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| Radiometry: Radiance law in non-uniform media | Prev topic | Next topic Fig. 1 |
The radiance law can be generalized to include media in which the refractive index depends on position (for example, Di Vita P and Vannucci 1975, Gershun A 1939). In the simplest of such cases, and by limiting ourselves to the real refractive index, m, the radiance law can be expressed as follows:
| LR / mR 2 = T LS / mS 2 | (1) |
where LS is the radiance emitted in a medium with the refractive index, mR, LR is the radiance received in a medium with the refractive index, mR, and T is the incidence angle- and polarization-dependent factor representing transmission of the radiation power through an interface between the two media.
To derive this generalized radiance law, consider a light source in a lossless uniform medium with a real refractive index mS, and a receiver located in a neighboring lossless uniform medium with a real refractive index mR. The media share an interface which affect the propagation of radiation by refraction and incomplete transmission of the radiation power. The power of radiation, dΦS, emitted by the source and intended to arrive at the receiver, and the power dΦR, ultimately arriving at the receiver are related as follows:
| dΦR = T dΦS | (2) |
By using the definition of radiance, we have:
| dΦR = LR dωR dApR | (3) |
| dΦS = LS dωS dApS | (4) |
where dApS and dApR are the projected, i.e. normal areas of the source and the receiver, respectively. In contrast to the derivation of the radiance law in uniform medium, and as shown in Fig. 1, we cannot now set dωS as dApR / R2, and dωR as dApS / R2, where R is the distance between the source and the receiver, because of the refraction of radiation at the interface between the two media. Instead, by using the results derived in the caption of Fig. 1, we have:
| dΦR = LR dApS mS2 R' -2 dApR | (5) |
| dΦS = LS dApR mR2 R' -2 dApS | (6) |
The substitution of Eq. 5 and Eq. 6 into Eq. 2 yields, after simple algebra, the generalized radiance law expressed in Eq. 1.
The above elementary derivation is one but several alternatives (see another derivation here). Di Vita P and Vannucci 1975 provide another optical proof of the generalized radiance law as well as prove that law on thermodynamics grounds.
See also Radiance law and Optical invariants.
| CITATION: Jonasz M. 2007. Radiometry: Radiance law in non-uniform media (www.tpdsci.com/Tpc/RdmRadERLwNU.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 19-Jan-2007 Modified: 06-Mar-2008 Peer-reviewed: PENDING |
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