Chapter 6Passive ModelockingAs we have seen in chapter 5thepulse width in an activelymodelocked laseris inverse proportional to the fourth root of the curvature in the loss modu-lation. In active modelocking one is limited to the speed of electronic signalgenerators. Therefore, this curvature can never be very strong. However, ifthe pulse can modulate the absorption on its own, the curvature of the ab-sorption modulationcan become large, or in other words the net gain windowgenerated by the pulse can be as short as the pulse itself. In this case, thenet gain window shortens with the pulse. Therefore, passively modelockedlasers can generate much shorter pulses than actively modelocked lasers.However, a suitable saturable absorber is required for passive modelock-ing. Depending on the ratio between saturable absorber recovery time and fi-nal pulse width, one may distinguish between the regimes of operation shownin Figure 6.1, which depicts the final steady state pulse formation process.In a solid state laser with intracavity pulse energies much lower than the sat-uration energy of the gain medium, gain saturation can be neglected. Thena fast saturable absorber must be present that opens and closes the net gainwindow generated by the pulse immediately before and after the pulse. Thismodelocking principle is called fast saturable absorber modelocking,see Fig-ure 6.1 a).In semiconductor and dye lasers usually the intracavity pulse energy ex-ceeds the saturation energy of the gain medium and so the the gain mediumundergoes saturation. A short net gain window can still be created, almostindependent of the recovery time of the gain, if a similar but unpumpedmedium is introduced into the cavity acting as an absorber with a somewhatlower saturation energy then the gain medium. For example, this can be225
Chapter 6 Passive Modelocking As we have seen in chapter 5 the pulse width in an actively modelocked laser is inverse proportional to the fourth root of the curvature in the loss modulation. In active modelocking one is limited to the speed of electronic signal generators. Therefore, this curvature can never be very strong. However, if the pulse can modulate the absorption on its own, the curvature of the absorption modulationcan become large, or in other words the net gain window generated by the pulse can be as short as the pulse itself. In this case, the net gain window shortens with the pulse. Therefore, passively modelocked lasers can generate much shorter pulses than actively modelocked lasers. However, a suitable saturable absorber is required for passive modelocking. Depending on the ratio between saturable absorber recovery time and fi- nal pulse width, one may distinguish between the regimes of operation shown in Figure 6.1, which depicts the final steady state pulse formation process. In a solid state laser with intracavity pulse energies much lower than the saturation energy of the gain medium, gain saturation can be neglected. Then a fast saturable absorber must be present that opens and closes the net gain window generated by the pulse immediately before and after the pulse. This modelocking principle is called fast saturable absorber modelocking, see Figure 6.1 a). In semiconductor and dye lasers usually the intracavity pulse energy exceeds the saturation energy of the gain medium and so the the gain medium undergoes saturation. A short net gain window can still be created, almost independent of the recovery time of the gain, if a similar but unpumped medium is introduced into the cavity acting as an absorber with a somewhat lower saturation energy then the gain medium. For example, this can be 225
