Photon Physics Week 4 6//13 1
Classical atom-field interaction Lorentz oscillator: Classical electron oscillator with frequency ω and damping constant γ Eqn of motion: Final result:
Classical atom-field interaction Lorentz oscillator: Classical electron oscillator with frequency ω and damping constant γ Define Scattering rate Compare with OBE for -level system: no saturation saturation with 3
Broadening mechanisms and lineshapes Homogeneous broadening mechanisms -affect all atoms equally Lorentzian lineshape natural broadening (lifetime broadening) collisional broadening (liquid, gas) phonon broadening (solid) Inhomogeneous broadening mechanisms- result of differences in transition frequency for different atoms Gaussian lineshape Doppler broadening (gas) site/strain broadening (solids) 4
Natural broadening Classical electron oscillator with frequency ω and damping constant γ (<<ω ) without applied field: = er( t) = er exp γt p t oscillating dipole radiates with amplitude proportional to dp/dt: cosω t Radiation from oscillating dipole has spectral intensity distribution: γ I( ω) = I g H ( ω ω ) g H ( ω ω ) = 1 π + ( γ ) ω ω Lorentzian 5
Natural broadening Radiation from oscillating dipole has spectral intensity distribution: I( ω) = I g H ( ω ω ) g H ( ω ω ) = 1 π + ( γ ) γ classical = ω ω e ω γ 6πε m e c 3 QM = 1 τ + 1 rad k τ = rad A kj + A kl l E j <E k γ transition between levels k and l E j <E l Lorentzian 6
Other homogeneous broadening effects Additional broadening can occur through collisions (gas, liquid) or vibrations (solids): γ I( ω) = I g H ( ω ω ) g H ( ω ω ) = 1 π ( ω ω ) + ( Δω ) Collisional broadening with mean collision time τ c : Lorentzian FWHM = Δω = γ+/τ c Note: τ c decreases with increasing pressure 7
Inhomogeneous broadening Broadening is due to distribution in transition frequencies. In gases this is mostly caused by Doppler broadening, i.e. the transition frequency depends on velocity: ω ω = v z c ω The Maxwellian distribution in velocities is a Gaussian, resulting in a Gaussian lineshape g D ( ω ω ) = Δω D ln π exp ω ω Δω D ln Δω D = ln ω c k B T M In solids the transition frequencies vary because atoms or ions have slightly varying environments. This effect is most pronounced in amorphous materials. Δω D reflects the distribution in frequencies. 8
Broadening mechanisms and lineshapes 9
Voigt profile 1
Voigt profile = g D ω c ω g V ω ω g H ( ω ω c )dω c Δω Ι /Δω Η = 5 Δω Ι /Δω Η = 1 Δω Ι /Δω Η =. Lorentzian Gaussian 11
Gain cross-section Homogeneous broadening Energy gained by beam= Net transition rate down x I( z,ω) I( z + δz,ω) [ I( z + δz,ω) I( z,ω) ]Aδω = δz [ N B 1 g H ( ω ω )ρ( ω,z)δω N 1 B 1 g H ( ω ω )ρ( ω,z)δω]aδz ω The z-dependence of I can then be written as: I z,ω z = [ N B 1 N 1 B 1 ]g H ( ω ω )ρ( ω,z) ω ω A = [ N B 1 N 1 B 1 ]g H ( ω ω )I( ω,z) ω c since I( ω,z) = ρ( ω,z)c I( z,ω) z σ = N * 1( ω ω ) = ω σ 1 ( ω ω )I( ω,z) c B g 1 H ( ω ω ) N * = N g g 1 N 1 gain cross-section population inversion density 1
