Abstract

A three-dimensional nonimaging concentrator is an optical system that transforms a given four-parametric manifold of rays reaching a surface (entry aperture) into another four-parametric manifold of rays reaching the receiver. A procedure of design of such concentrators is developed. In general, the concentrators use mirrors and inhomogeneous media (i.e., gradient-index media). Using this method, we obtain an axisymmetrical concentrator with flat-entry aperture and receiver, which collects every ray that forms, at the entry aperture, an angle lower than a given value with the normal to this surface. The concentrator has the maximum concentration allowed by the theorem of conservation of phase-space volume. This is the first known concentrator with such properties. The Welford-Winston edge-ray principle in 3-D geometry is proven under several assumptions. The linear compound parabolic concentrator is derived as a particular case of the procedure of design.

© 1986 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  2. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).
  3. J. C. Minano, “Refractive index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A. 2, 1821–1825 (1985).
    [CrossRef]
  4. J. C. Minano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826–1831 (1985).
    [CrossRef]
  5. See, for example, J. W. Leech, Classical Mechanics (Methuen, London, 1965).
    [CrossRef]
  6. V. Arnold, Les Méthodes Mathématiques de la Mécanique Classique (Mir, Moscow, 1974).
  7. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  8. J. C. Minano, “Cylindrical concentrators as a limit case of toroidal concentrators,” Appl. Opt. 23, 2017–2020 (1984).
    [CrossRef] [PubMed]
  9. See, for example, V. Arnold, Equations Différentielles Ordinaires (Mir, Moscow, 1974).
  10. See, for example, W. S. Massey, Algebraic Topology: An Introduction (Spanish translation) (Reverte, Barcelona, 1972); R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1978).
  11. It is a practical tool in mechanics to use phase spaces not homeo-morphic to IRn. See, for example, the case of the pendulum in Ref. 6.
  12. R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
    [CrossRef]
  13. R. Winston, W. T. Welford, “Ideal flux concentrators as shapes that do not disturb the geometrical vector flux field. A new derivation of the compound parabolic concentrator,” J. Opt. Soc. Am. 69, 536–539 (1979).
    [CrossRef]
  14. See Ref. 3. For a similar procedure in mechanics, see A. S. Galiullim, Inverse Problems of Dynamics (Mir, Moscow, 1984).
  15. The left-hand side of Eq. (8) should contain the term −ωt. Nevertheless, we are interested only in manifolds not depending on the parameter t.
  16. The trajectories of the rays in this medium have the properties of the “straight lines” in the Lovachevski geometry.
  17. See, for example, O. N. Stavroudis, The Optics of Rays, Wave-fronts and Caustics (Academic, New York, 1972).
  18. R. Winston, W. T. Welford, “Two-dimensional concentrators for inhomogeneous media,” J. Opt. Soc. Am. 68, 289–291 (1978).
    [CrossRef]

1985 (2)

J. C. Minano, “Refractive index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A. 2, 1821–1825 (1985).
[CrossRef]

J. C. Minano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826–1831 (1985).
[CrossRef]

1984 (1)

1979 (2)

1978 (1)

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Arnold, V.

See, for example, V. Arnold, Equations Différentielles Ordinaires (Mir, Moscow, 1974).

V. Arnold, Les Méthodes Mathématiques de la Mécanique Classique (Mir, Moscow, 1974).

Galiullim, A. S.

See Ref. 3. For a similar procedure in mechanics, see 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]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

Massey, W. S.

See, for example, W. S. Massey, Algebraic Topology: An Introduction (Spanish translation) (Reverte, Barcelona, 1972); R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1978).

Minano, J. C.

Stavroudis, O. N.

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

Welford, W. T.

Winston, R.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

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

J. C. Minano, “Refractive index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A. 2, 1821–1825 (1985).
[CrossRef]

Other (12)

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

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

See Ref. 3. For a similar procedure in mechanics, see A. S. Galiullim, Inverse Problems of Dynamics (Mir, Moscow, 1984).

The left-hand side of Eq. (8) should contain the term −ωt. Nevertheless, we are interested only in manifolds not depending on the parameter t.

The trajectories of the rays in this medium have the properties of the “straight lines” in the Lovachevski geometry.

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

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

V. Arnold, Les Méthodes Mathématiques de la Mécanique Classique (Mir, Moscow, 1974).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

See, for example, V. Arnold, Equations Différentielles Ordinaires (Mir, Moscow, 1974).

See, for example, W. S. Massey, Algebraic Topology: An Introduction (Spanish translation) (Reverte, Barcelona, 1972); R. Abraham, J. E. Marsden, Foundations of Mechanics, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1978).

It is a practical tool in mechanics to use phase spaces not homeo-morphic to IRn. See, for example, the case of the pendulum in Ref. 6.

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Figures (10)

Fig. 1
Fig. 1

The edge rays of a 3-D axisymmetrical concentrator. This manifold is composed of all the rays reaching the entry aperture forming an angle α0 with the z axis together with the rays reaching the boundaries of the entry aperture and forming an angle less than α0 with the z axis. A similar description of the mainfold can be given at the receiver, replacing α0 with α1.

