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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References

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  1. S. Nemoto, I. Makimoto, “Refractive-index distribution for a prescribed ray path,” J. Opt. Soc. Am. 69, 450 (1979).
    [CrossRef]
  2. H. A. Buchdahl, “Rays in gradient-index media: separable systems,” J. Opt. Soc. Am. 63, 46 (1973) (this research is an analysis focused to solve inverse problems);K. Maeda, J. Hamasaki, “A method of determining the refractive-index profile of a lenslike medium,” J. Opt. Soc. Am. 67, 1672 (1977).
    [CrossRef]
  3. A. S. Galiullim, Inverse Problems of Dynamics (Mir, Moscow, 1984).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  5. J. A. Arnaud, “Application of the mechanical theory of light to fiber optics,” J. Opt. Soc. Am. 65, 174 (1975).
    [CrossRef]
  6. See, for example, O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).
  7. W. H. Southwell, “Sine-wave optical paths in gradient-index media,” J. Opt. Soc. Am. 61, 1715 (1971).
    [CrossRef]
  8. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  9. J. C. Miñano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826 (1985).
    [CrossRef]
  10. See, for example, J. W. Leech, Classical Mechanics (Methuen, London, 1965).
    [CrossRef]
  11. In this example ω= constant turns out to be a first integral. Note that if we took ω as ω= [p2− f2(x, t)] R(x, t, p), where R is an arbitrary function, then we would obtain the same solution, but ω= constant would not be a first integral. Only ω= 0 would still be a particular integral.

1985 (1)

1979 (1)

1975 (1)

1973 (1)

1971 (1)

Arnaud, J. A.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Buchdahl, H. A.

Galiullim, A. S.

A. S. Galiullim, Inverse Problems of Dynamics (Mir, Moscow, 1984).

Leech, J. W.

See, for example, J. W. Leech, Classical Mechanics (Methuen, London, 1965).
[CrossRef]

Makimoto, I.

Miñano, J. C.

Nemoto, S.

Southwell, W. H.

Stavroudis, O. N.

See, for example, O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

Welford, W. T.

W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

Winston, R.

W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

Other (6)

See, for example, J. W. Leech, Classical Mechanics (Methuen, London, 1965).
[CrossRef]

In this example ω= constant turns out to be a first integral. Note that if we took ω as ω= [p2− f2(x, t)] R(x, t, p), where R is an arbitrary function, then we would obtain the same solution, but ω= constant would not be a first integral. Only ω= 0 would still be a particular integral.

W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).

See, for example, O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, New York, 1972).

A. S. Galiullim, Inverse Problems of Dynamics (Mir, Moscow, 1984).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (3)

Fig. 1
Fig. 1

Surface of the space xt representing the one-parameter manifold of rays of sinusoidal trajectory in the xt plane with constant amplitude a (x axis) and constant period 2π/b (t axis).

Fig. 2
Fig. 2

The surface ω = 0 connects the closed lines C0 and C1, which are contained in the planes t = t0 and t = t1, respectively. The surface is closed between t0tt1.

Fig. 3
Fig. 3

Rectangular tube in the xtp space. The trajectories in this space of three rays belonging to ω = 0 are represented assuming that n = (1 + K12)1/2 if |x| < K2 and n = 1 if |x| > K2, i.e., assuming that a(t) = in Eq. (24).

Equations (24)

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= d x / d t .
ω ( x , t , d x d t ) = 0 .
F x = d ( F ) d t = F x + F t + F ,
= ( 1 + 2 ) ( n x n n t n ) ,
d ω d t = ω x + ω t + ω ,
ω x + ω t + ω ( 1 + 2 ) ( n x n n t n ) = l ( x , t , ) ,
f x f f t + ( 1 + f 2 ) ( n x n n t n f ) = 0 ,
f x f f t ( 1 + f 2 ) ( n x n + n t n f ) = 0 ,
n x n = f x f 1 + f 2 , n t n = f t f ( 1 + f 2 ) .
f ( x , t ) = j ( x ) g ( t ) .
n = k [ 1 + j 2 ( x ) g 2 ( t ) ] 1 / 2 g ( t ) ,
p n ( 1 + 2 ) 1 / 2 = ± k j ( x ) ,
x = a sin ( b t + c ) ; ψ ( x , t , c ) x a sin ( b t + c ) = 0 ,
ω 2 b 2 ( a 2 x 2 ) = 0 .
n = k ( 1 b 2 + a 2 x 2 ) 1 / 2 .
p ( 1 + 2 ) 1 / 2 n ( x , t ) .
E = R ( C 0 ) d x d p = R ( C 1 ) d x d p = R ( C t ) d x d p ,
E = R ( C t ) n ( 1 + 2 ) 3 / 2 d x d .
ω x H p ω p H x = ω t .
H = ( n 2 p 2 ) 1 / 2 .
ω x p + ω p n x n + ( n 2 p 2 ) 1 / 2 ω t = l ( x , t , p ) ,
n 2 = f 2 + [ f t d x + a ( t ) ] 2 ,
p = f = V x f t d x + a ( t ) = V t ,
n = [ f 2 ( x ) + a 2 ( t ) ] 1 / 2 ,

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