226CHAPTER6.PASSIVEMODELOCKINGImageremovedduetocopyrightrestrictionsPleasesee:Kartner,F.X.andU.Keller."Stabilizationofsoliton-likepulseswitha slow saturableabsorber."OpticsLetters20(1990):16-19Figure 6.1: Pulse-shaping and stabilization mechanisms owing to gain andloss dynamics in passively mode-locked lasers: (a) using only a fast saturableabsorber; (b) using a combination of gain and loss saturation; (c) using asaturableabsorber with afiniterelaxationtimeandsolitonformationarranged for by stronger focusing in the absorber medium than in the gainmedium. Then the absorber bleaches first and opens a net gain window,that is closed by the pulse itself by bleaching the gain somewhat later, seeFigure 6.1 b). This principle of modelocking is called slow-saturable absorbermodelocking.When modelocking of picosecond and femtosecond lasers with semicon-ductor saturable absorbers has been developed it became obvious that evenwith rather slow absorbers, showing recovery times of a few picoseconds, onewas able to generate sub-picosecond pulses resulting in a significant net gainwindow after the pulse, see Figure 6.1 c). From our investigation of activemodelocking in the presence of soliton formation, we can expect that such asituation may still be stable up to a certain limit in the presence of strongsoliton formation. This is the case and this modelocking regime is calledsoliton modelocking, since solitary pulse formation due to SPM and GDDshapes the pulse to a stable sech-shape despite the open net gain windowfollowing the pulse
226 CHAPTER 6. PASSIVE MODELOCKING Figure 6.1: Pulse-shaping and stabilization mechanisms owing to gain and loss dynamics in passively mode-locked lasers: (a) using only a fast saturable absorber; (b) using a combination of gain and loss saturation; (c) using a saturable absorber with a finite relaxation time and soliton formation. arranged for by stronger focusing in the absorber medium than in the gain medium. Then the absorber bleaches first and opens a net gain window, that is closed by the pulse itself by bleaching the gain somewhat later, see Figure 6.1 b). This principle of modelocking is called slow-saturable absorber modelocking. When modelocking of picosecond and femtosecond lasers with semiconductor saturable absorbers has been developed it became obvious that even with rather slow absorbers, showing recovery times of a few picoseconds, one was able to generate sub-picosecond pulses resulting in a significant net gain window after the pulse, see Figure 6.1 c). From our investigation of active modelocking in the presence of soliton formation, we can expect that such a situation may still be stable up to a certain limit in the presence of strong soliton formation. This is the case and this modelocking regime is called soliton modelocking, since solitary pulse formation due to SPM and GDD shapes the pulse to a stable sech-shape despite the open net gain window following the pulse. Kartner, F. X., and U. Keller. "Stabilization of soliton-like pulses with a slow saturable absorber." Optics Letters 20 (1990): 16-19. Image removed due to copyright restrictions. Please see:
2276.1.SLOWSATURABLEABSORBERMODELOCKING6.1Slow Saturable Absorber Mode LockingDue to the small cross section for stimulated emission in solid state laserstypical intracavity pulse energies are much smaller than the saturation energyof thegain.Therefore,weneglected the effectof gain saturation dueto onepulse sofar, the gain only saturates with the average power. However, thereare gain media which have large gain cross sections like semiconductors anddyes, see Table 4.1, and typical intracavity pulse energies may become largeenough to saturate the gain considerably in a single pass. In fact, it is thiseffect, which made the mode-locked dye laser so sucessful. The model for theslow saturableabsorber mode lockinghas to take into account the changeof gain in the passage of one pulse [1,2].In the following,we consider amodelocked laser,thatexperiencesinoneround-tripa saturablegain and aslowsaturableabsorber.Inthedyelaser,bothmediaaredyeswithdifferentsaturation