh x)a(x) µ 1 −1 ¶ , (3.45) f (+) p (x) = −2jτ w σ3ft(x) = 2i w tanh xa(x) µ 1 −1 ¶ , (3.46) f (+) t (x) = 2jτ w σ3fp(x) = 2τ 2 w xa(x) µ 1 1 ¶ . (3.47) Now, the unity can be decomposed into two projections, one onto the continuum and one onto the perturbation of the soliton variables [28] δ(x − x0 ) = Z ∞ −∞ dk h |fk >< f (+) k | + | ¯fk >< ¯f (+) k | i + |fw >< f (+) w | + |fθ >< f (+) θ | (3.48) + |fp >< f (+) p | + |ft >< f (+) t |. Any deviation ∆A can be decomposed into a contribution that leads to a soliton with a shift in the four soliton paramters and a continuum contribution ac ∆A(z0 ) = ∆w(z0 )fw + ∆θ(z0 )fθ + ∆p(z0 )fp + ∆t(z0 )ft + ac(z0 ). (3.49)