132CHAPTER 4.LASERDYNAMICS (SINGLE-MODE)The larger this product the larger is the small signal gain go achievable witha certain laser material. Table 4.1From Eq.(2.145) and (4.4) we find the following relationship between theinteraction crosssection ofatransitionanditsmicroscopicparameters likelinewidth, dipole moment and energy relaxation rate2T2[MhfuT=2Z间This equation tells us that broadband laser materials naturally do showsmaller gain cross sections, if the dipole moment is the same.4.2Built-up of Laser Oscillation and Contin-uous Wave OperationIf Puac < P Psat = Esat/TL, than g = go and we obtain from Eq.(4.10),neglecting PuacdPdt(4.15)=2(g0-1);PTRorP(t) = P(0)e2(90-1)(4.16)The laser power builts up from vaccum fluctuations until it reaches the sat-uration power, when saturation of the gain sets in within the built-up timeTRTRPsatAeffTR(4.17)TB =—2(go - 1)" Pvac2 (go - 1)OTLSome time after the built-up phase the laser reaches steady state, with thesaturated gain and steady state power resulting from Eqs.(4.9-4.10), neglect-ing in the following the spontaneous emission, and for = 0 :go=l(4.18)9s=1+(4.19)Ps = Psat
132 CHAPTER 4. LASER DYNAMICS (SINGLE-MODE) The larger this product the larger is the small signal gain g0 achievable with a certain laser material. Table 4.1 From Eq.(2.145) and (4.4) we find the following relationship between the interaction cross section of a transition and its microscopic parameters like linewidth, dipole moment and energy relaxation rate σ = hfL IsatT1 = 2T2 ~2ZF |M ˆ E | 2 | ˜ˆ E| 2 . This equation tells us that broadband laser materials naturally do show smaller gain cross sections, if the dipole moment is the same. 4.2 Built-up of Laser Oscillation and Continuous Wave Operation If Pvac ¿ P ¿ Psat = Esat/τ L, than g = g0 and we obtain from Eq.(4.10), neglecting Pvac dP P =2(g0 − l) dt TR (4.15) or P(t) = P(0)e 2(g0−l) t TR . (4.16) The laser power builts up from vaccum fluctuations until it reaches the saturation power, when saturation of the gain sets in within the built-up time TB = TR 2 (g0 − l) ln Psat Pvac = TR 2 (g0 − l) ln Aef fTR στ L . (4.17) Some time after the built-up phase the laser reaches steady state, with the saturated gain and steady state power resulting from Eqs.(4.9-4.10), neglecting in the following the spontaneous emission, and for d dt =0: gs = g0 1 + Ps Psat = l (4.18) Ps = Psat ³g0 l − 1 ´ , (4.19)
4.3.STABILITYANDRELAXATIONOSCILLATIONS133Imageremovedduetocopyrightrestrictions.Please see:Keller,U.,UltrafastLaserPhysics,InstituteofQuantumElectronicsSwissFederalInstituteofTechnologyETHHonggerberg—HPT,CH-8093Zurich,SwitzerlandFigure 4.3: Built-up of laser power from spontaneous emission noise.4.3Stability and Relaxation OscillationsHow does the laser reach steady state, once a perturbation has occured?(4.20)g =gs+△gP =P+△P(4.21)SubstitutionintoEqs.(4.9-4.10)andlinearizationleadstoPaAgd△P(4.22)+2-dtTRdg19s-AP(4.23)AgdtEsatTstim=六(1+)is the stimulated lifetime. The perturbationswherePsTstimdecayorgrowlikeAP△Po(4.24)Ag△gowhich leads to the system of equations (using gs = l)2元△Po△PoTR0(4.25)TRAgoEsat22TTstim
4.3. STABILITY AND RELAXATION OSCILLATIONS 133 Figure 4.3: Built-up of laser power from spontaneous emission noise. 4.3 Stability and Relaxation Oscillations How does the laser reach steady state, once a perturbation has occured? g = gs + ∆g (4.20) P = Ps + ∆P (4.21) Substitution into Eqs.(4.9-4.10) and linearization leads to d∆P dt = +2 Ps TR ∆g (4.22) d∆g dt = − gs Esat ∆P − 1 τ stim ∆g (4.23) where 1 τ stim = 1 τ L ¡ 1 + Ps P sat¢ is the stimulated lifetime. The perturbations decay or grow like µ ∆P ∆g ¶ = µ ∆P0 ∆g0 ¶ est. (4.24) which leads to the system of equations (using gs = l) A µ ∆P0 ∆g0 ¶ = Ã −s 2 Ps TR − TR Esat2τp − 1 τ stim − s !µ ∆P0 ∆g0 ¶ = 0. (4.25) 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:
134CHAPTER4.LASERDYNAMICS (SINGLE-MODE)There is only a solution, if the determinante of the coefficient matrix vanishes,i.e.Ps(4.26)0EsatTpwhich determines the relaxation rates or eigen frequencies of the linearizedsystem1Ps1(4.27)81/2 =2TstimEsatTpetIntroducingthepumpparaneterP8,which tells us howoften we+pump the laser over threshold, the eigen frequencies can be rewritten as14(r-1)Tstim(4.28)1 ±,S1/22TstimrTp(r - 1)7(4.29)+2TL2TLTLTpThereareseveral conclusionstodraw:. (i): The stationary state (0, go) for go < I and (Ps, gs) for go > I arealways stable, i.e. Re[s) <0.(i): For lasers pumped above threshold, r > 1, the relaxation ratebecomes complex,i.e.therearerelaxation oscillations11(4.30)S1/2=2TstimTstimTpwith frequency wr equal to the geometric mean of inverse stimulatedlifetime and photon life time1(4.31)WRTstimTpThere is definitely a parameter range of pump powers for laser withlong upper state lifetimes, ie. /T
