EE485 Introduction to Photonics Laser Operation 1. Rate equations 2. Gain media 3. Steady-state laser operation 4. Laser line broadening 5. Pulsed operation Reading: Pedrotti3, Chapter 26 Ref: Verdeyen, “Laser Electronics,” 3rd ed. Sec. 9.4-9.5 Transition Cross Section Stimulated emission cross section σ=Bg21 ( ν ') h ν '/ c Absorption cross section σ=abs (/)gg21 σ Absorption coefficient α = −σ()NN21 − dI = −αI Effective absorption and emission cross sections of ytterbium- dz doped germanosilicate glass, as used in the cores of ytterbium-doped fibers. Exercise: The cross section σ, for a transition from the ground state to an excited state that is resonant with an EM field of λ = 808 nm, for a neodymium (Nd) atom doped into a YAG (yttrium aluminum garnet) crystal is ~3 × 10 cm2. Assume that the dopant density of Nd in the YAG crystal is 1020 atoms/cm3 and that the YAG crystal itself−20 is transparent to 808-nm light. A diode laser with λ = 808 nm is used to pump an Nd:YAG laser rod. (a) Estimate the small-signal absorption coefficient for the Nd:YAG crystal. (b) Estimate the depth of penetration into to the Nd:YAG crystal at which the intensity of the diode laser beam attenuates to 1/e of its initial value. (c) One can define saturation intensity , for an absorptive medium as the intensity for which the loss coefficient α is reduced by a factor of 2 from its small-signal value. Find the expression for , . 퐼푆 푎푎푎 퐼푆 푎푎푎 Lih Y. Lin 2 Four-Level System Rate equations: dN3 σPPI = −κ33N −() NN3 − 0 dt hνP dN σI 2 =κN −κ N −() NN − dt 32 3 2 2 hν' 2 1 dN σI 1 =κN +κ N −κ N +() NN − dt 31 3 21 2 10 1 hν' 2 1 dN0 σPPI =κ30N 3 +κ 20 N 2 +κ 10 N 1 +() NN3 − 0 dt hνP d (NNNN+++) =0 dt 0123 κ3 =κ 32 +κ 31 +κ 30 κ2 =κ 21 +κ 20 Gain coefficient γ=σ()NN21 − =σ Ninv dI = γI The gain coefficient depends on intensity I since the population dz inversion is, in general, dependent on the intensity. Lih Y. Lin Exercise: 4-level amplifying system 8 8 κ32 = 10 /s, κ21 = 1000/s, κ10 = 10 /s, κ30 = κ31 = κ20 ~ 0 -19 2 -18 2 σP = 3x10 cm , σ = 10 cm λ30 = 400 nm, λ21 = 600 nm (in free space) 26 3 NT = 1.5x10 /m (a) Rate equations for the population densities of the levels. (b) Steady-state small-signal population inversion as a function of the pump intensity. (c) Pump intensity IP required to sustain a steady-state population inversion. (d) Pump intensity IP required to sustain a small-signal gain coefficient of 0.01/cm. (e) Pump intensity IP required to sustain a small-signal gain coefficient of 1/cm. (f) Compare N0 to N1, N2, and N3 for the pump intensities in (d) and (e). (g) Use the ideal four-level gain medium relation together with the definition of the effective density under undepleted pump approximation, estimate the pump intensity required to sustain a small-signal gain coefficient of 0.01/cm and 1/cm. Compare these results to those obtained in (d) and (e). Lih Y. Lin 4 Steady-State Ring Laser Output Consider a ring laser cavity. Mirrors M1 and M2 have reflectances R1 = R2. Let the gain medium be homogeneously broadened and have length L = 10 cm and a 2 saturation intensity (at the lasing frequency) of Is = 2000 W/cm . (a) Find the threshold gain coefficient if R1 = R2 = 1, R3 = 0.95, and T3=0.05. (b) If the small-signal gain coefficient is twice the threshold value, find the intensity of the output. (c) Assuming that the laser beam has cross sectional area A = 0.1 cm2, find the