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 3-level laser Laser gain saturation Upper-level laser Transient rate equations Upper-level laser Three-level laser Solve rate-equations in steady-state Introduce the upper-level model Solve rate-equations under transients

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

4-level laser Rate equations: 𝑒𝑂4 𝑒𝑢 = 𝑋

𝑞 𝑂1 − 𝑂4 − 𝑂4/𝜐41

𝑒𝑂3 𝑒𝑢 = 𝑂4 𝜐43 − 𝑂3 𝜐3 𝑒𝑂2 𝑒𝑢 = 𝑂4 𝜐42 + 𝑂3 𝜐32 − 𝑂2 𝜐21 Atom conservation: 𝑂1 + 𝑂2 + 𝑂3 + 𝑂4 = 𝑂 “Optical approximation”, ℏ𝜕/𝑙𝐶𝑈 ≪ 1 Pumping - Decay Decay In/Out Same No thermal occupancy

4-level laser At steady state: 𝑂3 = 𝜐3 𝜐43 𝑂4 𝑂2 = 𝜐21 𝜐32 + 𝜐43𝜐21 𝜐42𝜐3 𝑂3 ≡ 𝛾𝑂3 For a good laser: 𝛿42≈ 0 (𝑗. 𝑓. 𝜐42 → ∞), → 𝛾 ≈ 𝜐21 𝜐32 Fluorescent quantum efficiency, 𝜃 ≡ 𝜐4 𝜐43 ⋅ 𝜐3 𝜐𝑠𝑏𝑒 Define beta No direct decay into lev2 → Useful photons: from 4 -> upper laser * From upper laser that lase

4-level laser Population inversion, 𝑂3 − 𝑂2 𝑂 = 1 − 𝛾 𝜃𝑋

𝑞𝜐𝑠𝑏𝑒

1 + 1 + 𝛾 + 2𝜐43 𝜐𝑠𝑏𝑒 𝜃𝑋

𝑞𝜐𝑠𝑏𝑒

For a good laser: 𝜐43 ≪ 𝜐𝑠𝑏𝑒 𝛾 ≈ 𝜐21/𝜐32 → 0 𝜃 → 1 ⇒ 𝑂3 − 𝑂2 𝑂 ≈ 𝑋

𝑞𝜐𝑠𝑏𝑒

1 + 𝑋

𝑞𝜐𝑠𝑏𝑒

• Short lev 4 lifetime
• Short lower lev lifetime
• High fluorescent quantum efficiency
• Red curves

Calculate the pop. Inv.

3-level laser 𝑒𝑂3 𝑒𝑢 = 𝑋

𝑞 𝑂1 − 𝑂3 − 𝑂3

𝜐3 𝑂1 + 𝑂2 + 𝑂3 = 𝑂 𝜃 = 𝜐3 𝜐32 𝜐21 𝜐𝑠𝑏𝑒 𝛾 = 𝑂3 𝑂2 = 𝜐32 𝜐21 𝑒𝑂2 𝑒𝑢 = 𝑂3 𝜐32 − 𝑂2 𝜐21 Rate equations: Atom conservation: As before, for 3-level BUT lower level is GROUND level Pumping – decay decays as before As before Different!

3-level laser At steady state, 𝑂2 − 𝑂1 𝑂 = 1 − 𝛾 𝜃𝑋

𝑞𝜐𝑠𝑏𝑒 − 1

1 + 2𝛾 𝜃𝑋

𝑞𝜐𝑠𝑏𝑒 + 1

Requirements for pop. inversion: 𝛾 < 1 𝑋

𝑞𝜐𝑠𝑏𝑒 ≥ 1 𝜃 1−𝛾

For a good laser, 𝛾 → 0 𝜃 → 1 𝑂2 − 𝑂1 𝑂 ≈ 𝑋

𝑞𝜐𝑠𝑏𝑒 − 1

𝑋

𝑞𝜐𝑠𝑏𝑒 + 1

No pumping NEGATIVE pop. Inv.

• >

As before New Red curves

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

Upper-level laser Lasing between two levels high above ground-level 𝑒𝑂3 𝑒𝑢 𝑞𝑣𝑛𝑞 = 𝑋

𝑞 𝑂0 − 𝑂3

Assuming, 𝑂0 ≈ 𝑂 ≫ 𝑂3 and pump efficiency, 𝜃𝑞, 𝑒𝑂3 𝑒𝑢 𝑞𝑣𝑛𝑞 ≈ 𝜃𝑞𝑋

𝑞𝑂 ≡ 𝑆𝑞

Rate equations: 𝑒𝑂2 𝑒𝑢 = 𝑆𝑞 − 𝑋

𝑡𝑗𝑕 𝑂2 − 𝑂1 − 𝛿2𝑂2

𝑒𝑂1 𝑒𝑢 = 𝑋

𝑡𝑗𝑕 𝑂2 − 𝑂1 + 𝛿21𝑂2 − 𝛿1𝑂1

Pump into upper lev. most atoms in ground- state Signal included

Upper-level laser At steady state: 𝑂1 = 𝑋

𝑡𝑗𝑕 + 𝛿21

𝑋

𝑡𝑗𝑕 𝛿1 + 𝛿20 + 𝛿1𝛿2

𝑆𝑞 𝑂2 = 𝑋

𝑡𝑗𝑕 + 𝛿1

𝑋

𝑡𝑗𝑕 𝛿1 + 𝛿20 + 𝛿1𝛿2

𝑆𝑞 No atom conservation! For example, changing pump changes N

Upper-level laser Population inversion: Δ𝑂21 = 𝑂2 − 𝑂1 = 𝛿1 − 𝛿21 𝛿1𝛿2 ⋅ 𝑆𝑞 1 + 𝛿1 + 𝛿20 𝛿1𝛿2 𝑋

𝑡𝑗𝑕

Define the small-signal population inversion, Δ𝑂0 =

𝛿1−𝛿21 𝛿1𝛿2 𝑆𝑞 and the effective

recovery time, 𝜐𝑓𝑔𝑔= 𝜐2 1 + 𝜐1

𝜐20 the expression becomes:

ΔN21 = Δ𝑂0 1 1 + 𝑋

𝑡𝑗𝑕𝜐𝑓𝑔𝑔

For a good laser: 𝛿2 ≈ 𝛿21 𝛿20 ≈ 0 → ΔN21 ≈ 𝑆𝑞 𝜐2 − 𝜐1 ⋅ 1 1 + 𝑋

𝑡𝑗𝑕𝜐2

The pop. Inv. Saturates as the signal increases

• Prop. To pump-rate and lifetimes,

saturation behavior

Upper-level laser

• Condition for obtaining inversion,

𝜐1/𝜐21 < 1 i.e. fast relaxation from lower level and slow relaxation from upper level

• Small-signal gain,

Δ𝑂0 ∼ 𝑆𝑞 ⋅ 𝜐2 1 − 𝜐1/𝜐21 i.e. small-signal gain is proportional to the pump-rate times a reduced upper-level lifetime

• Saturation behavior,

Δ𝑂21 = Δ𝑂0 ⋅ 1 1 + 𝑋

𝑡𝑗𝑕𝜐𝑓𝑔𝑔

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

Assume: No signal (𝑋

𝑡𝑗𝑕 = 0), fast lower-level relaxation (𝑂1 ≈ 0),

𝑒𝑂2 𝑢 𝑒𝑢 = 𝑆𝑞 𝑢 − 𝛿2𝑂2 𝑢

The upper level population becomes,

𝑂2 𝑢 = 𝑆𝑞 𝑢′ 𝑓−𝛿2(𝑢−𝑢′)𝑒𝑢′

𝑢 −∞

Applying a square pulse,

𝑂2 𝑈

𝑞 = 𝑆𝑞0𝜐2(1 − 𝑓−𝑈

𝑞/𝜐2)

