## Abstract

The one-time ray-tracing optimization method is a fast way to design LED illumination systems [Opt. Express **22**, 5357 (2014) [CrossRef] ]. The method optimizes the performance of LED illumination systems by modifying the LEDs’ luminous intensity distribution curve (LIDC) with a freeform lens, instead of modifying the illumination system structure. In finding the LEDs’ LIDC for optimizing the illumination system’s performance, the LEDs’ LIDC found by means of a general gradient descent method can be trapped in a local solution. This study develops a matrix operation method to directly find the global solution of the LEDs’ LIDC for the optimization of the illumination system’s performance for any initial design of an illumination system structure. As compared with the gradient descent method, using the proposed characteristic matrix operation method to find the best LEDs’ LIDC reduces the cost in time by several orders of magnitude. The proposed characteristic matrix operation method ensures that the one-time ray-tracing optimization method is an efficient and reliable method for designing LED illumination systems.

© 2016 Optical Society of America

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### Equations (22)

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(1)
$$F=\left[\begin{array}{cccc}{F}_{1,1}& {F}_{1,2}& \dots & {F}_{1,N}\\ {F}_{2,1}& {F}_{2,2}& \dots & {F}_{2,N}\\ \vdots & \vdots & \vdots & \vdots \\ {F}_{M,1}& {F}_{M,2}& \dots & {F}_{M,N}\end{array}\right]$$
(2)
$$G=\left[\begin{array}{cccc}{g}_{1},& {g}_{2},& \mathrm{...}& {g}_{M}\end{array}\right],$$
(3)
$$P=GF=\left[\begin{array}{cccc}{p}_{1},& {p}_{2},& \mathrm{...}& {p}_{N}\end{array}\right],$$
(4)
$${F}_{\text{Nor}}=\left[\begin{array}{cccc}{F}_{1,1}^{}/{\varphi}_{1}& {F}_{1,2}^{}/{\varphi}_{1}& \mathrm{...}& {F}_{1,N}^{}/{\varphi}_{1}\\ {F}_{2,1}^{}/{\varphi}_{2}& {F}_{2,2}^{}/{\varphi}_{2}& \mathrm{...}& {F}_{2,N}^{}/{\varphi}_{2}\\ \vdots & \vdots & \vdots & \vdots \\ {F}_{M,1}^{}/{\varphi}_{M}& {F}_{M,2}^{}/{\varphi}_{M}& \mathrm{...}& {F}_{M,N}^{}/{\varphi}_{M}\end{array}\right],$$
(5)
$${G}_{\text{Nor}}=\left[\begin{array}{cccc}{}_{n}{g}_{1},& {}_{n}{g}_{2},& \mathrm{...}& {}_{n}{g}_{M}\end{array}\right],$$
(6)
$${P}_{Nor}={G}_{Nor}{F}_{Nor}=\left[\begin{array}{cccc}{}_{n}{p}_{1},& {}_{n}{p}_{2},& \mathrm{...}& {}_{n}{p}_{N}\end{array}\right],$$
(7)
$${}_{n}{p}_{j}={\displaystyle \sum {}_{i=1}^{M}\left({}_{n}{g}_{i}{F}_{i,j}/{\varphi}_{i}\right)}.$$
(8)
$${p}_{\text{avg}}=\left({\displaystyle \sum {}_{j=1}^{N}{}_{n}p{}_{j}}\right)/N=\left({\displaystyle \sum {}_{j=1}^{N}\left({\displaystyle \sum {{}_{i=1}^{M}}_{n}{g}_{i}{F}_{i,j}/{\varphi}_{i}}\right)}\right)/N=1.$$
(9)
$${T}_{Nor}=\left[\begin{array}{cccc}{}_{n}t{}_{1},& {}_{n}t{}_{2},& \mathrm{...}& {}_{n}t{}_{N}\end{array}\right],$$
(10)
$${D}_{\text{F}}=\left[\begin{array}{cccc}{F}_{1,1}^{}/{\varphi}_{1}-{}_{n}t{}_{1}& {F}_{1,2}^{}/{\varphi}_{1}-{}_{n}t{}_{2}& \mathrm{...}& {F}_{1,N}^{}/{\varphi}_{1}-{}_{n}t{}_{N}\\ {F}_{2,1}^{}/{\varphi}_{2}-{}_{n}t{}_{1}& {F}_{2,2}^{}/{\varphi}_{2}-{}_{n}t{}_{2}& \mathrm{...}& {F}_{2,N}^{}/{\varphi}_{2}-{}_{n}t{}_{N}\\ \vdots & \vdots & \vdots & \vdots \\ {F}_{M,1}^{}/{\varphi}_{M}-{}_{n}t{}_{1}& {F}_{M,2}^{}/{\varphi}_{M}-{}_{n}t{}_{2}& \mathrm{...}& {F}_{M,N}^{}/{\varphi}_{M}-{}_{n}t{}_{N}\end{array}\right],$$
(11)
$${G}_{Nor}{D}_{F}\times {\left({G}_{Nor}{D}_{F}\right)}^{T}={G}_{Nor}{D}_{F}{D}_{F}{}^{T}{G}_{Nor}{}^{T}=\frac{{\displaystyle \sum {}_{j=1}^{N}{\left({}_{n}{p}_{j}{-}_{n}{t}_{j}\right)}^{2}}}{{p}_{avg}{}^{2}}=N{\left(CV\right)}^{2}$$
(12)
$${G}_{k}=\left[\begin{array}{cccc}{g}_{k1},& {g}_{k2},& \mathrm{...}& {g}_{kM}\end{array}\right],$$
(13)
$${G}_{k}{D}_{F}{D}_{F}{}^{T}{G}_{k}{}^{T}={\lambda}_{k}{}^{}\left({G}_{k}{G}_{k}{}^{T}\right)=N{\left(CV\right)}_{k}{}^{2}.$$
(14)
$${G}_{global}={\displaystyle \sum _{k=1}^{M}{a}_{k}{G}_{k}},$$
(15)
$$\sum _{k=1}^{M}{a}_{k}}=1.$$
(16)
$${\left(CV\right)}^{2}={G}_{global}{D}_{F}{D}_{F}{}^{T}{G}_{global}{}^{T}/N={\displaystyle \sum _{k=1}^{M}{a}_{k}{}^{2}}{\left(CV\right)}_{k}{}^{2}$$
(17)
$${\left(CV\right)}^{2}={\displaystyle \sum _{k=1}^{M-1}{a}_{k}{}^{2}}{\left(CV\right)}_{k}{}^{2}+{\left(1-{\displaystyle \sum _{k=1}^{M-1}{a}_{k}}\right)}^{2}{\left(CV\right)}_{M}{}^{2}$$
(18)
$$\frac{\partial {\left(CV\right)}^{2}}{\partial {a}_{k}}=2{a}_{k}{\left(CV\right)}_{k}{}^{2}+2\left({\displaystyle \sum _{l=1}^{M-1}{a}_{l}}-1\right){\left(CV\right)}_{M}{}^{2}=0.$$
(19)
$${a}_{k}=\frac{\left(1-{\displaystyle \sum _{l=1}^{M-1}{a}_{l}}\right){\left(CV\right)}_{M}{}^{2}}{{\left(CV\right)}_{k}{}^{2}}=\frac{{a}_{M}{\left(CV\right)}_{M}{}^{2}}{{\left(CV\right)}_{k}{}^{2}}.$$
(20)
$${a}_{1}:{a}_{2}:\mathrm{...}:{a}_{M}=\frac{1}{{\left(CV\right)}_{1}{}^{2}}:\frac{1}{{\left(CV\right)}_{2}{}^{2}}:\mathrm{...}:\frac{1}{{\left(CV\right)}_{M}{}^{2}}.$$
(21)
$$P=G{\text{'}}_{global}F,$$
(22)
$$G{\text{'}}_{global}=\left[\begin{array}{cccc}{g}_{k1}/{\phi}_{1},& {g}_{k2}/{\phi}_{2},& \mathrm{...}& {g}_{kM}/{\phi}_{M}\end{array}\right].$$