## Abstract

The edge-ray principle of nonimaging optics states that nonimaging devices can be designed by the mapping of edge rays from the source to the edge of the target. However, in most nonimaging reflectors, including the compound parabolic concentrator (CPC), at least part of the radiation undergoes multiple reflections, some rays even appear to be reflected infinitely many times, and closer examination reveals that some edge rays of the source are not mapped onto the edge of the target even though the CPC is indeed ideal in two dimensions. Using a topological approach, we refine the formulation of the edge-ray principle to ensure its validity for all configurations. We present two different versions of the general principle. The first involves the boundaries of the different zones corresponding to a different number of reflections. The second version is stated in terms of only a single reflection, but it involves the addition of an auxiliary region of phase space. We discuss the use of the edge-ray principle as a design procedure for nonimaging devices. The CPC is used to illustrate all steps of the argument.

© 1994 Optical Society of America

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### Equations (12)

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(1)
$$E(X\cup Y)=E(X)+E(Y)\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\hspace{0.17em}X\cap Y=\varnothing .$$
(2)
$$m(x)\ne x\iff x\in R.$$
(3)
$$M(x)={m}^{n}(x)\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{if}\hspace{0.17em}{m}^{n+1}(x)={m}^{n}(x).$$
(4)
$$\begin{array}{lll}S={\cup}_{i}{S}_{i}\hfill & \text{with}\hspace{0.17em}{S}_{i}\cap {S}_{j}=\varnothing \hfill & \text{for}\hspace{0.17em}i\ne j,\hfill \\ T={\cup}_{i}{T}_{i}\hfill & \text{with}\hspace{0.17em}{T}_{i}\cap {T}_{j}=\varnothing \hfill & \text{for}\hspace{0.17em}i\ne j,\hfill \\ {T}_{i}=m({S}_{i}){\forall}_{i}.\hfill & \hspace{0.17em}\hfill & \hspace{0.17em}\hfill \end{array}$$
(5)
$${D}_{n,k}\cap {D}_{i,j}=\varnothing \mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\hspace{0.17em}(n,k)\ne (i,j).$$
(6)
$${\cup}_{n=1}^{\infty}\hspace{0.17em}{\cup}_{k=1}^{n}\hspace{0.17em}{D}_{n,k}\hspace{0.17em}{\cup}_{k=1}^{\infty}\hspace{0.17em}{D}_{\infty ,k}\subseteq ({T}_{>0}\cup A).$$
(7)
$$E({D}_{n,k})=E({S}_{n})={E}_{n}{\forall}_{k\le n},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}E({D}_{\infty ,k})=E({S}_{\infty})={E}_{\infty}{\forall}_{k},$$
(8)
$$\sum _{n=1}^{\infty}\sum _{k=1}^{n}{E}_{n}+\sum _{k=1}^{\infty}{E}_{\infty}\le E(A)+E({T}_{>0}).$$
(9)
$$A={\cup}_{n=1}^{\infty}{m}^{n}({S}_{>0})\backslash T.$$
(10)
$$m({S}_{>0}\cup A)={\cup}_{n=1}^{\infty}\hspace{0.17em}{m}^{n}({S}_{>0})={T}_{>0}\cup A.$$
(11)
$$\sum _{n=1}^{\infty}n{E}_{n}\le E(A)+E({T}_{>0}).$$
(12)
$$\u3008n\u3009=\sum _{n=1}^{\infty}n{E}_{n}/\sum _{n=1}^{\infty}{E}_{n}\le 1+E(A)/E({S}_{>0}).$$