intensities or with different focusing into the dye jets so that gainand loss may show different saturation energies. The relaxation equation ofthe gain, in the limit of a pulse short compared with its relaxation time, canbe approximated by[A(t)2dg(6.1)dtELThe coefficient Er is the saturation energy of the gain. Integration of theequation shows, that the gain saturates with the pulse energy E(t)dt/A(t)2(6.2)E(t) =TR/2when passing the gain(6.3)g(t) = gi exp[-E(t)/EL]where gi is the initial small signal gain just before the arrival of the pulse. Asimilar equation holds for the loss of the saturable absorber whose response(loss) is represented by q(t)(6.4)q(t) = qo exp[-E(t)/EA]
6.1. SLOW SATURABLE ABSORBER MODE LOCKING 227 6.1 Slow Saturable Absorber Mode Locking Due to the small cross section for stimulated emission in solid state lasers, typical intracavity pulse energies are much smaller than the saturation energy of the gain. Therefore, we neglected the effect of gain saturation due to one pulse sofar, the gain only saturates with the average power. However, there are gain media which have large gain cross sections like semiconductors and dyes, see Table 4.1, and typical intracavity pulse energies may become large enough to saturate the gain considerably in a single pass. In fact, it is this effect, which made the mode-locked dye laser so sucessful. The model for the slow saturable absorber mode locking has to take into account the change of gain in the passage of one pulse [1, 2]. In the following, we consider a modelocked laser, that experiences in one round-trip a saturable gain and a slow saturable absorber. In the dye laser, both media are dyes with different saturation intensities or with different focusing into the dye jets so that gain and loss may show different saturation energies. The relaxation equation of the gain, in the limit of a pulse short compared with its relaxation time, can be approximated by dg dt = −g |A(t)| 2 EL (6.1) The coefficient EL is the saturation energy of the gain. Integration of the equation shows, that the gain saturates with the pulse energy E(t) E(t) = Z t −TR/2 dt|A(t)| 2 (6.2) when passing the gain g(t) = gi exp [−E(t)/EL] (6.3) where gi is the initial small signal gain just before the arrival of the pulse. A similar equation holds for the loss of the saturable absorber whose response (loss) is represented by q(t) q(t) = q0 exp [−E(t)/EA] (6.4)
228CHAPTER6.PASSIVEMODELOCKINGwhere Ea is the saturation energy of the saturable absorber. If the back-ground loss is denoted by l, the master equation of mode-locking becomes1a4= [gi (exp(-E(t)/EL)A -lA-TROT(6.5)40 exp(-E(t)/EA) A+%AHere, we have replaced the fltering action of the gain Dg = asproduced by a separate fixed filter.An analytic solution to this integro-differential equation can be obtained with one approximation: the exponen-tials are expanded to second order. This is legitimate if the population deple-tions of the gain and saturable absorber media are not excessive. Consideroneoftheseexpansions:1-(E(t)/EA)+(E(t)/EA)qo exp(-E(t)/EA) ~go(6.6)We only consider the saturable gain and loss and the finite gain bandwidth.Than the master equation is given byOA(T,t)02A(T,t)(6.7)[g(t) - q(t) -1+ Di%TROTThefilter dispersion, D =1/2?, effectivelymodels the finite bandwidthof the laser, that might not be only due to the finite gain bandwidth, butincludes all bandwidth limiting effects in a parabolic approximation. Sup-pose the pulse is a symmetric function of time. Then the first power of theintegral gives an antisymmetric function of time, its square is symmetric.An antisymmetric function acting on the pulse A(t) causes a displacement.Hence, the steady state solution does not yield zero for the change per pass,the derivative must be equated to a time shift At of the pulse. WhenTPATthis is done one can confirm easily that A(t) = A。 sech(t/t) is a solution of(6.6) with constraints on its coefficients. Thus we, are looking for a "steadystate" solution A(t,T) = A。 sech( + α).Note, that α is the fraction ofthe pulsewidth, the pulse is shifted in each round-trip due to the shaping byloss and gain. The constraints on its coefficients can be easily found using
