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 3-level laser Solve rate-equations in steady-state
Laser gain saturation Upper-level laser Introduce the upper-level model
Transient rate equations Upper-level laser Solve rate-equations under Three-level laser transients
Atomic transitions
Energy-level diagram of Nd:YAG
Simplify into -> 4-level laser
Rate equations: Pumping - Decay 푑푁4 = 푊 푁 − 푁 − 푁 /휏 푑푡 푝 1 4 4 41
푑푁3 푁4 푁3 = − Decay In/Out 푑푡 휏43 휏3
푑푁2 푁4 푁3 푁2 = + − Same 푑푡 휏42 휏32 휏21
Atom conservation: 푁1 + 푁2 + 푁3 + 푁4 = 푁
“Optical approximation”, ℏ휔/푘퐵푇 ≪ 1 No thermal occupancy
4-level laser
At steady state:
휏3 푁3 = 푁4 휏43 Define beta 휏21 휏43휏21 푁2 = + 푁3 ≡ 훽푁3 휏32 휏42휏3
For a good laser: No direct decay into lev2 훾42≈ 0 (푖. 푒. 휏42 → ∞),
휏21 → → 훽 ≈ 휏32
Fluorescent quantum efficiency, 휏4 휏3 휂 ≡ ⋅ 휏43 휏푟푎푑 Useful photons: from 4 -> upper laser * From upper laser that lase 4-level laser Calculate the pop. Inv. Population inversion,
푁3 − 푁2 1 − 훽 휂푊푝휏푟푎푑 = 푁 2휏43 1 + 1 + 훽 + 휂푊푝휏푟푎푑 휏푟푎푑
For a good laser: 휏43 ≪ 휏푟푎푑 - Short lev 4 lifetime 훽 ≈ 휏21/휏32 → 0 - Short lower lev lifetime 휂 → 1 - High fluorescent quantum efficiency
푁3 − 푁2 푊푝휏푟푎푑 ⇒ ≈ 푁 1 + 푊푝휏푟푎푑- Red curves
3-level laser As before, for 3-level Rate equations: BUT lower level is GROUND level Pumping – decay
푑푁3 푁3 = 푊 푁 − 푁 − 푑푡 푝 1 3 휏 3 푑푁2 푁3 푁2 decays = − 푑푡 휏32 휏21
Atom conservation:
푁1 + 푁2 + 푁3 = 푁 as before
휏3 휏21 휂 = As before 휏32 휏푟푎푑
푁3 휏32 훽 = = Different! 푁2 휏21
3-level laser No pumping NEGATIVE pop. Inv. At steady state, ->
푁2 − 푁1 1 − 훽 휂푊푝휏푟푎푑 − 1 = 푁 1 + 2훽 휂푊 푝휏푟푎푑 + 1
Requirements for pop. inversion: 훽 < 1 As before 1 푊푝휏푟푎푑 ≥ 휂 1− 훽 New
For a good laser,
훽 → 0
휂 → 1
푁2 − 푁1 푊푝휏푟푎푑 − 1 ≈ 푁 푊푝휏푟푎푑 + 1 Red curves
Population inversion
All else equal: 3-level requires more pumping Upper-level laser
Lasing between two levels high above ground-level
푑푁3 Pump into upper lev. = 푊푝 푁0 − 푁3 푑푡 푝푢푚푝
Assuming, 푁0 ≈ 푁 ≫ 푁3 and pump efficiency, 휂푝, 푑푁3 most atoms in ground- ≈ 휂푝 푊푝푁 ≡ 푅 푝 state 푑푡 푝푢푚푝
Rate equations:
Signal included 푑푁2 = 푅 − 푊 푁 − 푁 − 훾 푁 푑푡 푝 푠푖푔 2 1 2 2
푑푁1 = 푊 푁 − 푁 + 훾 푁 − 훾 푁 푑푡 푠푖푔 2 1 21 2 1 1
Upper-level laser
At steady state:
푊푠푖푔 + 훾21 푁1 = 푅푝 푊푠푖푔 훾1 + 훾20 + 훾1훾2
푊푠푖푔 + 훾1 푁2 = 푅푝 푊푠푖푔 훾1 + 훾20 + 훾1훾2
No atom conservation!
