## Abstract

For many tasks in illumination and collection it is necessary that the acceptance angle vary along the reflector. If the acceptance-angle function is known, then the reflector profile can be calculated as a functional of it. The total flux seen by an observer from a source of uniform brightness (radiance) is proportional to the sum of the view factor of the source and its reflection. This allows one to calculate the acceptance-angle function that one needs to produce a certain flux distribution and thereby to construct the reflector profile. We demonstrate the method for several examples, including finite-size sources with reflectors directly joining the source.

© 1993 Optical Society of America

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### Equations (22)

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(1)
$$d\ll {R}_{0}\ll D.$$
(2)
$$[\text{d}\hspace{0.17em}\text{log}(R)]/\text{d}\varphi =\text{tan}(\alpha ).$$
(3)
$$\alpha =(\varphi -\theta )/2.$$
(4)
$$\text{log}[R(\varphi )/{R}_{0}]={\int}_{0}^{\alpha (\varphi )}\text{tan}[\alpha (\varphi )]\text{d}\varphi $$
(5)
$$R(\varphi )={R}_{0}\hspace{0.17em}\text{exp}\left\{{\int}_{0}^{\alpha (\varphi )}\text{tan}[\alpha (\varphi )]\text{d}\varphi \right\}.$$
(6)
$${P}^{o}(\theta )={P}^{s}(\theta )+\mid M\mid {P}^{s}(\varphi ).$$
(7)
$${M}_{m}=\frac{\text{d}\varphi}{\text{d}\theta}.$$
(8)
$${M}_{s}=\frac{\text{d}{\mu}_{1}}{\text{d}{\mu}_{2}}=\frac{\text{sin}(\varphi )}{\text{sin}(\theta )},$$
(9)
$${M}_{t}={M}_{s}{M}_{m}=\frac{\text{d}\hspace{0.17em}\text{cos}(\varphi )}{\text{d}\hspace{0.17em}\text{cos}(\theta )}.$$
(10)
$${\varphi}^{\text{max}}+{\theta}^{\text{max}}=\pi .$$
(11)
$$\text{cos}\hspace{0.17em}\theta +\left|\text{cos}(\varphi )\frac{\text{d}\varphi}{\text{d}\theta}\right|=\frac{a}{{\text{cos}}^{2}(\theta )}.$$
(12)
$$\text{sin}(\varphi )=1-\mid \text{tan}(\theta )-\text{sin}(\theta )\mid .$$
(13)
$$R\hspace{0.17em}\text{sin}(2\alpha )={P}^{o}(\theta ).$$
(14)
$$\frac{\text{d}\alpha}{\text{d}\theta}=\frac{\text{sin}(2\alpha )}{2}\frac{\text{d}\hspace{0.17em}\text{log}[{P}^{o}(\theta )]}{\text{d}\theta}-{\text{sin}}^{2}(\alpha ).$$
(15)
$${P}_{T}=R({\theta}_{T}){\int}_{2{\alpha}_{T}}^{\pi}\text{sin}\hspace{0.17em}\gamma \text{d}\gamma =R({\theta}_{T})[1+\text{cos}(2{\alpha}_{T})].$$
(16)
$$R({\theta}_{T})=\frac{{P}^{o}({\theta}_{T})}{\text{sin}(2{\alpha}_{T})}.$$
(17)
$$\frac{1+\text{cos}(2{\alpha}_{T})}{\text{sin}(2{\alpha}_{T})}=\frac{1}{{P}^{o}({\theta}_{T})}{\int}_{{\theta}^{\text{max}}}^{{\theta}_{T}}{P}^{o}(\psi )\text{d}\psi =B({\theta}_{T}).$$
(18)
$$B(0)=\frac{1}{{P}^{o}(0)}{\int}_{{\theta}^{\text{max}}}^{0}{P}^{o}(\psi )\text{d}\psi =1.$$
(19)
$$2\alpha =\text{arccos}[({B}^{2}-1)/({B}^{2}++1)].$$
(20)
$$\varphi (\theta )=\theta +2\alpha .$$
(21)
$$R(\theta )={P}^{o}(\theta )[({B}^{2}+1)/2B].$$
(22)
$$R(\varphi )={R}_{o}\frac{1}{1+\text{cos}(\varphi )}.$$