188CHAPTER5.ACTIVEMODELOCKING5.5.1Stability ConditionWe want to show, that a stable soliton can exist in the presence of themodelocker and gain dispersion if the ratio between the negative GDD andgain dispersion is sufficiently large. From (5.61) we obtain the equations ofmotion for the soliton parameters and the continuum by carrying out thescalar product with the corresponding adjoint functions. Specifically, for thesoliton energy we getaw元2gMw(5.63)TROT322-224f(+)|Rac>+<We see that gain saturation does not lead to a coupling between the solitonand the continuum tofirst order in theperturbation,becausethey are or-thogonal to each other in the sense of the scalar product (3.36). This alsomeans that to first order the total field energy is contained in the soliton.Thus to zero order the stationary soliton energy Wo = 2Agt is determinedby the condition that the saturated gain is equal to the total loss due to thelinear loss l, gain filtering and modulator loss72g(5.64)MwMT?g-1:24302g-2with the saturated gain90(5.65)g=1+ Wo/ELLinearization around this stationary value gives for the soliton perturbations
188 CHAPTER 5. ACTIVE MODE LOCKING 5.5.1 Stability Condition We want to show, that a stable soliton can exist in the presence of the modelocker and gain dispersion if the ratio between the negative GDD and gain dispersion is sufficiently large. From (5.61) we obtain the equations of motion for the soliton parameters and the continuum by carrying out the scalar product with the corresponding adjoint functions. Specifically, for the soliton energy we get TR ∂W ∂T = 2 µ g − l − g 3Ω2 gτ 2 − π2 24Mω2 Mτ 2 ¶ W (5.63) + < f (+) w |Rac > . We see that gain saturation does not lead to a coupling between the soliton and the continuum to first order in the perturbation, because they are orthogonal to each other in the sense of the scalar product (3.36). This also means that to first order the total field energy is contained in the soliton. Thus to zero order the stationary soliton energy W0 = 2A2 0τ is determined by the condition that the saturated gain is equal to the total loss due to the linear loss l, gain filtering and modulator loss g − l = π2 24Mω2 Mτ 2 + g 3Ω2 gτ 2 (5.64) with the saturated gain g = g0 1 + W0/EL . (5.65) Linearization around this stationary value gives for the soliton perturbations
1895.5.ACTIVEMODELOCKINGWITHSOLITONFORMATIONgWoTROT322T2(1+Wo/EL)CEL元2Mw?△W+<f(+)|Ra>(5.66)+12e<f(+)|Ra>TR(5.67)OT4gap(5.68)+|Ra.>TRaT3022T2元2atMw%?t+2D/ApTROT6f(+)|Rac >(5.69)+<and forthe continuumweobtain0g(k)TR=jΦo(k2 +1)g(k)+< f(+)|Ra>OT+< f(+R(ao(r)+△wfu+pf,)>-< f(+|MwM sin(wMTr)ao(a) > .△t(5.70)Thus the action of the active modelocker and gain dispersion has severaleffects.First,themodelockerleadstoa restoring force in thetiming of thesoliton (5.69).Second,the gain dispersion and theactivemodelockerlead tocoupling between the perturbed soliton and the continuum which results ina steady excitation of the continuum.However, as we will see later, the pulse width of the soliton, which can bestabilized by the modelocker, is not too far from the Gaussian pulse widthby only active mode locking. Then relation(5.71)WMT<1gTis fulflled. The weak gain dispersion and the weak active modelocker onlycouples the soliton to thecontinuum,but to first order the continuum doesnot couple back to the soliton. Neglecting higher order terms in the matrixelements of eq(5.70) [6] results in a decoupling of the soliton perturbationsfrom the continuum in (5.66) to (5.70). For a laser far above threshold, i.e
5.5. ACTIVE MODE LOCKING WITH SOLITON FORMATION 189 TR ∂∆W ∂T = 2Ã − g (1 + W0/EL) µW0 EL + 1 3Ω2 gτ 2 ¶ + π2 12 Mω2 Mτ 2 ! ∆W+ < f (+) w |Rac > (5.66) TR ∂∆θ ∂T = < f (+) θ |Rac > (5.67) TR ∂∆p ∂T = − 4g 3Ω2 gτ 2 ∆p + < f (+) p |Rac > (5.68) TR ∂∆t ∂T = −π2 6 Mω2 Mτ 2 ∆t + 2|D|∆p + < f (+) t |Rac > (5.69) and for the continuum we obtain TR ∂g(k) ∂T = jΦ0(k2 + 1)g(k)+ < f (+) k |Rac > + < f (+) k |R (a0(x) + ∆w fw + ∆p fp) > − < f (+) k |MωM sin(ωMτx)a0(x) > .