output power of the laser. (d) Assuming that the overall efficiency of this laser system is 5%, find the pump power required to operate this laser system. Lih Y. Lin 5 Exercise: Four-Mirror Ring Laser Cavity Analyze the intensity change during a round trip in a four-mirror ring laser cavity. Lih Y. Lin 6 Two-mirror Linear Laser Cavity Threshold gain 11 γth =α+ ln 2L RR12 Output intensity 1 γ 0 (2L )− ln TI2 S RR12 Iout = 2 (1−+RR12 )(1 R2 / R 1 ) 2 Exercise: Assume saturation intensity IS = 10 W/cm and α = 0. Design a laser cavity by picking a high reflectivity R1 for the back mirror and a cavity length L, then write a computer program to plot Iout versus R2 for R2 ranging from 0.1 to 1. Do this consider two situations: (1) The laser medium is pumped at a rate that leads to a small-signal gain γ0 coefficient 5 times of the threshold gain (therefore it’s a function of R2). (2) The laser medium is pumped at a rate that leads to a fixed small-signal gain coefficient γ0 5 times of the threshold gain value when R2= 0.2. Discuss the difference in the results. Lih Y. Lin 7 Homogeneous and Inhomogeneous Broadening ∆ν H 2 ν= 4ln(2) −4ln(2)[( ν−ν0 )/ ∆νD ] g() 22 ge()ν= 2π ( ν − ν ) + ( ∆ν / 2) 2 0 H π∆ν D 1/2 8kTB ∆νD = ln(2) ν0 Mc2 + Exercise: A typical Ar laser has a cavity length of 1 m and a center wavelength λ0 = 488 nm. The atomic mass of an Ar atom is M = 6.64x10-26 kg. Assume the temperature of the gas under the operating condition is 3000 K. How many longitudinal modes can exist within the FWHM of the inhomogeneous broadened lineshape function? Lih Y. Lin 8 Relaxation Oscillation Laser turn-on and approach to steady state. I= h ν ' cNP dN σI 2 =RN −κ − NRNcNN = −κ −σ dt p2 22hν' 2 pP 2 22 2 dN V P g Γ = −ΓNPP + σ cN N2 : Cavity loss rate dt Vc VVgc/:Ratio of the gain cell length to the perimeter of the laser cavity σ=10−18 cm 2 Γ=1081 s− 71− κ=2 10 s γ0 /2 γ=th VVgc/= 0.3 Lih Y. Lin 9 Q-Switching Laser Q-switching techniques Ref: (1) Verdeyen, “Laser Electronics,” Prentice Hall (2) Siegman , “Lasers,” University Science Books Lih Y. Lin 10 Exercise: Q-Switched Laser dN N n PP= −1 dtτP n th dn n N =−⋅2 P dt nthτ P Assume a ruby laser (λ = 694.3 nm) to be pumped to four times threshold. For simplicity we assume equal degeneracy of the lasing states even though that is not true for ruby. We also assume a residual attenuation of 0.1 cm-1 by the switch medium even when it is in its high transmission state. The laser is operated at Q-switching mode. Calculate: (a) Threshold gain, inversion density and total # of inverted atoms at threshold (b) Maximum photon # in the cavity (c) Photon lifetime in the cavity (d) Output power at the maximum of the pulse Lih Y. Lin 11 Mode-Locked Laser Et( )= E00 (t)exp(i ω t ) +−(N 1)/2 Et()= ∑ En ()expi( t[ ω0 + n ωcn ) t +φ () t] −−(N 1)/2 −ω itω0 1 exp(iNc t ) Et()= Ee0 1−ω exp(itc ) 2 22 Et() E0 sin ( Ntωc / 2) It()= = 2 2ηη 2 sin ( ωct / 2) T=τ=RT 2/ dc 1 ∆≈t P ∆ν Lih Y. Lin 12 Mode-Locking RANDOMLOCKED phasesphase forfor allall thethe laserlaser modesmodes Out of phase Irradiance vs. Time In phase Out of phase Out of phase Time Time Lih Y. Lin 13 Bandwidth vs. Pulsewidth broadestbroader spectrumspectrum ∆ν∆τ = const. narrowbandwidth spectrum∆ν continuous waveduration∆τ (CW) pulsesshortest (mode pulses-locked) Lih Y. Lin 14 Various Types of Laser Output cw cw ML Power Power Time Time Q-switch Q-sw.ML Power Power Time Time Lih Y. Lin 15 .