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| Radiometry: Radiance law in uniform media | Prev topic | Next topic Fig. 1 |
Consider a source of radiation, generating radiance LS, and a receiver, experiencing radiance LR. Let the source and receiver be located in a lossless uniform medium with no other radiation sources. In this case the radiance is conserved:
| LR = LS | (1) |
This relationship is referred to as the radiance law. It is also known as the radiance theorem, or the law of conservation of radiance (for example, Köhler 1998, Mobley 1994).
The radiance law is the consequence of the law of energy conservation. Indeed, consider the power of radiation, dΦR, arriving at the receiver from the source. By using the definition of radiance, we obtain:
| dΦR = LR dωR dAR cosγR | (2) |
where, in reference to Fig. 1, dωR is the solid angle subtended by the source at the receiver, dAR is the area of the receiver, and γR is the angle between the normal to dAR and the line from the source to the receiver. Likewise, the power, dΦS, emitted by the source towards the receiver is expressed as follows:
| dΦS = LS dωS dAS cosγS | (3) |
where, in reference to Fig. 1, dωS is the solid angle subtended by the receiver at the source, dAS is the area of the source, and γS is the angle between the normal to dAS and the line from the source to the receiver.
If the source and receiver are located in a lossless medium with no other radiation sources, then, by definition, the power, dΦR, arriving at the receiver from the source equals the power, dΦS, emitted by the source towards the receiver. By using Eq. 2 and Eq. 3, this equality can be expressed as follows:
| LR dωR dAR cosγR = LS dωS dAS cosγS | (4) |
From the definition of the solid angle, dω, we have:
| dωR = dAS cosγS R -2 | (5) |
| dωS = dAR cosγR R -2 | (6) |
where R is the distance between the source and receiver. Hence, by combining Eq. 4 with Eq. 5 and Eq. 6, we have:
| LR dAS cosγS R -2 dAR cosγR = LS dAR cosγR R -2 dAS cosγS | (7) |
which simplifies to Eq. 1.
See also Radiance law in non-uniform media and Optical invariants.
| CITATION: Jonasz M. 2007. Radiometry: Radiance law in uniform media (www.tpdsci.com/Tpc/RdmRadERLw.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 18-Jan-2007 Modified: 23-Feb-2008 Peer-reviewed: PENDING |
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