Mie Theory, Cross-Sections, Efficiency Factors and Phase Function

Transcription

1 Appendix C Mie Theory, Cross-Sections, Efficiency Factors and Phase Function C.1 Mie Theory The concept behind the Mie Theory is explained in section.6.1. However, the mathematical derivation of the theory is complex and hence, only the results of the theory is presented here. The essence of Mie s theory are the two intensity distribution functions i 1 and i. i 1 = i = n + 1 P n (a (1) (cos θ) n n(n + 1) sinθ n + 1 n(n + 1) (a n d[p (1) n (cos θ)] dθ d[p (1) ) n (cos θ)] + b n dθ P (1) ) n (cos θ) + b n sinθ (C.1) (C.) 161

2 a n = ξ n(k 1 a)ξ n(k a) (η /η 1 )ξ n (k a)ξ n(k 1 a) ζ n (k 1 a)ξ n(k a) (η /η 1 )ξ n (k a)ζ n(k 1 a) b n = (η /η 1 )ξ n (k 1 a)ξ n(k a) ξ n (k a)ξ n(k 1 a) (η /η 1 )ζ n (k 1 a)ξ n(k a) ξ n (k a)ζ n(k 1 a) (C.3) (C.4) where the Hankel functions are[1] ζ n (k 1 r) = j n+1 exp(jk 1 r) ζ n(k 1 r) = j n exp(jk 1 r) ξ n (kr) = (πkr/) 1/ J n+1/ (kr) (C.5) (C.6) (C.7) and P (1) n (cos θ) are the associated Legendre functions of the first kind. For unpolarized light, ie. many light waves with different polarization, the intensity of the scattered light at the direction of interest is given by I(θ) = I (θ) + I (θ) = E λ 4π ( ) i1 + i (C.8) where E is the electric field of the primary wave and λ the wavelength. The functions presented above for i 1, i, a n and b n are not efficient if coded for computers without modifications. Techniques and codes for more efficient computation are presented by Dave[5] and Bohren and Huffman[1]. C. Absorption Besides scattering, some particles absorbs light. For example infrared light is absorbed by water. To take account for this, the refractive index of the scatterers are represented by a complex number in the Mie theory. The real part of the refractive index represents scattering loss and the imaginary part represents absorption loss. However, in this research, absorption is neglected. 16

3 C.3 Cross Sections, Efficiency Factors and Phase Function Cross sections, efficiency factors and phase function are coefficients that determine the scattering characteristics of a particle or a particulate medium. These functions are obtained from information of the scattering conditions (ie. particle size, particle refractive index, particle density and light wavelength) using Mie s theory. The standard coefficients are total cross section, volume total cross section, angular cross section, volume angular coefficient, efficiency factors and phase function. Before proceeding, there is a need to define the geometry of scattering. Consider a light beam incident on a small particle (not shown) at the centre of the unit sphere as shown in figure C.1. The incident light beam is coincident to the z-axis. In polar coordinates, the direction of a vector is represented by θ and φ. Y Incident beam Φ θ Z X Figure C.1: Geometry of angular scattering. It is sometimes convenient to represent the direction of the radiometric quantities in polar 163

4 coordinates. For example, suppose light is scattered into the direction represented by the unit vector â. The radiant intensity of the scattered light can either be denoted by I(θ, φ) or I(â). The surface integral of an arbitrary function F(θ, φ) over the unit sphere is denoted by S F(θ, φ)ds. In polar coordinates, S F(θ, φ)ds = π π 0 0 F(θ,φ)sin θdθdφ (C.9) C.3.1 Total Cross Section Cross section is a concept that is very useful in scattering theory. Consider a light beam incident on a particle. A portion of the radiant power would be scattered in all directions while the rest flows through it. The scattering of the incident light beam is analogous to a virtual cross section that scatters electromagnetic fields that is incident on it. The area of this virtual cross section is proportional to the amount of radiant power that is scattered. This virtual cross section is called the scattering cross section. Note that this virtual cross section is not equal to the physical cross section of the particle. There are three quantities under the heading of the total cross section total scattering cross section, total absorption cross section and total extinction cross section, denoted by σ sc, σ ab and σ ex respectively. The total scattering cross section can be defined as the ratio of the total radiant power scattered by a particle in all directions, to the radiant power incident on the particle. It is given by σ sc = λ π (n + 1)( a n + b n ) (C.10) The total absorption cross section can be defined as the ratio of the total radiant power absorbed by a particle, to the radiant power incident on the particle. Similarly, the total extinction cross section can be defined as the ratio of the radiant power removed from the incident beam by a particle, to the radiant power incident on the particle. In terms of the 164

5 intensity distribution functions, the total extinction cross section is given by, σ ex = λ π (n + 1)[Re(a n + b n )] (C.11) Since the total removed radiant power is the sum of the scattered and absorbed radiant power, the total absorption cross section is simply, σ ab = σ ex σ sc (C.1) C.3. Volume Total Cross Section In most practical cases, it is easier to measure the scattering for a volume of particulate media than a single particle. It is assumed that along the direction of the incident beam, non of the particles overlap each other. In this case, the cross section of the volume is the sum of the cross section of all the particles in that volume. The volume total scattering cross section, volume total absorption cross section and volume total extinction cross section are defined as the ratio of the total radiant power scattered, absorbed and removed respectively by a unit volume of particulate medium, to the radiant power incident on it. They are denoted by the symbols β sc, β ab and β ex respectively. Assuming a homogenous medium (ie. a medium where the particle density is constant throughout the medium), the volume total cross section is the product of the total cross section and the particle density of the medium. C.3.3 Angular Scattering Cross Section In situations where the angular properties of scattering is important, there is a need to know the radiant power scattered in the direction of interest rather than in all directions. Only the scattering effect has angular properties. Hence, these coefficients will only be used to describe the scattering properties of a particle or a particulate media. 165

6 The angular scattering cross section of a particle can be defined as the ratio of the radiant power scattered by the particle per unit solid angle at the direction of interest, to the total radiant power incident on the particle. It is denoted by the symbol σ sc (θ, φ), where (θ, φ) represents the direction of scattered beam. For unpolarized incident light, the angular scattering cross section is given by, σ sc (θ, φ) = λ 4π ( ) i1 + i (C.13) C.3.4 Volume Angular Scattering Cross Section The volume angular scattering cross section can be defined as the ratio of the radiant power per unit solid angle, scattered into the direction of interest by a unit volume of particulate medium, to the radiant power incident on the volume. The volume angular coefficient is denoted by the symbol β sc (θ,φ), where (θ, φ) represents the direction of the scattered beam. For a homogenous medium and unpolarized incident light, the volume angular coefficient is given by, β sc (θ, φ) = ρσ sc (θ, φ) = ρ λ 4π ( ) i1 + i (C.14) where ρ is the particle density of the medium. C.3.5 Efficiency Factors As mentioned earlier, the virtual cross section is not equal to the physical cross section. When there is a need to relate the virtual cross section to the physical cross section, the efficiency factors are used. The scattering, absorption and extinction efficiency are the ratio of their respective total cross section to the physical cross section of the particle (ie. πr where r is the radius of the particle). The scattering, absorption and extinction efficiency are denoted by the symbols Q sc, Q ab and Q ex respectively. 166

7 C.3.6 Phase Function The phase function is commonly used in angular scattering studies. The phase function in a direction of interest is the fraction of the total scattered intensity that is scattered into that direction. It is denoted by the symbol P(θ, φ) and is given by, P(θ, φ) = β(θ, φ) β sc /4π (C.15) where (θ, φ) represent the direction of the scattered beam. The phase function is normalized such that its surface integral over a unit sphere is unity, ie. 1 p(θ, φ)ds = 1 4π S (C.16) Sometimes, the incident beam is does not coincide with the z-axis in figure C.1. Thus, another direction is added in the phase function to represent the direction of the incident beam. This is denoted by P(θ A,φ A ; θ B,φ B ) = P( sˆ A ; sˆ B ) where (θ B,φ B ) or the unit vector sˆ B represent the direction of the incident beam and (θ A, φ A ) or the unit vector sˆ A represent the direction of the scattered beam. 167