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| Radiometry: Radiance law in non-uniform medium - an alternative derivation | Prev topic | Next topic Fig. 1 |
Consider the power of radiation, dΦ, passing through an elementary area dA at the interface between two media with different refraction indices. The energy conservation law for transmission of this power through the interface can be expressed (in reference to Fig. 1 and by using Eq. 1 in Radiance) as follows:
| LR dωR dA cosγR = T LS dωS dA cosγS | (1) |
where T ≤ 1 is the power transmission factor (see Power transmission).
The solid angles, dωS and dωR are expressed as follows:
| dωS = sinγS dγS dφ | (2) |
| dωR = sinγR dγR dφ | (3) |
where, by omitting the subscripts at dφ we indicate that this angle is the same in both media, as refraction does not affect the azimuthal orientation of a light ray.
From the refraction (Snell) law (for example, Hecht E 1987) we have:
| mS sinγS = mR sinγR | (4) |
and, by differentating the above equation:
| mS cosγS dγS = mR cosγR dγR | (5) |
Hence:
|
= |
|
= |
|
(6) |
and, as it follows from Eq. 1 and Eq. 6:
| 1 = |
|
= |
|
(7) |
which is the generalized radiance law.
| CITATION: Jonasz M. 2007. Radiometry: Radiance law in non-uniform media - an alternative derivation (www.tpdsci.com/Tpc/RdmRadERLwNUAlt.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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