Optimized Size and Shape of LSC

It is seen that the trajectory has an average ratio x = (ji/2 — t c)/cos t c much closer to 1 than to the upper limit (1/sin t c) = n. Hence, most of the fluorescence is emitted, for purely geometrical reasons, closer to the plane of the LSC than to the surface of the critical cone. A quite typical case for highly refracting glasses is cost ,. = 0.8, sint c = 0.6 where t c = 36° 52 and n = 1.6667. In such a case, the misdirection of the zig-zag motion on Fig. 2 only prolongates the horizontal projection, on the average, by 16%. 

Our typical case n = (5/3) provides N slightly below 0.3 (iyD). Since a spherical plate has (A(/Ae) = 0.25 (IVD) it cannot be avoided that the average number of consecutive total reflections N of trapped luminescence is about 1.2 to 0.8 times (Af/Ae). This illustrates the increasing problems with surface scratches and weak deviations from parallel planar surfaces involved in the otherwise desirable increase of the ratio (IVD). Since LSC are of little practical use before (A /Ae) is above 15, we cannot avoid at least 10 consecutive total reflections. 

An even more pervasive difficulty connected with the average photon trajectory (xiy 2) is absorption and scattering. We re-write Eq. (8) for an exponential absorption coefficient in the form of Beer s law for a series of coloured species having molar concentrations c and (at a given wave-number) molar extinction coefficients e  

Obviously, this linear approximation is not valid for high absorptivity. Thus, Eq. (23) vanishes at the crossing-point where the square bracket (multiplied by () is 0.4343 and Eq. (23) predicts half the original flux, when ( times the square bracket is 0.2171, whereas the exact Eq. (22) says that this product should be 0.3010. However, this discrepancy attenuates at lower absorptivities (I/I0) = 0.8 corresponds to 0.0869 in Eq. (23) but 0.0969 in Eq. (22). 

The preparation of transparent (inorganic or organic) glasses is the main problem for optical fibers. Since the Rayleigh scattering is proportional to the fourth power of the 

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