Quantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths 4.1 The Natural Line Shape 4.2 Collisional Broadening 4.3 Doppler Broadening 4.4 Einstein Treatment of Stimulated Processes Width and Shape of Spectral Lines 1 The Width and Shape of Spectral lines: Natural Line Shape Δ Ek = !Γk I- The Natural Line shape E •Discrete energy levels have finite k € radiative lifetimes and therefore are not infinitely sharp by application of E Heisenberg’s Uncertainty Principle Δ i = !Γi (HUP). The diagram illustrates this. The Ei angular frequency scale is used here, ie € photon energies are given by (with the conversion to frequency unit) The quantum mechanical line shape E ki = !ωki = hν ki The level widths are related to the for the radiative transition radiative lifetimes by HUP: between E and E is a Lorentzian function. k k ΔEk 1 €Δ Ek ⋅τk ≈ ! ⇒ = = ∑Aki The full width at half -maximum is : ! τ k i 1 1 The level widths are given by Γki = Γk + Γi = + τ k τ k ΔEk = !Γk and ΔEi = !Γi Γki 2π τk and τi are the total lifetimes Iki (ω) = I0 × 2 2 and inverse of the transition probabilities A (ω −ωki ) + Γki 4 ki Width and Shape of Spectral Lines 2 € € Spontaneous Emission Probability Aki • Aki= probability per unit • Spontaneous emission is a time for emission of a random process, i.e., one cannot predict when one single photon of energy ℏωki=hνki. atom will radiatively decay. It’s the Einstein A • The measurement of the coefficient. number of photons emitted by • Ei, Ni, gi represent the level a population of atoms over a energy, population and long period of time leads to the statistical weight notion of a radiative lifetime: At initial time t= 0, the population is N 0 respectively. k ( ) dN atoms decay in dt such that dN = − N t A dt Ek, Nk, gk k k ∑ k ( ) ki i< j % t ( Integrating between 0 and t: N k (t) = N k (0)exp'− * & τ k ) 1 τ k = ≡ Radiative lifetime A Ei, Ni, gi ∑ ki i< j Width and Shape of Spectral Lines 3 € Spontaneous Emission Probability Aki • Statistical weight of level Ek = 2Jk + 1 where Jk = total angular momentum of state ‘k’. • E.g., He 1s2p (1P1) state • (1P1) = (2S+1LJ), S = Total spin angular momentum, L = Total orbital angular momentum and J = Total angular momentum (L+S, L+S-1, …..L-S). • Here g = 2J+1 = 2(1) + 1 = 3 ! Width and Shape of Spectral Lines 4 The Width and Shape of Spectral lines Natural Line Shape If Ei is the fundamental level, then τ i → ∞. If E is resonance level just above the fundamental level, •Note: Atomic levels can k also de-excite non- 1 radiatively, e.g collisions, then Aki = . τ k emission of an electron (Auger effect). The total The Einstein Spontaneous emission coefficient Aki should transition probability is be modulated by a normalised line shape function such as thus the sum of radiative € terms and non-radiative I (ω) Γ 2π terms which can be L(ω) = ki = ki I (ω −ω )2 + Γ 2 4 written in the following 0 ki ki form: The transition probability that includes the line shape Total 1 1 A = + is thus A L( ). ki rad non−rad ki ω τ ki τ ki L(ω) is normalised such that € •Usually, one type of de- +∞ L(ω)dω = L(ω)dω = 1 excitation process will ∫−∞ ∫line € dominate I 0 = Total energy radiated in the line (normalisation factor) Width and Shape of Spectral Lines 5 € The Width and Shape of Spectral lines Line Broadening mechanisms in real line sources: Collisional broadening •In a real source, atoms (emitters) are subjected to the interaction forces (i.e., collisions) of neighbouring atoms, ions, electrons, etc…+ possibly external potentials. These perturb the potential energy of the radiating atom which results in a broadening of the line to a value greater than the natural line width. •This type of broadening may take different forms according to the exact nature of the interaction potential: Collisional broadening, Stark broadening, Pressure broadening •Detailed