Chapter 5Active Mode LockingFor simplicity, we assume, that the laser operates in the transverse fundamen-tal modes and, therefore, we only have to treat the longitudinal modes of thelaser similar to a simple plane parallel Fabry-Perot resonator (Figure: 5.1)We consider one polarization of the field only,however,as we will say laterfor some mode-locked laser polarization dynamics will become important.The task of mode-locking is to get as many of the longitudinal modeslasing in a phase synchronous fashion, such that the superposition of allmodes represents a pulse with a spatial extent much shorter than the cavity.The pulse will then propagate at the group velocity corresponding to thecenterfrequency of thepulse.XXX3Cavity Length, LFigure 5.1:Fabry-Perot resonator173
Chapter 5 Active Mode Locking For simplicity, we assume, that the laser operates in the transverse fundamental modes and, therefore, we only have to treat the longitudinal modes of the laser similar to a simple plane parallel Fabry-Perot resonator (Figure: 5.1). We consider one polarization of the field only, however, as we will say later for some mode-locked laser polarization dynamics will become important. The task of mode-locking is to get as many of the longitudinal modes lasing in a phase synchronous fashion, such that the superposition of all modes represents a pulse with a spatial extent much shorter than the cavity. The pulse will then propagate at the group velocity corresponding to the center frequency of the pulse. Figure 5.1: Fabry-Perot resonator 173
174CHAPTER5.ACTIVEMODELOCKING5.1The Master Equation of Mode Locking5Lets consider for the moment the cold cavity (i.e.there is only a simplelinear medium in the cavity no lasing). The most general solution for theintracavity field is a superpositon of left- and rightward running wavesZEnej(nt+Kn2)E(lef)(z,t) = Re ^ >(5.1)n=0andej(Ont-KnE(right)(z,t)= Re(5.2)The possible values for the wavenumbers are Kn = n/L, resulting from theboundary conditions on metallic mirrors or periodicity after one roundtrip inthe cavity. If the mirrors are perfectly reflecting, the leftward and rightwardmoving waves Eqs.(5.1) and (5.2) contain the same information and it issufficient to treat only one of them.Usually one of the cavity mirrors isnot perfectly refecting in order to couple out light, however, this can beconsidered a perturbation to the ideal mode structure.We consider the modes in Eq.(5.2) as a continuum and replace the sumby an integralE(K)e(2(K)t-K2)dKE(right)(z,t) (5.3)Be2TwithE(Km)= Em2L(5.4)Eq.(5.3) is similar to the pulse propagation discussed in chapter 2 and de-scribes the pulse propagation in the resonator. However, here it is ratheran initialvalueproblem,ratherthan aboundary valueproblem.Note,thewavenumbers of the modes are fixed, not the frequencies. To emphasize thiseven more, we introduce a new time variable T = t and a local time framet'=t-z/ug.o, instead of the propagation distance z, where ug.o is the groupvelocity at the central wave number Kno of the pulseaw(5.5)Ug.0ak
174 CHAPTER 5. ACTIVE MODE LOCKING 5.1 The Master Equation of Mode Locking Lets consider for the moment the cold cavity (i.e. there is only a simple linear medium in the cavity no lasing). The most general solution for the intracavity field is a superpositon of left- and rightward running waves E(lef t) (z, t) = Re (X∞ n=0 Eˆnej(Ωnt+Knz) ) , (5.1) and E(right) (z, t) = Re (X∞ n=0 Eˆnej(Ωnt−Knz) ) . (5.2) The possible values for the wavenumbers are Kn = nπ/L, resulting from the boundary conditions on metallic mirrors or periodicity after one roundtrip in the cavity. If the mirrors are perfectly reflecting, the leftward and rightward moving waves Eqs.(5.1) and (5.2) contain the same information and it is sufficient to treat only one of them. Usually one of the cavity mirrors is not perfectly reflecting in order to couple out light, however, this can be considered a perturbation to the ideal mode structure. We consider the modes in Eq.