Intense lasers: high peak power Part 1: amplification Bruno Le - PowerPoint PPT Presentation

Intense lasers: high peak power Part 1: amplification Bruno Le Garrec Directeur des Technologies Lasers du LULI LULI/Ecole Polytechnique, route de Saclay 91128 Palaiseau cedex, France bruno.le-garrec@polytechnique.edu 31/08/2016 Bruno Le Garrec page LPA school Capri 2017 1

What do you need for building a laser : An amplifying medium = an energy converter An electromagnetic radiation = an electromagnetic wave that propagates A resonant cavity = a set of mirrors facing each other = a « Fabry-Perot » cavity That’s true for both an oscillator and an amplifier. For many reasons, a Master Oscillator Power Amplifier (MOPA) is the most commonly used FREQUENCY Pre- SOURCE AMPLIFICATION CONVERSION - amplification 10 -9 J < 1 J 15 to 20 kJ 7,5 to 10 kJ Bruno Le Garrec page 2

Laser-matter interaction: the Blackbody radiation What is the power spectral density of the radiation emitted by a gas inside a box at a given temperature ? Wavelength Bruno Le Garrec page LPA school Capri 2017 3

Laser-matter interaction: the Blackbody radiation theory • Semi-classical model between electromagnetic radiation and a population of atoms: – The spectral energy density is defined as ρ ( ( ν ) = U( ν ) dN( ν )/ V the product of the average number of photons per energy mode times the photon energy times the modes density between ν and ν +d ν : 2 1 8 πν ( ) d h d ρ ν ν = ν ν h 3 ν c exp( ) 1 − kT – Photon energy – Average number of photons per mode – Photons density between ν and ν +d ν – That’s the Planck formula from the Blackbody radiation theory 3 1 8 h π ν ( ) d d ρ ν ν = ν h 3 ν c exp( ) 1 − kT Bruno Le Garrec page LPA school Capri 2017 4

Laser-matter interaction: the atomic system is made of many levels Any 2, 3 or 4 level system can be seen as a 2 level system: – Degeneracy g = 2m+1 (related to the number of sub-levels of a given kinetic momentum: orbital L, m L , total J=L+S, J, m J , F=J+I, F, m F ) – Homogeneous broadening related to the lifetime of the atomic system (free atoms , electrons in the crystal field, molecules): – Lorentzian Δ g ( ) ω = 2 ⎡ ⎤ 2 Δ 2 π ω − ω + ( ) 0 4 ⎢ ⎥ ⎣ ⎦ – Inhomogeneous broadening (Doppler effect of moving atoms and molecules) ( ) ) 2 2 ln 2 – Gaussian ω − ω g ( ) Exp ( 4 ln 2 ω = − 0 Δ π Δ Δ is the Full Width at Half Maximum (FWHM) of the line shape when Δ = 1/T rad +1/T non rad and g ( ω d ) = 1 ∫ ω Bruno Le Garrec page LPA school Capri 2017 5

Emission, Fluorescence, Phosphorescence White light Bruno Le Garrec page LPA school Capri 2017 6

Einstein’s coefficients (1) At thermodynamic equilibrium, each process going “down” must be balanced exactly by that going “up” and the transition probability can be written: P B ( ) t = ρ ν 0 j 0 j → Level j, g j • N j the population (or population density) of level i • N = N 0 + N j = cste Spontaneous Absorption Absorption emission • δ N 0 a = - N 0 B 0j ρ ( ν ) δ t Stimulated or induced emission • δ N 0 sti = + N j B j0 ρ ( ν ) δ t Spontaneous emission Level 0, g 0 • δ N 0 spont = + N j A j0 δ t The balance Δ N 0 = δ N 0 a + δ N 0 sti + δ N 0 spont =[(N j B j0 - N 0 B 0j ) ρ ( ν ) + N j A j0 ] δ t • • Δ N j = - Δ N 0 =[(N 0 B 0j -N j B j0 ) ρ ( ν ) - N j A j0 ] δ t • Commonly written dN j /dt, dN 0 /dt Bruno Le Garrec page LPA school Capri 2017 7

Einstein’s coefficients (2) Relationship between the coefficients : • Because N j ∝ g j exp-E j /kT and absorption = sum of emissions • g 0 B 0j = g j B j0 • A j0 / B j0 =8 π h ν 3 /c 3 • N j is the population (or n j the population density) of level i ρ ( ν ) d ν = ρ ( ω ) d ω then ρ ( ω ) = ρ ( ν ) /2 π • B j0 ρ ( ν ) / A j0 = n( ν ) is the number of photons per mode ρ ( ν ) = 8 πν 2 /c 3 h ν n( ν ) with thermal radiation included then n( ν ) = 1/Exp(h ν /kT)-1) one finds h ν >> kT in the optical domain and stimulated/spontaneous ≈ Exp-h ν /kT while in the thermal domain h ν << kT and stimulated/spontaneous ≈ kT/ h ν When the refraction index is n, v = c/n and one defines the (laser) intensity as I ν = c ρ ( ν ) /n • dN j /dt = (N 0 B 0j -N j B j0 ) ρ ( ν ) - N j A j0 • dN j /dt = (n I ν /c) A j0 (c 3 /n 3 )/ 8 π h ν 3 (N 0 g j / g 0 -N j ) - N j A j0 • dN j /dt = - A j0 ( λ 2 / 8 π n 2 )(I ν / h ν )(N j -N 0 g j / g 0 ) - N j A j0 Bruno Le Garrec page LPA school Capri 2017 8

