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Chapter 6. Laser: Theory and Applications

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Chapter 6. Laser: Theory and Applications Chapter 6. Laser: Theory and Applications Reading: Sigman, Chapter 6, 7, and 26 Bransden & Joachain, Chapter 15 Laser Basics Light Amplification by Stimulated Emission of Radiation hν = E − E E1 1 0 Stimulated emission hν E0 Population inversion (N1 > N0) ⇒ laser, maser 2 2 4π ⎛ e ⎞ I(ω ) 2 Transition rate W = ⎜ ⎟ 01 M (ω ) ∝ I(ω ) for stimulated emission 01 2 ⎜ ⎟ 2 01 01 01 m c ⎝ 4πε0 ⎠ ω01 Incident light intensity Pumping (optical, electrical, etc.) for population inversion Gain medium High reflector Out coupler Optical cavity Longitudinal Modes in an Optical Cavity EM wave in a cavity Boundary condition: λ cτ πc L = m = m = m m = 1, 2, 3, … 2 2 ω 2L c π c ⇒ λ = ,ν = m, ω = m m 2L L 2L Round-trip time of flight: T = = mτ c Typical laser cavity: L = 1.5 m, λ = 0.75 µm 2L 3 m : T = = =10−8 sec =10 nsec : c 3×108 m / sec 1 ⇒ ν = =108 Hz =100 MHz R T 2L 3 m L m = = = 4×106 = 4 milion !! λ 0.75×10−6 m Single mode 2 2 I(t) = cosω0t = cost Spectrum Intensity 1 Frequency Intensity -100 -50 0 50 100 Time Cavity Quality Factors, Qc End mirror L Out coupler R ≈ 1 T = 1 ~ 5 % Energy loss by reflection, transmission, etc. ∞ −δct /T −iω0t E(ω) = E0e e dt ∫0 E e−δct /T e−iω0t 0 E = 0 i(ω −ω0 ) +δc /T t E(ω) 2 Lorentzian Emission spectrum 2 2 E δ c ω E(ω) = 0 ~ = 2 2 2 T Q (ω −ω ) +δ /T c 0 c ω ω0 Energy of circulating EM wave, Icirc(t) ⎡ t ⎤ 2L T = Icirc (t) = Icirc (0)×exp⎢−δ c ⎥, : round-trip time of flight ⎣ T ⎦ c Number of round trips in t ⎡ ω ⎤ ⇒ Icirc (t) = Icirc (0)×exp⎢− t⎥ ⎣ Qc ⎦ Q-factor of a RLC circuit ωT 4π L 1 ω ωL Q = = Q = = where c ∆ω R δ c λ δ c Typical laser cavity: L = 1 m, λ = 0.8 µm, δc = 0.01 (~1% loss/RT) 4π 1 Q = ≈1.6×109 ∆ω ≈106 Hz c 0.8×10−6 0.01 Two Level Rate Equations and Saturation N2 E2 Two Level Rate Equation W N γ N W N dN (t) dN (t) 21 2 21 2 12 1 1 = − 2 dt dt E1 = −W12 N1(t) + (W21 + γ 21)N2 (t) N1 1 = −W12 []N1(t) − N2 (t) + γ 21N2 (t) γ 21 = : non-radiative decay rate T1 Total number of atoms: N = N1(t) + N2 (t) = constant Population difference: ∆N(t) = N1(t) − N2 (t) d ∆N(t) = −2W []N (t) − N (t) + 2γ N (t) dt 12 1 2 21 2 = −2W12 []N1(t) − N2 (t) −γ 21[N1(t) − N2 (t) − N1(t) − N2 (t)] d ∆N(t) − N ∆N(t) = −2W12∆N(t) − dt T1 Steady-State Atomic Response: Saturation d ∆N(t) − N ∆N(t) = 0 = −2W12∆N(t) − dt T1 N ∆Nss 1 1 ∆N = ∆Nss = = , Wsat ≡ 1+ 2W12T N 1+W12 /Wsat 2T1 1.2 Gain coefficient 1.0 