Pumping and population inversion - Laser amplification Gustav - PowerPoint PPT Presentation

Pumping and population inversion - Laser amplification Gustav Lindgren 2015-02-12

Contents Part I: Laser pumping and population inversion Steady state laser pumping and population inversion 4-level laser Solve rate-equations in steady-state 3-level laser Laser gain saturation Introduce the upper-level model Upper-level laser Transient rate equations Solve rate-equations under Upper-level laser transients Three-level laser

Atomic transitions Energy-level diagram of Nd:YAG Simplify into ->

4-level laser Rate equations: Pumping - Decay 𝑒𝑂 4 𝑒𝑢 = 𝑋 𝑞 𝑂 1 − 𝑂 4 − 𝑂 4 /𝜐 41 𝑒𝑂 3 𝑒𝑢 = 𝑂 4 − 𝑂 3 Decay In/Out 𝜐 43 𝜐 3 𝑒𝑂 2 𝑒𝑢 = 𝑂 4 + 𝑂 3 − 𝑂 2 Same 𝜐 42 𝜐 32 𝜐 21 Atom conservation: 𝑂 1 + 𝑂 2 + 𝑂 3 + 𝑂 4 = 𝑂 “Optical approximation”, ℏ𝜕/𝑙 𝐶 𝑈 ≪ 1 No thermal occupancy

4-level laser At steady state: 𝑂 3 = 𝜐 3 𝑂 4 𝜐 43 Define beta 𝜐 21 + 𝜐 43 𝜐 21 𝑂 2 = 𝑂 3 ≡ 𝛾𝑂 3 𝜐 32 𝜐 42 𝜐 3 For a good laser: No direct decay into lev2 𝛿 42 ≈ 0 (𝑗. 𝑓. 𝜐 42 → ∞) , → 𝛾 ≈ 𝜐 21 → 𝜐 32 Fluorescent quantum efficiency, 𝜃 ≡ 𝜐 4 ⋅ 𝜐 3 𝜐 43 𝜐 𝑠𝑏𝑒 Useful photons: from 4 -> upper laser * From upper laser that lase

4-level laser Calculate the pop. Inv. Population inversion, 𝑂 3 − 𝑂 2 1 − 𝛾 𝜃𝑋 𝑞 𝜐 𝑠𝑏𝑒 = 1 + 1 + 𝛾 + 2𝜐 43 𝑂 𝜐 𝑠𝑏𝑒 𝜃𝑋 𝑞 𝜐 𝑠𝑏𝑒 For a good laser: 𝜐 43 ≪ 𝜐 𝑠𝑏𝑒 - Short lev 4 lifetime 𝛾 ≈ 𝜐 21 /𝜐 32 → 0 - Short lower lev lifetime 𝜃 → 1 - High fluorescent quantum efficiency 𝑋 𝑞 𝜐 𝑠𝑏𝑒 ⇒ 𝑂 3 − 𝑂 2 ≈ 𝑂 1 + 𝑋 𝑞 𝜐 𝑠𝑏𝑒 - Red curves

3-level laser As before, for 3-level Rate equations: Pumping – decay BUT lower level is GROUND level 𝑒𝑂 3 𝑞 𝑂 1 − 𝑂 3 − 𝑂 3 𝑒𝑢 = 𝑋 𝜐 3 𝑒𝑂 2 𝑒𝑢 = 𝑂 3 − 𝑂 2 decays 𝜐 32 𝜐 21 Atom conservation: 𝑂 1 + 𝑂 2 + 𝑂 3 = 𝑂 as before 𝜃 = 𝜐 3 𝜐 21 As before 𝜐 32 𝜐 𝑠𝑏𝑒 𝛾 = 𝑂 3 = 𝜐 32 Different! 𝑂 2 𝜐 21