228 CHAPTER 6. PASSIVE MODELOCKING where EA is the saturation energy of the saturable absorber. If the background loss is denoted by l, the master equation of mode-locking becomes 1 TR ∂ ∂T A = [gi (exp (−E(t)/EL)) A − lA− q0 exp (−E(t)/EA)] A + 1 Ω2 f ∂2 ∂t2A (6.5) Here, we have replaced the filtering action of the gain Dg = 1 Ω2 f as produced by a separate fixed filter. An analytic solution to this integrodifferential equation can be obtained with one approximation: the exponentials are expanded to second order. This is legitimate if the population depletions of the gain and saturable absorber media are not excessive. Consider one of these expansions: q0 exp (−E(t)/EA) ≈ q0 ∙ 1 − (E(t)/EA) + 1 2 (E(t)/EA) 2 ¸ . (6.6) We only consider the saturable gain and loss and the finite gain bandwidth. Than the master equation is given by TR ∂A(T, t) ∂T = ∙ g(t) − q(t) − l + Df ∂2 ∂t2 ¸ A(T,t). (6.7) The filter dispersion, Df = 1/Ω2 f , effectively models the finite bandwidth of the laser, that might not be only due to the finite gain bandwidth, but includes all bandwidth limiting effects in a parabolic approximation. Suppose the pulse is a symmetric function of time. Then the first power of the integral gives an antisymmetric function of time, its square is symmetric. An antisymmetric function acting on the pulse A(t) causes a displacement. Hence, the steady state solution does not yield zero for the change per pass, the derivative 1 TR ∂A ∂T must be equated to a time shift ∆t of the pulse. When this is done one can confirm easily that A(t) = Ao sech(t/τ ) is a solution of (6.6) with constraints on its coefficients. Thus we, are looking for a "steady state" solution A(t, T) = Ao sech( t τ + α T TR ).Note, that α is the fraction of the pulsewidth, the pulse is shifted in each round-trip due to the shaping by loss and gain. The constraints on its coefficients can be easily found using
2296.1.SLOWSATURABLEABSORBERMODELOCKINGthe following relations for the sech-pulseWdt)A(t)P1 +tanh((6.8)E(t)TT1(2+ tan(+ + a ) -cda(/E(t)2(6.9)TRatT-)A(t, T)A(t,T) = -αtanh((6.10)TROT+0TR1 21 - 2sech2(= +(6.11)A(t,T),显A(t,T) =十022-2TBsubstituing them into the master equation (6.5) and collecting the coefficientsin front of the different temporal functions. The constant term gives thenecessarysmall signalgain1WW1W(6.12)1+912EL2EA222EAThe constant in front of the odd tanh-function delivers the timing shift perround-tripW△tWW(6.13)n2E-042E,2E2EAnd finallythe constant infront ofthesech?-functiondeterminesthepulsewidth122W2qoi(6.14)72E-E8Theseequations haveimportant implications.Consider firsttheeguation forthe inverse pulsewidth, (6.14). In order to get a real solution, the right handside has to be positive. This implies that qo/E > gi/E. The saturableabsorber must saturate more easily, and, therefore more strongly, than thegain medium in order to open a net window of gain (Figure 6.2).This was accomplished in a dye laser system by stronger focusing intothe saturable absorber-dye jet (Reducing the saturation energy for the sat-urable absorber) than into the gain-dye jet (which was inverted, i.e. optically
