Lecture 10. Lidar Effective Cross-Section vs. Convolution

Transcription

1 Lecture 10. Lidar Effective Cross-Section vs. Convolution q Introduction q Convolution in Lineshape Determination -- Voigt Lineshape (Lorentzian Gaussian) q Effective Cross Section for Single Isotope -- Na Resonance Fluorescence Lidar q Effective Cross Section for multiple Isotopes -- K Resonance Fluorescence Lidar q Convoluted Rayleigh Doppler Lineshape -- Influence of Laser Lineshape 1

2 3 Introduction 1 0 Absorption & Fluorescence Elastic Scattering (Rayleigh) Inelastic Scattering (Raman) Here, 0, 1, and are real energy states, while 3 is a virtual state. q Resonance fluorescence contains two single-photon processes: Absorption of a photon by an atom, and spontaneous emission of another photon by the atom. Fluorescence can be quenched by collisions. q Rayleigh scattering or Raman scattering is a two-photon process, in which two photons are involved simultaneously. Therefore, Rayleigh/ Raman scattering has much smaller scattering cross-section than absorption, about & orders of magnitude smaller, respectively.

3 Atomic Absorption Lineshape Determined by Convolution + I(ω) = I 0 n( ω!,ω 0 )g Lorentzian ( ω!,ω)d ω! 3

4 Natural Linewidth Determined by Finite Radiative Lifetime Usually only the ground state can have infinite radiative lifetime, therefore infinite thin energy level. Excited states all have finite radiative lifetime, therefore finite energy level widths. 4

5 Natural Linewidth Determined by Finite Radiative Lifetime Infinite lifetime à harmonic oscillator à narrow linewidth Finite radiative lifetime à damped oscillator à natural linewidth 5

6 Atomic Absorption Cross-Section q σ abs is proportional to the probability of a single-frequency photon being absorbed by an atom: λ g σ abs (ν,ν 0 ) = A k ki 8πn g g A (ν,ν 0 ) (10.1) i where A ki is the spontaneous transition probability per unit time, i.e., the Einstein A coefficient; g k and g i are the degeneracy factors for the upper and lower energy levels k and i, respectively; λ is the wavelength; n is the refraction index, and g A is the absorption lineshape. ν and ν o are the laser frequency and the central frequency of the atomic absorption line, respectively. q For single atom, the absorption lineshape g A is determined by natural linewidth and collisional broadening, which has Lorentzian shape Δν g A (ν,ν 0 ) = g H (ν,ν 0 ) = H & π ( ν ν 0 ) + Δν H / '( where Δν H is the homogeneous broadened linewidth. ( ) ) * + (10.) Refer to our textbook Chapter 5 and references therein 6

7 Absorption Cross Section Under Doppler Effects q For many atoms in thermal equilibrium, the lineshape is the integration of the Doppler broadening (Gaussian shape) with the Lorentzian natural lineshape. The absorption lineshape g A is given by Voigt integration + g A (ω) = g Gaussian (ω 0, ω!)g Lorentzian (ω, ω!)d ω! 7

8 Absorption Cross Section Under Doppler Effects q When ignoring the Lorentzian natural linewidth (~10 MHz), the Doppler broadening (~1 GHz) dominates the lineshape. The statically averaged absorption cross section for each atomic transition line is then given by a Gaussian % [ σ abs (ν,ν o ) = σ o exp ν(1 V R /c) ν o] ( ' * (10.3) ' & σ D * ) where σ o = 1 e πσ D 4ε o m e c f ik (10.4) and σ D = σ o peak absorption cross section (m or cm ) e charge of electron; m e mass of electron ε o electric constant; c speed of light f ik absorption oscillator strength; V R radial velocity k B T Mλ 0 (10.5) σ D Doppler broadened line width k B Boltzmann constant; T temperature M mass of atom; λ o wavelength 8

9 Resonance Fluorescence Lidar: Effective Cross Section q Refer to the textbook Section for effective cross section σ eff q Resonance fluorescence contains two single-photon processes: Absorption of a photon by an atom, and spontaneous emission of another photon by the atom. q Because resonance fluorescence is isotropic, the differential backscatter cross-section dσ/dω can be replaced by the total effective scattering cross-section σ eff divided by 4π. q The total effective scattering cross-section σ eff is defined as the ratio of the average photon number scattered by an individual atoms (in all directions) to the total incident photon number per unit area. q The σ eff is determined by the convolution of the atomic absorption cross-section σ abs and the laser spectral lineshape g L (ν). q The absorption cross-section σ abs is defined as the ratio of the average absorbed single-frequency photons per atom to the total incident photons per unit area. 9

