142CHAPTER4.LASERDYNAMICS(SINGLE-MODE4.4.3Theory of Active Q-SwitchingWe want to get some insight into the pulse built-up and decay of the activelyQ-switched pulse. We consider the ideal situation, where the loss of the lasercavity can beinstantaneously switched fromn ahigh value to alowvalue,i.ethe quality factor is switched from a low value to a high value, respectively(Figure: 4.11)Q-switchLaser outputpulseCavity LossN;n(t) α P(t)g(t) α N(t)I α Nth(t)Nf.Pulse OutputPumpingIntervalIntervalFigure 4.1l:Acitve Q-Switching dynamics assuming an instantaneousswitching [16]Figure by MIT OCW.Pumping Interval:During pumping with a constant pump rate Rp, proportional to the smallsignal gain go, the inversion is built up. Since there is no field present, thegainfollowsthesimpleequation:dg- 90(4.33)dgsTLorg(t) = go(1 - e-t/TL),(4.34)
142 CHAPTER 4. LASER DYNAMICS (SINGLE-MODE) 4.4.3 Theory of Active Q-Switching We want to get some insight into the pulse built-up and decay of the actively Q-switched pulse. We consider the ideal situation, where the loss of the laser cavity can be instantaneously switched from a high value to a low value, i.e. the quality factor is switched from a low value to a high value, respectively (Figure: 4.11) Figure 4.11: Acitve Q-Switching dynamics assuming an instantaneous switching [16]. Pumping Interval: During pumping with a constant pump rate Rp, proportional to the small signal gain g0, the inversion is built up. Since there is no field present, the gain follows the simple equation: d dtg = −g − g0 τ L , (4.33) or g(t) = g0(1 − e−t/τ L ), (4.34) Figure by MIT OCW. Pumping Interval Cavity Loss Q-switch t Laser output pulse g(t) ∝ N(t) I ∝ Nth(t) nL(t) ∝ P(t) Nf Ni Pulse Output Interval
Q-SWITCHING1434.4.PulseBuilt-up-Phase:Assuming an instantaneous switching of the cavity losses we look for anapproximate solution to therate equations starting of with the initial gainorinversiongi=hfN2i/(2Esat)=hfLN:/(2Esat),wecansavelyleavetheindex away since there is only an upper state population. We further assumethat during pulse built-up the stimulated emission rate is the dominate termchanging the inversion. Then the rate equations simplify tordgP(4.35)dt9Esatpa2(g - l) PP(4.36)-dtTRresulting indP2Esat(4.37)dgTR1.9Weusethefollowing inital conditionsfortheintracavitypowerP(t=O)=0and initial gain g(t = 0) = gi = r.l. Note, r means how many times the laseris pumped above threshold after the Q-switch is operated and the intracavitylosseshavebeen reduced tol.Then4.37canbe directlysolved and we obtain2Esatg(t)gi-g(t)+ Iln P(t) =(4.38)TRgiFrom this equation we can deduce the maximum power of the pulse, sincethe growth of the intracavity power will stop when the gain is reduced to thelosses, g(t)=l, (Figure 4.11)21Esat (r -1- Inr)Pmax(4.39)TREat (r -1 Inr).(4.40)TpThis is the first important quantity of the generated pulse and is shownnormalized in Figure 4.12
4.4. Q-SWITCHING 143 Pulse Built-up-Phase: Assuming an instantaneous switching of the cavity losses we look for an approximate solution to the rate equations starting of with the initial gain or inversion gi = hfLN2i/(2Esat) = hfLNi/(2Esat), we can savely leave the index away since there is only an upper state population. We further assume that during pulse built-up the stimulated emission rate is the dominate term changing the inversion. Then the rate equations simplify toτ d dtg = − gP Esat p (4.35) d dtP = 2(g − l) TR P, (4.36) resulting in dP dg = 2Esat TR µ l g − 1 ¶ . (4.37) We use the following inital conditions for the intracavity power P(t = 0) = 0 and initial gain g(t = 0) = gi = r ·l. Note, r means how many times the laser is pumped above threshold after the Q-switch is operated and the intracavity losses have been reduced to l. Then 4.37 can be directly solved and we obtain P(t) = 2Esat TR µ gi − g(t) + l ln g(t) gi ¶ . (4.38) From this equation we can deduce the maximum power of the pulse, since the growth of the intracavity power will stop when the gain is reduced to the losses, g(t)=l, (Figure 4.11) Pmax = 2lEsat TR (r − 1 − ln r) (4.39) = Esat τ p (r − 1 − ln r). (4.40) This is the first important quantity of the generated pulse and is shown normalized in Figure 4.12