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Radiative transfer equation and transmission measurement

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Transmission of light through a turbid medium is measured with a transmissometer, also referred to as an attenuation meter. It consists of a unidirectional light source and a detector viewing the light source through a layer of the medium, within a small acceptance solid angle. The light source points in a direction with a unit vector ξ₀, and generates radiance, L_d:

Ld(z = 0, ξ) = Ed0 δ(ξ - ξ0)

(1)

at z = 0. The detector acceptance solid angle, Δω, is centered about the direction ξ₀. In the above equation, ξ is the unit vector indicating an observation direction and δ is the Dirac delta function. E_d0 is the irradiance at z = 0. The use of irradiance in Eq. 1 is implied by the dimension of the delta function, as applicable here, i.e. sr ^-1. Suffix d indicates the "direct", as opposed to the "scattered" radiance component. With the use of the Lambert law, the transmission, T, can be used to calculate the attenuation coefficient, c, of a medium (c = - lnT ). Such use of the Lambert law for a turbid medium requires certain assumptions that we will discuss here.

If one assumes, for simplicity, that the light source axis points in the +z direction, then d/dr in the Eq. 1 in Radiative transfer equation (RTE) can be replaced by (k · ξ) d/dz, where k is the unit vector of the z-axis. Thus, that equation can be re-written as follows:

(k · ξ) dL(z, ξ) / dz = -cL(z, ξ) + L*(z, ξ)

(2)

where L is the radiance, z is the distance, and c is the attenuation coefficient. The term -cL(z, ξ) represents the loss of radiance due to the attenuation of light. L^* is the path function (Eq. 2 in RTE), representing the gain of radiance per unit distance, due to re-scattering of the three-dimensional radiance field, L, at z in the direction ξ.

Integration of Eq. 2 over z from 0 to z_m yields a radiance distribution L(z_m, ξ) at the detector, where z_m is the thickness of the turbid medium layer located between the light source and the detector. The power received by the transmissometer's detector is the double integral of that radiance distribution over the detector area and acceptance solid angle, Δω, about the z-axis. The integral of L(z_m, ξ) over the detector area is of little interest here. However, the integral over the acceptance angle is crucial for understanding how the latter affects the accuracy of retrieving the attenuation coefficient from the transmission measurement.

See also: Measuring attenuation of light

CITATION: Jonasz M. 2006. Radiative transfer equation and transmission measurement (www.tpdsci.com/Tpc/RTETsm.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 19-Jan-2006 Modified: 28-Jun-2009 Peer-reviewed: 12-Feb-2007

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