Einstein’s approach (3) Transition (lifetime) broadening: • General case ρ ( ν ) and g( ν ) : • A becomes A’ = A g( ν ) and B becomes B’ = B g( ν ) • Different cases to be considered : • Narrow transition g( ν ) << ρ ( ν ) then g( ν ) = δ ( ν−ν 0 ) • Broad transition g( ν ) >> ρ ( ν ) then ρ ( ν ) =I ν /c = I 0 g( ν −ν 0 ) /c Relationship between the coefficients: • Spontaneous emission isotropic and un-polarized • Stimulated emission : transmitted wave has the same frequency, is in the same direction, and has the same polarization as the incident wave. • There is a relation between gain and intensity Bruno Le Garrec page LPA school Capri 2017 9

Amplification (1) One writes the intensity balance Δ I = I transmitted + I spontaneous - I incident as a function of the Einstein’s coefficients in a two-level atomic system (1, 2) for a given medium thickness Δ z Δ I = h ν B 21 I ν /c g( ν ) N 2 Δ z - h ν B 12 I ν /c g( ν ) N 1 Δ z +h ν A 21 Δν g( ν ) N 2 Δ z d Ω /4 π one direction acceptance cone Δ I / Δ z = h ν B 21 (N 2 –N 1 g 2 / g 1 ) g( ν ) I ν + h ν A 21 Δν g( ν ) Δ z d Ω /4 π There is gain if : Transmitted Incident intensity • N 2 > N 1 g 2 / g 1 intensity with d Ω With a « noise » contribution Δ z Level 2, g 2 even without incident light Polarizer filter Level 1, g 1 Bruno Le Garrec page LPA school Capri 2017 10

Amplification (2) The gain factor reads: dI ν /dz = A 21 ( λ 2 / 8 π n 2 )g( ν )(N 2 –N 1 g 2 / g 1 )I ν = γ 0 ( ν ) I ν This γ 0 ( ν ) or g 0 ( ν ) is the small signal gain when I incident is small compared to a so-called « saturation » value I saturation . The first part of dI ν /dz is the transition cross section. There is a difference between stimulated emission cross section and absorption cross section. γ 0 ( ν ) = A 21 ( λ 2 / 8 π n 2 )g( ν )(N 2 –N 1 g 2 / g 1 ) σ se = A 21 ( λ 2 / 8 π n 2 )g( ν ) σ ab = A 21 ( λ 2 / 8 π n 2 )g( ν ) g 2 / g 1 Δ N = N 2 –N 1 g 2 / g 1, so far : γ 0 ( ν ) = σ se Δ N When dI ν /dz can be integrated over z then: I ν (z) = I ν (0) Exp[ γ 0 ( ν ) z] G 0 ( ν ) = Exp[ γ 0 ( ν ) z] = I ν (z )/ I ν (0) is the gain. Another very important factor is the saturation fluence : F sat =h ν / σ Bruno Le Garrec page LPA school Capri 2017 11

Population inversion (1) As soon as : N 2 > N 1 g 2 / g 1 or γ 0 ( ν ) >0, there is population inversion or populations are said to be “inverted” When there are relations between Einstein’s coefficients or rate equations Population inversion ⇔ amplification At thermodynamic equilibrium, level populations are given by the Maxwell-Boltzmann relationship: N j ∝ g j exp-E j /kT If E2 > E1, then N 2 /g 2 < N 1 /g 1 E So far: • N 2 /g 2 > N 1 /g 1 is an abnormal state of affairs . Level 2, g 2 This state has to be sustained to compensate for emission losses. • The extracted energy E = Δ N h ν tells us that anytime 1 photon is emitted ⇔ Level 1, g 1 the atom “goes” from E 2 to E 1 N/g Bruno Le Garrec page LPA school Capri 2017 12

Population inversion (2) : 2-level system • In a 2-level system, population inversion is impossible 2 2 m 1 1 «closed» 2-level «open» 2-level • The probability to empty level 2 is always greater than that to empty level 1. In the « open » case, le upper level will be progressively drained to the meta-stable level and the lower level will be “depleted”: this process is called « optical pumping ». Bruno Le Garrec page LPA school Capri 2017 13

Population inversion (2) : 3-level and 4-level systems • According to selection rules between levels (parity, Δ L, Δ J, Δ F= 0, ± 1), absorption, spontaneous or stimulated emission are or are not possible between any set of 2 levels. 2 3 1 2 1 0 0 3 levels 4 levels • Non radiative transitions are possible: collisions (gas), crystal vibrations. These transitions can allow fast population transfers between neighbor levels. Bruno Le Garrec page LPA school Capri 2017 14

The laser was born in 1960, May 16 th . • Maiman has used a flash lamp (GE FT-506 model) inside a simple aluminum tube. • The rod has a 0.95 cm diameter (3/8 inch) and a 1.9 cm length (3/4 inch) with end faces coated with silver. • On one face, the central part of the silver coating is removed in order to let the radiation escape from the rod. Bruno Le Garrec page LPA school Capri 2017 15