(5.2) as a continuum and replace the sum by an integral E(right) (z, t) = 1 2π Re ½Z ∞ K=0 Eˆ(K)ej(Ω(K)t−Kz) dK¾ (5.3) with Eˆ(Km) = Eˆm2L. (5.4) Eq.(5.3) is similar to the pulse propagation discussed in chapter 2 and describes the pulse propagation in the resonator. However, here it is rather an initial value problem, rather than a boundary value problem. Note, the wavenumbers of the modes are fixed, not the frequencies. To emphasize this even more, we introduce a new time variable T = t and a local time frame t 0 = t − z/υg,0, instead of the propagation distance z, where υg,0 is the group velocity at the central wave number Kn0 of the pulse υg,0 = ∂ω ∂k ¯ ¯ ¯ ¯ k=0 = µ∂k ∂ω¶−1 ¯ ¯ ¯ ¯ ¯ ω=0 . (5.5)
1755.1.THEMASTEREQUATIONOFMODELOCKINGFor introduction of a slowly varying envelope, we shift the frequency andwavenumber by the center frequency wo = no and center wave numberko = Knok=K-Kno,(5.6)(5.7)w(k) = 2(Kno +k)-no,E(k) = E(Kno +k),(5.8)The temporal evolution of the pulse is than determined by1.E(k)e;(a(k)t-k2) dkj(wot-koz)E(right)(z,t) =Re(5.9)2元Analogous to chapter 2, we define a slowly varying field envelope, that isalready normalized to the total power flow in the beamAeff1E(k)e;(k)t-k2) dk.A(z,t) =(5.10)2Z02元With the retarded time t' and time T, we obtain analogous to Eq. (2.184).AefiE(k)ei((k)-ug,ok)T+kug.ot dk.A(T,t) =(5.11)2Z02元whichcanbewrittenasQA(T,t)A(T,t),TR(5.12)i>OTOtI(GDD)n=22with thedispersion coefficients per resonator round-trip Tr=Ua.02Lan-1ug(k)Dn :(5.13)nlug,tOkn-The dispersion coefficients (5.13) look somewhat suspicious, however, it isnot difficult to show, that they are equivalent to derivatives of the roundtripphase r(2) = n(2)2L in the resonator at the center frequency1 angm(2)Dn=(5.14)n!02nl0=wo
5.1. THE MASTER EQUATION OF MODE LOCKING 175 For introduction of a slowly varying envelope, we shift the frequency and wavenumber by the center frequency ω0 = Ωn0 and center wave number k0 = Kn0 k = K − Kn0 , (5.6) ω(k) = Ω(Kn0 + k) − Ωn0 , (5.7) Eˆ(k) = Eˆ(Kn0 + k), (5.8) The temporal evolution of the pulse is than determined by E(right) (z, t) = 1 2π Re (Z ∞ −Kn0→−∞ Eˆ(k)ej(ω(k)t−kz) dk) ej(ω0t−k0z) . (5.9) Analogous to chapter 2, we define a slowly varying field envelope, that is already normalized to the total power flow in the beam A(z, t) = rAef f 2Z0 1 2π Z ∞ −∞ Eˆ(k)ej(ω(k)t−kz) dk. (5.10) With the retarded time t 0 and time T, we obtain analogous to Eq. (2.184). A(T,t0 ) = rAef f 2Z0 1 2π Z ∞ −∞ Eˆ(k)ej((ω(k)−υg,0k)T +kυg,0t0 dk. (5.11) which can be written as TR ∂A(T,t0 ) ∂T ¯ ¯ ¯ ¯ (GDD) = j X∞ n=2 Dn µ −j ∂n ∂t0 ¶n A(T, t0 ), (5.12) with the dispersion coefficients per resonator round-trip TR = 2L υg,0 Dn = 2L n!υn+1 g,0 ∂n−1υg(k) ∂kn−1 ¯ ¯ ¯ ¯ ¯ k=0 . (5.13) The dispersion coefficients (5.13) look somewhat suspicious, however, it is not difficult to show, that they are equivalent to derivatives of the roundtrip phase φR(Ω) = Ω c n(Ω)2L in the resonator at the center frequency Dn = − 1 n! ∂nφ(n) R (Ω) ∂Ωn ¯ ¯ ¯ ¯ ¯ Ω=ω0 , (5.14)
176CHAPTER5.ACTIVEMODELOCKINGSofar, only the lossless resonator is treated. The gain and loss can be mod-elled by adding a term likeA(T,t)= -IA(T,t)(5.15)TRaTI(loss)where l is the amplitude loss per round-trip. In an analogous manner we canwrite for thegainA(T,t)A(T,t),(5.16)TR--1OTat/asnwhere g(T) is the gain and and D, is the curvature of the gain at the maxi-mumoftheLorentzian lineshape.g(T)Dg=(5.17)22Dg is the gain dispersion. g(T) is an average gain, which can be computedThe dis-from the rate equation valid for each unit cell in the resonator.tributed gain obeys the equation[A(z, t)2Og(z,t)g - 9o(5.18)otELTLAeff,T the upper state lifetimewhere E is the saturation energy E=and o the gain cross section. For typical solid-state lasers, the intracavitypulse energy is much smaller than the saturation energy. Therefore, the gainchanges within one roundtrip are small. Furthermore, we assume that thegain saturates spatially homogeneous, g(z,t') = g(t'). Then, the equation forthe average gain g(T) can be found by averageing (5.18) over one round-tripandweobtainW(T)0g(T)g - 90(5.19)'ELTRaTTLwhere W(T) is the intracavity pulse energy at time t = TTR/2IA(T,t)’dt ~A(T,t)Pdt(5.20)W(T) 2
176 CHAPTER 5. ACTIVE MODE LOCKING Sofar, only the lossless resonator is treated. The gain and loss can be modelled by adding a term like TR ∂A(T, t0 ) ∂T ¯ ¯ ¯ ¯ (loss) = −lA(T, t0 ) (5.15) where l is the amplitude loss per round-trip. In an analogous manner we can write for the gain TR ∂A(T,t0 ) ∂T ¯ ¯ ¯ ¯ (gain) = µ g(T) + Dg ∂2 ∂t02 ¶ A(T,t0 ), (5.16) where g(T) is the gain and and Dg is the curvature of the gain at the maximum of the Lorentzian lineshape. Dg = g(T) Ω2 g (5.17) Dg is the gain dispersion. g(T) is an average gain, which can be computed from the rate equation valid for each unit cell in the resonator. The distributed gain obeys the equation ∂g(z, t) ∂t = −g − g0 τ L − g |A(z, t)| 2 EL , (5.18) where EL is the saturation energy EL = hνL 2∗σL Aef f , τ L the upper state lifetime and σL the gain cross section. For typical solid-state lasers, the intracavity pulse energy is much smaller than the saturation energy. Therefore, the gain changes within one roundtrip are small. Furthermore, we assume that the gain saturates spatially homogeneous, g(z, t0 ) = g(t 0 ). Then, the equation for the average gain g(T) can be found by averageing (5.18) over one round-trip and we obtain ∂g(T) ∂T = −g − g0 τ L − g W(T) ELTR , (5.19) where W(T) is the intracavity pulse energy at time t = T W(T) = Z TR/2 t0=−TR/2 |A(T,t0 )| 2 dt0 ≈ Z ∞ −∞ |A(T,t0 )| 2 dt0 . (5.20)
1775.2.ACTIVEMODELOCKINGBYLOSSMODULATIONHighOutputCouplerReflectorAOMGainFigure 5.2:Actively modelocked laser with an amplitude modulator(Acousto-Optic-Modulator).Taking all effects into account, the linear ones: loss, dispersion, gain andgain dispersion, as well as the nonlinear ones like saturable absorption andself-phase modulation, weend up with the master equation of modelockingaA(T,t)ar-IA(T,t)+ZA(T,t)TROTotn=2102(5.21)A(T,t)+ g(T22 0t/2q(T,t)A(T,t) - js|A(T,t)?A(T,t)To keep notation simple, we replace t' by t again. This equation was firstderived by Haus [4] under the assumption of small changes in pulse shapeper round-trip and per element passed within one round-trip.5.2Active Mode Locking by Loss Modula-tionActive mode locking was first investigated in 1970 by Kuizenga and Siegmanusing a gaussian pulse analyses, which we want to delegate to the exercises[3]. Later in 1975 Haus [4] introduced the master equation approach (5.21),We follow the approach of Haus, because it also shows the stability of thesolution.We introduce a loss modulator into the cavity,for example an acousto-optic modulator, which periodically varias the intracavity loss according to
5.2. ACTIVE MODE LOCKING BY LOSS MODULATION 177 Figure 5.2: Actively modelocked laser with an amplitude modulator (Acousto-Optic-Modulator). Taking all effects into account, the linear ones: loss, dispersion, gain and gain dispersion, as well as the nonlinear ones like saturable absorption and self-phase modulation, we end up with the master equation of modelocking TR ∂A(T,t0 ) ∂T = −lA(T,t0 ) + j X∞ n=2 Dn µ j ∂n ∂t ¶n A(T, t0 ) + g(T) µ 1 + 1 Ω2 g ∂2 ∂t02 ¶ A(T,t0 ) (5.21) − q(T, t0 )A(T,t0 ) − jδ|A(T, t0 )| 2 A(T,t0 ). To keep notation simple, we replace t 0 by t again. This equation was first derived by Haus [4] under the assumption of small changes in pulse shape per round-trip and per element passed within one round-trip. 5.2 Active Mode Locking by Loss Modulation Active mode locking was first investigated in 1970 by Kuizenga and Siegman using a gaussian pulse analyses, which we want to delegate to the exercises [3]. Later in 1975 Haus [4] introduced the master equation approach (5.21). We follow the approach of Haus, because it also shows the stability of the solution. We introduce a loss modulator into the cavity, for example an acoustooptic modulator, which periodically varias the intracavity loss according to
