the Rayleigh cross section at θ=π (Eq. 3.7) is constant and referred to as the Rayleigh scattering ratio Lray:
Lray= σ
The Rayleigh backscatter coefficient βray, which is used for the evaluation of the intensity profiles measured with LIDAR, relates the physical parameters of the scattering molecules to the signal intensity. It is the product of the differential Rayleigh cross section with the altitude dependent number density of the scatterers N(z):
βray(λ,z) = dσ
raysca(λ,θ= π)
dΩ N(z). (3.10)
3.1.2. Raman Scattering
Raman scattering theory describes the vibrational Raman lines connected with a change in the principal quantum number and hence a wavelength shift. The differential vibrational Raman cross sectiondσramsca/dΩ can be calculated for the sum of the first polarization states of the Stokes lines of a harmonically oscillating molecule. Including the bands with
∆n= +1 (Stokes) and ∆J= 0,±2 [Inaba, 1976] one yields:
The primed parameters are equivalent to αpol and γ considered for Rayleigh scattering, λ0 is the wavelength of the incident radiation and λram is the wavelength of the Stokes vibrationally scattered photons, respectively. The temperature dependence can be neglected. kB denotes the Boltzmann constant. The differential Raman cross section for both polarization directions for photons at 532 nm scattered at N2 molecules amounts to 4.5·10−27m2sr−1. Hence, it is three orders of magnitude smaller than the Rayleigh cross section, which amounts to6.22·10−24m2sr−1. The Raman backscatter coefficient βram is calculated as
βram(λ0,λram,z) = dσ
ramsca (λ0,λram,θ= π)
dΩ N(z). (3.12)
The considered Raman wavelengths λram can be found in Tab. 5.1.
3.2. Particulate/Mie Scattering
Scattering events with x≈1 are often called Lorenz-Mie scattering. Based on Maxwell’s equations [Liou, 2002], the solution for the interaction of a plane wave with an isotropic homogeneous sphere is applicable to spherical aerosols and cloud droplets. The scattering efficiency Qmie depends on the radius r of the scattering particle, the Mie scattering
0 20 40 60 80 100 120 140 160 180
Rayleigh Figure 3.2: T-matrix
computations of
cross section σmie, the wavelength of the incident radiation λ as well as on the index of refractionm and is given by the following expansion:
Qmie = σmie
πr2(r,λ,m) =c1x4(1+c2x2+c3x4+. . .). (3.13) In the case of nonabsorbing particles, the coefficients are given by:
c1 = 8 The leading term equals the contribution associated with Rayleigh scattering, replacing 1/N=V= 4π r3/3. As for molecules x is ∼10−3 in the visible, the higher order terms can be neglected. For aerosols and clouds, the scattered intensity primarily depends on the particle size rather than on the wavelength. Therefore, clouds and nonabsorbing aerosols in the atmosphere appear to be white or at least brighten up the blue sky from pure Rayleigh scattering.
The scattered intensity I can also be described by a scattering phase function f(θ), which can be computed from the Lorenz-Mie theory for spheres:
f(θ) = I(θ) I0
4πz2
σmie , (3.14)
where z is the distance between the particle and the observer. The phase function for spheroidal, cylindrical and other particles can be calculated by numerical implementations of approximate formulations. For instance, the T-matrix approach [Mishchenko et al., 1996] is a generalization of the Mie theory to calculate the extinction by non-spherical particles using a spherical wave function expansion. In Figure 3.2, the scattering phase functions for some size parameters and refractive indices are shown. While molecular scattering is almost independent from the observation angle (see Eq. A.3), cloud droplets and aerosols show a strong peak in the forward direction as well as characteristic peaks at certain angles including the backscattering direction.
3.3. ABSORPTION Again, a Mie backscatter coefficient βaer can be defined. The index "aer" is used as an equivalent to "mie" to point out that Mie scattering is usually caused by aerosol particles.
Since particles with different radii show different scattering efficiencies, the cross sections have to be weighted with the particle size distribution dn(r)/dr:
βaer(λ,z) =
Z ∞
0 drσmiesca(r,λ,m)dn(r) dr
=
Z ∞
0 drπr2Qmie(r,λ,m)dn(r)
dr . (3.15)
3.3. Absorption
Atmospheric molecules and particles also act as light-absorbing species, depending on the wavelength of the incident light. Besides the main absorbing gases (H2O, CO2, ozone (O3), and O2) which absorb photons in ultraviolet (UV), visual, and infrared (IR) regions of the solar spectrum, trace contaminants such as carbon monoxide (CO) and the oxides of nitrogen show discrete absorption frequencies. Absorbing particles are characterized by a complex index of refractionm. The higher the imaginary part of the refractive index, the higher the absorption. Since the extinction is comprised of absorption and scattering, trace gas and aerosol absorption must be considered if the detected wavelength coincides with an atmospheric absorption band. As extinction is produced by particles in all size ranges, the extinction coefficient α(z,λ) splits up into the scattering and absorption contributions:
α(z,λ) = αray(z,λ) +αaer(z,λ) (3.16)
= αray,sca(z,λ) +αray,abs(z,λ) +αaer,sca(z,λ) +αaer,abs(z,λ). (3.17)
3.4. Polarization
The polarization of light describes the orientation of the wave’s electric field vector. For light traveling in free space, the polarization is perpendicular to the wave’s direction of propagation. One distinguishes between random polarization, linear polarization when the electric field is oriented in a single direction, and circular or elliptical polarization when it rotates as the wave travels. The state of polarization can be converted to any other state, e.g. through a scattering process. The ratio of scattered radiation with original linear polarization to radiation which is polarized perpendicularly (⊥) to the incident radiation is called depolarization. The fraction of depolarized radiation due to Rayleigh scatteringδray is given by the ratio of the depolarization factors for parallel (k) polarized incident light Toutin (see Tab. A.1 in Appendix A) to the different polarization directions of the scattered light. It includes the Cabannes line and the rotational Raman lines:
δray= T
k
⊥ Tkk
= 3e
4e+45 ≈ 0.0144 . (3.18)
If scattering by the Cabannes line is considered exclusively, the value of δray is reduced to 0.00365 [Young, 1980].
Considering particulate scattering, the degree of depolarization is determined by the particles’ size and shape. For spherical, homogeneous (with respect to the refractive index) particles no depolarization is expected as the Mie scattering in backwards direction does not change the polarization of the incident radiation for reasons of symmetry [Liou, 2002]. Small deviations from the backwards direction, however, can induce significant depolarization values [Beyerle et al., 1995]. Also, non-spherical or inhomogeneous particles can change the polarization of the incident radiation significantly. For non-spherical particles of a size comparable to the incident laser wavelength other scattering theories, e.g., the T-Matrix approach [Mishchenko and Travis, 1998] have to be applied. For large particles (50 <x< 100) scattering is described by ray-tracing theory. It can be used to calculate the depolarization induced by spheres and simple ice crystals.