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Consider the measurement of transmission in a purely light-absorbing medium. The path function, L*, Eq. 2 in Radiative transfer equation (RTE) and transmission measurement, vanishes in such a medium because the scattering function of the medium vanishes. Thus, with a unidirectional light source aligned with the z-axis, located at z = 0, and emitting radiance Ld (Eq. 1 in RTE and transmission measurement), the radiance field, L, in the medium vanishes in all other directions. Hence,
| L(z, ξ) = Ed(z) δ(ξ - k) | (1) |
where ξ is the unit vector along an observation direction, k is the unit vector of the z-axis, Ed is the direct irradiance, and δ is the Dirac delta function. Integration of L(z, ξ), over the acceptance angle, Δω, of the transmissometer detector about the z-axis, yields Ed(z). Thus, by integrating Eq. 2 in RTE and transmission measurement over Δω, one obtains:
| dEd(z) / dz = -aEd(z) | (2) |
which, after an additional integration over the detector area, yields the differential form of the Lambert law. Note that k · ξ in Eq. 2 in RTE and transmission measurement becomes k · k = 1. Also note that the absorption coefficient, a, has replaced the attenuation coefficient, c = a + b, used in the general-form RTE, because the scattering coefficient, b, vanishes in the present case.
In the case of non-negligible scattering of light, radiance L(z, ξ) in Eq. 2 in RTE and transmission measurement, can be expressed as a sum of the direct radiance, Ld, and the scattered radiance, Ls, as follows:
| L(z, ξ) | = Ld(z, ξ) + Ls(z, ξ) | |
| = Ed(z) δ(ξ - k) + Ls(z, ξ) | (3) |
Hence, the RTE can be rewritten from its general form (Eq. 2 in RTE and transmission measurement) as follows:
| (k · ξ) dL(z, ξ) / dz | = -cL(z, ξ) + ∫4π L(z, ξ') β(ξ', ξ) dω(ξ') | |
| = -cL(z, ξ) + | ||
| Ed(z) ∫4π δ(ξ' - k) β(ξ', ξ) dω(ξ') + | ||
| ∫4π Ls(z, ξ') β(ξ', ξ) dω(ξ') | ||
| = -cL(z, ξ) + Ed(z) β(k, ξ) + | ||
| ∫4π Ls(z, ξ') β(ξ', ξ) dω(ξ') | (4) |
where for simplicity we expanded L only inside of the integral. We also assume that the medium is uniform, which is a reasonable assumption when measuring transmission of light. Hence the scattering function, β, is independent of position.
In the case of single scattering, the re-scattered light is negligible, hence the last term of the sum on the right side of Eq. 4 can be neglected. However, even with this simplification it is clear that unless a contribution of the term Ed(z) β(k, ξ) in the right side of Eq. 4 is somehow limited, that equation does not lead to the Lambert law.
The contribution of this terms to the right hand side of Eq. 4 can be nulled by integrating Eq. 4 over Δω, as it is done by the transmissometer detector, and going to the limit of Δω = 0. In contrast, the integral over Δω of L = Ld + Ls = Ed(z) δ(k - ξ) + Ls(z, ξ) does not vanish in the limit of Δω = 0 and yields Ed(z). Again, after integrating over the detector area one obtains the differential form of the Lambert law. Hence, the operational applicability of the Lambert law to a turbid medium clearly requires the minimization of the acceptance angle of the transmissometer detector.
Similar arguments can be given for the case of multiple scattering of light, with an exception that the full RTE (Eq. 4) must be used.
See also: Measuring attenuation of light
| CITATION: Jonasz M. 2006. Radiative transfer equation and Lambert's law (www.tpdsci.com/Tpc/RTELmbLw.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 14-Feb-2007 Modified: 22-Jan-2008 Peer-reviewed: 12-Feb-2007 |
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