Chapter 4Laser Dynamics (single-mode)Before we start to look into the dynamics of a multi-mode laser, we shouldrecall the technically important regimes of operation of a"single-mode" laser.The term"single-mode" is set in apostrophes, since it doesn't have to bereally single-mode. There can be several modes running, for example due tospatial holeburning, but in an incoherent fashion, so that only the averagepower of the beam matters. For a more detailed account on single-modelaser dynamics and Q-Switching the following references are recommended[1][3][16][4][5] 4.1Rate EquationsIn section 2.5, we derived for the interaction of a two-level atom with a laserfield propagating to the right the equations of motion (2.171) and (2.172),which are given here again:Nh010(4.1)A(z,t) =4T,E, (2, t) A(2,t),32aOt[A(z, t)2w-wo(4.2)ww(z,t)TIEswhere Ti is the energy relaxation rate, Ug the group velocity in the hostmaterial where the two level atoms are embedded, Es = I,Ti, the saturationfuence JJ/cm?, of the medium.and I, the saturation intensity according to127
Chapter 4 Laser Dynamics (single-mode) Before we start to look into the dynamics of a multi-mode laser, we should recall the technically important regimes of operation of a ”single-mode” laser. The term ”single-mode” is set in apostrophes, since it doesn’t have to be really single-mode. There can be several modes running, for example due to spatial holeburning, but in an incoherent fashion, so that only the average power of the beam matters. For a more detailed account on single-mode laser dynamics and Q-Switching the following references are recommended [1][3][16][4][5]. 4.1 Rate Equations In section 2.5, we derived for the interaction of a two-level atom with a laser field propagating to the right the equations of motion (2.171) and (2.172), which are given here again: µ ∂ ∂z + 1 vg ∂ ∂t¶ A(z, t) = N~ 4T2Es w (z, t) A(z, t), (4.1) w˙ = −w − w0 T1 + |A(z, t)| 2 Es w(z, t) (4.2) where T1 is the energy relaxation rate, vg the group velocity in the host material where the two level atoms are embedded, Es = IsT1, the saturation fluence [J/cm2] , of the medium.and Is the saturation intensity according to 127
128CHAPTER4.LASERDYNAMICS(SINGLE-MODE)Eq.(2.145)ME2TT,ZFIs :h2Ewhich relates the saturation intensity to the microscopic parameters of thetransition like longitudinal and transversal relaxation rates as well as thedipole moment of the transition.TN.I-hf'gAEffVN1VgFigure4.l:Rateequationsforthetwo-level atomIn many cases it is more convenient to normalize (4.1) and (4.2) to thepopulations in level e and g or 2 and 1, respectively, N2 and Ni, and thedensity of photons, nL, in the mode interacting with the atoms and travelingat the corresponding group velocity, g, see Fig. 4.1. The intensity I in amode propagating at group velocity ug with a mode volume V is related tothe numberof photons Nstored in themode with volumeVbyNL-1(4.3)I=hfL7-hfiniug,2*VUg2*where hfi is the photon energy. 2* = 2 for a linear laser resonator (thenonly half of the photons are going in one direction), and 2* = 1 for a ringlaser. In this first treatment we consider the case of space-independent rateequations, i.e. we assume that the laser is oscillating on a single mode andpumping and mode energy densities are uniform within the laser material.Withtheinteractioncrosssectiongdefined ashfi(4.4)a2*I,T
128 CHAPTER 4. LASER DYNAMICS (SINGLE-MODE) Eq.(2.145) Is = ⎡ ⎢ ⎣ 2T1T2ZF ~2 ¯ ¯ ¯ M ˆ E ¯ ¯ ¯ 2 ¯ ¯ ¯ ˆ E ¯ ¯ ¯ 2 ⎤ ⎥ ⎦ −1 , which relates the saturation intensity to the microscopic parameters of the transition like longitudinal and transversal relaxation rates as well as the dipole moment of the transition. Figure 4.1: Rate equations for the two-level atom In many cases it is more convenient to normalize (4.1) and (4.2) to the populations in level e and g or 2 and 1, respectively, N2 and N1, and the density of photons, nL, in the mode interacting with the atoms and traveling at the corresponding group velocity, vg, see Fig. 4.1. The intensity I in a mode propagating at group velocity vg with a mode volume V is related to the number of photons NL stored in the mode with volume V by I = hfL NL 2∗V vg = 1 2∗hfLnLvg, (4.3) where hfL is the photon energy. 