1835.4.ACTIVEMODELOCKINGWITHADDITIONALSPMthe carrier, which leads to a chirp on the steady state pulse. We can againuse the master equation to study this type of modelocking. All that changesis that the modulation becomes imaginary, i.e. we have to replace M by jMin Eq.(5.22)02OATROTg(T) + D(5.40)l - jM (1 - cos(wmt))OtThe imaginary unit can be pulled through much of the calculation and wearrive at the same Hermite Gaussian eigen solutions (5.26,5.27), however, theparameter Ta becomes Ta and is now complex and not quite the pulse widthTa=V-j/Dg/M.(5.41)The ground mode or stationary solution is given byW.e-(1+)Ao(t) =(5.42)2nV元n!awith Ta = Dg/M, as before. We end up with chirped pulses. How doesthe pulse shortening actually work, because the modulator just puts a chirpon the pulse, it does actually not shorten it? One can easily show, that if aGaussian pulse with chirp parameter βAo(t)~e-力(1+jp)(5.43)has a chirp β > 1, subsequent filtering is actually shortening the pulse.5.4Active Mode Locking with Additional SPMDue to the strong focussing of the pulse in the gain medium also additionalself-phase modulation can become important. Lets consider the case of anactively mode-locked laser with additional SPM, see Fig. 5.7. One can writedown the corresponding master equationOA82[(T) + D, % - - M,t j0lAr] (5.44)TROTUnfortunately, there is no analytic solution to this equation.But it is notdifficult to guess what will happen in this case. As long as the SPM is not
5.4. ACTIVE MODE LOCKING WITH ADDITIONAL SPM 183 the carrier, which leads to a chirp on the steady state pulse. We can again use the master equation to study this type of modelocking. All that changes is that the modulation becomes imaginary, i.e. we have to replace M by jM in Eq.(5.22) TR ∂A ∂T = ∙ g(T) + Dg ∂2 ∂t2 − l − jM (1 − cos(ωMt))¸ A. (5.40) The imaginary unit can be pulled through much of the calculation and we arrive at the same Hermite Gaussian eigen solutions (5.26,5.27), however, the parameter τ a becomes τ 0 a and is now complex and not quite the pulse width τ 0 a = p4 −j 4 q Dg/Ms. (5.41) The ground mode or stationary solution is given by A0(t) = s Ws 2n√πn!τ 0 a e − t2 2τ2 a √1 2 (1+j) , (5.42) with τ a = p4 Dg/Ms as before. We end up with chirped pulses. How does the pulse shortening actually work, because the modulator just puts a chirp on the pulse, it does actually not shorten it? One can easily show, that if a Gaussian pulse with chirp parameter β A0(t) ∼ e − t2 2τ2 a √1 2 (1+jβ) , (5.43) has a chirp β > 1, subsequent filtering is actually shortening the pulse. 5.4 Active Mode Locking with Additional SPM Due to the strong focussing of the pulse in the gain medium also additional self-phase modulation can become important. Lets consider the case of an actively mode-locked laser with additional SPM, see Fig. 5.7. One can write down the corresponding master equation TR ∂A ∂T = ∙ g(T) + Dg ∂2 ∂t2 − l − Mst 2 − jδ|A| 2 ¸ A. (5.44) Unfortunately, there is no analytic solution to this equation. But it is not difficult to guess what will happen in this case. As long as the SPM is not
184CHAPTER5.ACTIVEMODELOCKING21口g/1+dt2Galn- M(1-cos mt)poSPM-18IA12Figure5.7:Activemode-lockingwithSPMexcessive, the pulses will experience additional self-phase modulation, whichcreates a chirp on the pulse.Thus onecan make an ansatz with a chirpedGaussian similar to (5.43)for the steady state solution of the master equation(5.44)Ao(t) = Ae~最(+i)+wT/Tr(5.45)Note, we allow for an additional phase shift per roundtrip , because theadded SPM does not leavethephase invariant after one round-trip.This isstill a steady state solution for theintensity envelope.Substitution into themaster equation usingtheintermediateresult0212(1+jp)2_1 (1 +jp)Ao(t)(5.46)2Ao(t) Tleads to(1+j)*-(1+jp)(5.47)jtAo(t)-1+DT-M,t2-j81AI2e-%Ao(t)