178CHAPTER5.ACTIVEMODELOCKINGImageremoveddueto copyright restrictionsPleaseseeKeller,U.UltrafastLasrPhyicsInstitutefQuantumlectronicsSwissFederalInstituteofechnoogETHHonggerberg—HPT,CH-8093Zurich,Switzerland.Figure 5.3: Schematic representation of the master equation for an activelymode-locked laser.q(t) = M (1 - cos(wmt). The modulation frequency has to be very preciselytuned to the resonator round-trip time, wM = 2/Tr, see Fig.5.2. Themodelocking process is then described by the master equation02OA[g(T) + D f2-1 - M (1 -cos(wmt)A.(5.22)TRaTneglecting GDD and SPM. The equation can be interpreted as the total pulseshaping due to gain, loss and modulator, see Fig.5.3.If we fix the gain in Eq. (5.22) at its stationary value, what ever it mightbe, Eq.(5.22) is a linear p.d.e, which can be solved by separation of variables.Thepulses, we expect, will have a width much shorter than the round-triptime Tr. They will be located in the minimum of the loss modulation wherethe cosine-function can beapproximated by a parabola and we obtain02OAMst(5.23)g-1+DgATR90t2OTM, is the modulation strength, and corresponds to the curvature of the lossmodulation in the time domain at the minimum loss point9Dg(5.24)MwMs(5.25)2
178 CHAPTER 5. ACTIVE MODE LOCKING Figure 5.3: Schematic representation of the master equation for an actively mode-locked laser. q(t) = M (1 − cos(ωMt)). The modulation frequency has to be very precisely tuned to the resonator round-trip time, ωM = 2π/TR, see Fig.5.2. The modelocking process is then described by the master equation TR ∂A ∂T = ∙ g(T) + Dg ∂2 ∂t2 − l − M (1 − cos(ωMt))¸ A. (5.22) neglecting GDD and SPM. The equation can be interpreted as the total pulse shaping due to gain, loss and modulator, see Fig.5.3. If we fix the gain in Eq. (5.22) at its stationary value, what ever it might be, Eq.(5.22) is a linear p.d.e, which can be solved by separation of variables. The pulses, we expect, will have a width much shorter than the round-trip time TR. They will be located in the minimum of the loss modulation where the cosine-function can be approximated by a parabola and we obtain TR ∂A ∂T = ∙ g − l + Dg ∂2 ∂t2 − Mst 2 ¸ A. (5.23) Ms is the modulation strength, and corresponds to the curvature of the loss modulation in the time domain at the minimum loss point Dg = g Ω2 g , (5.24) Ms = Mω2 M 2 . (5.25) 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:
1795.2.ACTIVEMODELOCKINGBYLOSSMODULATIONThe differential operator on the right side of (5.23) corresponds to the Schrodinger-Operator of the harmonic oscillator problem. Therefore, the eigen functionsof this operator are the Hermite-GaussiansAn(T,t) = An(t)e^nT/TR(5.26)Wn-Hn(t/Ta)e-2r(5.27)An(t)2nV元n!Tawhere Ta defines the width of the Gaussian. The width is given by the fourthroot of the ratio between gain dispersion and modulator strengthTa= /Dg/M..(5.28)Note, from Eq.(5.26)we can follow, that the gain per round-trip of eacheigenmode is given by An (or in general the real part of >n),which are givenby1an = n -1 - 2MsTa(n +(5.29)The corresponding saturated gain for each eigen solution is given by1(5.30)9n1+PLTRwhere Wn is the energy of the corresponding solution and Pr = EL/TL thesaturation power of the gain. Eq. (5.29) shows that for given g the eigensolution with n = O, the ground mode, has the largest gain per roundtrip.Thus, if there is initially a field distribution which is a superpostion of alleigen solutions, the ground mode will grow fastest and will saturate the gainto avaluegs =I + M.Ta.(5.31)Such that Ao = 0 and consequently all other modes will decay since An < o forn ≥ 1. This also proves the stability of the ground mode solution [4]. Thusactive modelocking without detuning between resonator round-trip time andmodulator period leads to Gaussian steady state pulses with a FWHM pulsewidth(5.32)△tFWHM = 2ln 2Ta = 1.66Ta