Define the pump efficiency,

𝜃𝑞 = 𝑂2 𝑢 = 𝑈

𝑞

𝑆𝑞0𝑈

𝑞

= 1 − 𝑓−𝑈

𝑞/𝜐2

𝑈

𝑞/𝜐2

As for instance before a Q-switched pulse ^Pop. In upper lev per pump-photon

3-level laser: pulses Assume: No signal (𝑋

𝑡𝑗𝑕 = 0),

Fast upper-level relaxation (𝜐3 ≈ 0), 𝑒𝑂1 𝑒𝑢 = − 𝑒𝑂2 𝑒𝑢 ≈ −𝑋

𝑞 𝑢 𝑂1 𝑢 + 𝑂2 𝑢

𝜐 𝑒 𝑒𝑢 Δ𝑂 𝑢 = − 𝑋

𝑞 𝑢 + 1

𝜐 Δ𝑂 𝑢 + 𝑋

𝑞 𝑢 − 1

𝜐 𝑂 Square pulse: Δ𝑂 𝑢 𝑂 = (𝑋

𝑞𝜐 − 1) − 2𝑋 𝑞 𝜐 ⋅ exp [− 𝑋 𝑞𝜐 + 1 𝑢/𝜐]

𝑋

𝑞𝜐 + 1

If pump pulse duration is short (𝑈

𝑞 ≪ 𝜐),

and the pumping rate is high (𝑋

𝑞𝜐 ≫ 1

Δ𝑂 𝑈

𝑞

𝑂 ≈ 1 − 2𝑓−𝑋

𝑞𝑈 𝑞

3-level laser from prev, no signal Integrate to get pop. Inv: Simple model-agrees with experiment!

Summary Steady state laser pumping and population inversion 4-level laser 𝑂3 − 𝑂2 𝑂 ≈ 𝑋

𝑞𝜐𝑠𝑏𝑒

1 + 𝑋

𝑞𝜐𝑠𝑏𝑒

3-level laser

𝑂2−𝑂1 𝑂

≈

𝑋

𝑞𝜐𝑠𝑏𝑒 − 1

1 + 𝑋

𝑞𝜐𝑠𝑏𝑒

Laser gain saturation Upper-level laser, saturation behavior Δ𝑂21 = Δ𝑂0 ⋅

1 1+𝑋𝑡𝑗𝑕𝜐𝑓𝑔𝑔

Transient rate equations Upper-level laser 𝜃𝑞 = 𝑂2 𝑢 = 𝑈

𝑞

𝑆𝑞0𝑈

𝑞

= 1 − 𝑓−𝑈

𝑞/𝜐2

𝑈

𝑞/𝜐2

Three-level laser

Δ𝑂 𝑈

𝑞

𝑂

≈ 1 − 2𝑓−𝑋

𝑞𝑈 𝑞

Difference between three and four-level systems, and why four-level systems are superior Saturation intensity is independent of the pumping-rate ”i.e. The signal intensity

needed to reduce the pop. Inv. To half its initial value doesn’t depend on the pumping rate”