10 Na Atomic Energy Levels q Na (sodium) is probably one of the most well studied alkali species. 3 Na is the only stable isotope for Na, and it has hyperfine structures. Refer to our textbook Chapter 5 and references therein 10

11 Na Atomic Parameters 11

12 Doppler-Limited Na Spectroscopy q Doppler-broadened Na absorption cross-section is approximated as a Gaussian with rms width σ D 1 e f 6 ' σ abs (ν) = A n exp [ν n ν(1 V R ) /c)] πσ D 4ε 0 m e c n=1 ( σ D *, + (10.6) q Assume the laser lineshape is a Gaussian with rms width σ L q The effective cross-section is the convolution of the atomic absorption cross-section and the laser lineshape σ eff (ν) = where 1 πσ e e f 4ε 0 m e c 6 n=1 ' A n exp [ν n ν(1 V R /c)] * ), ( σ e + σ e = σ D + σ σ D = L (10.8) and k B T Mλ 0 (10.5) (10.7) 1

13 Effective Cross-Section q The effective cross section is a convolution of the atomic absorption cross section and the laser line shape. When the laser has finite spectral linewidth with lineshape (its intensity distribution along frequency) of g L (ν,ν L )dν =1 (10.9) 0 q The effective cross-section σ eff is given by the convolution of σ abs with the laser lineshape g L : σ eff (ν L,ν o ) = + q If the laser spectral lineshape is a Gaussian shape: 1 & g L (ν,ν L ) = exp (ν ν L ) ) ( + πσ L ' σ L * σ abs (ν,ν o )g L (ν,ν L )dν (10.11) (10.10) q then σ eff can be written as σ eff (ν L,ν 0 ) = % σ D σ 0 exp ' ν L (1 V R /c) ν 0 σ D +σ ' L & (σ D +σ L ) [ ] ( * * ) (10.1) 13

14 Effective Cross-Section for K Atoms total effective cross section x m K lidar effective cross laser FWHM=70MHz frequency offset from the line center (MHz) 15 K 00 K 75 K Atomic K energy levels Reference our textbook Chapter 5 q K (potassium) is another alkali species, and it has several stable isotopes ( 39 K, 40 K, and 41 K) with hyperfine structures. 14

15 Effective Cross-Section for K Atoms q Absorption cross section of K atom s D1 line is given by σ abs (ν) = 41 A=39. 0 / IsotopeAbdn(A) 10 1 πσ D e f 4ε 0 m e c 4 n=1 ' A n exp [ν n ν(1 V R /c)] * 0 ), 3 ( σ D + 40 Isotope abundance: % ( 39 K), % ( 40 K), 6.730% ( 41 K) Line strength: An = 5/16, 1/16, 5/16, 5/16 Oscillator strength: f, Doppler broadening: σ D q The effective total scattering cross section of K atom s D1 line is the convolution of the absorption cross section and the laser lineshape. Under the assumption of Gaussian lineshape of the laser, it is given by σ eff (ν) = where 41 A=39. 0 / IsotopeAbdn(A) 10 1 πσ e e f 4ε 0 m e c 4 n=1 ' A n exp [ν n ν(1 V R /c)] * 0 ), 3 ( σ e + 40 k B T σ e = σ D + σ σ L (11.11) and D = (11.1) Mλ 0 Refer to our textbook Chapter 5 and references therein (11.9) (11.10) 15

16 Convoluted Rayleigh Doppler Lineshape q If scattering is produced with a single-frequency laser beam and without pressure broadening, Rayleigh scattering from atmosphere molecules experiences pure Doppler effects (Doppler shift and Doppler broadening) and exhibits a Gaussian shape. q When the laser has finite spectral linewidth with lineshape, the return Rayleigh Doppler lineshape is a convolution of the pure Doppler Gaussian with the laser line shape. q If the laser spectral lineshape is a Gaussian, g L (ν,ν L ) = σ eff (ν L,ν o ) = + 1 & exp (ν ν L ) ) ( + πσ L ' σ L * σ abs (ν,ν o )g L (ν,ν L )dν where the laser spectral lineshape is normalized: g L (ν,ν L )dν =1 0 then the convolution leads to another Gaussian with RMS linewidth σ convoluted = σ D +σ L 16