Fig. 2
Fig. 2

Physical meaning of u, υ, and w. α = cos−1 (au/n); β = cos−1 (/n); γ = cos−1 (cw/n).

Fig. 3
Fig. 3

The edge-ray theorem for 2-D optical systems. T(∂R0) [= T(∂R1)] is a surface formed by the trajectories, in the extended phase space xzp, of the rays of the manifold ∂R0.

Fig. 4
Fig. 4

The inverse theorem. The surface T(∂R0) is given by the equation ω0 = 0.

Fig. 5
Fig. 5

Geometry of a concentrator with curved-entry aperture and receiver.

Fig. 6
Fig. 6

The mirror placed in a plane of symmetry of the trajectories in the xyz space of the rays of the manifold does not modify this manifold.

Fig. 7
Fig. 7

Entry aperture, receiver, and lines of the geometrical vector flux field in the axisymmetrical concentrator.

Fig. 8
Fig. 8

i lines and k lines in a θ = constant plane of the axisymmetrical concentrator.

Fig. 9
Fig. 9

Refractive-index distribution for an axisymmetrical concentrator that collects every ray that forms, at the entry aperture, an angle with the normal to this surface less than 30°. Also shown are two possible profiles of the mirror. When a mirror is used, the rays arriving at the boundaries of the entry aperture and receiver do not belong to the edge-ray manifold (these boundaries are occupied by the mirror).

Fig. 10
Fig. 10

i lines, k lines, and the z axis for the linear symmetric concentrator.

Equations (37)

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d x d z = H p , d y d z = H q , d p d z = H x , d q d z = H y ,
d x d t = H p * , d y d t = H q * , d z d t = H r * , d p d t = H x * , d q d t = H y * , d r d t = H z * ,
d i d t = H u * , d j d t = H υ * , d h d t = H w * , d u d t = H i * , d υ d t = H j * , d w d t = H k * ,
( p , q , r ) = ( u , v , w ) ( i x i y i z j x j y j z k x k y k z ) .
R 0 d x d y d p d q = R i d x d y d p d q = R i d x d y d p d q ,
x = x ( x 0 , y 0 , p 0 , q 0 , z ) y = y ( x 0 , y 0 , p 0 , q 0 , z ) p = p ( x 0 , y 0 , p 0 , q 0 , z ) q = q ( x 0 , y 0 , p 0 , q 0 , z )
n = ( sin 2 α + e 2 c 2 ) 1 / 2
ω i H u * ω i H i * + ω j H υ * ω υ H j * + ω k H w * ω w H k * = f ,
e i a 2 u + e k c 2 w ( n n k a a k u 2 b b k h 2 c c k w 2 ) = f / 2 .
n n k + b k b n 2 + e i a 2 u + e k c 2 w + ( a a k b k b a 2 ) u 2 + ( c c k b k b c 2 ) w 2 = f / 2 .
when ω = 0 ; then w = e and f = 0.
n n k + b h b n 2 + ( c c k b k b c 2 ) e 2 + e k c 2 e + e i a 2 u + ( a a k b k b a 2 ) u 2 = 0.
e i = 0 , a a k b k b a 2 = 0 , n n k + b k b n 2 + ( c c k b k b c 2 ) e 2 + e k c 2 e = 0.
e = E ( k ) ,
b 2 a 2 = [ d M ( i ) d i ] 2 ,
n 2 = c 2 E 2 ( k ) + b 2 N 2 ( i ) ,
| grad M | 2 = 1 ρ 2 .
M ρ 2 + M τ 2 = 1 ρ 2 .
τ = k + R ( k ) [ R 2 ( k ) ρ 2 ] 1 / 2 .
k τ = 1 [ 1 + R R R ( R 2 ρ 2 ) 1 / 2 ]
k ρ = ρ / ( R 2 ρ 2 ) 1 / 2 [ 1 + R R R ( R 2 ρ 2 ) 1 / 2 ]
i = 2 R ρ R + ( R 2 p 2 ) 1 / 2 exp ( 0 k d k R ) .
i = ρ r exp ( 0 1 d k R )
C g = [ exp ( 0 1 d k R ) ] 2 .
R ( k ) = 0.03 k 2 ( 1 k ) 2 ,
E ( k ) = n r { 1 [ exp ( 0 k d k R ) ] 2 / C g } 1 / 2 ,
N ( i ) = i n r / C g 1 / 2 , n r = 1.5.
m = M 2 ( i ) ,
c 2 = L 2 ( k ) ( 1 a 2 / m ) ,
n 2 = 1 + c 2 N 2 ( k ) ,
| grad G | 2 + | grad F | 2 = 1 ,
grad F · grad G = 0 ,
| grad ( F ± G ) | = 1.
M = ln ( R ) ln [ R ρ + ( R 2 ρ 2 1 ) 1 / 2 ] + 0 k d k R .
M τ = k r R [ R ' R ' R ( R 2 ρ 2 ) 1 / 2 + 1 ]
M ρ = k ρ R [ R ' R ' R ( R 2 ρ 2 ) 1 / 2 + 1 ] + R ρ ( R 2 ρ 2 ) 1 / 2
M τ 2 = 1 R 2 , M ρ 2 = 1 ρ 2 1 R 2 ,

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