Short pulses are needed to

• btain high pumping

efficiency These simple models give good agreement with reality

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

Wave propagation in an atomic medium Maxwell’s equations: 𝛼𝑦𝑭 = −𝑘𝜕𝑪 𝛼𝑦𝑰 = 𝑲 + 𝑘𝜕𝑬 Constitutive relations: 𝑪 = 𝜈𝑰 𝑲 = 𝜏𝑭 𝑬 = 𝜗𝑭 + 𝑸𝑏𝑢 = 𝜗 1 + 𝝍𝑏𝑢 𝑭 Vector field of the form: 𝝑 𝒔, 𝑢 = 1 2 𝑭 𝒔 𝑓𝑘𝜕𝑢 + 𝑑. 𝑑 Assume a spatially uniform material (𝛼 ⋅ 𝐹 = 0), and apply 𝛼 × to get the wave equation: 𝛼2 + 𝜕2𝜈𝜗 1 + 𝜓 𝑏𝑢 − 𝑘𝜏 𝜕𝜗 𝐹 𝑦, 𝑧, 𝑨 = 0 Material parameters: 𝜈 – magnetic permeability 𝜏 – ohmic losses 𝜗 – dielectric permittivity (not counting atomic transitions) 𝜓𝑏𝑢(𝜕) – resonant susceptibility due to laser transitions

Plane-wave approximation Consider a plane wave, 𝜖2𝐹 𝜖𝑦2 , 𝜖2𝐹 𝜖𝑧2 ≪ 𝜖2𝐹 𝜖𝑨2 i.e. 𝛼2 → 𝑒2

𝑒𝑨2

The equation reduces to: 𝑒𝑨

2 + 𝜕2𝜈𝜗 1 + 𝜓

𝑏𝑢 − 𝑘𝜏 𝜕𝜗 𝐹 𝑨 = 0 <- Ok approx. If wavefront is flat

Plane-wave approximation Without losses: 𝑒𝑨

2 + 𝜕2𝜈𝜗 𝐹

𝑨 = 0, Assume solutions on the form: 𝐹 𝑨 = 𝑑𝑝𝑜𝑡𝑢 ⋅ 𝑓−Γ𝑨 ⇒ Γ2 + 𝜕2𝜈𝜗 𝐹 = 0 The allowed values for Γ are, Γ = ±𝑘𝜕 𝜈𝜗 ≡ ±𝑘𝛾 With the solution, 𝝑 𝑨, 𝑢 = 1 2 𝐹+𝑓𝑘 𝜕𝑢−𝛾𝑨 + 𝐹+

∗𝑓−𝑘 𝜕𝑢−𝛾𝑨

+ 1 2 𝐹−𝑓𝑘 𝜕𝑢+𝛾𝑨 + 𝐹−

∗𝑓−𝑘 𝜕𝑢+𝛾𝑨

The free space propagation constant, 𝛾, may be written: 𝛾 = 𝜕 𝜈𝜗 = 𝜕 𝑑 = 2𝜌 𝜇 Different beta from last chapter First, no losses

Plane-wave approximation With laser action and losses: 𝑒𝑨

2 + 𝛾2 1 + 𝜓

𝑏𝑢 − 𝑘𝜏 𝜕𝜗 𝐹 𝑨 = 0 Assuming 𝐹 𝑨 = 𝑑𝑝𝑜𝑡𝑢 ⋅ 𝑓−Γ𝑨, as before: Γ = 𝑘𝛾 1 + 𝜓 𝑏𝑢

′

𝜕 + 𝑘𝜓 𝑏𝑢

′′ 𝜕 − 𝑘𝜏/𝜕𝜗

Normally, 𝜓 𝑏𝑢 𝜕 ,

𝜏 𝜕𝜗 ≪ 1:

Γ ≈ 𝑘𝛾 1 + 1 2 𝜓𝑏𝑢

′

𝜕 + 𝑘 2 𝜓𝑏𝑢

′′ 𝜕 − 𝑘𝜏

2𝜕𝜗 = ≡ 𝑘𝛾 + 𝑘Δ𝛾𝑛 𝜕 − 𝛽𝑛 𝜕 + 𝛽0 The final solution becomes: 𝝑 𝑨, 𝑢 = 𝑆𝑓𝐹 0 exp 𝑘𝜕𝑢 − 𝑘 𝛾 + 𝛦𝛾𝑛 𝜕 𝑨 + 𝛽𝑛 𝜕 − 𝛽0 𝑨 1 + 𝜀 ≈ 1 + 𝜀 2 Put losses back Expand the sqrt Define four terms