134 CHAPTER 4. LASER DYNAMICS (SINGLE-MODE) There is only a solution, if the determinante of the coefficient matrix vanishes, i.e. s µ 1 τ stim + s ¶ + Ps Esatτ p = 0, (4.26) which determines the relaxation rates or eigen frequencies of the linearized system s1/2 = − 1 2τ stim ± sµ 1 2τ stim ¶2 − Ps Esatτ p . (4.27) Introducing the pump parameter r =1+ Ps Psat , which tells us how often we pump the laser over threshold, the eigen frequencies can be rewritten as s1/2 = − 1 2τ stim à 1 ± j s 4 (r − 1) r τ stim τ p − 1 ! , (4.28) = − r 2τ L ± j s (r − 1) τ Lτ p − µ r 2τ L ¶2 (4.29) There are several conclusions to draw: • (i): The stationary state (0, g0) for g0 < l and (Ps, gs) for g0 > l are always stable, i.e. Re{si} < 0. • (ii): For lasers pumped above threshold, r > 1, the relaxation rate becomes complex, i.e. there are relaxation oscillations s1/2 = − 1 2τ stim ± j s 1 τ stimτ p . (4.30) with frequency ωR equal to the geometric mean of inverse stimulated lifetime and photon life time ωR = s 1 τ stimτ p . (4.31) There is definitely a parameter range of pump powers for laser with long upper state lifetimes, i.e. r 4τ L < 1 τp
4.3.STABILITYANDRELAXATIONOSCILLATIONS135. If the laser can be pumped strong enough, i.e. r can be made largeenough so that the stimulated lifetime becomes as short as the cavitydecay time, relaxation oscillations vanish.The physical reason for relaxation oscillations and later instabilities is.that the gain reacts to slow on the light field, i.e. the stimulated lifetime islong in comparison with the cavity decay time.Example: diode-pumped Nd:YAG-Laser入o = 1064 nm, α = 4.10-20cm2, Aeff = 元(100μm × 150μm),r = 50TL=1.2ms,l=1%,TR=10nsFrom Eq.(4.4) we obtain:.kwhfL2=3.9Isat,Psat=IsatAeff=1.8W,Ps=91.5Wcm2OTL1 = 24μs, Tp= 1μs,wR= 2.105s-1.TstimrTstimTpFigure 4.4 shows the typically observed fluctuations of the output of a solid-state laser with long upperstate life time of several 100 μs in the time andfrequency domain.One can alsodefinea qualityfactorfor therelaxation oscillations by theratioof imaginary to real part of the complex eigenfrequencies 4.294TL(r-1)D1mwhich can be as large a several thousand for solid-state lasers with longupper-state lifetimes in the millisecond range
4.3. STABILITY AND RELAXATION OSCILLATIONS 135 • If the laser can be pumped strong enough, i.e. r can be made large enough so that the stimulated lifetime becomes as short as the cavity decay time, relaxation oscillations vanish. The physical reason for relaxation oscillations and later instabilities is, that the gain reacts to slow on the light field, i.e. the stimulated lifetime is long in comparison with the cavity decay time. Example: diode-pumped Nd:YAG-Laser λ0 = 1064 nm, σ = 4 · 10−20cm2 , Aef f = π (100µm × 150µm), r = 50 τ L = 1.2 ms, l = 1%, TR = 10ns From Eq.(4.4) we obtain: Isat = hfL στ L = 3.9 kW cm2 , Psat = IsatAef f = 1.8 W, Ps = 91.5W τ stim = τ L r = 24µs, τ p = 1µs, ωR = s 1 τ stimτ p = 2 · 105 s−1 . Figure 4.4 shows the typically observed fluctuations of the output of a solidstate laser with long upperstate life time of several 100 µs in the time and frequency domain. One can also define a quality factor for the relaxation oscillations by the ratio of imaginary to real part of the complex eigen frequencies 4.29 Q = s 4τ L τ p (r − 1) r2 , which can be as large a several thousand for solid-state lasers with long upper-state lifetimes in the millisecond range
136CHAPTER 4.LASERDYNAMICS (SINGLE-MODE)Image removed due to copyright restrictionsPlease see:Keller,U.,UltrafastLaserPhysics, InstituteofQuantum Electronics,SwissFederal Institute ofTechnologyETHHonggerberg—HPT,CH-8093Zurich,SwitzerlandFigure 4.4:Typically observed relaxation oscillations in time and frequencydomain.4.4Q-SwitchingThe energy stored in the laser medium can be released suddenly by increasingthe Q-value of the cavity so that the laser reaches threshold. This can bedone actively, for example by quickly moving one of the resonator mirrors inplace or passively by placing a saturable absorber in the resonator [1, 16].Hellwarth was first to suggest this method only one year after the invention of
136 CHAPTER 4. LASER DYNAMICS (SINGLE-MODE) Figure 4.4: Typically observed relaxation oscillations in time and frequency domain. 4.4 Q-Switching The energy stored in the laser medium can be released suddenly by increasing the Q-value of the cavity so that the laser reaches threshold. This can be done actively, for example by quickly moving one of the resonator mirrors in place or passively by placing a saturable absorber in the resonator [1, 16]. Hellwarth was first to suggest this method only one year after the invention of 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: