Lectures 14-15 Light scattering and absorption by atmospheric particulates. Part 2: Scattering and absorption by individual spherical particles. Objectives: 1. Maxwell equations. Wave equation. Dielectrical constants of a medium. 2. Mie-Debye theory. 3. Optical properties of an ensemble of spherical particles. Required Reading: L02: 5.2, 3.3.2 Additional/Advanced Reading: Bohren, G.F., and D.R. Huffmn, Absorption and scattering of light by small particles. John Wiley&Sons, 1983. (Mie theory derivation is given on pp.82-114)
1. Maxwell equations. Wave equation. Dielectrical constants of a medium. r Maxwell equations connect the five basic quantities the electric vector, E , magnetic r r r vector, H , magnetic induction, B , electric displacement, D , and electric current r density, j : (in cgs system) r r 1 ∂D 4π r ∇ × H = + j c ∂t c r r − 1 ∂ B ∇ × E = [14.1] c ∂ t r ∇ • D = 4πρ r ∇ • B = 0 where c is a constant (wave velocity); and ρ is the electric charge density. To allow a unique determination of the electromagnetic field vectors, the Maxwell equations must be supplemented by relations which describe the behavior of substances under the influence of electromagnetic field. They are r r r r r r j = σ E D = ε E B = µ E [14.2] where σ is called the specific conductivity; ε is called the dielectrical constant (or the permittivity), and µ is called the magnetic permeability.
1 Depending on the value of σ, the substances are divided into: conductors: σ ≠ 0 (i.e., σ is NOT negligibly small), (for instance, metals)
dielectrics (or insulators): σ = 0 (i.e., σ is negligibly small), (for instance, air, aerosol and cloud particulates)
Let consider the propagation of EM waves in a medium which is (a) uniform, so that ε has the same value at all points; (b) isotropic, so that ε is independent of the direction of propagation; (c) non-conducting (dielectric), so that σ = 0 and therefore j =0; (d) free from charge, so that ρ =0. With these assumptions the Maxwell equations reduce to r r ε ∂E ∇ × H = c ∂t r r − µ ∂ H ∇ × E = [14.3] c ∂ t r ∇ • E = 0 r ∇ • H = 0 Eliminating E and H in the first two equations in [14.3] and using the vector theorem, we have r r εµ ∂ 2 E ∇ 2 E = [14.4] c 2 ∂ t 2 r r εµ ∂ 2 H ∇ 2 H = c 2 ∂ t 2 The above equation are standard equations of wave motion for a wave propagating with a velocity c v = [14.5] εµ where c is the speed of light in vacuum.
NOTE: for vacuum: µ = 1 and ε =1 in cgs units, but in SI system µο and ε0 are constants
such that c = 1/ ε o µ o .
2 • For most substances (including the air) µ is unity. Thus, the electrical properties of a medium is characterized by the dielectrical constant ε.
Refractive index (or optical constants) of a medium is defined as
m = ε [14.6] assuming that µ=1. NOTE: Strictly speaking, ε in Eq.[14.6] is the relative permittivity of medium (here it is relative to vacuum).
Refractive index:
9 The refractive index m=mr +imi is commonly expressed as a complex number.
The nonzero imaginary part mi of the refractive index is responsible for absorption of the wave as it propagates through the medium; whereas the real part
mr of the refractive index relates to the velocity of propagation of the EM wave.
9 The refractive index is a strong function of the wavelengths. Each substance has a specific spectrum of the refractive index (see figures 5.6-5.7, Lecture 5)
9 Particles of different sizes, shapes and indices of refraction will have different scattering and absorbing properties.
9 Aerosol particles often consist of several chemical species (called the internal mixture). There are several approaches (called mixing rules) to calculate the
effective refractive index me of the internally mixed particles using the refractive indices of the individual species (see Lecture 5)
3 ¾ Scattering domains: Rayleigh scattering: 2πr/λ <<1 and m is arbitrary (applies to scattering by molecules and small aerosol particles); 2πr Rayleigh-Gans scattering: m −1 << 1and m −1 << 1 (not useful for atmospheric λ application); Mie-Debye scattering: 2πr/λ and m are both arbitrary but for spheres only (applies to scattering by aerosol and cloud particles) Geometric optics: 2πr/λ >>1 and m is real (applies to scattering by large cloud droplets).
Figure 14.1 Relationship between particle size, radiation wavelength and scattering behavior for atmospheric particles. Diagonal dashed lines represent rough boundaries between scattering regimes.