3-level laser No pumping NEGATIVE pop. Inv. At steady state, -> 1 − 𝛾 𝜃𝑋 𝑞 𝜐 𝑠𝑏𝑒 − 1 𝑂 2 − 𝑂 1 = 𝑞 𝜐 𝑠𝑏𝑒 + 1 𝑂 1 + 2𝛾 𝜃𝑋 Requirements for pop. inversion: 𝛾 < 1 As before 1 𝑋 𝑞 𝜐 𝑠𝑏𝑒 ≥ 𝜃 1−𝛾 New For a good laser, 𝛾 → 0 𝜃 → 1 ≈ 𝑋 𝑞 𝜐 𝑠𝑏𝑒 − 1 𝑂 2 − 𝑂 1 𝑞 𝜐 𝑠𝑏𝑒 + 1 𝑂 𝑋 Red curves

Population inversion All else equal: 3-level requires more pumping

Upper-level laser Lasing between two levels high above ground-level 𝑒𝑂 3 Pump into upper lev. = 𝑋 𝑞 𝑂 0 − 𝑂 3 𝑒𝑢 𝑞𝑣𝑛𝑞 Assuming, 𝑂 0 ≈ 𝑂 ≫ 𝑂 3 and pump efficiency, 𝜃 𝑞 , 𝑒𝑂 3 most atoms in ground- ≈ 𝜃 𝑞 𝑋 𝑞 𝑂 ≡ 𝑆 𝑞 state 𝑒𝑢 𝑞𝑣𝑛𝑞 Rate equations: Signal included 𝑒𝑂 2 𝑒𝑢 = 𝑆 𝑞 − 𝑋 𝑡𝑗𝑕 𝑂 2 − 𝑂 1 − 𝛿 2 𝑂 2 𝑒𝑂 1 𝑒𝑢 = 𝑋 𝑡𝑗𝑕 𝑂 2 − 𝑂 1 + 𝛿 21 𝑂 2 − 𝛿 1 𝑂 1

Upper-level laser At steady state: 𝑋 𝑡𝑗𝑕 + 𝛿 21 𝑂 1 = 𝑆 𝑞 𝑋 𝑡𝑗𝑕 𝛿 1 + 𝛿 20 + 𝛿 1 𝛿 2 𝑋 𝑡𝑗𝑕 + 𝛿 1 𝑂 2 = 𝑆 𝑞 𝑋 𝑡𝑗𝑕 𝛿 1 + 𝛿 20 + 𝛿 1 𝛿 2 No atom conservation! For example, changing pump changes N

Upper-level laser The pop. Inv. Saturates as the signal increases Population inversion: 𝑆 𝑞 𝛿 1 − 𝛿 21 Δ𝑂 21 = 𝑂 2 − 𝑂 1 = ⋅ 1 + 𝛿 1 + 𝛿 20 𝛿 1 𝛿 2 𝑋 𝑡𝑗𝑕 𝛿 1 𝛿 2 𝛿 1 −𝛿 21 Define the small-signal population inversion, Δ𝑂 0 = 𝛿 1 𝛿 2 𝑆 𝑞 and the effective recovery time, 𝜐 𝑓𝑔𝑔 = 𝜐 2 1 + 𝜐 1 𝜐 20 the expression becomes: 1 ΔN 21 = Δ𝑂 0 1 + 𝑋 𝑡𝑗𝑕 𝜐 𝑓𝑔𝑔 For a good laser: 𝛿 2 ≈ 𝛿 21 𝛿 20 ≈ 0 1 → ΔN 21 ≈ 𝑆 𝑞 𝜐 2 − 𝜐 1 ⋅ 1 + 𝑋 𝑡𝑗𝑕 𝜐 2 Prop. To pump-rate and lifetimes, saturation behavior

Upper-level laser • Condition for obtaining inversion, 𝜐 1 /𝜐 21 < 1 i.e. fast relaxation from lower level and slow relaxation from upper level • Small-signal gain, 𝜐 2 Δ𝑂 0 ∼ 𝑆 𝑞 ⋅ 1 − 𝜐 1 /𝜐 21 i.e. small-signal gain is proportional to the pump-rate times a reduced upper-level lifetime • Saturation behavior, 1 Δ𝑂 21 = Δ𝑂 0 ⋅ 1 + 𝑋 𝑡𝑗𝑕 𝜐 𝑓𝑔𝑔 i.e. the saturation intensity depends only on the signal intensity and the effective lifetime , not on the pumping rate.

