## Abstract

The optical performance of axisymmetric radiation concentrators and
illuminators that are derived when two-dimensional edge-ray designs are
rotated about their optic axis is investigated. Of particular
interest are devices with spherical and cylindrical absorbers or light
sources, for which the inherent ray rejection can be
substantial. From the principle of etendue (phase-space)
conservation, a lower bound for ray rejection can be
established. With computer ray tracing, we demonstrate that this
bound underestimates the actual ray rejection by only a few percent at
most. Hence, to a good approximation, it can be used as an equality
in analytic predictions of characteristic efficiency-concentration
curves. By designing for absorbers or sources with a bald spot, the
full range of efficiency and flux concentration values can be realized
and the trade-off between them can be quantified. The optical
performance of these edge-ray designs is also compared against
fundamental upper bounds on the flux concentration and efficiency of
axisymmetric devices.

© 1998 Optical Society of America

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### Equations (21)

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(1)
$${C}_{2\mathrm{D}}=\frac{\mathrm{aperture}\mathrm{width}}{\mathrm{absorber}\mathrm{perimeter}}$$
(2)
$$C_{2\mathrm{D}}{}^{max}=\frac{1}{sin\left({\mathrm{\theta}}_{a}\right)},$$
(3)
$${C}_{3\mathrm{D}}=\frac{\mathrm{aperture}\mathrm{area}}{\mathrm{absorber}\mathrm{area}}$$
(4)
$$C_{3\mathrm{D}}{}^{max}=\frac{1}{{sin}^{2}\left({\mathrm{\theta}}_{a}\right)}$$
(5)
$${C}_{2\mathrm{D}}=\frac{D}{d\left(\mathrm{\pi}-\mathrm{\phi}\right)},$$
(6)
$${C}_{3\mathrm{D}}={\left[\frac{D}{2dcos\left(\mathrm{\phi}/2\right)}\right]}^{2},$$
(7)
$${f}_{\mathrm{abs}}={cos}^{2}\left(\mathrm{\phi}/2\right).$$
(8)
$${C}_{2\mathrm{D}}=\frac{D}{2\left(h-{h}_{0}+r\right)},$$
(9)
$${C}_{3\mathrm{D}}=\frac{{D}^{2}}{4r\left[2\left(h-{h}_{0}\right)+r\right]},$$
(10)
$${f}_{\mathrm{abs}}=\frac{2\left(h-{h}_{0}\right)+r}{2h+r}.$$
(11)
$$\mathrm{\eta}=\frac{C_{2\mathrm{D}}{}^{2}}{{C}_{3\mathrm{D}}},$$
(12)
$$\mathrm{\eta}={\left[\frac{2cos\left(\mathrm{\phi}/2\right)}{\mathrm{\pi}-\mathrm{\phi}}\right]}^{2}.$$
(13)
$$\mathrm{\eta}=\frac{r\left[2\left(h-{h}_{0}\right)+r\right]}{{\left(h-{h}_{0}+r\right)}^{2}}.$$
(14)
$$\mathrm{\eta}=\frac{\u3008{C}_{\mathrm{flux}}\u3009}{{C}_{3\mathrm{D}}{f}_{\mathrm{abs}}}.$$
(15)
$$\u3008{C}_{\mathrm{flux}}\u3009=C_{2\mathrm{D}}{}^{2}{f}_{\mathrm{abs}}.$$
(16)
$${C}_{2\mathrm{D}}sin\left({\mathrm{\theta}}_{a}\right)\le 1.$$
(17)
$${C}_{\mathrm{rel}}=\u3008{C}_{\mathrm{flux}}\u3009{sin}^{2}\left({\mathrm{\theta}}_{a}\right)={\left[{C}_{2\mathrm{D}}sin\left({\mathrm{\theta}}_{a}\right)\right]}^{2}{f}_{\mathrm{abs}}.$$
(18)
$$\mathrm{\phi}=\mathrm{\pi}-\frac{Dsin\left({\mathrm{\theta}}_{a}\right)}{d},$$
(19)
$${h}_{0}=\frac{2\left(h+r\right)-Dsin\left({\mathrm{\theta}}_{a}\right)}{2}.$$
(20)
$$P=I\mathrm{\eta}{A}_{\mathrm{ap}}=I\mathrm{\eta}{C}_{3\mathrm{D}}{A}_{\mathrm{abs}}=I\mathrm{\eta}{C}_{3\mathrm{D}}A_{\mathrm{abs}}{}^{\mathrm{full}}{f}_{\mathrm{abs}}.$$
(21)
$$\frac{P}{{\mathit{IC}}_{3\mathrm{D}}A_{\mathrm{abs}}{}^{\mathrm{full}}}=\mathrm{\eta}{f}_{\mathrm{abs}},$$