1. Nonlinear Optics Lab. Hanyang Univ. Chapter 3. Classical Theory of Absorption 3.1 Introduction Visible color of an object : Selective absorption, Scattering, Transmission 1) Absorption in gases : line spectrum - Electronic resonance : UV region ex) white daylight : N 2 or O 2 does not absorb at visible frequency - Molecular vibration : IR region - Molecular rotation : Microwave region 2) Absorption in liquids or sold : broad band spectrum ex) Green water : absorption in the red portion ex) Red dye : strong absorption in the blue or UV ex) Metal (free electron, plasma freq. : UV region) - Shine <= Visible frequencies are completely reflected - Characteristic color <= absorption of bound electrons ex) Most good insulators : usually transparent in the visible, but opaque in the UV ex) Semiconductors : usually absorbs visible, transparent in the IR => defects or impurities : modify the absortion spectra

2. Nonlinear Optics Lab. Hanyang Univ. 3.2 Absorption and the Lorentz model Absorption (?) => Frictional force that damps out dipole oscillations => Equation of motion ; 3.3 Complex Polarization and Index of Refraction sol) where, Origin of friction : collision (Section 3.9, 3.10)

3. Nonlinear Optics Lab. Hanyang Univ. - Dipole moment, In case of many electrons ; Complex refractive index - Polarizability, Polarizability (2.3.13)

4. Nonlinear Optics Lab. Hanyang Univ. i) Intensity where, Electric field in the medium : absorption coefficient ii) Refractive Index

5. Nonlinear Optics Lab. Hanyang Univ. 3.4 Polarizability and Index of Refraction near a Resonance Most gain media of lasers are composed of a background material and a small amount of resonant material. ex) gas : background gas + active gas medium ex) solid (ruby) : Al2O3 + Cr+3 ex) liquid (dye) : ethanol + dye Thus we can write, (nearly const.)

6. Nonlinear Optics Lab. Hanyang Univ. Near the resonance ( ),

7. Nonlinear Optics Lab. Hanyang Univ. 3.5 Lorentzian Atoms and Radiation in Cavities Electric field in a cavity is not the form of a traveling wave but a standing wave, where, Similarly as previous sections, where, Maxwell wave equation for a cavity including cavity loss, where, : conductivity

8. Nonlinear Optics Lab. Hanyang Univ. Near resonance approxiamtion ; : cavity pulling : threshold gain (*)

9. Nonlinear Optics Lab. Hanyang Univ. (*) Relations 1), 2) are correct except the sign of g If, then field amplification is possible, But, from (3.5.9), : negative This means that cannot be satisfied. That is, a classical laser theory based on the linear electron oscillator model is not possible. And, it requires a quantum mechanical treatment of light-matter interaction to understand how can be made positive.

10. Nonlinear Optics Lab. Hanyang Univ. 3.6 The Absorption Coefficient [Lorentzian Lineshape] (3.3.25) => Near the resonance, where,

11. Nonlinear Optics Lab. Hanyang Univ. Lineshape Function ; : Lorentzian lineshape : FWHM In general, where, is the integrated absorption coefficient Normalization :

12. Nonlinear Optics Lab. Hanyang Univ. 3.7 Oscillator Strength Absorption Cross section : From (3.6.17) and (3.6.23),: universal value Oscillator strength, As can be seen in Table 3.1 for the integrated cross section of hydrogen atom, actual values of do not agree with the calculated values by classical theory. This problem of classical theory could be patched by introducing oscillator strength assigned empirically. * Quantum theory removes this defect of Lorentz’s model.

13. Nonlinear Optics Lab. Hanyang Univ. 3.11 Doppler Broadening - 1842, C. J. Doppler, Predicted - 1854, C.H.D. Buys Ballot, Demonstrated (trumpet in a moving train) To an atom moving with velocity away from a source of radiation of frequency, the frequency of the radiation appears to be shifted : Atomic velocity distribution : : Maxwell-Boltzman distribution Thus, the atom with resonance frequency will absorb radiation near to the frequency :

14. Nonlinear Optics Lab. Hanyang Univ. Since the absorption rate must be proportional to, we may write the Doppler line shape function as : Doppler line shape function : FWHM

15. Nonlinear Optics Lab. Hanyang Univ. 3.12 The Voigt Profile Collisional broadening : Homogeneous => Lorentzian Doppler Broadening : Inhomogeneous => Gaussian In general, we cannot characterize an absorption lineshape of a gas as a pure Lorentzian or a pure Gaussian. Voigt profile : Described the absorption lineshape when both collision broadening and Doppler broadening must be taken into account. where,

16. Nonlinear Optics Lab. Hanyang Univ. : Voigt profile where, : complementary error function

17. Nonlinear Optics Lab. Hanyang Univ. Limit case of Voigt profile : nearly pure Lorentzian : nearly pure Gaussian ii) General case : numerical calculation i) where, : error function of complex argument

18. Nonlinear Optics Lab. Hanyang Univ. Example) Na vapor : If P Doppler regime Absorption coefficient,