2∗ = 2 for a linear laser resonator (then only half of the photons are going in one direction), and 2∗ = 1 for a ring laser. In this first treatment we consider the case of space-independent rate equations, i.e. we assume that the laser is oscillating on a single mode and pumping and mode energy densities are uniform within the laser material. With the interaction cross section σ defined as σ = hfL 2∗IsT1 , (4.4)
1294.1.RATEEQUATIONSand multiplying Eq. (??) with the number of atoms in the mode, we obtaind(Ma- M) --(=N) - (Na- ~) gn1 + R(4.5)dtT1Note, Ugnz is the photon flux, thus o is the stimulated emission cross sectionbetween theatoms and thephotons.Rpis thepumping rateintotheupperlaser level. A similar rate equation can be derived for the photon densitydnL++ 4agv [N (mL+ 1) - Ninl] .(4.6)nLdtL VgTpHere, Tp is the photon lifetime in the cavity or cavity decay time and theone in Eq(4.6) accounts for spontaneous emission which is equivalent tostimulated emission by one photon occupying the mode. V. is the volume ofthe active gain medium.For a laser cavity with a semi-transparent mirrorwith transmission T, producing a small power loss 2l = - ln(1 - T) ~ T (forsmall T)per round-trip in the cavity, the cavity decay time is Tp=21/Tr,if TR = 2*L/co is the roundtrip-time in linear cavity with optical length 2Lor a ring cavity with optical length L. The optical length L is the sum of theoptical length in the gain medium ngrouplg and the remaining free space cavitylength la.Internal losses can be treated in a similar way and contribute tothe cavity decay time. Note, the decay rate for the inversion in the absenceof a field, 1/Ti, is not only due to spontaneous emission, but is also a result ofnon radiative decay processes. See for example the four level system shownin Fig. 4.2. In the limit, where the populations in the third and first levelare zero, because of fast relaxation rates, i.e. T32, Tio → 0, we obtaindN2(4.7)N2gu.N2ni+RdtTLdnLeggN2 (nL+1).(4.8)dtntLVgTpwhere TL = T2r is the lifetime of the upper laser level. Experimentally, thephoton number and the inversion in a laser resonator are not
4.1. RATE EQUATIONS 129 and multiplying Eq. (??) with the number of atoms in the mode, we obtain d dt(N2 − N1) = −(N2 − N1) T1 − σ (N2 − N1) vgnL + Rp (4.5) Note, vgnL is the photon flux, thus σ is the stimulated emission cross section between the atoms and the photons. Rp is the pumping rate into the upper laser level. A similar rate equation can be derived for the photon density d dtnL = −nL τ p + lg L σvg Vg [N2 (nL + 1) − N1nL] . (4.6) Here, τ p is the photon lifetime in the cavity or cavity decay time and the one in Eq.(4.6) accounts for spontaneous emission which is equivalent to stimulated emission by one photon occupying the mode. Vg is the volume of the active gain medium. For a laser cavity with a semi-transparent mirror with transmission T, producing a small power loss 2l = − ln(1− T) ≈ T (for small T) per round-trip in the cavity, the cavity decay time is τ p = 2l/TR , if TR = 2∗L/c0 is the roundtrip-time in linear cavity with optical length 2L or a ring cavity with optical length L. The optical length L is the sum of the optical length in the gain medium ngroup g lg and the remaining free space cavity length la. Internal losses can be treated in a similar way and contribute to the cavity decay time. Note, the decay rate for the inversion in the absence of a field, 1/T1, is not only due to spontaneous emission, but is also a result of non radiative decay processes. See for example the four level system shown in Fig. 4.2. In the limit, where the populations in the third and first level are zero, because of fast relaxation rates, i.e. T32, T10 → 0, we obtain d dtN2 = −N2 τ L − σvgN2nL + Rp (4.7) d dtnL = −nL τ p + lg L σvg Vg N2 (nL + 1). (4.8) where τ L = T21 is the lifetime of the upper laser level. Experimentally, the photon number and the inversion in a laser resonator are not