6.1. SLOW SATURABLE ABSORBER MODE LOCKING 229 the following relations for the sech-pulse E(t) = Z t −TR/2 dt|A(t)| 2 = W 2 µ 1 + tanh(t τ + α T TR ) ¶ (6.8) E(t) 2 = µW 2 ¶2 µ 2 + 2tanh(t τ + α T TR ) − sech2 ( t τ + α T TR ) ¶ (6.9) TR ∂ ∂T A(t, T) = −α tanh(t τ + α T TR )A(t, T) (6.10) 1 Ω2 f ∂2 ∂t2A(t, T) = 1 Ω2 f τ 2 µ 1 − 2sech2 ( t τ + α T TR ) ¶ A(t, T), (6.11) substituing them into the master equation (6.5) and collecting the coefficients in front of the different temporal functions. The constant term gives the necessary small signal gain gi " 1 − W 2EL + µ W 2EL ¶2 # = l + q0 " 1 − W 2EA + µ W 2EA ¶2 # − 1 Ω2 f τ 2 . (6.12) The constant in front of the odd tanh −function delivers the timing shift per round-trip α = ∆t τ = gi " W 2EL − µ W 2EL ¶2 # − q0 " W 2EA − µ W 2EA ¶2 # . (6.13) And finally the constant in front of the sech2 -function determines the pulsewidth 1 τ 2 = Ω2 fW2 8 µ q0 E2 A − gi E2 L ¶ (6.14) These equations have important implications. Consider first the equation for the inverse pulsewidth, (6.14). In order to get a real solution, the right hand side has to be positive. This implies that q0/E2 A > gi/E2 L. The saturable absorber must saturate more easily, and, therefore more strongly, than the gain medium in order to open a net window of gain (Figure 6.2). This was accomplished in a dye laser system by stronger focusing into the saturable absorber-dye jet (Reducing the saturation energy for the saturable absorber) than into the gain-dye jet (which was inverted, i.e. optically
230CHAPTER6.PASSIVEMODELOCKINGSaturableAbsorberGainTrLossNetgainGain0TRTimeDvnamics ofalasermode-lockedwithaslowsaturableabsorberFigure 6.2: Dynamics of a laser mode-locked with a slow saturable absorber.Figure by MIT OCW
230 CHAPTER 6. PASSIVE MODELOCKING Figure 6.2: Dynamics of a laser mode-locked with a slow saturable absorber. Figure by MIT OCW. 0 Time Gain Loss Gain Saturable Absorber Net gain Dynamics of a laser mode-locked with a slow saturable absorber. TR TR
2316.1.SLOWSATURABLEABSORBERMODELOCKINGpumped). Equation (6.12) makes a statement about the net gain before pas-sage of the pulse. The net gain before passage of the pulse is1W/9-0-123-22EL2E(6.15)WM1o2EA2EAUsing condition (6.14) this can be expressed as(6.16)9i-q0-1:12EA2ELO2TThis gain is negative since the effect of the saturable absorber is larger thanthat of the gain. Since the pulse has the same exponential tail after passageas before, one concludes that the net gain after passage of the pulse is thesame as before passage and thus also negative. The pulse is stable againstnoise build-up both in its front and its back. This principle works if theratio between the saturation energies for the saturable absorber and gainXp = Ea/Ep is very small. Then the shortest pulsewidth achievable with agivensystemis4EA-.2(6.17)Vqo2,WVo2The greater sign comes from the fact that our theory is based on the ex-pansion of the exponentials, which is only true for < 1. If the filterdispersion 1/? that determines the bandwidth of the system is again re-placed by an average gain dispersion g/ and assuming g = go. Note thatthemodelocking principle ofthe dyelaserisa very faszinating one due tothe fact that actually non of the elements in the system is fast. It is the in-terplay between two media that opens a short window in time on the scale offemtoseconds. The media themselves just have to be fast enough to recovercompletely between one round trip, ie. on a nanosecond timescale.Over the last fifteen years, the dye laser has been largely replaced bysolid state lasers, which offer even more bandwidth than dyes and are on topof that much easier to handle because they do not show degradation overtime. With it came the need for a different mode locking principle, since thesaturation energy of these broadband solid-state laser media are much higher
