6.2.FASTSATURABLEABSORBERMODELOCKING235where G is the net energy gain per roundtrip, which vanishes when steadystate is reached [3]. Substitution of the master equation into (6.30) with(secha)d =2,(6.32)4(secha) dr(6.33)3'd22dd (secha) dasechrdr(6.34)sechr3:drresults inDf2+40/2(6.35)G(gs,W) =gs-lo -3T2Df02/A0/2 = 9s - 0 +(6.36)= gs- lo + #T2with the saturated gaingo(6.37)gs(W)1+PLTREquation (6.36)togetherwith (6.28)determinesthepulse energyDi90gs(W)=lo1+T2(w)?= lo -(6.38)16DgFigure 6.4 shows the time dependent variation of gain and loss in a lasermodelocked with a fast saturable absorber on a normalized time scale Here,we assumed that the absorber saturates linearly with intensity up to a max-imum value go = A2. If this maximum saturable absorption is completelyexploited see Figure 6.5.The minimum pulse width achievable with a givensaturable absorption qo results from Eq.(6.26)Dio(6.39)2=2to be[21(6.40)Vqoay
6.2. FAST SATURABLE ABSORBER MODE LOCKING 235 where G is the net energy gain per roundtrip, which vanishes when steady state is reached [3]. Substitution of the master equation into (6.30) with Z ∞ −∞ ¡ sech2 x ¢ dx = 2, (6.32) Z ∞ −∞ ¡ sech4 x ¢ dx = 4 3 , (6.33) − Z ∞ −∞ sechx d2 dx2 (sechx) dx = Z ∞ −∞ µ d dxsechx ¶2 dx = 2 3 . (6.34) results in G(gs, W) = gs − l0 − Df 3τ 2 + 2 3 γ|A0| 2 (6.35) = gs − l0 + 1 2 γ|A0| 2 = gs − l0 + Df τ 2 = 0 (6.36) with the saturated gain gs(W) = g0 1 + W PLTR (6.37) Equation (6.36) together with (6.28) determines the pulse energy gs(W) = g0 1 + W PLTR = l0 − Df τ 2 = l0 − (γW) 2 16Dg (6.38) Figure 6.4 shows the time dependent variation of gain and loss in a laser modelocked with a fast saturable absorber on a normalized time scale Here, we assumed that the absorber saturates linearly with intensity up to a maximum value q0 = γA2 0. If this maximum saturable absorption is completely exploited see Figure 6.5.The minimum pulse width achievable with a given saturable absorption q0 results from Eq.(6.26) Df τ 2 = q0 2 , (6.39) to be τ = r 2 q0 1 Ωf . (6.40)
236CHAPTER6.PASSIVEMODELOCKINGImageremovedduetocopyrightrestrictions.Please see:Kartner,F.X.,and U.Keller."Stabilizationof soliton-likepulses witha slow saturableabsorber."OpticsLetters20 (1990):16-19Figure 6.4: Gain and loss in a passively modelocked laser using a fast sat-urable absorber on a normalized time scale = t/t. The absorber is assumedto atrae inaly with inesity acoding toa(4) o(1 - 4F),.edoq(IA/ 2)01IA/2140 /2Figure 6.5: Saturation characteristic of an ideal saturable absorber
236 CHAPTER 6. PASSIVE MODELOCKING Figure 6.4: Gain and loss in a passively modelocked laser using a fast saturable absorber on a normalized time scale x = t/τ . The absorber is assumed to saturate linearly with intensity according to q(A) = q0 ³ 1 − |A| 2 A2 0 ´ . Figure 6.5: Saturation characteristic of an ideal saturable absorber 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: