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