Upper-level laser: Transient rate equation As for instance before a Q-switched pulse Assume: No signal ( 𝑋 𝑡𝑗𝑕 = 0 ), fast lower-level relaxation ( 𝑂 1 ≈ 0 ), 𝑒𝑂 2 𝑢 = 𝑆 𝑞 𝑢 − 𝛿 2 𝑂 2 𝑢 𝑒𝑢 The upper level population becomes, 𝑢 𝑆 𝑞 𝑢 ′ 𝑓 −𝛿 2 (𝑢−𝑢 ′ ) 𝑒𝑢′ 𝑂 2 𝑢 = −∞ Applying a square pulse, 𝑞 = 𝑆 𝑞0 𝜐 2 (1 − 𝑓 −𝑈 𝑞 /𝜐 2 ) 𝑂 2 𝑈 Define the pump efficiency, = 1 − 𝑓 −𝑈 𝑞 /𝜐 2 𝜃 𝑞 = 𝑂 2 𝑢 = 𝑈 𝑞 𝑆 𝑞0 𝑈 𝑈 𝑞 /𝜐 2 𝑞 ^Pop. In upper lev per pump-photon

3-level laser: pulses 3-level laser from prev, no signal Assume: No signal ( 𝑋 𝑡𝑗𝑕 = 0 ), Fast upper-level relaxation ( 𝜐 3 ≈ 0 ), 𝑒𝑂 1 𝑒𝑢 = − 𝑒𝑂 2 𝑞 𝑢 𝑂 1 𝑢 + 𝑂 2 𝑢 𝑒𝑢 ≈ −𝑋 𝜐 𝑒 𝑞 𝑢 + 1 𝑞 𝑢 − 1 𝑒𝑢 Δ𝑂 𝑢 = − 𝑋 𝜐 Δ𝑂 𝑢 + 𝑋 𝜐 𝑂 Integrate to get pop. Inv: Square pulse: = (𝑋 𝑞 𝜐 − 1) − 2𝑋 𝑞 𝜐 ⋅ exp [− 𝑋 𝑞 𝜐 + 1 𝑢/𝜐] Δ𝑂 𝑢 𝑂 𝑋 𝑞 𝜐 + 1 If pump pulse duration is short ( 𝑈 𝑞 ≪ 𝜐 ), and the pumping rate is high ( 𝑋 𝑞 𝜐 ≫ 1 Simple model-agrees with experiment! Δ𝑂 𝑈 𝑞 ≈ 1 − 2𝑓 −𝑋 𝑞 𝑈 𝑞 𝑂

Summary Steady state laser pumping and population inversion 4-level laser 𝑂 3 − 𝑂 2 𝑋 𝑞 𝜐 𝑠𝑏𝑒 Difference between three ≈ 𝑂 1 + 𝑋 𝑞 𝜐 𝑠𝑏𝑒 and four-level systems, and 3-level laser why four-level systems are 𝑋 𝑞 𝜐 𝑠𝑏𝑒 − 1 𝑂 2 −𝑂 1 superior ≈ 𝑞 𝜐 𝑠𝑏𝑒 𝑂 1 + 𝑋 Saturation intensity is Laser gain saturation independent of the Upper-level laser, saturation behavior 1 pumping-rate ”i.e. The signal intensity Δ𝑂 21 = Δ𝑂 0 ⋅ 1+𝑋 𝑡𝑗𝑕 𝜐 𝑓𝑔𝑔 needed to reduce the pop. Inv. To half its initial value doesn’t depend on the pumping rate” Short pulses are needed to Transient rate equations obtain high pumping Upper-level laser = 1 − 𝑓 −𝑈 𝑞 /𝜐 2 𝜃 𝑞 = 𝑂 2 𝑢 = 𝑈 efficiency 𝑞 𝑆 𝑞0 𝑈 𝑈 𝑞 /𝜐 2 𝑞 Three-level laser These simple models give good agreement with Δ𝑂 𝑈 𝑞 ≈ 1 − 2𝑓 −𝑋 𝑞 𝑈 𝑞 reality 𝑂