Propagation Factors 𝝑 𝑨, 𝑢 = 𝑆𝑓𝐹 0 exp 𝑘𝜕𝑢 − 𝑘 𝛾 + 𝛦𝛾𝑛 𝜕 𝑨 + 𝛽𝑛 𝜕 − 𝛽0 𝑨 Linear phase-shift (Red) 𝛾 – Plane-wave propagation constant, fundamental phase variation, Nonlinear phase-shift (Blue) Δ𝛾𝑛 𝜕 – Additional atomic phase- shift due to population inversion → Total phase-shift (Green) Gain (Orange) 𝛽𝑛 𝜕 – Atomic gain

• r loss coefficient (due to transitions)

Background 𝛽0 – Ohmic background loss Phase-shift loss/gain

Exceptions Larger atomic gain or absorption effects The results so far are based on the assumption that 𝜓 𝑏𝑢 −

𝑘𝜏 𝜕𝜗 ≪ 1, however, there are a

few situations of interest where this doesn’t hold:

• Absorption in metals and semiconductors, at frequencies higher than the bandgap

energy, the effective conductivity, 𝜏 can become very large

• Absorption on strong resonance lines in metal vapor, the transitions are very strongly

allowed giving a high absorption per unit length For expanding the sqrt Big sigma Big chi

The paraxial wave equation The full wave-equation, 𝛼2 + 𝛾2 1 + 𝜓 𝑏𝑢 − 𝑘𝜏 𝜕𝜗 𝑭 𝒔 = 0 New ansatz, 𝐹 𝒔 ≡ 𝑣 𝒔 𝑓−𝑘𝛾𝑨 Insert into the wave equation,

𝜖2𝑣 𝜖𝑦2 + 𝜖2𝑣 𝜖𝑧2 + 𝜖2𝑣 𝜖𝑨2 − 2𝑘𝛾 𝜖𝑣 𝜖𝑨 − 𝛾2𝑣

𝑓−𝑘𝛾𝑨 = 0 Assume that 𝑣 𝒔 changes slowly along the z-direction,

𝜖2𝑣 𝜖𝑨2 ≪ 2𝛾 𝜖𝑣 𝜖𝑨

and

𝜖2𝑣 𝜖𝑨2 ≪ 𝜖2𝑣 𝜖𝑦2 , 𝜖2𝑣 𝜖𝑧2

⇒ 𝛼𝑢

2𝑣

− 2𝑘𝛾 𝜖𝑣 𝜖𝑨 + 𝛾2 𝜓 𝑏𝑢 − 𝑘𝜏 𝜕𝜗 𝑣 = 0

Want to handle transverse variations change const -> u(r) ”Paraxial wave equation”

Diffraction- and propagation effects Rewrite: 𝜖𝑣 𝒔 𝜖𝑨 = − 𝑘 2𝛾 𝛼𝑢

2𝑣

𝒔 − [𝛽0−𝛽𝑛 + 𝑘Δ𝛾𝑛]𝑣 𝒔 𝛽0, 𝛽𝑛 𝜕 , and 𝑘Δ𝛾𝑛 𝜕 are defined as before Two terms: Diffraction, − 𝑘

2𝛾 𝛼𝑢 2

Ohmic and atomic gain/loss, −[𝛽0−𝛽𝑛 + 𝑘Δ𝛾𝑛]

Separate terms –> Independent to first order, gain and phase-shift results are the same as for plane waves

Re-write the equation Four terms, same as before Reproduce the plane-wave result

Laser amplification The laser gain after length L: 𝑕 𝜕 ≡ 𝐹

𝑀 𝐹

In terms of intensity, 𝐻 𝜕 = 𝐽 𝑀 𝐽 0 = 𝐹 𝑀

2

𝐹

2 = exp 2𝛽𝑛 𝜕 𝑀 − 2𝛽0𝑀 = exp

[𝛾𝜓′′ 𝜕 𝑀 − 𝜏 𝜗𝑑 𝑀] For most lasers, 𝛽0 ≪ 𝛽𝑛 ⇒

𝐻 𝜕 ≈ exp (𝛾𝜓′′ 𝜕 𝑀)