∆t (5.70) Thus the action of the active modelocker and gain dispersion has several effects. First, the modelocker leads to a restoring force in the timing of the soliton (5.69). Second, the gain dispersion and the active modelocker lead to coupling between the perturbed soliton and the continuum which results in a steady excitation of the continuum. However, as we will see later, the pulse width of the soliton, which can be stabilized by the modelocker, is not too far from the Gaussian pulse width by only active mode locking. Then relation ωMτ ¿ 1 ¿ Ωgτ (5.71) is fulfilled. The weak gain dispersion and the weak active modelocker only couples the soliton to the continuum, but to first order the continuum does not couple back to the soliton. Neglecting higher order terms in the matrix elements of eq.(5.70) [6] results in a decoupling of the soliton perturbations from the continuum in (5.66) to (5.70). For a laser far above threshold, i.e
190CHAPTER5.ACTIVEMODELOCKINGWo/E, >> 1, gain saturation always stabilizes the amplitude perturbationand eqs.(5.67) to (5.69) indicate for phase, frequency and timing fluctuations.This is in contrast to the situation in a soliton storage ring where the laseramplifier compensating for the loss in the ring is below threshold [14].By inverse Fourier transformation of (5.70) and weak coupling, we obtainfor the associated function of the continuum102aGTR-1+ j+量(1-jD.)aTOt2-M (1 - cos(wMt)G + F-1/ <f(t)|Rao(a) >(5.72)-<f(+)|MwM sin(wMTr)ao() > △twhere Dn is the dispersion normalized to the gain dispersionDn=|D|2g/g(5.73)Note, that the homogeneous part of the equation of motion for the continuum,which governs the decay of the continuum, is the same as the homogeneouspart of the equation for the noise in a soliton storage ring at the positionwhere no soliton or bit is present [14]. Thus the decay of the continuum isnot affected by the nonlinearity.but there is a continuous excitation of thecontinuum by the soliton when the perturbing elements are passed by thesoliton.Thus under the above approximations the question of stability ofthe soliton solution is completely governed by the stability of the continuum(5.72).As we can see from (5.72) the evolution of the continuum obeysthe active mode locking equation with GVD but with a value for the gaindetermined by (5.64). In the parabolic approximation of the cosine, we obtainagain the Hermite Gaussians as the eigensolutions for the evolution operatorbut the width of these eigensolutions is now given byTe=TaV(1-jDn)(5.74)and the associated eigenvalues are入m=jΦo+g-l-MwmTaV(1-jDn)(m+(5.75)
190 CHAPTER 5. ACTIVE MODE LOCKING W0/EL >> 1, gain saturation always stabilizes the amplitude perturbation and eqs.(5.67) to (5.69) indicate for phase, frequency and timing fluctuations. This is in contrast to the situation in a soliton storage ring where the laser amplifier compensating for the loss in the ring is below threshold [14]. By inverse Fourier transformation of (5.70) and weak coupling, we obtain for the associated function of the continuum TR ∂G ∂T = ∙ g − l + jΦ0 + g Ω2 g (1 − jDn) ∂2 ∂t2 −M (1 − cos(ωMt)) ¸ G + F −1 ½ < f (+) k |Ra0(x) > (5.72) − < f (+) k |MωM sin(ωMτx)a0(x) > ∆t ¾ where Dn is the dispersion normalized to the gain dispersion Dn = |D|Ω2 g/g. (5.73) Note, that the homogeneous part of the equation of motion for the continuum, which governs the decay of the continuum, is the same as the homogeneous part of the equation for the noise in a soliton storage ring at the position where no soliton or bit is present [14]. Thus the decay of the continuum is not affected by the nonlinearity, but there is a continuous excitation of the continuum by the soliton when the perturbing elements are passed by the soliton. Thus under the above approximations the question of stability of the soliton solution is completely governed by the stability of the continuum (5.72). As we can see from (5.72) the evolution of the continuum obeys the active mode locking equation with GVD but with a value for the gain determined by (5.64). In the parabolic approximation of the cosine, we obtain again the Hermite Gaussians as the eigensolutions for the evolution operator but the width of these eigensolutions is now given by τ c = τ a p4 (1 − jDn) (5.74) and the associated eigenvalues are λm = jΦ0 + g − l − Mω2 Mτ 2 a p(1 − jDn)(m + 1 2 ). (5.75)