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| Radiometry: Radiance law in non-uniform media | Parent topic | Back |
Fig. 1. A two-dimensional sketch of the geometry of the radiance law in non-uniform media with real refractive indices mS and mR, separated by a (locally) plane interface. The layout of the drawing implies that mS > mR, without limiting the generality of the conclusions. Notation: dω - elementary solid angle, γ - incidence angle, subscripts S and R denote the receiver and source variables, respectively. For simplicity, the source and receiver areas dAp are those projected on the plane perpendicular to the line linking the source and receiver. Other variables are self-explanatory or will be discussed further in this caption. The shaded areas denote the solid angles of the source and receiver, as they are refracted by the interface between the two media. Thin gray lines in the upper panel outline the source solid angle in a medium with the refractive index mS in the entire space between the source and receiver. The axial symmetry of this sketch simplifies the presentation but does not limit the generality of the conclusions.
Consider the upper panel. The solid angle dωS which contains radiation emitted by the source towards the receiver can be expressed approximately as πγS2 (see Solid angle). This can be further expanded as follows: π(mRhR)2 / R' 2, where R' = RSmR + RRmS, where we substituted the angle for its sine in the Snell law of refraction because all angles are assumed to be much smaller than 1 rad. Hence dωS = dApR mR2 / R' 2. The thick orange line on the right side shows the apparent receiver size as viewed from the source. Likewise, in reference to the lower panel, the solid angle, dωR, from which radiation arrives at the receiver can be expressed as follows: dωR = dApS mS2 / R' 2. The thick orange line on the left side shows the apparent source size as viewed from the receiver. Note that if the refractive indices on both sides of the interface are the same, dωR = dApS / R 2 and dωS = dApR / R 2, where R = RS + RR, as in the uniform medium case discussed in Radiance law.
| 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: 07-Mar-2008 Peer-reviewed: PENDING |
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