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)