5.2. ACTIVE MODE LOCKING BY LOSS MODULATION 179 The differential operator on the right side of (5.23) corresponds to the SchrödingerOperator of the harmonic oscillator problem. Therefore, the eigen functions of this operator are the Hermite-Gaussians An(T,t) = An(t)eλnT /TR , (5.26) An(t) = s Wn 2n√πn!τ a Hn(t/τ a)e − t2 2τ2 a , (5.27) where τ a defines the width of the Gaussian. The width is given by the fourth root of the ratio between gain dispersion and modulator strength τ a = 4 q Dg/Ms. (5.28) Note, from Eq. (5.26) we can follow, that the gain per round-trip of each eigenmode is given by λn (or in general the real part of λn), which are given by λn = gn − l − 2Msτ 2 a(n + 1 2 ). (5.29) The corresponding saturated gain for each eigen solution is given by gn = 1 1 + Wn PLTR , (5.30) where Wn is the energy of the corresponding solution and PL = EL/τ L the saturation power of the gain. Eq. (5.29) shows that for given g the eigen solution with n = 0, the ground mode, has the largest gain per roundtrip. Thus, if there is initially a field distribution which is a superpostion of all eigen solutions, the ground mode will grow fastest and will saturate the gain to a value gs = l + Msτ 2 a. (5.31) such that λ0 = 0 and consequently all other modes will decay since λn < 0 for n ≥ 1. This also proves the stability of the ground mode solution [4]. Thus active modelocking without detuning between resonator round-trip time and modulator period leads to Gaussian steady state pulses with a FWHM pulse width ∆tFWHM = 2 ln 2τ a = 1.66τ a. (5.32)
180CHAPTER5.ACTIVEMODELOCKINGThe spectrum of the Gaussian pulse is given byAo(t)eiut dtAo(w) =(5.33)VVrWnTae-sp(5.34)and its FWHM is1.66(5.35)△fFWHM2元TaTherfore, the time-bandwidth product of the Gaussian is(5.36)AtFWHM·△fFWHM=0.44.The stationary pulse shape of the modelocked laser is due to the parabolicloss modulation (pulse shortening) in the time domain and the parabolicfiltering (pulse stretching) due to the gain in the frequency domain, see Figs.5.4 and 5.5. The stationary pulse is achieved when both effects balance.Since external modulation is limited to electronic speed and the pulse widthdoes only scale with the inverse square root of the gain bandwidth activelymodelocking typically only results in pulse width in the range of 10-100ps.(a) 14-Loss Modulation, q(t) / MPulse"n'e'9lo1M.83.6AOM2Loss.4seo200-0.4-0.20.00.20.4Time, t/TFigure 5.4: (a) Loss modulation gives pulse shortening in each roundtrip
180 CHAPTER 5. ACTIVE MODE LOCKING The spectrum of the Gaussian pulse is given by A˜0(ω) = Z ∞ −∞ A0(t)eiωtdt (5.33) = q√πWnτ ae− (ωτa)2 2 , (5.34) and its FWHM is ∆fFWHM = 1.66 2πτ a . (5.35) Therfore, the time-bandwidth product of the Gaussian is ∆tFWHM · ∆fFWHM = 0.44. (5.36) The stationary pulse shape of the modelocked laser is due to the parabolic loss modulation (pulse shortening) in the time domain and the parabolic filtering (pulse stretching) due to the gain in the frequency domain, see Figs. 5.4 and 5.5. The stationary pulse is achieved when both effects balance. Since external modulation is limited to electronic speed and the pulse width does only scale with the inverse square root of the gain bandwidth actively modelocking typically only results in pulse width in the range of 10-100ps. Figure 5.4: (a) Loss modulation gives pulse shortening in each roundtrip
