## Abstract

The theory of two-dimensional (2-D) nonimaging concentrators is reviewed. A method for designing 2-D concentrators with inhomogeneous media, in which the refractive-index distribution is an output, is given. This method allows one to choose the shape of the mirrors when they are needed. As a result of the method we obtain a concentrator with maximum theoretical concentration formed by the composition of triangles of three different indices of refraction. The edge-ray principle is proven under several specific assumptions.

© 1985 Optical Society of America

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### Equations (24)

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(1)
$${n}^{2}={{U}_{x}}^{2}+{{U}_{t}}^{2},$$
(2)
$${n}^{2}={{L}_{x}}^{2}+{{L}_{t}}^{2},$$
(3)
$${n}^{2}={{F}_{x}}^{2}+{{F}_{t}}^{2}+{{G}_{x}}^{2}+{{G}_{t}}^{2},$$
(4)
$${F}_{x}{G}_{x}+{F}_{t}{G}_{t}=0,$$
(5)
$$\begin{array}{cccc}{G}_{x}={p}_{0},& {G}_{t}=0,& {F}_{x}=0,& {F}_{t}={({{n}_{0}}^{2}-{{p}_{0}}^{2})}^{1/2};\end{array}$$
(6)
$$\begin{array}{cccc}{G}_{x}={p}_{1},& {G}_{t}=0,& {F}_{x}=0,& {F}_{t}={({{n}_{1}}^{2}-{{p}_{1}}^{2})}^{1/2}.\end{array}$$
(7)
$$J=K(x,t)\times ({F}_{x},{F}_{t}).$$
(8)
$$\begin{array}{cc}G={p}_{0}x,& F=\frac{{A}_{e}}{2}{p}_{0}+{q}_{0}t.\end{array}$$
(9)
$$\begin{array}{ll}2G\hfill & =\left(x+\frac{{A}_{e}}{2}\right){p}_{0}+t{q}_{0}-\frac{{A}_{e}+{A}_{c}}{2}{p}_{0}\hfill \\ \hfill & -{q}_{0}+{n}_{c}{\left[{\left(x+\frac{{A}_{c}}{2}\right)}^{2}+{\left(1-t\right)}^{2}\right]}^{1/2},\hfill \\ 2F\hfill & =\left(x+\frac{{A}_{e}}{2}\right){p}_{0}+t{q}_{0}+\frac{{A}_{e}+{A}_{c}}{2}{p}_{0}\hfill \\ \hfill & +{q}_{0}-{n}_{c}{\left[{\left(x+\frac{{A}_{c}}{2}\right)}^{2}+{\left(1-t\right)}^{2}\right]}^{1/2}.\hfill \end{array}$$
(10)
$$\begin{array}{ll}2G\hfill & ={n}_{c}{\left[{\left(\frac{{A}_{c}}{2}+x\right)}^{2}+{\left(1-t\right)}^{2}\right]}^{1/2}\hfill \\ \hfill & -{n}_{c}{\left[{\left(\frac{{A}_{c}}{2}-x\right)}^{2}+{\left(1-t\right)}^{2}\right]}^{1/2},\hfill \\ 2F\hfill & =({A}_{e}+{A}_{c}){p}_{0}+2{q}_{0}-{n}_{c}{\left[{\left(\frac{{A}_{c}}{2}-x\right)}^{2}+{\left(1-t\right)}^{2}\right]}^{1/2}\hfill \\ \hfill & -{n}_{c}{\left[(\frac{{A}_{c}}{2}+{\left(1-t\right)}^{2}\right]}^{1/2},\hfill \end{array}$$
(11)
$$\frac{{A}_{e}+{A}_{c}}{2}=\frac{{p}_{0}}{{q}_{0}},$$
(12)
$$\frac{{A}_{e}}{{A}_{c}}=\frac{1}{sin\alpha}=\frac{{n}_{c}}{{p}_{0}}.$$
(13)
$$({F}_{x},{F}_{t})\cdot ({F}_{x}+{G}_{x},{F}_{t}+{G}_{t})=({F}_{x},{F}_{t})\cdot ({F}_{x}-{G}_{x},{F}_{t}-{G}_{t})={{F}_{x}}^{2}+{{F}_{t}}^{2}>0,$$
(15)
$$\frac{E}{2{A}_{e}}$$
(16)
$$\frac{-2b({A}_{e}-{A}_{c})}{4a+({A}_{e}-{A}_{c}){A}_{e}}$$
(17)
$$\frac{4b}{4a+({A}_{e}-{A}_{c}){A}_{e}}$$
(18)
$$\frac{E}{2[{A}_{e}-a({A}_{e}-{A}_{c})]}$$
(19)
$$\frac{E({A}_{e}-{A}_{c})}{4[{A}_{e}-a({A}_{e}-{A}_{c})]}$$
(20)
$$\frac{b[4(1-a)-({A}_{e}-{A}_{c}){A}_{c}]}{[4a+({A}_{e}-{A}_{c}){A}_{e}](1-a)}$$
(21)
$$\frac{E}{2{A}_{c}}$$
(22)
$$\frac{-2b({A}_{e}-{A}_{c})}{4a+({A}_{e}-{A}_{c}){A}_{e}}$$
(23)
$$\frac{4b}{4a+({A}_{e}-{A}_{c}){A}_{e}}$$
(24)
$$J=2{(\frac{{{G}_{x}}^{2}+{{G}_{t}}^{2}}{{{F}_{x}}^{2}+{{F}_{t}}^{2}})}^{1/2}({F}_{x},{F}_{t})=2(-{G}_{t},{G}_{x}).$$