Topics in Particle and Dispersion Science

Radiative transfer equation (RTE)

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The radiative transfer equation (RTE) quantifies the change of the three-dimensional radiance field (light field) in a medium due to absorption, scattering, and emission of light. See, for example, Thomas GE and Stamnes 1999, Mobley CD 1994, Preisendorfer RW 1961, for a concise introduction. In its steady-state version for a medium with no internal light source, the RTE can be expressed as follows:

dL(r, ξ) / dr = -cL(r, ξ) + L*(r, ξ)

(1)

where L is radiance, L^* is the path function, r is a position vector, ξ is a direction unit vector, and d/dr = (∂x/∂r) (∂/∂x) + (∂y/∂r) (∂/∂y) + (∂z/∂r) (∂/∂z), i.e. a scalar (dot) product of the direction vector ξ and the nabla (del, gradient) operator, and where x, y, and z are Cartesian coordinates. For simplicity, it is assumed here, that the medium does not contain secondary light sources.

The left-hand side of the RTE represents the rate of change of radiance, L, with distance, r, along direction ξ. The first term on the right-hand side represents the radiance lost per unit distance, due to the attenuation of light, as represented by the attenuation coefficient, c. The second term on the right-hand side, the scattering contribution term, L^*, called the path function, or source function, represents the radiance gained per unit distance through scattering of the radiance field at r into direction ξ. Thus, the path function is an integral of a product of radiance and the volume scattering function of the medium over the whole range of the solid angle, ω, i.e. over 4π:

L*(r, ξ) = ∫4π L(r, ξ') β(r, ξ', ξ) dω(ξ')

(2)

where β(r, ξ', ξ) is the scattering function at r for light incident from direction ξ' and scattered into direction ξ. If the medium contains secondary light sources (for example, thermal radiation of the medium needs to be accounted for), the path (source) function needs to include additional terms that represent contributions of these sources to radiance, L(r, ξ).

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

HISTORY: Published: 18-Jan-2006 Modified: 12-Feb-2007 Peer-reviewed: 12-Oct-2006

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