Nonlinear Optics Lab . Hanyang Univ.
3. Widths and Profiles of Spectral Lines
Spectral lines in absorption or emission spectra are never strictly monochromatic, but has a finite width. Origins of the finite line-width ?
Line profile : the function of I(n) in the vicinity of n0
FWHM(full width at half maximum) :
,
(3.2)
Nonlinear Optics Lab . Hanyang Univ.
Excited electron damped harmonic oscillator with (m, k, w)
3.1 Natural linewidth
(3.4) where,
Real solution :
where,
(3.3)
If (small damping), then and can neglect the second term;
(3.5)
Nonlinear Optics Lab . Hanyang Univ.
3.1.1 Lorentzian line profile of the emitted radiation
Fourier transformation : (3.6)
In the vicinity of the central frequency w0 where
(3.7)
(3.8)
(3.9)
: Lorentzian profile
Nonlinear Optics Lab . Hanyang Univ.
Normalizations
a)
(3.10) • FWHM : ,
• Intensity profile :
• ,
b) , FWHM :
(3.10b) • Intensity profile :
•
where,
(3.11)
(3.10d)
Nonlinear Optics Lab . Hanyang Univ.
3.1.2 Relation between linewidth and lifetime
(3.3) (3.12)
Inserting from (3.5) with neglecting terms with (3.13)
(3.14)
(3.15)
Decay time : (Section 2.6)
; Classical damping factor, g Einstein A coefficient
Nonlinear Optics Lab . Hanyang Univ.
3.1.3 Natural linewidth of absorbing transitions
Intensity decrease dI of a plane wave passing in z-direction through an absorbing sample ; (3.21)
where,
: absorption coefficient
(3.22)
( for )
When the absorption coefficient does not depend on the intensity (for sufficiently small intensity), (Beer’s law) (3.23)
Nonlinear Optics Lab . Hanyang Univ.
Classical forced harmonic oscillator model for the absorption coefficient (3.24)
solution ; (3.25) where,
Induced dipole moment : (3.26)
Macroscopic polarization : (3.27, 28)
Except for ferromagnetic materials the relative permeability m~1, and n=e1/2 (3.29)
(3.30)
Nonlinear Optics Lab . Hanyang Univ.
Complex refractive index
(3.31, 32) If n~1, n2-1 = (n+1)(n-1) ~ 2(n-1)
Plane wave passing through the medium,
(3.33)
Intensity passing through the medium,
(3.34)
I I
0e
z (3.23)
(3.35)Nonlinear Optics Lab . Hanyang Univ.
(3.36a)
(3.37a)
(Kramers-Kronig dipersion relations)
In near resonance,
(3.36a)
(3.37a)
ex1) Na D1 line [3S1/2 – 3P3/2(t=16 ns)] : dnn=1/2pt=10 MHz ex2) Typical molecular transition (t=1 ms)] : dnn=160 Hz ex3) H Forbidden line [1S – 2S (t=8.23 s)] : dnn=0.02 Hz
Nonlinear Optics Lab . Hanyang Univ.
Generally the spectrum line with natural linewidth cannot be observed since it is completely
concealed by other broadening effects. One of the major contribution is the Doppler width, which is due to the Thermal motion of the absorbing/emitting molecules in a low pressure gas sample.
3.2 Doppler width
Emitting molecule with a velocity, Detecting frequency :
•
(3.38)
Absorbing molecule with a velocity,
•
Wave frequency in the frame of the moving molecule :
Absorption frequency : (3.39a)
Nonlinear Optics Lab . Hanyang Univ.
Maxwellian velocity distribution : (3.40)
where, : total density of molecules in level Ei : most probable velocity
, and
(3.41) Intensity profile :
(Gaussian, Doppler profile) (3.42)
Halfwidth : (3.43a)
(3.44)
ex1) Lyman line [H-atom, 2p-1s] : dnD=5.6 GHz
•
ex2) Na D line [3p-3s] : dnD=1.7 GHz ex3) CO2 vib. line : dnD=56 MHz
Bigger than the natural linewidth by about two orders of magnitude
Nonlinear Optics Lab . Hanyang Univ.
Lorentzian and Gaussian Profiles
Line intensity approaches zero for large arguments (n-n0) much faster for a Gaussian line profile than for a Lorentzian line profile.
It is therefore possible to obtain information about the Lorentzian profile from the extreme line wings even if the Doppler width is much larger than the natural line width. Noise spectroscopy
Nonlinear Optics Lab . Hanyang Univ.
Voigt profile
Not all molecules with a velocity emit/absorb at the same frequency
Doppler spectrum cannot be strictly represented by a pure Gaussian
Frequency response of these molecules with the same velocity is represented by a Lorentzian profile
(3.45)
(3.46)
where,