For example, changing pump changes N Upper-level laser The pop. Inv. Saturates as the signal increases Population inversion: 훾1 − 훾21 푅푝 Δ푁21 = 푁2 − 푁1 = ⋅ 훾1훾2 훾1 + 훾20 1 + 푊푠푖푔 훾1훾2
훾1−훾21 Define the small-signal population inversion, Δ푁0 = 푅푝 and the effective 훾1훾2 휏1 recovery time, 휏푒푓푓= 휏2 1 + the expression becomes: 휏20
1 ΔN21 = Δ푁0 1 + 푊푠푖푔휏푒푓푓
For a good laser: 훾2 ≈ 훾21 훾20 ≈ 0 1 → ΔN21 ≈ 푅푝 휏2 − 휏1 ⋅ 1 + 푊푠푖푔휏2
Prop. To pump-rate and lifetimes, saturation behavior Upper-level laser
• Condition for obtaining inversion, 휏1/휏21 < 1
i.e. fast relaxation from lower level and slow relaxation from upper level
• Small-signal gain, 휏2 Δ푁0 ∼ 푅푝 ⋅ 1 − 휏1/휏21 i.e. small-signal gain is proportional to the pump-rate times a reduced upper-level lifetime
• Saturation behavior, 1 Δ푁21 = Δ푁0 ⋅ 1 + 푊푠푖푔휏푒푓푓 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 (푊푠푖푔 = 0), fast lower-level relaxation (푁1 ≈ 0), 푑푁2 푡 = 푅 푡 − 훾 푁 푡 푑푡 푝 2 2
The upper level population becomes, 푡 ′ ′ −훾2(푡−푡 ) 푁2 푡 = 푅푝 푡 푒 푑푡′ −∞
Applying a square pulse, −푇푝/휏2 푁2 푇푝 = 푅푝0휏2(1 − 푒 )
Define the pump efficiency, −푇푝/휏2 푁2 푡 = 푇푝 1 − 푒 휂푝 = = 푅푝0푇푝 푇푝/휏2
^Pop. In upper lev per pump-photon 3-level laser: pulses 3-level laser from prev, no signal Assume: No signal (푊푠푖푔 = 0), Fast upper-level relaxation (휏3 ≈ 0),
푑푁1 푑푁2 푁2 푡 = − ≈ −푊 푡 푁 푡 + 푑푡 푑푡 푝 1 휏
푑 1 1 Δ푁 푡 = − 푊 푡 + Δ푁 푡 + 푊 푡 − 푁 푑푡 푝 휏 푝 휏
Integrate to get pop. Inv: Square pulse: Δ푁 푡 (푊푝휏 − 1) − 2푊푝 휏 ⋅ exp [− 푊푝휏 + 1 푡/휏] = 푁 푊푝휏 + 1
If pump pulse duration is short (푇푝 ≪ 휏), and the pumping rate is high (푊푝휏 ≫ 1 Simple model-agrees with experiment! Δ푁 푇푝 ≈ 1 − 2푒−푊푝푇푝 푁
Summary
Steady state laser pumping and population inversion 4-level laser 푁3 − 푁2 푊푝휏푟푎푑 ≈ Difference between three 푁 1 + 푊푝휏푟푎푑 and four-level systems, and 3-level laser why four-level systems are 푁 −푁 푊 휏 − 1 2 1 ≈ 푝 푟푎푑 superior 푁 1 + 푊푝휏푟푎푑
Laser gain saturation Saturation intensity is Upper-level laser, saturation behavior independent of the 1 pumping-rate ”i.e. The signal intensity Δ푁21 = Δ푁0 ⋅ 1+푊푠푖푔휏푒푓푓 needed to reduce the pop. Inv. To half its initial value doesn’t depend on the pumping rate” Transient rate equations Short pulses are needed to Upper-level laser obtain high pumping −푇푝/휏2 푁2 푡 = 푇푝 1 − 푒 efficiency 휂푝 = = 푅푝0푇푝 푇푝/휏2 Three-level laser These simple models give
good agreement with Δ푁 푇 푝 ≈ 1 − 2푒−푊푝푇푝 reality 푁 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: 훻푥푬 = −푗휔푩 훻푥푯 = 푱 + 푗휔푫 Material parameters: Constitutive relations: 휇 – magnetic permeability 푩 = 휇푯 휎 – ohmic losses 푱 = 휎푬 휖 – dielectric permittivity (not counting 푫 = 휖푬 + 푷푎푡 = 휖 1 + 흌푎푡 푬 atomic transitions)
휒푎푡(휔) – resonant susceptibility due Vector field of the form: to laser transitions 1 흐 풓, 푡 = 푬 풓 푒푗휔푡 + 푐. 푐 2
Assume a spatially uniform material (훻 ⋅ 퐸 = 0), and apply 훻 × to get the wave equation:
푗휎 훻2 + 휔2휇휖 1 + 휒 − 퐸 푥, 푦, 푧 = 0 푎푡 휔휖
Plane-wave approximation
Consider a plane wave, <- Ok approx. If wavefront is flat 휕2퐸 휕2퐸 휕2퐸 , ≪ 휕푥2 휕푦2 휕푧2
푑2 i.e. 훻2 → 푑푧2
The equation reduces to: 푗휎 푑2 + 휔2휇휖 1 + 휒 − 퐸 푧 = 0 푧 푎푡 휔휖 Plane-wave approximation
Without losses: First, no losses
2 2 푑푧 + 휔 휇휖 퐸 푧 = 0,
Assume solutions on the form: 퐸 푧 = 푐표푛푠푡 ⋅ 푒−Γ푧
⇒ Γ2 + 휔2휇휖 퐸 = 0
The allowed values for Γ are, Γ = ±푗휔 휇휖 ≡ ±푗훽
With the solution, 1 1 흐 푧, 푡 = 퐸 푒푗 휔푡−훽푧 + 퐸∗ 푒−푗 휔푡−훽푧 + 퐸 푒푗 휔푡+훽푧 + 퐸∗ 푒−푗 휔푡+훽푧 2 + + 2 − −
The free space propagation constant, 훽, may be written: 휔 2휋 Different beta 훽 = 휔 휇휖 = = 푐 휆 from last chapter Plane-wave approximation Put losses back With laser action and losses:
2 2 푗휎 푑푧 + 훽 1 + 휒 푎푡 − 퐸 푧 = 0 휔휖
−Γ푧 Assuming 퐸 푧 = 푐표푛푠푡 ⋅ 푒 , as before: ′ ′′ Γ = 푗훽 1 + 휒 푎푡 휔 + 푗휒 푎푡 휔 − 푗휎/휔휖
휎 Normally, 휒 휔 , ≪ 1: Expand the sqrt 푎푡 휔휖
1 ′ 푗 ′′ 푗휎 훿 Γ ≈ 푗훽 1 + 휒푎푡 휔 + 휒푎푡 휔 − = 1 + 훿 ≈ 1 + 2 2 2휔휖 2 ≡ 푗훽 + 푗Δ훽푚 휔 − 훼푚 휔 + 훼0
Define four terms The final solution becomes:
흐 푧, 푡 = 푅푒퐸 0 exp 푗휔푡 − 푗 훽 + 훥훽푚 휔 푧 + 훼푚 휔 − 훼0 푧
Propagation Factors
흐 푧, 푡 = 푅푒퐸 0 exp 푗휔푡 − 푗 훽 + 훥훽푚 휔 푧 + 훼푚 휔 − 훼0 푧 Phase-shift loss/gain Linear phase-shift (Red) 훽 – Plane-wave propagation constant, fundamental phase variation,
Nonlinear phase-shift (Blue) Δ훽푚 휔 – Additional atomic phase- shift due to population inversion
→ Total phase-shift (Green)
Gain (Orange) 훼푚 휔 – Atomic gain or loss coefficient (due to transitions)
Background 훼0 – Ohmic background loss
Exceptions
