## Abstract

The rubric tailored edge-ray designs (TED’s) refers to the procedure for tailoring lighting reflectors to produce a prescribed flux distribution for an extended Lambertian source while ensuring maximum radiative efficiency (no radiation being returned to the source). Most TED studies to date have been restricted to the case in which the two edges of the image of the source in the reflectors are bound by a source edge ray and a reflector edge. The extension to the more general, and challenging, solution in which both edges of the image can be bound by rays from opposite edges of the source was recently begun by Ries and Winston [J. Opt. Soc. Am. A **11,** 1260–1264 (1994)] but was described in detail only for one particular design. We show that there are four topologically distinct classes of such reflectors; we derive the governing differential equations and obtain the solution in analytical form. Our results are illustrated for the case of uniform far-field illuminance production with symmetric configurations in two dimensions. Relative to earlier TED’s, these new devices can offer increased uniform core regions and superior glare control, although they are somewhat less compact. We offer a comprehensive analysis of the geometric properties, flux-map characteristics, and limitations of these new TED’s.

© 1996 Optical Society of America

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### Equations (18)

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(1)
$$\frac{\mathrm{d}\phantom{\rule{0.2em}{0ex}}\text{ln}\left(r\right)}{\mathrm{d}\varphi}=\text{tan}\left(\text{\alpha}\right)\pm \frac{t}{r},$$
(2)
$$2\text{\alpha}=\varphi -\text{\theta ,}$$
(3)
$$L\left(\text{\theta}\right)=\frac{E\left(\text{\theta}\right)}{{\text{cos}}^{2}\left(\text{\theta}\right)}.$$
(4)
$$L\left(\text{\theta}\right)=\frac{L\left(0\right)}{{\text{cos}}^{2}\left(\text{\theta}\right)}.$$
(5)
$$L\left(\text{\theta}\right)={L}_{\text{source}}\left(\text{\theta}\right)+{L}_{\text{involute}}\left(\text{\theta}\right)+{L}_{\text{TED}}\left(\text{\theta}\right).$$
(6)
$$\begin{array}{c}{L}_{\text{source}}\left(\text{\theta}\right)+{L}_{\text{involute}}\left(\text{\theta}\right)=2t+2t\phantom{\rule{0.2em}{0ex}}\text{sin}\left({\text{\beta}}_{L}+\text{\theta}\right)\\ \begin{array}{cc}+2t\phantom{\rule{0.2em}{0ex}}\text{sin}\left({\text{\beta}}_{R}-\text{\theta}\right)& \left|\text{\theta}\right|\le {\text{\theta}}_{d},\end{array}\end{array}$$
(7)
$$\text{sin}\left({\text{\theta}}_{d}\right)=\frac{1}{1+\frac{g}{t}}.$$
(8)
$$\begin{array}{ll}{L}_{\text{TED}}\left(\text{\theta}\right)=& \left[r+\frac{t}{\text{tan}\left(\text{\alpha}\right)}\right]\text{sin}\left(2\text{\alpha}\right)\\ & -\left[{r}_{b}+\frac{t}{\text{tan}\left({\text{\alpha}}_{b}\right)}\right]\text{sin}\left(2{\text{\alpha}}_{b}\right),\end{array}$$
(9)
$$\begin{array}{cc}{L}_{\text{TED}}\left(\text{\theta}\right)=& \left[{r}_{b}+t\phantom{\rule{0.2em}{0ex}}\text{tan}\left({\text{\alpha}}_{b}\right)\right]\text{sin}\left(2{\text{\alpha}}_{b}\right)\\ & -\left[r+t\phantom{\rule{0.2em}{0ex}}\text{tan}\left(\text{\alpha}\right)\right]\text{sin}\left(2\text{\alpha}\right).\end{array}$$
(10)
$$p\left(\text{\theta}\right)=\left\{r+t{\left[\text{tan}\left(\text{\alpha}\right)\right]}^{\pm 1}\right\}\text{sin}\left(2\text{\alpha}\right),$$
(11)
$$\left\{\begin{array}{c}-1\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}\text{far-edge}\phantom{\rule{0.2em}{0ex}}\text{designs}\\ +1\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}\text{near-edge}\phantom{\rule{0.2em}{0ex}}\text{designs}\end{array}\phantom{\rule{0.5em}{0ex}}\right\}.$$
(12)
$$\begin{array}{cc}\frac{\text{d\alpha}}{\text{d\theta}}\left[1\pm \frac{2t\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.2em}{0ex}}\text{\alpha}}{p\left(\text{\theta}\right)}\right]=& \text{sin}\left(\text{\alpha}\right)\text{cos}\left(\text{\alpha}\right)\frac{\mathrm{d}\left[\text{ln}\left\{p\left(\text{\theta}\right)\right\}\right]}{\text{d\theta}}-{\text{sin}}^{2}\text{\alpha}\\ & +i\frac{2t}{p\left(\text{\theta}\right)}\left(\frac{\text{d\alpha}}{\text{d\theta}}+{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\text{\alpha}\right),\end{array}$$
(13)
$$\text{tan}\left(u\right)=\text{cot}\left(\text{\alpha}\right),$$
(14)
$$\begin{array}{r}\text{tan}\left(u\right)\left[p\left(\text{\theta}\right)-\left(1-i\right)\left(2t\right)\right]\pm \left(2t\right)u\\ -i\phantom{\rule{0.2em}{0ex}}\left(2t\right)\text{\theta}=P\left(\text{\theta}\right)-P\left({\text{\theta}}_{m}\right),\end{array}$$
(15)
$$P\left(\text{\theta}\right)={\displaystyle {\int}^{\theta}}p\left({\text{\theta}}^{\prime}\right)\mathrm{d}{\text{\theta}}^{\prime}$$
(16)
$${A}_{o}=\left[\text{sin}\left({\text{\beta}}_{o}\right)-\frac{{r}_{o}}{t}\text{cos}\left({\text{\beta}}_{o}\right)\right]\left(2t\right),$$
(17)
$$\frac{{r}_{o}}{t}=\frac{{r}_{d}}{t}+\left({\text{\beta}}_{o}-{\text{\beta}}_{d}\right).$$
(18)
$$\begin{array}{ll}f& =\left[{\displaystyle {\int}_{0}^{{\text{\theta}}_{c}}\frac{1}{{\text{cos}}^{2}\left(\text{\theta}\right)}\mathrm{d}\left(\text{\theta}\right)}\right]\phantom{\rule{0.2em}{0ex}}\left[\frac{{A}_{o}}{\text{area}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}\text{source}}\right]\\ & =\text{tan}\left({\text{\theta}}_{c}\right)\left(\frac{\left\{\text{sin}\left({\text{\beta}}_{o}\right)-\left[\frac{{r}_{d}}{t}+\left({\text{\beta}}_{o}-{\text{\beta}}_{d}\right)\right]\text{cos}\left({\text{\beta}}_{o}\right)\right\}}{\text{\pi}}\right).\end{array}$$