Condition for optical gain and gain coefficient N * = N g g 1 N 1 > gain N * = N g g 1 N 1 < absorption I( z,ω) z α ω ω = N * σ 1 ( ω ω )I ω,z = α ω ω I ω,z depends in general on intensity through the intensity dependence of N, but for low intensities it is constant and denoted by α ( ω ω ), the smallsignal gain coefficient. In that case the intensity grows exponentially with z I( ω,z) = I ω, [ ] exp α ( ω ω )z gain coefficient 13
Frequency dependence of gain cross-section σ 1 ( ω ω ) = ω c B 1 1 π = π c ω A 1 1 π σ 1 = ω c B 1 ω ω Δω + ( Δω ) ω ω Δω + ( Δω ) πδω = πc ω Δω A 1 Lorentzian lineshape Gaussian lineshape σ 1 ( ω ω ) = ω c B 1 14 Δω = π c ω A 1 Δω σ 1 = ω c B 1 Δω ln π exp ω ω Δω ln π exp ω ω Δω ln π = π 3 lnc ω Δω ln A 1 ln
Gain narrowing The growth of spectral intensity depends on frequency: faster growth near line center leads to gain narrowing normalized to input peak intensity normalized to output peak intensity 15
Some numbers for real lasers 16
Amplification of narrowband radiation Total intensity of beam I T ( z) = I( ω,z) dω I ω,z dω = N * σ z 1 ( ω ω )I ω,z d dz I ω,z [ ] dω = N * σ 1 ω L ω Growth of total beam intensity: di T dz = N * σ ( 1 ω L ω )I T ( z) I ω,z dω dω Assume radiation in narrow band around ω L 17
Rate equations dn dt = N 1 B 1 g H ( ω ω )ρ( ω)δω N B 1 g H ( ω ω )ρ( ω)δω N τ since g 1 B 1 = g B 1 dn = N g N 1 ω dt g 1 c B g 1 H ω ω dn dt c = N * σ 1 ( ω ω ) 1 ω I ( ω )δω N τ ω ρ ( ω )δω N τ Interaction with all frequencies in beam: dn dt = N * σ 1 ω ω ω I ( ω )δω N 1 τ dn dt = N * σ 1 ( ω L ω ) I T N ω L τ again, assume radiation in narrow band around ω L 18
Absorption cross-section and coefficient I( z,ω) z = N * σ 1 ( ω ω )I( ω,z) g 1 σ 1 ( ω ω ) = g σ 1 ( ω ω ) N * = N g g 1 N 1 < absorption I( z,ω) z abs cross-section = N 1 g 1 N σ 1 ( ω ω )I ω,z g abs coefficient = κ ω ω I ω,z N ** Absorption from ground state, low I: N 1 σ 1 ( ω ω ) κ ω ω I( ω,z) = I ω, I( z) = I [ ] [ z] exp κ( ω ω )z exp κ ω L ω Beer s law 19
Self-absorption and radiation trapping Emission lineshape can be distorted by self-absorption. This happens when absorption probability is high, so when optical thickness τ > 1: κ ω ω τ ω ω R thickness of sample Since absorption is frequency-dependent center of spectrum is affected more. Self-absorption also affects the measured emission lifetime 1 τ = 1 trap τ + ( g rad trap 1)A 1 g trap depends on sample geometry
Gain cross-section for inhomogeneous broadening Net transition rate down for group of atoms with central frequency ω c : [ ΔN ( ω c )δω c B 1 g H ( ω ω c )ρ( ω,z)δω ΔN 1 ( ω c )δω c B 1 g H ( ω ω c )ρ( ω,z)δω]aδz Net transition rate down for all groups of atoms: ΔN ( ω c )B 1 g H ( ω ω c ) dω c ΔN 1 ( ω c )B 1 g H ω ω c number density of atoms with center frequencies between ω c and ω c +δω c I( z,ω) z I( z,ω) z σ D 1 ω ω = ΔN ω c g g 1 ΔN 1 ω c = N * σ D 1 ( ω ω )I( ω,z) = ω c B 1 g D ω c ω g H ( ω ω c )dω c ΔN ω c g g H ( ω ω c )dω c dω c ρ( ω,z)δωaδz ω c B 1I ω,z g 1 ΔN 1 ω c = N * g D ω c ω Inhomogeneous gain cross-section 1