Calculate the gain from def. alpha exponential dependence on chi

Gain narrowing The gain, 𝐻 𝜕 ∼ exp 𝜓′′ 𝜕 → Narrower frequency dependence, i.e. ”Gain Narrowing” Laser gain & Atomic lineshape FWHM bandwidth is lowered by 30-40%

Absorbing media have opposite sign of 𝜓(𝜕) → ”Absorption broadening” Absorption broadening Uninverted population

• > absorption broadening

For a black-body: Δ𝑄𝑏𝑐𝑡 = 𝜏 ⋅ 𝐽 Thin slab, atomic densities N1 and N2, cross-sections 𝜏12 and 𝜏21 Net absorbed power: Δ𝑄𝑏𝑐𝑡 = 𝑂1𝜏12 − 𝑂2𝜏21 ⋅ 𝑄Δ𝑨 The cross-sections are related, 𝑕1𝜏12 = 𝑕2𝜏21 Convert power → intensity:

1 𝐽 𝑒𝐽 𝑒𝑨 = −Δ𝑂12𝜏21

Previously:

1 𝐽 𝑒𝐽 𝑒𝑨 = −2𝛽𝑛(𝜕)

Stimulated transition cross-sections Loss or gain can be calculated from cross-sections and atomic density

⇒ 2𝛽𝑛 𝜕 = Δ𝑂12𝜏21(𝜕)

Cross-section formula From previous page: 2𝛽𝑛 𝜕 = Δ𝑂12𝜏21(𝜕) From definition of 𝛽: 𝛽𝑛 𝜕 =

𝜌𝜓′′ 𝜕 𝜇

⇒ 2𝛽𝑛 𝜕 = Δ𝑂𝜏 𝜕 = 2𝜌 𝜇 𝜓′′(𝜕) Using the full expression for 𝜓: 𝜏 𝜕𝑏 = 3∗ 2𝜌 𝛿𝑠𝑏𝑒 Δ𝜕𝑏 𝜇2 ⋅ 1 1 + 2(𝜕 − 𝜕𝑏)/Δ𝜕𝑏 2 Cross-section estimation: Assume, Δ𝜕𝑏 ≡ 𝛿𝑠𝑏𝑒. (purely radiative transmission), 3 = 3∗, (all atoms aligned) ⇒ 𝜏𝑛𝑏𝑦 =

3𝜇2 2𝜌 ≈ 𝜇2 2 ≈ 10−9𝑑𝑛2

The cross section is far greater than the area of an atom, due to its internal resonance.

The cross-section is a function of wavelength, transition rate and lineshape Estimate the max cross-section

Real cross-sections Allowed transitions in gas and non-allowed in solids -> 𝝊𝒉𝒃𝒕 ≪ 𝝊𝒕𝒑𝒎𝒋𝒆

𝜇2 2 > Real cross-sections ≫ Atom area

Population difference saturation Gain saturation, In a medium, 𝑒𝐽 𝑒𝑨 = ±2𝛽𝑛𝐽 = ±Δ𝑂𝜏𝐽 As shown previously, Δ𝑂 saturates according to: Δ𝑂 = Δ𝑂0 ⋅ 1 1 + 𝑋𝜐𝑓𝑔𝑔 = Δ𝑂0 ⋅ 1 1 + 𝐽/𝐽𝑡𝑏𝑢 𝑂0 - unsaturated or small signal gain 𝐽𝑡𝑏𝑢 - saturation intensity Stimulated-transition probability, From before, Δ𝑄𝑏𝑐𝑡 = 𝑂1𝜏12 − 𝑂2𝜏21 ⋅ 𝑄Δ𝑨 From rate equation analysis,Δ𝑄𝑏𝑐𝑡 = 𝑋

