## Abstract

The conditions that a one-parameter manifold of rays must fulfill in order that a refractive-index distribution will exist for which the manifold is possible are analyzed. A method for calculating the refractive-index distribution is given for several cases. The problem is solved easily when the trajectories of the rays of the manifold do not cross, but it becomes more difficult when the rays are permitted to cross. Example are given of manifolds in which no more than two rays may cross at a given point. The Welford–Winston edge-ray principle, used in the design of nonimaging concentrators, is proved under several assumptions. The connection that exists between the problem treated in this paper and the synthesis of gradient-index two-dimensional nonimaging concentrators is discussed.

© 1985 Optical Society of America

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### Equations (24)

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(1)
$$\u1e8b=\text{d}x/\text{d}t.$$
(2)
$$\omega \left(x,t,\frac{\text{d}x}{\text{d}t}\right)=0.$$
(3)
$${F}_{x}=\frac{\text{d}({F}_{\u1e8b})}{\text{d}t}={F}_{\u1e8bx}\u1e8b+{F}_{\u1e8bt}+{F}_{\u1e8b\u1e8b}\u1e8d,$$
(4)
$$\u1e8d=(1+{\u1e8b}^{2})\hspace{0.17em}\left(\frac{{n}_{x}}{n}-\frac{{n}_{t}}{n}\u1e8b\right),$$
(5)
$$\frac{\text{d}\omega}{\text{d}t}={\omega}_{x}\u1e8b+{\omega}_{t}+{\omega}_{\u1e8b}\u1e8d,$$
(6)
$${\omega}_{x}\u1e8b+{\omega}_{t}+{\omega}_{\u1e8b}(1+{\u1e8b}^{2})\hspace{0.17em}\left(\frac{{n}_{x}}{n}-\frac{{n}_{t}}{n}\u1e8b\right)=l(x,t,\u1e8b),$$
(7)
$$-{f}_{x}f-{f}_{t}+(1+{f}^{2})\hspace{0.17em}\left(\frac{{n}_{x}}{n}-\frac{{n}_{t}}{n}f\right)=0,$$
(8)
$${f}_{x}f-{f}_{t}-(1+{f}^{2})\hspace{0.17em}\left(\frac{{n}_{x}}{n}+\frac{{n}_{t}}{n}f\right)=0,$$
(9)
$$\begin{array}{cc}\frac{{n}_{x}}{n}=\frac{{f}_{x}f}{1+{f}^{2}},& \frac{{n}_{t}}{n}=\frac{-{f}_{t}}{f(1+{f}^{2})}.\end{array}$$
(10)
$$f(x,t)=j(x)g(t).$$
(11)
$$n=k\frac{{[1+{j}^{2}(x){g}^{2}(t)]}^{1/2}}{g(t)},$$
(12)
$$p\equiv n\frac{\u1e8b}{{(1+{\u1e8b}^{2})}^{1/2}}=\pm kj(x),$$
(13)
$$\begin{array}{cc}x=asin(bt+c);& \psi (x,t,c)\equiv x-asin(bt+c)=0,\end{array}$$
(14)
$$\omega \equiv {\u1e8b}^{2}-{b}^{2}({a}^{2}-{x}^{2})=0.$$
(15)
$$n=k{\left(\frac{1}{{b}^{2}}+{a}^{2}-{x}^{2}\right)}^{1/2}.$$
(16)
$$p\equiv \frac{\u1e8b}{{(1+{\u1e8b}^{2})}^{1/2}}n(x,t).$$
(17)
$$E={\mathit{\int}}_{R({C}_{0})}\text{d}x\text{d}p={\mathit{\int}}_{R({C}_{1})}\text{d}x\text{d}p={\mathit{\int}}_{R({C}_{t})}\text{d}x\text{d}p,$$
(18)
$$E={\mathit{\int}}_{R({C}_{t})}\frac{n}{{(1+{\u1e8b}^{2})}^{3/2}}\text{d}x\text{d}\u1e8b.$$
(19)
$${\omega}_{x}{H}_{p}-{\omega}_{p}{H}_{x}=-{\omega}_{t}.$$
(20)
$$H=-{({n}^{2}-{p}^{2})}^{1/2}.$$
(21)
$${\omega}_{x}p+{\omega}_{p}{n}_{x}n+{({n}^{2}-{p}^{2})}^{1/2}{\omega}_{t}=l(x,t,p),$$
(22)
$${n}^{2}={f}^{2}+{[\mathit{\int}{f}_{t}\text{d}x+a(t)]}^{2},$$
(23)
$$\begin{array}{cc}p=f=\frac{\partial V}{\partial x}& \mathit{\int}{f}_{t}\text{d}x+a(t)=\frac{\partial V}{\partial t},\end{array}$$
(24)
$$n={[{f}^{2}(x)+{a}^{2}(t)]}^{1/2},$$