1815.2.ACTIVEMODELOCKINGBYLOSSMODULATION(b) 1.038 9(0M0.8GainGain,0.6Spectrump0.4eF0.20.0-0.50.00.5-1.01.0Frequency, 0/gFigure 5.5:(b) the finite gain bandwidth gives pulse broadening in eachroundtrip. For a certain pulse width there is balance between the two pro-cesses.For example: Nd:YAG; 21 = 2g = 10%, g = π△frwHM = 0.65 THz,M = 0.2, fm = 100 MHz, Dg = 0.24 ps2, M。= 4 . 1016g-1, Tp ~ 99 ps.With the pulse width (5.28),Eq.(5.31) can be rewritten in several ways=I+M.T2+gs =I+MsTa =1+-+(5.37)2T2Ta2which means that in steady state the saturated gain is lifted above the losslevel l, so that many modes in the laser are maintained above threshold.Thereis additional gain necessaryto overcometheloss of the modulator dueto thefinitetemporal width of the pulse and the gainfilter duetothe finitebandwidthofthepulse.UsuallyM.ra<1,gs-1(5.38)11since thepulses are much shorter than theround-triptime and the stationarypulse energy can therefore be computed from1(5.39)=9s1+
5.2. ACTIVE MODE LOCKING BY LOSS MODULATION 181 Figure 5.5: (b) the finite gain bandwidth gives pulse broadening in each roundtrip. For a certain pulse width there is balance between the two processes. For example: Nd:YAG; 2l = 2g = 10%, Ωg = π∆fFWHM = 0.65 THz, M = 0.2, fm = 100 MHz, Dg = 0.24 ps2, Ms = 4 · 1016s−1, τ p ≈ 99 ps. With the pulse width (5.28), Eq.(5.31) can be rewritten in several ways gs = l + Msτ 2 a = l + Dg τ 2 a = l + 1 2 Msτ 2 a + 1 2 Dg τ 2 a , (5.37) which means that in steady state the saturated gain is lifted above the loss level l, so that many modes in the laser are maintained above threshold. There is additional gain necessary to overcome the loss of the modulator due to the finite temporal width of the pulse and the gain filter due to the finite bandwidth of the pulse. Usually gs − l l = Msτ 2 a l ¿ 1, (5.38) since the pulses are much shorter than the round-trip time and the stationary pulse energy can therefore be computed from gs = 1 1 + Ws PLTR = l. (5.39)
182CHAPTER5.ACTIVEMODELOCKING1-MMMffno+1fno-1fnoFigure 5.6: Modelocking in the frequency domain: The modulator transversenergy from each mode to its neighboring mode, thereby redistributing en-ergy from the center to the wings of the spectrum. This process seeds andinjection locks neighboring modes.The name modelocking originates from studying this pulse formation processin the frequency domain. Note, the term-M [1 - cos(wMt)] Adoes generate sidebands on each cavity mode present according to-M [1 - cos(w Mt)] exp(jwnot)1exp(anot - wmt) -:2exp(j(wnot +wMt)-Mexp(jwnot)-1= M[exp(jnot)+xp(iwno-t)+ep(jwno+)]if the modulation frequency is the same as the cavity round-trip frequencyThe sidebands generated from each running mode is injected into the neigh-boring modes which leads to synchronisation and locking of neighboringmodes, i.e. mode-locking, see Fig.5.65.3Active Mode-Locking by Phase Modula-tionSide bands can also be generated by a phase modulator instead of an am-plitude modulator. However, the generated sidebands are out of phase with
182 CHAPTER 5. ACTIVE MODE LOCKING f 1-M n0-1 M M n0+1 f n0 f f Figure 5.6: Modelocking in the frequency domain: The modulator transvers energy from each mode to its neighboring mode, thereby redistributing energy from the center to the wings of the spectrum. This process seeds and injection locks neighboring modes. The name modelocking originates from studying this pulse formation process in the frequency domain. Note, the term −M [1 − cos(ωMt)] A does generate sidebands on each cavity mode present according to −M [1 − cos(ωMt)] exp(jωn0 t) = −M ∙ exp(jωn0 t) − 1 2 exp(j(ωn0 t − ωMt)) − 1 2 exp(j(ωn0 t + ωMt))¸ = M ∙ − exp(jωn0 t) + 1 2 exp(jωn0−1t) + 1 2 exp(jωn0+1t) ¸ if the modulation frequency is the same as the cavity round-trip frequency. The sidebands generated from each running mode is injected into the neighboring modes which leads to synchronisation and locking of neighboring modes, i.e. mode-locking, see Fig.5.6 5.3 Active Mode-Locking by Phase Modulation Side bands can also be generated by a phase modulator instead of an amplitude modulator. However, the generated sidebands are out of phase with