quantum mechanical calculations would show that the collisionally broadened line shape is also Lorentzian of FWHM Δωtotal = ΔΓNatural + ΔΓCollisional Width and Shape of Spectral Lines 6 € The Width and Shape of Spectral lines Line Broadening mechanisms in real line sources: Doppler broadening •A stationary atom of mass m •In a gas in equilibrium there exists a emits a photon of (angular) Maxwell-Boltzmann distribution of frequency ω velocities (controlled by the absolute 0. temperature T, see Y3 Stat.Phys). •The apparent frequency ω ’ 0 •The probability P(vz)dvz that an atom observed by a stationary observer has z-component of velocity between vz when the atom emits a photon ! and vz+dvz is: 1 moving at the relative velocity v : # & 2 # 2 & m mvz P(vz )dvz = % ( exp% − ( dvz ! 2πk T 2k T % v ⋅ rˆ ( $ B ' $ B ' ω' = ω 1− ! 0 0' * v & c ) •Thus the probability of detecting a € wave with angular frequency between ’ and ’+d ’ is given by the ' € ω0 ω0 ω0 ω0 following Gaussian distribution: rˆ ' 2 % ω −ω ( € ' ' % 2 ( ' ( 0 0 ) * ' P(ω0)dω0 = ' * exp −4 dω0 & πΔ ) ' Δ2 * € & ) Width and Shape of Spectral Lines 7 € € € The Width and Shape of Spectral lines Line Broadening mechanisms in real line sources: Doppler broadening Where Δ is the linewidth parameter, from which one obtains the FWHM Δω 1 (Doppler) (see " truncation of 2 Gaussian functions" in "Optical Resonator"): ω 0 2kT Δω 1 (Doppler) = Δ Ln2 = 2 Ln2 2 c m •A measurement of the Doppler width can thus provide an estimation for the temperature of the medium € •Doppler broadened lines are inhomogeneously broadened The total width of any spectral line emitted in a real source is thus: Δω total = ΔΓNatural + ΔΓCollisional + Δω 1 (Doppler) 2 Width and Shape of Spectral Lines 8 € Einstein Treatment of Stimulated Absorption and Emission • Thermal equilibrium of two-level Ek, Nk, gk atoms with radiation Spontaneous • Radiation in thermal equi., ie black- process body radiation model, is given by Stimulated processe s Planck’s radiation law (radiation energy density=energy per unit Ei, Ni, gi volume per unit interval frequency) 8πhν 3 1 Einstein Coefficients ρ(ν ,T ) = 3 hν c A = coefficient of e kT −1 ki spontaneous emission • The atomic populations are in Maxwell-Boltzmann equilibrium, Bik = coefficient of € maintained by a dynamic (stimulated) absorption equilibrium between emission and absorption processes Bki = coefficient of stimulated emission Width and Shape of Spectral Lines 9 Einstein Treatment of Stimulated Absorption and Emission • The upward and downward Maxwell -Boltzmann: transition rates are obtained in N g $ !ω ' terms of the populations and k = k exp&− ki ) coefficients: N i gi % kBT ( Upward rate: From above: dNi N B = −N B ρ k ikρ dt i ik = N i (Aki + Bki ρ) Downward rate: gkAki exp(−!ω ki kBT ) dNk ρ(ω ki ) = = −Nk (Aki + Bki ρ) gi Bik − gkBki exp(−!ω ki kBT ) dt Equilibrium (detailed balance): Using Planck’s radiation law for ρ(ω ki ) : g B = g B dNi dNk i ik k ki = = 0 2 3 dt dt π c gk Bik = Aki N B ρ = N A + B ρ ω 3 g i ik k ( kii ki ) ! ki i Width and Shape of Spectral Lines 10 € Einstein Treatment of Stimulated Absorption and Emission • To have equal rates of spontaneous and stimulated emission requires: A 8πh A = Bρ(ν 0) ⇒ = ρ(ν 0) = B λ3 −6 2 λ ≡ λ0 = 1 µm ⇒ cρ(ν 0) = I(ν 0) ≈ 5x10 W/m Hz For a (Lorentzian) spectral line of width Δν = 107 Hz 2 -2 -2 I(ν 0) = × I 0 ⇒ I 0 = 50 Wm = 5 mWcm πΔν • Stimulated emission typically negligible for laboratory line sources € Width and Shape of Spectral Lines 11 Elementary Interaction Processes N2 N2 N2 h 12 h 12 h 12 h 12 h 12 N1 N1 N1 • Stimulated • Stimulated • Spontaneous absorption: one emission: one emission: photon lost photon gained one photon • Attenuation • Amplification emitted at • Stimulated photon: random same direction and • Noise phaseWidth and Shape of Spectral Lines 12.