Quantum Electronics/Laser Physics Chapter 4 Line Shapes and Line Widths

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1 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

2 The Width and Shape of Spectral lines: Natural Line Shape I- The Natural Line shape Discrete energy levels have finite radiative lifetimes and therefore are not infinitely sharp by application of Heisenberg s Uncertainty Principle (HUP). The diagram illustrates this. The angular frequency scale is used here, ie photon energies are given by (with the conversion to frequency unit) E ki =!ω ki = hν ki The level widths are related to the radiative lifetimes by HUP: ΔE k τ k! ΔE k! = 1 = A ki τ k i The level widths are given by E k E i ΔE k =!Γ k ΔE i =!Γ i The quantum mechanical line shape for the radiative transition between E k and E k is a Lorentzian function. The full width at half - maximum is : Γ ki = Γ k + Γ i = τ k τ k ΔE k =!Γ k and ΔE i =!Γ i τ k and τ i are the total lifetimes and inverse of the transition probabilities A ki I ki (ω) = I 0 Γ ki 2π (ω ω ki ) 2 + Γ 2 ki 4 Width and Shape of Spectral Lines 2

3 Spontaneous Emission Probability A ki A ki = probability per unit time for emission of a photon of energy ħω ki =hν ki. It s the Einstein A coefficient. E i, N i, g i represent the level energy, population and statistical weight E k, Nrespectively. k, g k E i, N i, g i Spontaneous emission is a random process, i.e., one cannot predict when one single atom will radiatively decay. The measurement of the number of photons emitted by a population of atoms over a long period of time leads to the notion of a radiative lifetime: At initial time t= 0, the population is N k ( 0) dn k atoms decay in dt such that dn k = N k t Width and Shape of Spectral Lines 3 i< j ( ) A ki dt Integrating between 0 and t: N k ( t) = N ( k 0 )exp % t ( ' & τ * ) k τ k = i< j 1 A ki Radiative lifetime

4 Spontaneous Emission Probability A ki Statistical weight of level Ek = 2Jk + 1 where Jk = total angular moment of state k. E.g., He 1s2p ( 1 P1) state ( 1 P1) = ( 2S+1 LJ), 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

5 The Width and Shape of Spectral lines Natural Line Shape If E i is the fundamental level, then τ i. If E k is resonance level just above the fundamental level, then A ki = 1 τ k. The Einstein Spontaneous emission coefficient A ki should be modulated by a normalised line shape function such as L(ω) = I ki (ω) I 0 = Γ ki 2π (ω ω ki ) 2 + Γ ki 2 4 The transition probability that includes the line shape is thus A ki L(ω). L(ω) is normalised such that + L(ω) dω = L(ω) dω = 1 line I 0 = Total energy radiated in the line (normalisation factor) Note: Atomic levels can also de-excite nonradiatively, e.g collisions, emission of an electron (Auger effect). The total transition probability is thus the sum of radiative terms and non-radiative terms which can be written in the following form: A ki Total = 1 τ ki rad + 1 τ ki non rad Usually, one type of deexcitation process will dominate Width and Shape of Spectral Lines 5

6 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 Δω total = ΔΓ Natural + ΔΓ Collisional Width and Shape of Spectral Lines 6

7 The Width and Shape of Spectral lines Line Broadening mechanisms in real line sources: Doppler broadening A stationary atom of mass m emits a photon of (angular) frequency ω 0. The apparent frequency ω 0 observed by a stationary observer when the atom emits a photon! moving at the relative velocity : ω 0 ' ˆ r % = ω 0 ' 1 &! v r ˆ ( * c ) ω 0 '! v v In a gas in equilibrium there exists a Maxwell-Boltzmann distribution of velocities (controlled by the absolute temperature T, see Y3 Stat.Phys). The probability P(v z )dv z that an atom has z-component of velocity between v z and v z +dv z is: # m & P(v z )dv z = % ( $ 2πk B T ' Width and Shape of Spectral Lines # exp mv 2 & z % ( dv z $ 2k B T ' Thus the probability of detecting a wave with angular frequency between ω 0 and ω 0 +dω 0 is given by the following Gaussian distribution: % ' ' % 2 ( P(ω 0 )dω 0 = ' * exp 4 ω ' ( 0 ω 0 ) 2 ( ' * & πδ ) ' Δ 2 * dω ' 0 & )

8 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 2 (Doppler) (see " truncation of Gaussian functions" in "Optical Resonator"): Δω 1 2 (Doppler) = Δ Ln2 = 2 ω 0 c 2kT m Ln2 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 = ΔΓ Natural + ΔΓ Collisional + Δω 1 2 (Doppler) Width and Shape of Spectral Lines 8

9 Einstein Treatment of Stimulated Absorption and Emission Thermal equilibrium of two-level atoms with radiation Radiation in thermal equi., ie blackbody radiation model, is given by Planck s radiation law (radiation energy density=energy per unit volume per unit interval frequency) ρ( ν,t ) = 8πhν 3 1 c 3 hν e kt 1 The atomic populations are in Maxwell-Boltzmann equilibrium, maintained by a dynamic equilibrium between emission and absorption processes Spontaneous process Stimulated processe s Einstein Coefficients Width and Shape of Spectral Lines 9 E k, N k, g k E i, N i, g i A ki = coefficient of spontaneous emission B ik = coefficient of (stimulated) absorption B ki = coefficient of stimulated emission

10 Einstein Treatment of Stimulated Absorption and Emission The upward and downward transition rates are obtained in terms of the populations and coefficients: Upward rate: dn i = N i B ik ρ dt Downward rate: dnk = N k ( A ki + B ki ρ) dt Equilibrium (detailed balance): Maxwell - Boltzmann: N k N i = g $ k exp!ω ' ki & ) g i % k B T ( From above: N k N i = ρ( ω ki ) = B ik ρ A k i + B ki ρ ( ) g k A ki exp(!ω ki k B T ) g i B ik g k B ki exp!ω ki k B T ( ) ( ) Using Planck s radiation law for ρ ω ki dn i = dn k = 0 dt dt N i B ik ρ = N k ( A ki i + B ki ρ) g i B ik = g k B ki B ik = π 2 c 3!ω ki 3 g k g i A ki Width and Shape of Spectral Lines 10

11 Einstein Treatment of Stimulated Absorption and Emission To have equal rates of spontaneous and stimulated emission requires: A = Bρ( ν 0 ) A B = ρ ( ν 0) = 8πh λ 3 λ λ 0 = 1 µm cρ( ν 0 ) = I( ν 0 ) 5x10 6 W/m 2 Hz For a (Lorentzian) spectral line of width Δν = 10 7 Hz I ν 0 ( ) = 2 πδν I 0 I 0 = 50 Wm -2 = 5 mwcm -2 Stimulated emission typically negligible for laboratory line sources Width and Shape of Spectral Lines 11

12 Elementary Interaction Processes N 2 N 2 N 2 h 12 h 12 h 12 h 12 N 1 N 1 h 12 N 1 Stimulated absorption: one photon lost Attenuation Stimulated emission: one photon gained Amplification Stimulated photon: same direction and phase Spontaneous emission: one photon emitted at random Noise Width and Shape of Spectral Lines 12

Molecular spectroscopy

Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering