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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References

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  1. R. Winston, “Light collection within the framework of geometrical optics,” J. Opt. Soc. Am. 60, 245 (1970).
    [CrossRef]
  2. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978).
  3. R. Winston, W. T. Welford, “Two-dimensional concentrators for inhomogeneous media,” J. Opt. Soc. Am. 68, 289 (1978).
    [CrossRef]
  4. W. T. Welford, R. Winston, “Two-dimensional concentrators with refracting optics,” J. Opt. Soc. Am. 69, 917 (1979).
    [CrossRef]
  5. W. T. Welford, Aberrations of Symmetrical Optical System (Academic, New York, 1974).
  6. J. C. Miñano, “Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays,” J. Opt. Soc. Am. A 2, 1821 (1985).
    [CrossRef]
  7. In the case when the concentrator does not use mirrors, then ω= 0 must be a closed surface, and so the equation u(x, t) = l(x, t) must have as its solution two lines in the x−t plane that connect the boundaries of the entry aperture with those of the collector. It is obvious that Gx= 0 in those lines [Gx= (u− l)/2]. By application of Eq. (4) it is obtained that Gt or Ft must also be equal to zero in those lines. If Ft= 0 (and Gt≠ 0) the value of q for the rays of the upper manifold (q= Ft+ Gt) or the value of q for the rays of the lower manifold (q=Ft−Gt) does not fulfill the second assumption used for proving the edge-ray theorem. Therefore Gx= Gt= 0 in the lines u= l, and so they are level lines of G. In the case when the concentrator uses mirrors we shall see that, since these mirrors must reflect the upper manifold into the lower one and vice versa, they must be placed in level lines of G. Therefore we also conclude that two level lines of G must connect, in the x−t plane, the boundaries of the collector with those of the entry aperture.
  8. R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532 (1979).
    [CrossRef]
  9. 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 (1979).
    [CrossRef]
  10. The expression of J in terms of G and F is given byJ=2(Gx2+Gt2Fx2+Ft2)1/2(Fx,Ft)=2(−Gt,Gx).Note that the zero divergence of the vector field J is here reflected in the independence on the order of derivation of the second derivatives of G.
  11. A. Rabl, R. Winston, “Ideal concentrators for finite sources and restricted exit angles,” Appl. Opt. 15, 2880 (1976).
    [CrossRef] [PubMed]

1985 (1)

1979 (3)

1978 (1)

1976 (1)

1970 (1)

Miñano, J. C.

Rabl, A.

Welford, W. T.

Winston, R.

Appl. Opt. (1)

J. Opt. Soc. Am. (5)

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

Other (4)

In the case when the concentrator does not use mirrors, then ω= 0 must be a closed surface, and so the equation u(x, t) = l(x, t) must have as its solution two lines in the x−t plane that connect the boundaries of the entry aperture with those of the collector. It is obvious that Gx= 0 in those lines [Gx= (u− l)/2]. By application of Eq. (4) it is obtained that Gt or Ft must also be equal to zero in those lines. If Ft= 0 (and Gt≠ 0) the value of q for the rays of the upper manifold (q= Ft+ Gt) or the value of q for the rays of the lower manifold (q=Ft−Gt) does not fulfill the second assumption used for proving the edge-ray theorem. Therefore Gx= Gt= 0 in the lines u= l, and so they are level lines of G. In the case when the concentrator uses mirrors we shall see that, since these mirrors must reflect the upper manifold into the lower one and vice versa, they must be placed in level lines of G. Therefore we also conclude that two level lines of G must connect, in the x−t plane, the boundaries of the collector with those of the entry aperture.

The expression of J in terms of G and F is given byJ=2(Gx2+Gt2Fx2+Ft2)1/2(Fx,Ft)=2(−Gt,Gx).Note that the zero divergence of the vector field J is here reflected in the independence on the order of derivation of the second derivatives of G.

W. T. Welford, Aberrations of Symmetrical Optical System (Academic, New York, 1974).

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

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Figures (10)

Fig. 1
Fig. 1

The points of the xp plane enclosed by C0 represent the manifold of rays arriving at the entry aperture that reach the collector with an x coordinate and with an optical direction cosine p such that its representation in the xp plane is enclosed by C1. We also assume that every point enclosed by C1 corresponds to a ray that has its representation at t = 0 in a point enclosed by C0.

Fig. 2
Fig. 2

Surface ω(x, t, p) = 0 representing the trajectories of the extreme rays in the xtp space. The values of |p| at ω = 0 are lower than n(x, t) by hypothesis in the region 0 ≤ t < 1 (q >0). The three possible trajectories of a ray crossing t = 0 through a point enclosed by C0 are also shown. The hypothesis of the theorem ensures that the trajectory of a ray in the x, t, p space is continuous.

Fig. 3
Fig. 3

The edge-ray theorem when using mirrors. The surface representing the mirror in the xtp space is formed by parallels to the p axis, which cross the x, t plane in the mirror line. Region U is the one bounded by the mirror’s surface, by the planes t = 0 and t = 1, and by the upper and the lower manifolds.

Fig. 4
Fig. 4

Level lines of functions G and F in a CPC with 30° of semi-acceptance angle and n = 1. Line F = 0.19 is the concentrator’s entry aperture, and line F = 0.961 is its collector. The mirrors are placed in lines G = ±0.19.

Fig. 5
Fig. 5

Trajectories in the x, t plane of the rays in the upper and lower manifolds of the CPC.

Fig. 6
Fig. 6

Function G of a compound triangular concentrator (see Fig. 9).

Fig. 7
Fig. 7

Function F of a compound triangular concentrator.

Fig. 8
Fig. 8

Level lines of F and G in the CTC and definition of the regions A, B, C, and D.

Fig. 9
Fig. 9

Two compound triangular concentrators whose characteristics are given in Table 2. The trajectories of two extreme rays are also shown. The angles αu and α1 are the ones formed by a ray of the upper and lower manifolds, respectively, with the t axis, which is normal to the entry aperture and to the collector and goes from up to down {α = tan−1[(Fx ± Gx)/(Ft ± Gt)]}. When a ray of the upper manifold is reflected in the mirrors it becomes a ray of the lower manifold, and vice versa.

Fig. 10
Fig. 10

The optical medium to be designed in this concentrator occupies region B. In regions A and C several rays of the upper manifold (continuous line) and also of the lower one (dashed line) were drawn. The limits between regions AB and BC are lines F = constant. Several lines of G = constant are drawn in region B.

Tables (2)

Tables Icon

Table 1 Partial Derivatives of the Functions F and G in the Compound Triangular Concentrator for the x-Positive Side of the Concentrator (see Fig. 8)

Tables Icon

Table 2 Partial Derivatives of the Functions F and G, Indices of Refraction, Angles Formed by the Extreme Rays with the t Axis, and Geometrical Characteristics of the Concentrators of Fig. 9 (for the x-Positive Side of the Concentrators, i.e., the Left-Hand Side of Fig. 9)

Equations (24)

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n 2 = U x 2 + U t 2 ,
n 2 = L x 2 + L t 2 ,
n 2 = F x 2 + F t 2 + G x 2 + G t 2 ,
F x G x + F t G t = 0 ,
G x = p 0 , G t = 0 , F x = 0 , F t = ( n 0 2 p 0 2 ) 1 / 2 ;
G x = p 1 , G t = 0 , F x = 0 , F t = ( n 1 2 p 1 2 ) 1 / 2 .
J = K ( x , t ) × ( F x , F t ) .
G = p 0 x , F = A e 2 p 0 + q 0 t .
2 G = ( x + A e 2 ) p 0 + t q 0 A e + A c 2 p 0 q 0 + n c [ ( x + A c 2 ) 2 + ( 1 t ) 2 ] 1 / 2 , 2 F = ( x + A e 2 ) p 0 + t q 0 + A e + A c 2 p 0 + q 0 n c [ ( x + A c 2 ) 2 + ( 1 t ) 2 ] 1 / 2 .
2 G = n c [ ( A c 2 + x ) 2 + ( 1 t ) 2 ] 1 / 2 n c [ ( A c 2 x ) 2 + ( 1 t ) 2 ] 1 / 2 , 2 F = ( A e + A c ) p 0 + 2 q 0 n c [ ( A c 2 x ) 2 + ( 1 t ) 2 ] 1 / 2 n c [ ( A c 2 + ( 1 t ) 2 ] 1 / 2 ,
A e + A c 2 = p 0 q 0 ,
A e A c = 1 sin α = n c p 0 .
( F x , F t ) ( F x + G x , F t + G t ) = ( F x , F t ) ( F x G x , F t G t ) = F x 2 + F t 2 > 0 ,
b a
E 2 A e
2 b ( A e A c ) 4 a + ( A e A c ) A e
4 b 4 a + ( A e A c ) A e
E 2 [ A e a ( A e A c ) ]
E ( A e A c ) 4 [ A e a ( A e A c ) ]
b [ 4 ( 1 a ) ( A e A c ) A c ] [ 4 a + ( A e A c ) A e ] ( 1 a )
E 2 A c
2 b ( A e A c ) 4 a + ( A e A c ) A e
4 b 4 a + ( A e A c ) A e
J=2(Gx2+Gt2Fx2+Ft2)1/2(Fx,Ft)=2(Gt,Gx).

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