Larger atomic gain or absorption effects For expanding the sqrt 푗휎 The results so far are based on the assumption that 휒 − ≪ 1, however, there are a 푎푡 휔휖 few situations of interest where this doesn’t hold:
- Absorption in metals and semiconductors, at frequencies higher than the bandgap energy, the effective conductivity, 휎 can become very large Big sigma - Absorption on strong resonance lines in metal vapor, the transitions are very strongly allowed giving a high absorption per unit length Big chi
The paraxial wave equation Want to handle transverse variations The full wave-equation, 푗휎 훻2 + 훽2 1 + 휒 − 푬 풓 = 0 푎푡 휔휖
New ansatz, 퐸 풓 ≡ 푢 풓 푒 − 푗훽푧 change const -> u(r)
Insert into the wave equation, 2 2 2 휕 푢 휕 푢 휕 푢 휕푢 2 −푗훽푧 2 + 2 + 2 − 2푗훽 − 훽 푢 푒 = 0 휕푥 휕푦 휕푧 휕푧
푢 풓 Assume that changes slowly along the z-direction,
휕2푢 휕푢
2 ≪ 2훽 휕푧 휕푧 and 휕2푢 휕2푢 휕2푢
2 ≪ 2 , 2 휕푧 휕푥 휕푦 ”Paraxial wave equation”
휕푢 푗휎 2 2 ⇒ 훻푡 푢 − 2푗훽 + 훽 휒 푎푡 − 푢 = 0 휕푧 휔휖
Diffraction- and propagation effects
Re-write the equation Rewrite:
휕푢 풓 푗 2 = − 훻푡 푢 풓 − [훼0−훼푚 + 푗Δ훽푚]푢 풓 휕푧 2훽
훼0, 훼푚 휔 , and 푗Δ훽 푚 휔 are defined as before
Four terms, same as before Two terms:
푗 Diffraction, − 훻2 2훽 푡
Ohmic and atomic gain/loss, −[훼0−훼푚 + 푗Δ훽푚]
Separate terms –> Independent to first order, gain and phase-shift results are the same as for plane waves
Reproduce the plane-wave result
Laser amplification
Calculate the gain The laser gain after length L: 퐸 퐿 푔 휔 ≡ 퐸 0
In terms of intensity, from def. alpha 2 퐼 퐿 퐸 퐿 휎 퐺 휔 = = 2 = exp 2훼푚 휔 퐿 − 2훼0퐿 = exp [훽휒′′ 휔 퐿 − 퐿] 퐼 0 퐸 0 휖푐
exponential For most lasers, 훼 ≪ 훼 ⇒ 퐺 휔 ≈ exp (훽휒′′ 휔 퐿) 0 푚 dependence on chi
Gain narrowing
The gain, 퐺 휔 ∼ exp 휒′′ 휔 → Narrower frequency dependence, i.e. ”Gain Narrowing”
Laser gain & Atomic lineshape
FWHM bandwidth is lowered by 30-40% Absorption broadening
Absorbing media have opposite sign of 휒(휔) → ”Absorption broadening” Uninverted population -> absorption broadening Stimulated transition cross-sections
For a black-body: Δ푃푎푏푠 = 휎 ⋅ 퐼
Thin slab, atomic densities N1 and N2, cross-sections 휎12 and 휎21
Net absorbed power: Δ푃푎푏푠 = 푁1휎12 − 푁2휎21 ⋅ 푃Δ푧
The cross-sections are related, 푔 휎 = 푔 휎21 1 12 2 Loss or gain can be calculated from
cross-sections and atomic density Convert power → intensity: 1 푑퐼 = −Δ푁 휎 퐼 푑푧 12 21