4 2. Mie-Debye theory. NOTE: Mie-Debye theory is often called Mie theory or Lorentz-Mie theory. ------Mie theory outline: Assumptions: i) Particle is a sphere of radius a; ii) Particle is homogeneous (therefore it is characterized by a single refractive index
m=mr + imi at a given wavelength);
NOTE: Mie theory requires the relative refractive index = refractive index of a particle/refractive index of a medium. But for air m is about 1, so one needs to know the refractive index of the particle (i.e., refractive index of the material of which the particle is composed). NOTE: If a particle has complex chemical composition, the effective refractive index must be calculated at a given wavelength.
Strategy: r r 1) Seek a solution of a vector wave equation (Eq.[14.4]) for E and H r r ε ∂ 2 E ∇ 2 E = c 2 ∂ t 2 r r with the boundary condition that the tangential component of E and H be continuous across the spherical surface of a particle. Assumption on the spherical surface of a particle allows solving the vector equation analytically. 2) Re-write the wave equation in spherical coordinates and express electric field inside and outside sphere in a vector spherical harmonic expansions. NOTE: Mie theory calculates the electromagnetic field at all points in the particle (called internal field) and at all points of the homogeneous medium in which the particle is embedded. For all practical applications in the atmosphere, light scattering observations are carried out in the far-field zone (i.e., at the large distances from a sphere): 3) Apply boundary conditions – match transverse fields at sphere surface to obtain
scattered spherical wave Mie coefficients an and bn which don’t depend on the angles
5 but depend on size parameter x = 2πa/λ (a is the radius of the particle) and variable y= x m (m is refractive index of the particle).
4) Use series involving an and bn to obtain extinction and scattering efficiencies (Qe
and Qs).
5) Use series in Mie angular functions πn and τn to obtain scattering amplitude
functions S1(Θ) and S2(Θ), from which the scattering phase function is derived.
NOTE: Full derivation of Mie theory are given in L02, section 5.2 (and Bohren&Huffman 1983, pp.82-114).
Mie scattering amplitudes (also called scattering functions) derived from Mie theory are (see Eqs.5.2.78 in L02) ∞ 2 n + 1 S 1 (Θ ) = ∑ []a nπ n (cos Θ ) + b nτ n (cos Θ ) n =1 n (n + 1) ∞ 2 n + 1 [14.7] S 2 (Θ ) = ∑ []b nπ n (cos Θ ) + a nτ n (cos Θ ) n =1 n (n + 1)
where Mie coefficients an and bn are (see Eqs.5.2.74 in L02)
ψ n′ (y)ψ n (x) − mψ n (y)ψ n′ (x) mψ n′ (y)ψ n (x) −ψ n (y)ψ n′ (x) an (x, y) = bn (x, y) = [14.8] ψ n′ ( y)ξ n (x) − mψ n (y)ξ n′ (x) mψ n′ ( y)ξ n (x) −ψ n (y)ξ n′(x) here the prime denotes differentiation; x = 2πa/λ and y= x m;
πρ πρ (2) ψ (ρ) = J (ρ) and ξ (ρ) = H (ρ) where J n+1/ 2 (ρ) is the half-integral-order n 2 n+1/ 2 n 2 n+1/ 2
(2) spherical Bessel function and H n+1/ 2 is the half-integral-order Hankel function of the second kind;
and πn and τn are the Mie angular functions 1 π (cos Θ) = P1(cos Θ) n sin(Θ) n d τ (cos Θ) = P1 (cos Θ) [14.9] n dΘ n
1 where Pn are the associated Legendre polynomials (see Appendix E). ------
6 In the far-field zone (i.e., at the large distances r from a sphere), Mie theory gives the solution of the vector wave equation can be obtained as
s i ⎡El ⎤ exp( −ikr + ikz ) ⎡S 2 S 3 ⎤⎡El ⎤ ⎢ ⎥ = ⎢ ⎥ [14.10] s ikr ⎢S S ⎥ i ⎣⎢E r ⎦⎥ ⎣ 4 1 ⎦⎣⎢E r ⎦⎥ Eq.[14.10] is a fundamental equation of scattered radiation including polarization in the far field.
⎡S 2 (Θ) S 3 (Θ)⎤ ⎢ ⎥ is the amplitude scattering matrix (unitless) ⎣S 4 (Θ) S1 (Θ)⎦
For spheres: S3(Θ) = S4(Θ) = 0
Thus, for spheres Eq.[14.10] reduces to
s i ⎡El ⎤ exp( −ikr + ikz ) ⎡S 2 0 ⎤⎡El ⎤ ⎢ ⎥ = ⎢ ⎥ [14.11] s ikr ⎢ 0 S ⎥ i ⎣⎢E r ⎦⎥ ⎣ 1 ⎦⎣⎢E r ⎦⎥
where exp( ikz ) is the incident plane wave, and exp( −ikr ) is the outgoing scattered ikr wave.