6.1. SLOW SATURABLE ABSORBER MODE LOCKING 231 pumped). Equation (6.12) makes a statement about the net gain before passage of the pulse. The net gain before passage of the pulse is gi − q0 − l = − 1 Ω2 f τ 2 + gi " W 2EL − µ W 2EL ¶2 # −q0 " W 2EA − µ W 2EA ¶2 # . (6.15) Using condition (6.14) this can be expressed as gi − q0 − l = gi ∙ W 2EL ¸ − q0 ∙ W 2EA ¸ + 1 Ω2 f τ 2 . (6.16) This gain is negative since the effect of the saturable absorber is larger than that of the gain. Since the pulse has the same exponential tail after passage as before, one concludes that the net gain after passage of the pulse is the same as before passage and thus also negative. The pulse is stable against noise build-up both in its front and its back. This principle works if the ratio between the saturation energies for the saturable absorber and gain χP = EA/EP is very small. Then the shortest pulsewidth achievable with a given system is τ = 4 √q0Ωf EA W > 2 √q0Ωf . (6.17) The greater sign comes from the fact that our theory is based on the expansion of the exponentials, which is only true for W 2EA < 1. If the filter dispersion 1/Ω2 f that determines the bandwidth of the system is again replaced by an average gain dispersion g/Ω2 g and assuming g = q0. Note that the modelocking principle of the dye laser is a very faszinating one due to the fact that actually non of the elements in the system is fast. It is the interplay between two media that opens a short window in time on the scale of femtoseconds. The media themselves just have to be fast enough to recover completely between one round trip, i.e. on a nanosecond timescale. Over the last fifteen years, the dye laser has been largely replaced by solid state lasers, which offer even more bandwidth than dyes and are on top of that much easier to handle because they do not show degradation over time. With it came the need for a different mode locking principle, since the saturation energy of these broadband solid-state laser media are much higher
232CHAPTER6.PASSIVEMODELOCKINGthan the typical intracavity pulse energies. The absorber has to open andclose the net gain window.6.2Fast SaturableAbsorberModeLockingThe dynamics of a laser modelocked with a fast saturable absorber is againcovered by the master equation (5.21) [3]. Now, the losses q react instantlyon the intensity or power P(t) = |A(t)/2 of the fieldqoq(A) =(6.18)1+4F,PAwhere Pa is the saturation power of the absorber.There is no analyticsolution of the master equation (5.21) with the absorber response (6.18).Therefore, we make expansions on the absorber response to get analyticinsight. If the absorber is not saturated, we can expand the response (6.18)forsmallintensitiesq(A) = q0 - A/2,(6.19)with the saturable absorber modulation coefficient=qo/PA.The constantnonsaturated loss qo can be absorbed in the losses lo = I + qo. The resultingmaster equation is, see also Fig. 6.302aA(T,t)02A(T,t)(6.20)+A2+D2202-j8[A/2lo+ DfTROT
232 CHAPTER 6. PASSIVE MODELOCKING than the typical intracavity pulse energies. The absorber has to open and close the net gain window. 6.2 Fast Saturable Absorber Mode Locking The dynamics of a laser modelocked with a fast saturable absorber is again covered by the master equation (5.21) [3]. Now, the losses q react instantly on the intensity or power P(t) = |A(t)| 2 of the field q(A) = q0 1 + |A|2 PA , (6.18) where PA is the saturation power of the absorber. There is no analytic solution of the master equation (5.21) with the absorber response (6.18). Therefore, we make expansions on the absorber response to get analytic insight. If the absorber is not saturated, we can expand the response (6.18) for small intensities q(A) = q0 − γ|A| 2 , (6.19) with the saturable absorber modulation coefficient γ = q0/PA. The constant nonsaturated loss q0 can be absorbed in the losses l0 = l + q0. The resulting master equation is, see also Fig. 6.3 TR ∂A(T, t) ∂T = ∙ g − l0 + Df ∂2 ∂t2 + γ|A| 2 + j D2 ∂2 ∂t2−j δ|A| 2 ¸ A(T,t). (6.20)