Previously: ⇒ 2훼푚 휔 = Δ푁12휎21(휔) 1 푑퐼 = −2훼 (휔) 퐼 푑푧 푚
Cross-section formula
From previous page: 2훼푚 휔 = Δ푁12휎21(휔)
휋휒′′ 휔 From definition of 훼: 훼 휔 = 푚 휆 2휋 ⇒ 2훼 휔 = Δ푁휎 휔 = 휒′′(휔) 푚 휆 The cross-section is a function of Using the full expression for 휒: wavelength, transition rate and ∗ lineshape 3 훾푟푎푑 2 1 휎 휔푎 = 휆 ⋅ 2 2휋 Δ휔푎 1 + 2(휔 − 휔푎)/Δ휔푎
Cross-section estimation: Estimate the max cross-section ∗ Assume, Δ휔푎 ≡ 훾푟푎푑. (purely radiative transmission), 3 = 3 , (all atoms aligned)
3휆2 휆2 ⇒ 휎 = ≈ ≈ 10−9푐푚2 푚푎푥 2휋 2
The cross section is far greater than the area of an atom, due to its internal resonance. Real cross-sections
휆2 > Real cross-sections ≫ Atom area 2 Allowed transitions in gas and non-allowed in solids -> 흉품풂풔 ≪ 흉풔풐풍풊풅 Population difference saturation Gain saturation, In a medium, 푑퐼 = ±2훼 퐼 = ±Δ푁휎퐼 푑푧 푚
As shown previously, Δ푁 saturates according to: 1 1 Δ푁 = Δ푁0 ⋅ = Δ푁0 ⋅ 1 + 푊휏푒푓푓 1 + 퐼/퐼푠푎푡
푁0 - unsaturated or small signal gain 퐼푠푎푡 - saturation intensity As Δ푁 saturates, the gain saturates
Stimulated-transition probability,
From before, Δ푃푎푏푠 = 푁1휎12 − 푁2휎21 ⋅ 푃Δ푧
From rate equation analysis,Δ푃푎푏푠 = 푊12푁1 − 푊21푁2 퐴Δ푧ℏ휔푎
σI ⇒ W = ℏω
i.e. the cross-section, the intensity and the stimulated transition probability are interdependent Population difference saturation At the line center:
From previous slide, Δ푁0 Δ푁0휎 2훼푚(휔, 퐼) = = 1 + 퐼/퐼푠푎푡 1 + 휎휏푒푓푓/ℏ휔 퐼 Re-arrange prev expressions ℏ휔 → 퐼푠푎푡 ≡ 휎휏푒푓푓
i.e. 퐼 = 퐼푠푎푡: one incident photon within the cross-section per recovery time
Off center: The expression is modified by the lineshape, Δ푁0휎 2훼 (휔, 퐼) = 푚 퐼 1 1 + ⋅ 2 퐼푠푎푡 1 + 푦
Where y is the normalized frequency detuning, 푦 = 2 휔 − 휔0 /Δ휔
Frequency dependence due to, transition lineshape and saturation behavior Practical values From previous slide, ℏ휔 퐼푠푎푡 ≡ I-sat varies a lot between materials 휎휏푒푓푓
퐼푠푎푡 is an important parameter since little only little gain will be obtained once the intensity approaches this level.
Laser type: ℏ휔 휎 휏 퐼 푒푓푓 푠푎푡 Gas 10−19 퐽 10−ퟏퟑ 푐푚2 10−ퟔ 푠 1 푾/푐푚2
−19 −ퟏퟗ 2 −ퟑ 2 Solid-state 10 퐽 10 푐푚 10 푠 1 풌푾/푐푚
퐼푠푎푡 is independent of the pumping intensity, since neither 휎 nor 휏푒푓푓 are intensity dependent.