12𝑂1 − 𝑋 21𝑂2 𝐵Δ𝑨ℏ𝜕𝑏

⇒ W = σI ℏω i.e. the cross-section, the intensity and the stimulated transition probability are interdependent As Δ𝑂 saturates, the gain saturates

Population difference saturation At the line center: From previous slide, 2𝛽𝑛(𝜕, 𝐽) = Δ𝑂0 1 + 𝐽/𝐽𝑡𝑏𝑢 = Δ𝑂0𝜏 1 + 𝜏𝜐𝑓𝑔𝑔/ℏ𝜕 𝐽 → 𝐽𝑡𝑏𝑢 ≡ ℏ𝜕 𝜏𝜐𝑓𝑔𝑔 i.e. 𝐽 = 𝐽𝑡𝑏𝑢: one incident photon within the cross-section per recovery time Off center: The expression is modified by the lineshape, 2𝛽𝑛(𝜕, 𝐽) = Δ𝑂0𝜏 1 + 𝐽 𝐽𝑡𝑏𝑢 ⋅ 1 1 + 𝑧2 Where y is the normalized frequency detuning, 𝑧 = 2 𝜕 − 𝜕0 /Δ𝜕 Frequency dependence due to, transition lineshape and saturation behavior Re-arrange prev expressions

Practical values From previous slide, 𝐽𝑡𝑏𝑢 ≡ ℏ𝜕 𝜏𝜐𝑓𝑔𝑔 𝐽𝑡𝑏𝑢 is an important parameter since little only little gain will be obtained once the intensity approaches this level. 𝐽𝑡𝑏𝑢 is independent of the pumping intensity, since neither 𝜏 nor 𝜐𝑓𝑔𝑔 are intensity dependent. However, harder pumping does increase the small-signal gain Laser type: ℏ𝜕 𝜏 𝜐𝑓𝑔𝑔 𝐽𝑡𝑏𝑢 Gas 10−19 𝐾 10−𝟐𝟒 𝑑𝑛2 10−𝟕 𝑡 1 𝑿/𝑑𝑛2 Solid-state 10−19 𝐾 10−𝟐𝟘 𝑑𝑛2 10−𝟒 𝑡 1 𝒍𝑿/𝑑𝑛2

I-sat varies a lot between materials

Saturation in laser amplifiers As a signal passes through an amplifier, it grows exponentially with distance until the intensity approaches 𝐽𝑡𝑏𝑢 For a single pass amplifier: 1 𝐽(𝑨) 𝑒𝐽 𝑒𝑨 = 2𝛽𝑛 𝐽 = 2𝛽𝑛0 1 + 𝐽(𝑨)/𝐽𝑡𝑏𝑢 Integrating, 1 𝐽 + 1 𝐽𝑡𝑏𝑢 𝑒𝐽 =

𝐽=𝐽𝑝𝑣𝑢 𝐽=𝐽𝑗𝑜

2𝛽𝑛0 𝑒𝑨

𝑨=𝑀 𝑨=0

gives: ln 𝐽𝑝𝑣𝑢 𝐽𝑗𝑜 + 𝐽𝑝𝑣𝑢 − 𝐽𝑗𝑜 𝐽𝑡𝑏𝑢 = 2𝛽𝑛0𝑀 = ln 𝐻0 where, 𝐻0 is the small-signal gain

Calculate the saturation

Saturation in laser amplifiers The total gain, 𝐻 ≡ 𝐽𝑝𝑣𝑢 𝐽𝑗𝑜 = 𝐻0 ⋅ exp(− 𝐽𝑝𝑣𝑢 − 𝐽𝑗𝑜 𝐽𝑡𝑏𝑢 ) Is the unsaturated gain, reduced by a factor exponentially dependent on 𝐽𝑝𝑣𝑢 − 𝐽𝑗𝑜