130CHAPTER4.LASERDYNAMICS (SINGLE-MODE)N331322N2RP1211N1T1o0NoFigure 4.2: Vier-Niveau-Laservery convenient quantities, therefore, we normalize both equations to thelwaN,Trexperiencedbythelightandtheround-tripamplitudegaing=L21circulating intracavity power P=TAeffdgp9 - 90(4.9)dgEsatTLd12g(P+ Puac),(4.10)dtTRTpwithhfLEs(4.11)=IsAeffTL2*gPsatEsat/TL(4.12)=Puac(4.13)=hfLug/2*L=hfL/TR20,RpoTL(4.14)902Aeffcothe small signal round-trip gain of the laser.Note, the factor of two in frontof gain and loss is due to the fact, the g and I are gain and loss with respect toamplitude.Eq.(4.14) elucidates that the figure of merit that characterizes thesmall signal gain achievable with a certain laser material is the oTr-product
130 CHAPTER 4. LASER DYNAMICS (SINGLE-MODE) 3 0 1 2 N N N N 3 2 1 0 T T T 32 21 10 R p Figure 4.2: Vier-Niveau-Laser very convenient quantities, therefore, we normalize both equations to the round-trip amplitude gain g = lg L σvg 2Vg N2TR experienced by the light and the circulating intracavity power P = I · Aef f d dtg = −g − g0 τ L − gP Esat (4.9) d dtP = − 1 τ p P + 2g TR (P + Pvac), (4.10) with Es = IsAef f τ L = hfL 2∗σ (4.11) Psat = Esat/τ L (4.12) Pvac = hfLvg/2∗ L = hfL/TR (4.13) g0 = 2∗vgRp 2Aef f c0 στ L, (4.14) the small signal round-trip gain of the laser. Note, the factor of two in front of gain and loss is due to the fact, the g and l are gain and loss with respect to amplitude. Eq.(4.14) elucidates that the figure of merit that characterizes the small signal gain achievable with a certain laser material is the στ L-product
1314.1.RATEEQUATIONSWave-LinewidthCrossUpper-St.Refr.AfFWHM =SectionTypLaser MediumlengthLifetimeindex(THz)α (cm2)Xo(nm)TL (μs)n4.1·1019Nd3+:YAG1,0641,2000.210H1.821.3·10-19HNd3+:LSB1,062871.21.47 (ne)1.8.10-19Nd3+:YLF1,047450H0.3901.82 (ne)2.5.1019Nd3+:YVO4500.300H2.19 (ne)1,0644·10-203Nd3+ :glass1,054350H/I1.56·10-211,554H/I1.46Er3+:glass10,0002.10-20HRuby694.31,0000.061.763·1019HTi3+:Al,0331001.76660-11804.8.102080Cr3+:LiSAF67H1.4760-9601.3·102065Cr3+:LiCAF710-840170H1.43.3·10-20Cr3+:LiSGAF8880H1.4740-9301·10-131He-Ne632.80.7~10.00153·10-121Ar+515~10.070.00353·1018CO2H~110,6002,900,0000.0000603·10165H560-6400.00331.33Rhodamin-6G~10-1425H/l3-4450-30,000~0.002semiconductorsTable 4.1: Wavelength range, cross-section for stimulated emission, upper-state lifetime, linewidth, typ of lineshape (H-homogeneously broadened,I-inhomogeneously broadened) and index for some often used solid-statelaser materials, and in comparison with semiconductor and dye lasers
4.1. RATE EQUATIONS 131 Laser Medium Wavelength λ0(nm) Cross Section σ (cm2) Upper-St. Lifetime τ L (µs) Linewidth ∆fFWHM = 2 T2 (THz) Typ Refr. index n Nd3+:YAG 1,064 4.1 · 10−19 1,200 0.210 H 1.82 Nd3+:LSB 1,062 1.3 · 10−19 87 1.2 H 1.47 (ne) Nd3+:YLF 1,047 1.8 · 10−19 450 0.390 H 1.82 (ne) Nd3+:YVO4 1,064 2.5 · 10−19 50 0.300 H 2.19 (ne) Nd3+:glass 1,054 4 · 10−20 350 3 H/I 1.5 Er3+:glass 1,55 6 · 10−21 10,000 4 H/I 1.46 Ruby 694.3 2 · 10−20 1,000 0.06 H 1.76 Ti3+:Al2O3 660-1180 3 · 10−19 3 100 H 1.76 Cr3+:LiSAF 760-960 4.8 · 10−20 67 80 H 1.4 Cr3+:LiCAF 710-840 1.3 · 10−20 170 65 H 1.4 Cr3+:LiSGAF 740-930 3.3 · 10−20 88 80 H 1.4 He-Ne 632.8 1 · 10−13 0.7 0.0015 I ∼1 Ar+ 515 3 · 10−12 0.07 0.0035 I ∼1 CO2 10,600 3 · 10−18 2,900,000 0.000060 H ∼1 Rhodamin-6G 560-640 3 · 10−16 0.0033 5 H 1.33 semiconductors 450-30,000 ∼ 10−14 ∼ 0.002 25 H/I 3-4 Table 4.1: Wavelength range, cross-section for stimulated emission, upperstate lifetime, linewidth, typ of lineshape (H=homogeneously broadened, I=inhomogeneously broadened) and index for some often used solid-state laser materials, and in comparison with semiconductor and dye lasers
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 arealways stable, i.e. Re[s) 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 are always stable, i.e. Re{si} 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: