Topics in Particle and Dispersion Science

Transmission function in the small-angle approximation for a medium illuminated at normal incidence

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The solution of the RTE in the small-angle approximation, expressed in Eq. 11 of Solving RTE in the small-angle scattering approximation .. represents the total radiance, L, i.e. a sum of the direct (unscattered) and scattered radiances. We can obtain the scattered part by simply subtracting the direct part, L_dir, that is attenuated by a factor of exp(-τ), with τ being the optical thickness:

Ldir(κ, τ) = [E0 exp( -τ ) / (4π)] ∑ j = 0 to ∞ (2j + 1) Pj(κ)

(1)

where E₀ is the irradiance at τ = 0. Hence, the scattered radiance, L_sca, can be expressed as follows:

Lsca (κ, τ) =	[E0 / (4π)]
	∑ j = 0 to ∞ (2j + 1) [exp( -cj τ ) - exp( -τ )] Pj(κ)	(2)

Let us now consider a region of very small observation angles, so that acos(κ) ~ γ = θ (recall that γ' = 0 for the normal incidence assumed here, hence θ = γ - γ' = γ; see also Fig. 1). Then the following asymptotic relationship holds: P_j(κ) = J₀(α_jθ), where α_j = j + ½ and J₀ is the Bessel function of the first kind and order 0. Therefore, Eq. 2 can be rewritten as follows:

Lsca (κ, τ) =	[E0 exp( -τ ) / (2π)]
	∑ j = 0 to ∞ αj [exp( -ej τ - 1 ) - 1 ] J0(αjθ)	(3)

where, by using a definition of c_j (Eq. 7 in Solving RTE in the small-angle scattering approximation ..), one obtains:

ej	= 1 - cj
	= ω0 hj / (2j + 1)	(4)

The transmission, T, of a layer of the turbid medium can be generalized to a transmission function, dependent on the incidence and observation angles. This function can be expressed as follows:

T = πLsca / (µ0 E0 )

(5)

Hence:

T = [exp( -τ ) / 2] ∑ j = 0 to ∞ αj [exp( -ej τ - 1 ) - 1 ] J0(αjθ)

(6)

since I consider only the case of µ₀ = 1. This function does not depend on the azimuth angle, φ, due to the symmetry of the problem.

As noted by Kokhanowsky (2007f) this discussion is directly applicable to the evaluation of the effect of multiple light scattering on the measurement of transmission of light in (and hence of the attenuation coefficient of) turbid medium (see RTE and transmission measurement and Radiative transfer equation and Lambert's law).

Please also see Transmission function in the small-angle approximation (SAA) for a medium illuminated at normal incidence: SAA and exact solutions.

CITATION: Kokhanovsky A. A. 2007. Transmission function in the small-angle approximation for a medium illuminated at normal incidence (www.tpdsci.com/Tpc/RTSAANmlTsm.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 14-Feb-2007 Modified: 05-Feb-2007 Peer-reviewed: PENDING

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