6.2.FASTSATURABLEABSORBERMODELOCKING233Imageremovedduetocopyright restrictionsPleasesee:Keller,U,UtrafastLaserPhysics,Insttute ofQuantumElectronics,SwissFederal InstituteofTechnologyETHHonggerberg—HPT,CH-8093Zurich,SwitzerlandFigure 6.3: Schematic representation of the master equation for a passivelymodelocked laser with a fast saturable absorber.Eq. (6.20) is a generalized Ginzburg-Landau equation well known fromsuperconductivity with a rather complex solution manifold.6.2.1Without GDD and SPMWe consider first the situation without SPM and GDD,i.e. D2=d = 002A(T,t)12+AA(T,t).(6.21)g-lo+DTRaTUp to the imaginary unit, this equation is still very similar to the NSE. Tofind thefinal pulseshape and width,welook forthestationary solutionAs(T,t)=0.TROTSince the equation is similar to the NSE, we try the following ansatzA(T,t) = As(t) = Aosech(6.22)
6.2. FAST SATURABLE ABSORBER MODE LOCKING 233 Figure 6.3: Schematic representation of the master equation for a passively modelocked laser with a fast saturable absorber. Eq. (6.20) is a generalized Ginzburg-Landau equation well known from superconductivity with a rather complex solution manifold. 6.2.1 Without GDD and SPM We consider first the situation without SPM and GDD, i.e. D2=δ = 0 TR ∂A(T, t) ∂T = ∙ g − l0 + Df ∂2 ∂t2 + γ|A| 2 ¸ A(T, t). (6.21) Up to the imaginary unit, this equation is still very similar to the NSE. To find the final pulse shape and width, we look for the stationary solution TR ∂As(T, t) ∂T = 0. Since the equation is similar to the NSE, we try the following ansatz As(T, t) = As(t) = A0sech µ t τ ¶ . (6.22) Keller, U., Ultrafast Laser Physics, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH Hönggerberg—HPT, CH-8093 Zurich, Switzerland. Image removed due to copyright restrictions. Please see:
234CHAPTER6.PASSIVEMODELOCKINGNote, there isd(6.23)sechr=-tanhr sechr,drd2asecha = tanh2r secha- sech3r,dr2=(sechr-2 sech3r)(6.24)Substitution of ansatz (6.22)intothemaster equation (6.21),assuming steadystate,results in[-0) + [1-28ecr (]0+140P'sech? ()] Aosech ((6.25)Comparison of the coefficients with the sech- and sech3-expressions resultsin the conditions for the pulse peak intensity and pulse width and for thesaturated gainDf=Aol2,(6.26)-2Df(6.27)g = lo-72From Eq.(6.26) and with the pulse energy of a sech pulse, see Eq.(3.8), W :2|Ao/’T,4Di(6.28)7WEq.(6.28)is rather similar to the soliton width with the exception thatthe conservative pulse shaping effects GDD and SPM are replaced by gaindispersion and saturable absorption.The soliton phase shift per roundtrip isreplaced by the difference between the saturated gain and loss in Eq.(6.28)It is interestingto have a closer look on howthedifference between gain andloss P per round-trip comes about. From the master equation (6.21) we canderive an equation of motion for thepulse energy according toaoW(T)IA(T,t)P dt(6.29)TRTRROTOTadt(6.30)A(T,t)A(T,t) + c.c.TROT(6.31)2G(gs, W)W
234 CHAPTER 6. PASSIVE MODELOCKING Note, there is d dxsechx = − tanh x sechx, (6.23) d2 dx2 sechx = tanh2 x sechx − sech3 x, = ¡ sechx − 2 sech3 x ¢ . (6.24) Substitution of ansatz (6.22) into the master equation (6.21), assuming steady state, results in 0 = ∙ (g − l0) + Df τ 2 ∙ 1 − 2sech2 µ t τ ¶¸ +γ|A0| 2 sech2 µ t τ ¶¸ · A0sech µ t τ ¶ . (6.25) Comparison of the coefficients with the sech- and sech3-expressions results in the conditions for the pulse peak intensity and pulse width τ and for the saturated gain Df τ 2 = 1 2 γ|A0| 2 , (6.26) g = l0 − Df τ 2 . (6.27) From Eq.(6.26) and with the pulse energy of a sech pulse, see Eq.(3.8), W = 2|A0| 2τ, τ = 4Df γW . (6.28) Eq. (6.28) is rather similar to the soliton width with the exception that the conservative pulse shaping effects GDD and SPM are replaced by gain dispersion and saturable absorption. The soliton phase shift per roundtrip is replaced by the difference between the saturated gain and loss in Eq.(6.28). It is interesting to have a closer look on how the difference between gain and loss Df τ2 per round-trip comes about. From the master equation (6.21) we can derive an equation of motion for the pulse energy according to TR ∂W(T) ∂T = TR ∂ ∂T Z ∞ −∞ |A(T,t)| 2 dt (6.29) = TR Z ∞ −∞ ∙ A(T,t) ∗ ∂ ∂T A(T, t) + c.c.¸ dt (6.30) = 2G(gs, W)W, (6.31)