## Abstract

We establish a fundamental bound on the field of view over which
strictly uniform far-field irradiance can be achieved in symmetric
two-dimensional (2D, troughlike) and three-dimensional (3D,
conelike) illumination systems. Earlier results derived for
particular 2D devices are shown to be special cases of the general
formula. For a rotationally symmetric 3D luminaire with a
Lambertian disk light source and a prescribed uniform core region
half-angle θ_{c}, no more than
tan^{2}(θ_{c}) can be projected
within a uniform core region. Hence the efficiency with which such
illuminators can produce uniform flux is severely limited for many
problems of practical interest. Guided by the tailored edge-ray
device formalism for the design of 2D luminaires, we develop a 3D
reflector that produces extremely uniform far-field
illuminance.

© 1998 Optical Society of America

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### Equations (14)

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(1)
$$L\left(\mathrm{\theta}\right)=A{cos}^{-d}\left(\mathrm{\theta}\right),$$
(2)
$$F={\int}_{0}^{{\mathrm{\theta}}_{c}}L\left(\mathrm{\theta}\right)\mathrm{d}\mathrm{\theta},2\mathrm{D}\mathrm{luminaires},$$
(3)
$$F=2{\int}_{0}^{{\mathrm{\theta}}_{c}}L\left(\mathrm{\theta}\right)\phantom{\rule{0.5em}{0ex}}sin\left(\mathrm{\theta}\right)\mathrm{d}\mathrm{\theta}3\mathrm{D}\mathrm{luminaires}.$$
(4)
$$F=Atan\left({\mathrm{\theta}}_{c}\right),2\mathrm{D}\mathrm{luminaires},$$
(5)
$$F=A{tan}^{2}\left({\mathrm{\theta}}_{c}\right),\text{3D luminaires}.$$
(6)
$${\mathrm{\theta}}_{c}\le {tan}^{-1}\left(1/A\right),2\mathrm{D}\mathrm{luminaires},$$
(7)
$${\mathrm{\theta}}_{c}\le {tan}^{-1}\left[\sqrt{\left(1/\mathrm{A}\right)}\right],3\mathrm{D}\mathrm{luminaires}.$$
(8)
$$F=Atan\left({\mathrm{\theta}}_{c}\right)/sin\left({\mathrm{\psi}}_{max}\right),2\mathrm{D}\mathrm{luminaires},$$
(9)
$$F=A{tan}^{2}\left({\mathrm{\theta}}_{c}\right)/{sin}^{2}\left({\mathrm{\psi}}_{\mathrm{max}}\right),3\mathrm{D}\mathrm{luminaires}.$$
(10)
$${\mathrm{\theta}}_{c}\le {tan}^{-1}\left(sin\left({\mathrm{\psi}}_{max}\right)/A\right),2\mathrm{D}\mathrm{luminaires},$$
(11)
$${\mathrm{\theta}}_{c}\le {tan}^{-1}\left(sin\left({\mathrm{\psi}}_{max}\right)/\sqrt{A}\right),3\mathrm{D}\mathrm{luminaires}.$$
(12)
$$L\left(\mathrm{\theta}\right)={cos}^{-p}\left(\mathrm{\theta}\right),$$
(13)
$${L}_{3\mathrm{D}}\left(\mathrm{\theta}\right)=\left(\mathrm{const}.\right){D}_{L}{D}_{T}={cos}^{-3}\left(\mathrm{\theta}\right).$$
(14)
$${cos}^{-p}\left(\mathrm{\theta}\right){cos}^{-p}\left(\mathrm{\theta}\right)={cos}^{-3}\left(\mathrm{\theta}\right),$$