Contents Part II: Laser amplification Wave propagation in atomic media Solve wave-eqs Plane-wave approximation The paraxial wave equation Single-pass laser amplification Gain narrowing Transition cross-sections See effects on the gain Gain saturation Power extraction

Wave propagation in an atomic medium Maxwell’s equations: 𝛼𝑦𝑭 = −𝑘𝜕𝑪 𝛼𝑦𝑰 = 𝑲 + 𝑘𝜕𝑬 Material parameters: Constitutive relations: 𝜈 – magnetic permeability 𝑪 = 𝜈𝑰 𝜏 – ohmic losses 𝑲 = 𝜏𝑭 𝜗 – dielectric permittivity (not counting 𝑬 = 𝜗𝑭 + 𝑸 𝑏𝑢 = 𝜗 1 + 𝝍 𝑏𝑢 𝑭 atomic transitions) 𝜓 𝑏𝑢 (𝜕) – resonant susceptibility due Vector field of the form: to laser transitions 𝝑 𝒔, 𝑢 = 1 2 𝑭 𝒔 𝑓 𝑘𝜕𝑢 + 𝑑. 𝑑 Assume a spatially uniform material ( 𝛼 ⋅ 𝐹 = 0 ), and apply 𝛼 × to get the wave equation: 𝑏𝑢 − 𝑘𝜏 𝛼 2 + 𝜕 2 𝜈𝜗 1 + 𝜓 𝑦, 𝑧, 𝑨 = 0 𝐹 𝜕𝜗

Plane-wave approximation <- Ok approx. If Consider a plane wave, wavefront is flat 𝜖 2 𝐹 𝜖𝑦 2 , 𝜖 2 𝐹 𝜖𝑧 2 ≪ 𝜖 2 𝐹 𝜖𝑨 2 i.e. 𝛼 2 → 𝑒 2 𝑒𝑨 2 The equation reduces to: 𝑏𝑢 − 𝑘𝜏 2 + 𝜕 2 𝜈𝜗 1 + 𝜓 𝑨 = 0 𝑒 𝑨 𝐹 𝜕𝜗

Plane-wave approximation Without losses: First, no losses 2 + 𝜕 2 𝜈𝜗 𝐹 𝑨 = 0, 𝑒 𝑨 Assume solutions on the form: 𝑨 = 𝑑𝑝𝑜𝑡𝑢 ⋅ 𝑓 −Γ𝑨 𝐹 ⇒ Γ 2 + 𝜕 2 𝜈𝜗 𝐹 = 0 The allowed values for Γ are, Γ = ±𝑘𝜕 𝜈𝜗 ≡ ±𝑘𝛾 With the solution, 𝝑 𝑨, 𝑢 = 1 + 1 2 𝐹 + 𝑓 𝑘 𝜕𝑢−𝛾𝑨 + 𝐹 + 2 𝐹 − 𝑓 𝑘 𝜕𝑢+𝛾𝑨 + 𝐹 − ∗ 𝑓 −𝑘 𝜕𝑢+𝛾𝑨 ∗ 𝑓 −𝑘 𝜕𝑢−𝛾𝑨 The free space propagation constant, 𝛾, may be written: 𝛾 = 𝜕 𝜈𝜗 = 𝜕 𝑑 = 2𝜌 Different beta 𝜇 from last chapter