## Abstract

We report on tailored reflector design methods that allow the placement of general illumination patterns onto a target plane. The use of a new integral design method based on the edge-ray principle of nonimaging optics gives much more compact reflector shapes by eliminating the need for a gap between the source and the reflector profile. In addition, the reflectivity of the reflector is incorporated as a design parameter. We show the performance of design for constant irradiance on a distant plane, and we show how a leading-edge-ray method may be used to achieve general illumination patterns on nearby targets.

© 1996 Optical Society of America

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### Equations (12)

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(1)
$$\frac{\text{d}R}{\text{d\varphi}}=R\text{\hspace{0.17em}\hspace{0.17em}tan}\left[\frac{\text{\varphi}+\text{\delta}(R,\text{\varphi})-\text{\theta}(R,\text{\varphi})}{2}\right].$$
(2)
$$X=R\text{\hspace{0.17em}\hspace{0.17em}sin\hspace{0.17em}\hspace{0.17em}\varphi}+\text{tan\hspace{0.17em}\hspace{0.17em}\theta}\left(R\text{\hspace{0.17em}\hspace{0.17em}cos\hspace{0.17em}\hspace{0.17em}\varphi}+D\right).$$
(4)
$$P(0)\ge {P}_{2}(X)/f(X),\text{\hspace{1em}}X>0,$$
(5)
$$f(Y)>{P}_{3}(Y)/P(0),$$
(6)
$$\frac{\text{d}}{\text{d}X}\frac{{P}_{3}(X)}{f(X)}\le 0,$$
(7)
$$P(X)={\displaystyle \int W(\text{\lambda})\text{d\lambda}{\displaystyle {\int}_{\text{\theta}}^{{\text{\theta}}_{s}}B(\text{\lambda},\text{\beta})\text{cos\hspace{0.17em}\hspace{0.17em}\beta d\beta \rho}}{(\text{\lambda})}^{n(\text{\beta})},}$$
(8)
$$P(X)=\frac{I\left[{A}_{s}(\text{\theta})+\text{\rho}{A}_{r}(\text{\theta})\right]{\text{cos}}^{2}\text{\hspace{0.17em}\hspace{0.17em}\theta}}{4\text{\pi}rD},$$
(9)
$${A}_{s}(\text{\theta})=2r,$$
(10)
$${A}_{r}(\text{\theta})=R\text{\hspace{0.17em}\hspace{0.17em}sin}\left(\text{\varphi}+\text{\theta}\right)-r.$$
(11)
$$\frac{\text{d}}{\text{d\theta}}{\left[{A}_{s}(\text{\theta})+{A}_{r}(\text{\theta})\right]{\text{cos}}^{2}(\text{\theta})\mid}_{\text{\theta}=0}\le 0.$$
(12)
$$I=P(0){\displaystyle {\int}_{-{X}_{\mathrm{max}}}^{{X}_{\mathrm{max}}}\text{d}x=2DP(0)\text{tan\hspace{0.17em}\hspace{0.17em}}{\text{\theta}}_{\mathrm{max}}.}$$