Saturation in laser amplifiers 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. Extracted Intensities Output Intensities

Gain starts at G0 and declines towards 0 dB

Power-extraction Define the extracted intensity, 𝐽𝑓𝑦𝑢𝑠 ≡ 𝐽𝑝𝑣𝑢 − 𝐽𝑗𝑜 = ln 𝐻0 𝐻 ⋅ 𝐽𝑡𝑏𝑢 As the amplifier saturates, 𝐻 → 1, 𝐽𝑏𝑤𝑏𝑗𝑚 = lim

𝐻→1 ln 𝐻0

𝐻 ⋅ 𝐽𝑡𝑏𝑢 = ln 𝐻0 ⋅ 𝐽𝑡𝑏𝑢 Using ln 𝐻0 = 2𝛽𝑛0𝑀: 𝐽𝑏𝑤𝑏𝑗𝑚 = 2𝛽𝑛0𝑀 ⋅ 𝐽𝑡𝑏𝑢 = Δ𝑂0𝜏𝑀 ⋅ ℏ𝜕 𝜏𝜐𝑓𝑔𝑔 And re-writing: 𝐽𝑏𝑤𝑏𝑗𝑚 𝑀 ≡ 𝑄𝑏𝑤𝑏𝑗𝑚 𝑊 = Δ𝑂0ℏ𝜕 𝜐𝑓𝑔𝑔 i.e. the maximum available power is the total inversion energy, Δ𝑂0ℏ𝜕, once every recovery time, 𝜐𝑓𝑔𝑔

Calculate the extracted power

Power-extraction efficiency Full power extraction requires complete saturation of the amplifier which gives low small-signal gain. Define the extraction efficiency, 𝜃𝑓𝑦𝑢𝑠 ≡ 𝐽𝑓𝑦𝑢𝑠 𝐽𝑏𝑤𝑏𝑗𝑚 = ln G0 − ln 𝐻 ln 𝐻0 = 1 − 𝐻𝑒𝐶 𝐻0,𝑒𝐶 i.e. there’s a tradeoff between effiency and small-signal gain for single- pass amplifiers.

Summary Wave propagation in atomic media Plane-wave approximation, 𝑒𝑨

2 + 𝛾2 1 + 𝜓

𝑏𝑢 −

𝑘𝜏 𝜕𝜗

𝐹 𝑨 = 0 The paraxial wave equation,

𝜖𝑣 𝒔 𝜖𝑨

= −

𝑘 2𝛾 𝛼𝑢 2𝑣

𝒔 − [𝛽0−𝛽𝑛 + 𝑘Δ𝛾𝑛]𝑣 𝒔 Single-pass laser amplification Gain narrowing, 𝐻 𝜕 ∼ exp 𝜓′′ 𝜕 Transition cross-sections, 2𝛽𝑛 𝜕 = Δ𝑂12𝜏21(𝜕) Gain saturation, 𝐻 = 𝐻0 ⋅ exp(− 𝐽𝑝𝑣𝑢 − 𝐽𝑗𝑜 𝐽𝑡𝑏𝑢 ) Power extraction, 𝜃𝑓𝑦𝑢𝑠 = 1 − 𝐻𝑒𝐶 𝐻0,𝑒𝐶

𝛾 – Plane-wave propagation constant, fundamental phase variation, Δ𝛾𝑛 𝜕 – Additional atomic phase- shift due to population inversion 𝛽𝑛 𝜕 – Atomic gain

• r loss coefficient (due to transitions)

𝛽0 – Ohmic background loss Diffraction and gain are separate to first order

30-40% lower FWHM bandwidth Low signal gain is saturated by a factor exponentialy dependent

• n the intensity difference

Loss/gain can be calculated from atomic density and cross-sections There’s a tradeoff between efficiency and small-signal gain