Black body radiation and line shapes (Ch. 13) Apr. 15, 2017

Atomic radiation based on traditional approach from blackbody radiation (Laser Electrionics 2nd Ed. Verdeyen)

• Radiation from a hot object: Black body is also a perfect radiation absorber.
Wien’s law (Classic result) gives peak emission wavelengthλpeak=2.8978×103(mK)/T(K)
• Relating spectral energy densityρ(ν,T)(J/m3Hz) and intensity spectrumu(ν,T)(W/m2Hz) :coρ=u
u andρ can be expressed in terms ofλ , i.e.ρ(λ,T) ,u(λ,T)
Notice that intensityI=u(λ,T)dλ orI=u(ν,T)dν
For blackbody, radiation only emit to the front making an angle with the optical axis and its intensity spectrumuBB(λ,T)=ρ(λ,T)co/4
• Rayleigh-Jeans catastrophe: Energy densityρ(ν)dν=kTM(ν)dν=kT8πn3ν2dν/co3 Energy density increases with freq without upper limit, not reflecting experimental results.
• Quantization of energy: Planck assigned probability density for each discrete energy following Boltzmann distributionehν/kT and compute average energy level<E>=Σl=0lhνelhν/(kT)Σl=0ehν/(kT)=hνehν/(kT)1
Energy densityρ(ν)dν=hνehν/(kT)1(8πn3ν2co3)dν
IntensityuBB(λ)dλ=2πh3co2λo5hehν/(kT)1dλ
• Spontaneous emission: No input, noise-like, nondirectivedN1dt=dN2dt=A21N2g2
• Stimulated emission: Input photons, direction, output following input’s polarization, freq and phase, i.e. light amplification,dN1dt=dN2dt=B21N2g2ρ(ν)
• Absorption: freq and polarization dependent,dN2dt=dN1dt=B12N1g1ρ(ν)
• Einstein’s approach relating radiative processes: Detailed balance -- emission rate = absorption rate under thermal equilibrium, i.e.dN1dt|emission+dN1dt|absorption=0
AsT ,ρ(ν) andB21g2=B12g1

ρ(ν)=A21B211eh(ν2ν1)/(kT)1A21B21=8πn3ν3hco3

InterpretA21=1/tsp as spontaneous emission rate wheretsp is the spontaneous emission lifetime.
Note:gi=2Ji+1 for i = 1, 2 whereJi is the angular momentum of the energy level.
• More complete approach: The text book consider probability density or rate of spon. emission for a modepsp=cσ(ν)/V(s1) where V is the volume andσ(ν) is the transition cross section inm2
Probability of emission between t and t+dt ispspΔt The spatial distribution ofσ depending on the angleθ between dipole moment of the atom and the field,σ=σmaxcos2θ
The probability density or rate of stimulated emission and absorption for a mode can be defined similarly, except that these processes have number of photons n, i.e.Pab=Pst=ncσ(ν)/V=Wi
• Relationship between lines shapeg(ν) andσ(ν) : After normalizingσ(ν) by oscillator strengthS=0σ(ν)dν ,g(ν)dν=σ(ν)dν/S will give the probability of spon. emission betweenν andν+Δν
Hence,g(ν) is a probability density function, i.e.0g(ν)dν=1 It peaks atνo (the transition freq.) and has FWHMΔν
Notice that field’s bandwidth fromρ(ν) is limited by that of the cavityδν
• General rate equation for light amplification:dN2/dt=A21N20g(ν)dν+B12N10ρ(ν)g(ν)dνB21N20ρ(ν)g(ν)dν
Einstein consideredδνΔν and treatedg(ν) like delta function.
Book concentrated on the laser caseΔνδν andρ(ν)=ρνδ(νν)

dN2dt=A21N2σ(ν)Iνhν[N2g2g1N1]=A21N2Wi[N2g2g1N1]

whereσ(ν)=A21λo2g(ν)/(8πn2) ,φ=Iν/(hν) andWi=φσ(ν)
• Total spon. emission to all modes: Consider spatial averageσ¯(ν)=σmax(ν)/3 ,Psp=0cVσ¯(ν)VM(ν)dν For blackbody, we have Einstein case whereg(ν) is like a delta function.Psp=1/tspM(νo)cS¯ whereS¯=0σ¯(ν)dν=λo2/(8πtspn2) andσ¯(ν)=λo2g(ν)/(8πtspn2)
• Stimulated emission relating to blackbody radiation:Wi=0ρ(ν)Vhνcσ(ν)Vdν
Blackbody is broadbandδνΔνg(ν)δ(ννo)

Wi=n¯/tsp

where number of photons per moden¯=λo3ρ(νo)/(8πn3h)
Line broadening: Uncertainty of energy levelsΔE=ΔE1+ΔE2=h2π(1τ1+1τ2) whereτ is transit lifetime.
In another words,Δν=12π(1τ1+1τ2)
• Homogeneous broadening: The linewidth is widened by an effect that applies equally to all atoms, e.g. collisional (pressure) broadening in gas, lifetime broadening (radiative lifetime), and phonon broadening.
Lifetime of the excited atoms withN2 population isτ2 which followsτ21=τNR1+τR1 whereτNR is nonradiative lifetime from processes not generating light andτR is radiative lifetime from light generating processes.
• Lineshape function for homogeneous broadening: Since decay in excited atoms or power corresponds to fielde(t)=eoet/(2τ2)cosωot , spectral intensity|e(ω)|2 ~1(ωωo)2+(γ/2)2 from Fourier transform whereγ=1/τ2
This remind us the Lorentzian lineshapeg(ν)=Δν2π1(ννo)2+(Δν/2)2 whereΔν=γ/(2π) andg(νo)=2/(Δνπ) Alsoσ¯o=λo22πn212πtspΔν Ifτ2 is entirely radiative,σ¯o=λo22πn2
• Characteristics of homogeneous broadening: Line shape can change with radiation (photons) and will scale down as a result of saturation, i.e.g(νo) decreases.
• Collisional broadening: Collision interrupts phase and decreases the coherent lifetime.
Probability of finding collision free atoms=p(t)dt=1τcet/τcdt whereτc = mean time between collisions.
For a gaseous mixture of type m and n, freq of collisionfcol=1τc=N<σv>=Nmσ[8kTπ(1Mm+1Mn)] where v is mean velocity,σ is cross section (function of v), N is the number density,Nm is the number density of type m,Mm is mass of type m atom andMn is mass of type n atom.
Compute correlation of field and take Fourier transform to find intensity

<e(t)e(t+τ)>=eo2cosωoτe|τ|/τc

I ~ Fourier transform of<e(t)e(t+τ)>

=1(ωωo)2+(1/τc)2

Lorentzian width =2/(2πτc)
Overall lineshape function (homogeneous):g(ν)=Δνtotal2π1(ννo)2+(Δνtotal/2)2 whereΔνtotal=12π(2τc+1τ2)


Last Modified: Nov. 15, 2017
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