in laser materials 0.8 α m ∝ ∆N ∆N ss N 0.6 α α (I) = m0 0.4 m Saturation 1+ I / I sat Intensity, I=I 0.2 sat 0.0 0.01 0.1 1 10 Signal strength, 2W12T Transient Two-Level Solutions d ∆N(t) − N ⎛ 1 ⎞ N ⎜ ⎟ ∆N(t) = −2W12∆N(t) − = −⎜2W12 + ⎟∆N(t) + dt T1 ⎝ T1 ⎠ T1 ⎡ ⎛ 1 ⎞ ⎤ N ∆N(t) = ∆N ss + Aexp⎢− ⎜2W12 + ⎟t⎥, ∆N = ⎜ T ⎟ ss ⎣ ⎝ 1 ⎠ ⎦ 1+ 2W12T Initial population difference at t = 0: ∆N(0) = ∆N ss + A N ⎡ N ⎤ ⎡ t ⎤ ∆N(t) = + ⎢∆N(0) − ⎥ exp⎢− ()1+ 2W12T1 ⎥ 1+ 2W12T ⎣ 1+ 2W12T ⎦ ⎣ T1 ⎦ 1.0 2W12T1 = 0 0.5 Transient saturation behavior ∆N(t) 1 0.5 following sudden turn-on of an N 2 applied signal 4 0.0 0123 t /T1 Two-Level Systems with Degeneracy Stimulated transition rates: g1W12 = g2W21 N2 g Rate equation: E2 2 dN (t) dN (t) 1 = − 2 = −W N (t) + (W + γ )N (t) dt dt 12 1 21 21 2 E1 g1 Population ⎛ g ⎞ N ∆N(t) ≡ ⎜ 2 ⎟N (t) − N (t) 1 difference: ⎜ ⎟ 1 2 ⎝ g1 ⎠ 1 Effective signal-stimulated transition probability: W ≡ ()W +W eff 2 12 21 d ∆N(t) − N Rate equation: ∆N(t) = −2Weff ∆N(t) − dt T1 Atomic time constants: T1 and T2 T1 : longitudinal (on-diagonal) relaxation time population recovery or energy decay time T2 : dephasing time, transverse (off-diagonal) relaxation time time constant for dephasing of coherent macroscopic polarization Steady-State Laser Pumping and Population Inversion Four-level pumping analysis Rate equation for level 4 E4 τ43 (fast) E 3 pump transition probability, W14 = W41 = Wp pumping τ 32 dN W (slow) 4 = W (N − N ) − (γ + γ + γ )N p dt p 1 4 43 42 41 4 = Wp (N1 − N4 ) − N4 /τ 4 , E2 1 τ (fast) where ≡ γ 4 = γ 43 + γ 42 + γ 41 E1 21 τ 4 Steady state population: W τ N = p 4 N ≈ W τ N , if W τ <<1 4 1+W τ 1 p 4 1 p 4 p 4 Normalized pumping rate Rate equations for level 2 and 3 at steady state dN3 N4 N3 τ 3 = γ 43 N4 − (γ 32 + γ 31)N3 = − N3 = N4 dt τ 43 τ 3 τ 43 E4 τ43 (fast) τ >>τ so that N >> N In a good laser system, 3 43 3 4 E3 pumping τ32 Wp (slow) dN N N N 2 = γ N + γ N −γ N = 4 + 3 − 2 E dt 42 4 32 3 21 2 τ τ τ 2 42 32 21 τ (fast) E1 21 ⎛τ τ τ ⎞ τ τ τ ⎜ 21 43 21 ⎟ 21 43 21 N2 = ⎜ + ⎟N3 = β N3 where β ≡ + ⎝τ 32 τ 42τ 3 ⎠ τ 32 τ 42τ 3 : population inversion on the 3 → 2 transition If β < 1, N2 < N3 In a good laser system, γ 42 ≈ 0 so that the level 4 will relax primarily into the level 3. τ N τ β ≈ 21 condition for population inversion β ≡ 2 ≈ 21 <<1 τ 32 N3 τ 32 Fluorescent quantum efficient The number of fluorescent photons spontaneously emitted on the laser transition divided by the number of pump photons absorbed on the pump transitions when the laser material is below threshold γ γ τ τ η = 43 × rad = 4 × 3 γ 4 γ 3 τ 43 τ rad Fraction of the total atoms excited to Fraction of the total decay out of level level 4 relax directly into the level 3 3 is purely radiative decay to level2 Four level population inversion N = N1 + N2 + N3 + N4 N N − N (1− β )ηW τ 4-level system 3 2 = p rad N 1+ []1+ β + 2τ 43 /τ rad ηWpτ rad ∆Nth N 0 ∆ In a good laser system, τ rad >>τ 43 , β →0. 3-level system N − N (1− β )ηW τ ηW τ -N 3 2 ≈ p rad ≈ p rad N 1+ (1+ β )ηWpτ rad 1+ηWpτ rad 0123 Wpτrad Laser gain saturation analysis N3 E3 γ32 Pumping transition E2 N 2 dN3 = Wp (N0 − N3 ) ≈ Wp N0 ≈ Wp N pumping stimulated dt pump γ21 W Wp sig E Effective pumping rate: Rp =η p Wp N0 1 N1 γ quantum efficiency E3 → E2 E0 1 N0≈N Rate equations for laser levels 1 and 2: γ 2 = γ 21 + γ 20 Wsig + γ 21 dN2 N1 = Rp = Rp −Wsig (N2 − N1) −γ 2 N2 dt Wsig (γ 1 + γ 20 ) + γ 1γ 2 W + γ dN1 sig 1 = Wsig (N2 − N1) + γ 21N2 −γ 1N1 N2 = Rp dt Wsig (γ 1 + γ 20 ) + γ 1γ 2 Gain saturation behavior ⎛ γ −γ ⎞ 1 ⎜ 1 21 ⎟ ∆N21 = N2 − N1 = ⎜ ⎟Rp × ⎝ γ 1γ 2 ⎠ 1+ [](γ 1 + γ 20 ) /γ 1γ 2 Wsig 1 = ∆N0 1+Wsigτ eff ⎛ γ −γ ⎞ ⎛ τ ⎞ Small signal or unsaturated ⎜ 1 21 ⎟ ⎜ 1 ⎟ ∆N0 = ⎜ ⎟Rp = ⎜1− ⎟× Rpτ 2 population inversion ⎝ γ 1γ 2 ⎠ ⎝ τ 21 ⎠ 1 γ γ ⎛ τ ⎞ 1 2 or ⎜ 1 ⎟ Effective recovery time = τ eff =τ 2 ⎜1+ ⎟ τ eff γ 1 + γ 20 ⎝ τ 20 ⎠ 1 If γ20 ≈ 0, γ2 ≈γ21 ∆N21 = Rp (τ 2 −τ1)× 1+Wsigτ 2 • Population inversion requires τ21 > τ1. • ∆N0 ∝ Rp ×τ 2 (1−τ1 /τ 2 ) • If τ1 → 0, τeff ≈τ2. • The saturation intensity of the inverted population is independent of Rp. Wave Propagation in an Atomic Medium Wave equation in a laser medium 2 2 [∇ +ω µε()1+ χat − iσ /ωε ]E(x, y, z) = 0 atomic Ohmic loss susceptibility Plane wave approximation 2 ⎡ d 2 ⎤ ⎢ 2 + β ()1+ χat − iσ /ωε ⎥E(z) = 0 ⎣dz ⎦ ω 2π n “Free-space” propagation constant: β = ω µε = n = c λ −Γz Propagation factor: E(z) = E0e 2 2 Γ = −β ()1+ χat − iσ /ωε Γ = iβ 1+ χat − iσ /ωε = iβ 1+ χ′(ω) + iχ′′(ω) − iσ /ωε Usually, χ , − iσ /ωε <<1 a at χ (ω) = at ω 2 −ω 2 + iΓω ⎡ 1 1 σ ⎤ a Γ ≈ iβ 1+ χ′(ω) + i χ′′(ω) − i ω 2 −ω 2 + iΓω ⎢ 2 2 2ωε ⎥ ()a ⎣ ⎦ = a 2 ω 2 −ω 2 + Γ2ω 2 1 1 σ ()a = iβ + i βχ′(ω) − βχ′′(ω) + χ′(ω) 2 2 2 2 2ε v a()ω −ωa = 2 ω 2 −ω 2 + Γ2ω 2 1 ()a = iβ + i ∆βm (ω) −α m (ω) +α0 aγ ω 2 + i 2 2 2 2 2 ()ω −ωa + γ ω Propagation of a +z traveling wave χ′′(ω) E(z,t) = Re E0 exp{iω t − i[β + ∆βm (ω)]z + [α m (ω) −α0 ]z} Phase shift Gain by atomic transiton by atomic transition + ohmic loss The effects of ohmic losses and atomic transition are included.
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