184 CHAPTER 5. ACTIVE MODE LOCKING Figure 5.7: Active mode-locking with SPM excessive, the pulses will experience additional self-phase modulation, which creates a chirp on the pulse. Thus one can make an ansatz with a chirped Gaussian similar to (5.43) for the steady state solution of the master equation (5.44) A0(t) = Ae− t2 2τ2 a (1+jβ)+jΨT /TR (5.45) Note, we allow for an additional phase shift per roundtrip Ψ, because the added SPM does not leave the phase invariant after one round-trip. This is still a steady state solution for the intensity envelope. Substitution into the master equation using the intermediate result ∂2 ∂t2A0(t) = ½ t 2 τ 4 a (1 + jβ) 2 − 1 τ 2 a (1 + jβ) ¾ A0(t). (5.46) leads to jΨA0(t) = ½ g − l + Dg ∙ t 2 τ 4 a (1 + jβ) 2 − 1 τ 2 a (1 + jβ) ¸ (5.47) −Mst 2 − jδ |A| 2 e − t2 τ2 a ¾ A0(t)
5.4.ACTIVEMODELOCKINGWITHADDITIONALSPM185To find an approximate solution we expand the Gaussian in the bracket,which is a consequency of the SPM to first order in the exponent.t2(1+ jp)-Mt2- j8|Al2(1 +jp)? -j=g-l+DT1P(5.48)This has to be fulflled for all times, so we can compare coefficients in frontof the constant terms and the quadratic terms, which leads to two complexconditions.This leads tofour equations for the unknown pulsewidth Tachirp β, round-trip phase and the necessary excess gain g - l. With thenonlinear peak phase shift due to SPM, o = [Al?. Real and Imaginaryparts of the quadratic terms lead toDo (1 - β2) - Ms,0(5.49)T4D.do02B-(5.50)Ta'T4and the constant terms give the excess gain and the additional round-tripphase.Dg(5.51)g-T亚D(5.52)The first two equations directly give the chirp and pulse width.dotaβ(5.53)2DgD.T4(5.54)M.+oHowever, one has to note, that this simple analysis does not give any hinton the stability of these approximate solution.Indeed computer simulationsshow, that after an additional pulse shorting of about a factor of 2 by SPMbeyond the pulse width already achieved by pure active mode-locking on itsown, the SPM drives the pulses unstable [5].This is one of the reasonswhy very broadband laser media, like Ti:sapphire, can not simply generate
5.4. ACTIVE MODE LOCKING WITH ADDITIONAL SPM 185 To find an approximate solution we expand the Gaussian in the bracket, which is a consequency of the SPM to first order in the exponent. jΨ = g − l + Dg ∙ t 2 τ 4 a (1 + jβ) 2 − 1 τ 2 a (1 + jβ) ¸ − Mst 2 − jδ |A| 2 µ 1 − t 2 τ 2 a ¶ . (5.48) This has to be fulfilled for all times, so we can compare coefficients in front of the constant terms and the quadratic terms, which leads to two complex conditions. This leads to four equations for the unknown pulsewidth τ a, chirp β, round-trip phase Ψ and the necessary excess gain g − l. With the nonlinear peak phase shift due to SPM, φ0 = δ |A| 2 . Real and Imaginary parts of the quadratic terms lead to 0 = Dg τ 4 a ¡ 1 − β2¢ − Ms, (5.49) 0=2β Dg τ 4 a + φ0 τ 2 a , (5.50) and the constant terms give the excess gain and the additional round-trip phase. g − l = Dg τ 2 a , (5.51) Ψ = Dg ∙ − 1 τ 2 a β ¸ − φ0. (5.52) The first two equations directly give the chirp and pulse width. β = −φ0τ 2 a 2Dg (5.53) τ 4 a = Dg Ms + φ2 0 4Dg . (5.54) However, one has to note, that this simple analysis does not give any hint on the stability of these approximate solution. Indeed computer simulations show, that after an additional pulse shorting of about a factor of 2 by SPM beyond the pulse width already achieved by pure active mode-locking on its own, the SPM drives the pulses unstable [5]. This is one of the reasons, why very broadband laser media, like Ti:sapphire, can not simply generate
186CHAPTER5.ACTIVEMODELOCKINGn, dt?Verstarker-VerlustDeiDa?M1-cos@mt)atSPM-18IA12Figure5.8:Acitvemode-lockingwithadditional solitonformationfemtosecond pulses via active modelocking. The SPM occuring in the gainmedium for very short pulses drives the modelocking unstable. Additionalstabilization measures have to be adopted. For example the addition ofnegativegroupdelaydispersionmightleadtostablesolitonformationinthepresence of the active modelocker.5.5Active Mode Locking with Soliton For-mationExperimental results with fiber lasers [8, 9, 11] and solid state lasers [10]indicated that soliton shaping in the negative GDD regime leads to pulsestabilization and considerable pulse shorting.With sufficient negative dis-persion and self-phase modulation in the system and picosecond or evenfemtosecond pulses, it is possible that the pulse shaping due to GDD andSPM is much stronger than due to modulation and gain filtering, see Fig.5.8. The resulting master equation for this case isOA102% - 1 - M (1 - cos(wmt) - j8|AIP A. (5.55)g + (D- jIDI),LRaT