Fundamental extinction formula (or optical theorem) gives the extinction cross section of a particle 4π σ = Re[ S (0 0 )] [14.12] e k 2 1,2 But for the forward direction (i.e. Θ =00) from Eq.[14.7], we have
∞ 0 0 1 S1 (0 ) = S 2 (0 ) = ∑ (2n + 1)(an + bn ) [14.13] 2 n=1 Thus, extinction cross section is related to scattering in forward direction.
7 ¾ Efficiencies (or efficiency factors) for extinction, scattering and absorption are defined as σ σ σ Q = e Q = s Q = a [14.14] e πa 2 s πa 2 a πa 2 where πa2 is the particle area projected onto the plane perpendicular to the incident beam. Mie efficiency factors are derived from the Mie scattering amplitude 2 ∞ Q ( 2 n 1) Re[ a b ] [14.15] e = 2 ∑ + n + n x n = 1
∞ 2 2 2 Q = ( 2 n + 1)[ a + b ] [14.16] s 2 ∑ n n x n = 1 and the absorption efficiency can be calculated as
Q a = Q e − Q s [14.17]
Figure 14.2 Examples of Qe and Qa calculated with Mie theory for several refractive indexes.
8 ¾ Scattering phase matrix Recall definition of Stokes parameters (see Lecture 13), which uniquely characterize the
electromagnetic waves. Let I0, Q0, U0 and V0 be the Stokes parameters of incident field and I, Q, U and V be the Stokes parameters of scattered radiation
⎡I ⎤ ⎡I o ⎤ ⎢ ⎥ ⎢ ⎥ Q σ Q ⎢ ⎥ s ⎢ o ⎥ [14.18] = 2 P ⎢U ⎥ 4πr ⎢U 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣V ⎦ ⎣Vo ⎦ where P is the scattering phase matrix.
⎡P11 P12 0 0 ⎤ ⎢P P 0 0 ⎥ P = ⎢ 12 22 ⎥ [14.19] ⎢ 0 0 P33 − P34⎥ ⎢ ⎥ ⎣ 0 0 P34 P44 ⎦ where each element depends on the scattering angle (1/r2 is from solid angle)
For spheres: P22 = P11 and P 44 = P33
NOTE: In general, for a particle of any shape, the scattering phase matrix consists of 16 independent elements, but for a sphere this number reduces to four.
Thus for spheres, Eq.[14.18] reduces to
⎡I ⎤ ⎡ P11 P12 ⎤ ⎡I o ⎤ ⎢ ⎥ ⎢Q ⎥ ⎢P P ⎥ Q ⎢ ⎥ σ s ⎢ 12 11 ⎥ ⎢ o ⎥ [14.20] = 2 ⎢U ⎥ 4πr ⎢ P33 − P34 ⎥ ⎢U 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣V ⎦ ⎣ P34 P33 ⎦ ⎣V o ⎦ where each element of the scattering phase matrix is expressed via the scattering
amplitudes S1(Θ) and S2(Θ)
4π * * P11 = 2 [S 1 S 1 + S 2 S 2 ] 2 k σ s
4 π * * [14.21] P12 = 2 [S 2 S 2 − S 1 S 1 ] 2 k σ s
9 4 π * * P33 = 2 [S 2 S 1 + S 1 S 2 ] 2 k σ s 4 π * * − P34 = 2 [S 1 S 2 − S 2 S 1 ] 2 k σ s
P11(Θ) =P(Θ) is the scattering phase function defined in Lecture 13.
Figure 14.3 Examples of scattering phase functions calculated with Mie theory for several size parameter for nonabsorbing spheres.
10 Some highlights of Mie scattering results: • Extinction efficiency vs. size parameter x (assuming NO ABSORPTION):
4 1) small in Rayleigh limit: Q e ∝ x
2) largest Qe when particles and wavelength have similar size
3) Qe -> 2 in the geometric limit ( x → ∞ ) 4) Oscillations (see Fig.14.3) from interference of transmitted and diffracted waves • Period in x of interference oscillations depends on the refractive index. Absorption reduces interference oscillations and kills ripple structure. • Scattering and absorption efficiencies vs. size parameter with ABSORPTION:
As x → ∞ : Q s → 1 and , entering rays are absorbed inside particle. Smaller imaginary part of the refractive index requires larger particle to fully absorb internal rays. • Scattering phase function: forward peak height increases dramatically with x. For single particles – number of oscillations in P(Θ) increases with x.
NOTE: Several Mie codes are freely available to the scientific community (see http://atol.ucsd.edu/~pflatau/scatlib/scatterlib.htm) How a Mie code works:
1) Compute coefficients an and bn (see Eq. [14. 8]) for n=1,…N from size parameter x and index of refraction m (uses recursion relations for the spherical Bessel functions). Estimate of the total number: N = x + 4x1/3+2
2) Compute Qe and Qs from an and bn .
3) Compute scattering amplitude functions S1(Θ) and S2(Θ) at desired scattering angles from an and bn and Mie angular functions πn (Θ) and τn (Θ) (πn and τn from recursion).
Compute phase matrix elements P11(Θ), P12(Θ), P33(Θ), and P34(Θ) from S1(Θ) and
S2(Θ). 4) If optical properties of an ensemble of particles are required, one needs to perform an integration over the size distribution function.
11 3. Optical properties of an ensemble of spherical particles. 9 Mie theory gives the extinction, scattering and absorption cross-sections (and efficiencies) and the scattering phase matrix of a single spherical particle. NOTE: Recall Lecture 5 where the particle size distributions were introduced for atmospheric aerosols and clouds. If the particles characterized by a size distribution N(r), the volume extinction, scattering and absorption coefficients (in units LENGTH-1) are calculated as
rmax β = σ (r ) N (r )dr e ∫ e rmin
rmax β = σ ( r ) N (r )dr [14.22] s ∫ s rmin
rmax β = σ ( r ) N ( r ) dr a ∫ a rmin where σ is the corresponding cross section of a particle of radius r and N(r) is the particle size distribution (e.g., in units m-3µm-1).
Single scattering albedo (unitless) if defined as
β s ω 0 = [14.23] β e 9 The single scattering albedo gives the percentage of light which will be scattered in a single scattered event.
Scattering phase function is
4π rmax P (Θ ) = []S S * + S S * N (r )dr [14.24] 2k 2 β ∫ 1 1 2 2 s rmin
rmax P (Θ)σ N(r)dr ∫ r s or P(Θ) = rmin [14.25] βs
12 Asymmetry parameter is defined as the first moment of the scattering phase function
1 1 g = ∫ P(cosΘ)cos(Θ)d(cosΘ) [14.26] 2 −1 g = 0 for equal forward and backward scattering; g = 1 for totally forward scattering
The Henyey-Greenstein scattering phase function is a model phase function, which is often used in radiative transfer calculations:
1 − g 2 P (Θ) = [14.27] HG (1 + g 2 − 2g cos Θ) 3 / 2 where g is the asymmetry parameter.
Optical properties of cloud particles:
• For many practical applications, the optical properties of water clouds are parameterized as a function of the effective radius and liquid water content (LWC). The effective radius is defined as
r 3 N ( r ) dr r = ∫ [14.28] e ∫ r 2 N ( r ) dr where N(r) is the particle size distribution (e.g., in units m-3µm-1). The liquid water content (LWC) was defined in Lecture 5 (see eq.[5.10]): 4 LWC = ρ V = ρ πr3N(r)dr [14.29] w 3 w ∫ Using that the extinction coefficient of cloud droplets is
β = σ ( r ) N ( r ) dr = Q π r 2 N ( r ) dr e ∫ e ∫ e
and that Qe ≈ 2 for water droplets at solar wavelengths, we have
3 LWC [14.30] β e ≈ 2 re ρ w
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Figure 14.4 Single scattering albedo (or co-albedo = 1- ω0) as a function of wavelength for water and ice spheres of varying sizes.
14 ¾ Effective optical properties of an atmospheric layer consisting of gas, and aerosol and/or cloud particles: Effective (also called total) optical depth:
M M A A τ λ = τ a,λ +τ s,λ +τ a,λ +τ s,λ [14.31]
M M where τ a,λ and τ s,λ are optical depth due to absorption by gases and molecular (Rayleigh) scattering, respectively;
A A τ a,λ and τ s,λ are optical depth due to absorption and scattering by aerosol (and/or cloud) particles, respectively.
Effective single scattering albedo:
M A τ s,λ + τ s,λ ω 0,λ = [14.32] τ λ Effective scattering phase function:
M M A A τ s,λ Pλ (Θ) + τ s,λ Pλ (Θ) Pλ (Θ) = M A [14.33] τ s,λ + τ s,λ Effective asymmetry parameter:
A A τ s,λ g λ g λ = M A [14.34] τ s,λ + τ s,λ
15