However, harder pumping does increase the small-signal gain Saturation in laser amplifiers Calculate the saturation As a signal passes through an amplifier, it grows exponentially with distance until the intensity approaches 퐼푠푎푡
For a single pass amplifier: 1 푑퐼 2훼푚0 = 2훼푚 퐼 = 퐼(푧) 푑푧 1 + 퐼(푧)/퐼푠푎푡
Integrating, 퐼=퐼표푢푡 푧=퐿 1 1 + 푑퐼 = 2훼푚0 푑푧 퐼 퐼푠푎푡 퐼=퐼푖푛 푧=0
gives: 퐼표푢푡 퐼표푢푡 − 퐼푖푛 ln + = 2훼푚0퐿 = ln 퐺0 퐼푖푛 퐼푠푎푡
where, 퐺0 is the small-signal gain
Saturation in laser amplifiers
The total gain,
퐼표푢푡 퐼표푢푡 − 퐼푖푛 퐺 ≡ = 퐺0 ⋅ exp(− ) 퐼푖푛 퐼푠푎푡
Is the unsaturated gain, reduced by a factor exponentially dependent on 퐼표푢푡 − 퐼푖푛 Saturation in laser amplifiers Output Intensities
Extracted Intensities
Gain starts at G0 and declines towards 0 dB
Plotting the normalized output intensity vs the normalized input intensity, it is clear that the transmission tends to unity as the input intensity is increased -The amplifier becomes transparent. Power-extraction Define the extracted intensity, 퐺0 퐼 ≡ 퐼 − 퐼 = ln ⋅ 퐼 Calculate the extracted power 푒푥푡푟 표푢푡 푖푛 퐺 푠푎푡
As the amplifier saturates, 퐺 → 1,
퐺0 퐼푎푣푎푖푙 = lim ln ⋅ 퐼푠푎푡 = ln 퐺0 ⋅ 퐼푠푎푡 퐺→1 퐺
Using ln 퐺0 = 2훼푚0퐿: ℏ휔 퐼푎푣푎푖푙 = 2훼푚0퐿 ⋅ 퐼푠푎푡 = Δ푁0휎퐿 ⋅ 휎휏푒푓푓 And re-writing: 퐼푎푣푎푖푙 푃푎푣푎푖푙 Δ푁0ℏ휔 ≡ = 퐿 푉 휏푒푓푓
i.e. the maximum available power is the total inversion energy, Δ푁0ℏ휔, once every recovery time, 휏푒푓푓 Power-extraction efficiency
Full power extraction requires complete saturation of the amplifier which gives low small-signal gain.
Define the extraction efficiency, 퐼푒푥푡푟 ln G0 − ln 퐺 퐺푑퐵 휂푒푥푡푟 ≡ = = 1 − 퐼푎푣푎푖푙 ln 퐺0 퐺0,푑퐵
i.e. there’s a tradeoff between effiency and small-signal gain for single- pass amplifiers. Summary 훽 – Plane-wave propagation constant, fundamental phase variation,
Wave propagation in atomic media Δ훽푚 휔 – Additional atomic phase- Plane-wave approximation, shift due to population inversion 푗휎 푑2 + 훽2 1 + 휒 − 퐸 푧 = 0 푧 푎푡 휔휖 훼푚 휔 – Atomic gain or loss coefficient (due to transitions) The paraxial wave equation, 휕푢 풓 푗 = − 훻2푢 풓 − [훼 −훼 + 푗Δ훽 ]푢 풓 훼 – Ohmic background loss 휕푧 2훽 푡 0 푚 푚 0
Diffraction and gain are separate Single-pass laser amplification to first order Gain narrowing, 퐺 휔 ∼ exp 휒′′ 휔 30-40% lower FWHM bandwidth
Transition cross-sections, Loss/gain can be calculated from atomic density and cross-sections 2훼푚 휔 = Δ푁12휎21(휔)
Gain saturation, Low signal gain is saturated by a 퐼표푢푡 − 퐼푖푛 퐺 = 퐺0 ⋅ exp(− ) factor exponentialy dependent 퐼푠푎푡 on the intensity difference
Power extraction, There’s a tradeoff between 퐺푑퐵 efficiency and small-signal gain 휂푒푥푡푟 = 1 − 퐺0,푑퐵