186 CHAPTER 5. ACTIVE MODE LOCKING VerstärkerVerlust Modulator + 1 Ω 2 2 2 g d dt g 1 - l - (1- M cosωm t) SPM 2 - iδ A GVD 0 2 2 t iD Figure 5.8: Acitve mode-locking with additional soliton formation femtosecond pulses via active modelocking. The SPM occuring in the gain medium for very short pulses drives the modelocking unstable. Additional stabilization measures have to be adopted. For example the addition of negative group delay dispersion might lead to stable soliton formation in the presence of the active modelocker. 5.5 Active Mode Locking with Soliton Formation Experimental results with fiber lasers [8, 9, 11] and solid state lasers [10] indicated that soliton shaping in the negative GDD regime leads to pulse stabilization and considerable pulse shorting. With sufficient negative dispersion and self-phase modulation in the system and picosecond or even femtosecond pulses, it is possible that the pulse shaping due to GDD and SPM is much stronger than due to modulation and gain filtering, see Fig. 5.8. The resulting master equation for this case is TR ∂A ∂T = ∙ g + (Dg − j |D|) ∂2 ∂t2 − l − M (1 − cos(ωMt)) − jδ|A| 2 ¸ A. (5.55)
5.5.ACTIVEMODELOCKINGWITHSOLITONFORMATION187For the case, that soliton formation takes over, the steady state solution asoliton plusa continuum contributionA(T,t) = (a(r)ejpt + ae(T,t) e-jo(5.56)with+2Da(r)=Asech(r), and rp(T')dT' - to)(5.57)where a is the continuum contribution. The phase is determined byD((T))a,(T) = o(T) -(5.58)TRwhereby we always assume that the relation between the soliton energy andsoliton width is maintained (3.9)[D] A(T)?(5.59)T(T)22WealsoallowforacontinuouschangeinthesolitonamplitudeA or energyW= 2A?r and the soliton variables phase o, carrier frequency p and timingto. o is the soliton phase shift per roundtrip[D](5.60)oHowever,we assume that the changes in carrier frequency, timing andphasestaysmall.Introducing (5.56)into(5.55)weobtain accordingtothesoliton perturbation theory developed in chapter 3.5Oacow0A0apAtTROT+OTOTaTO(5.61)= ΦoL(ac+△pf,)+R(a+△pf,+ac)-MwMsin(wMTr)△ta(r)Thelastterm arisesbecausetheactivemodelockerbreaks thetimeinvarianceof the system and leads to a restoring force pushing the soliton back to itsequilibrium position. L, R are the operators of the linearized NSE and ofthe activemode locking scheme, respectively1R=(5.62)l - M (1 - cos(wMT)),2220r2The vectorsfw,fo,f,and f, describe the change in the soliton when the solitonenergy, phase, carrier frequency and timing varies
5.5. ACTIVE MODE LOCKING WITH SOLITON FORMATION 187 For the case, that soliton formation takes over, the steady state solution a soliton plus a continuum contribution A(T,t) = ¡ a(x)ejpt + ac(T, t) ¢ e−jθ (5.56) with a(x) = A sech(x), and x = 1 τ (t + 2D Z T 0 p(T0 )dT0 − t0) (5.57) where ac is the continuum contribution. The phase is determined by θ(T) = θ0(T) − D TR Z T 0 µ 1 τ (T0 )2 − p(T0 ) 2 ¶ dT0 , (5.58) whereby we always assume that the relation between the soliton energy and soliton width is maintained (3.9) |D| τ (T)2 = δA(T)2 2 . (5.59) We also allow for a continuous change in the soliton amplitude A or energy W = 2A2τ and the soliton variables phase θ0, carrier frequency p and timing t0. φ0 is the soliton phase shift per roundtrip φ0 = |D| τ 2 . (5.60) However, we assume that the changes in carrier frequency, timing and phase stay small. Introducing (5.56) into (5.55) we obtain according to the soliton perturbation theory developed in chapter 3.5 TR ∙ ∂ac ∂T + ∂W ∂T fw + ∂∆θ ∂T fθ + ∂∆p ∂T fp + ∂∆t ∂T ft ¸ = φ0L(ac + ∆pfp) + R(a + ∆pfp + ac) (5.61) −MωM sin(ωMτx)∆t a(x) The last term arises because the active modelocker breaks the time invariance of the system and leads to a restoring force pushing the soliton back to its equilibrium position. L, R are the operators of the linearized NSE and of the active mode locking scheme, respectively R = g µ 1 + 1 Ω2 gτ 2 ∂2 ∂x2 ¶ − l − M (1 − cos(ωMτx)), (5.62) The vectors fw,fθ,fp and ft describe the change in the soliton when the soliton energy, phase, carrier frequency and timing varies