Abstract

A new family of three-dimensional ideal nonimaging concentrators with rotational symmetry is presented. The flow-line concentrator and the cone concentrator are particular cases of this family. First, we looked for elliptic bundles of rays, i.e., bundles such that the subset of rays passing through any point of the space forms a cone with an elliptic base (this search was done with the Lorentz geometry formalism.) Second, the concentrators, defined by their reflectors and receiver shapes, were derived from these elliptic bundles with the flow-line design method.

© 1996 Optical Society of America

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References

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  1. R. Winston, H. Ries, “Nonimaging reflectors as functionals of the acceptance angle,” J. Opt. Soc. Am. A 10, 1902–1908 (1993).
    [CrossRef]
  2. H. Ries, R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994).
    [CrossRef]
  3. A. Rabl, “Reflector design for illumination with extended sources: the basic solution,” in Nonimaging Optics: Maximum Efficiency Light Transfer II, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2016, 66–77 (1993).
    [CrossRef]
  4. J. M. Gordon, P. Kashin, A. Rabl, “Nonimaging reflectors for efficient uniform illumination,” Appl. Opt. 31, 6027–6035 (1992).
    [CrossRef] [PubMed]
  5. Juan C. Minano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1986).
    [CrossRef]
  6. G. W. Forbes, I. M. Bassett, “An axially symmetric variable-angle non-imaging transformer,” Opt. Acta 29, 1283–1297 (1982).
    [CrossRef]
  7. I. M. Bassett, G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
    [CrossRef]
  8. R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
    [CrossRef]
  9. R. Winston, W. T. Welford, “Ideal flux concentrators as shapes that do not disturb the geometrical vector flux field, and new derivation of the compound parabolic concentrator,” J. Opt. Soc. Am. 69, 536–539 (1979).
    [CrossRef]
  10. B. O’Neill, Semi-Reimannian Geometry with Applications to Relativity (Academic, New York, 1983), pp. 1–96.
  11. J. C. Minano, “Optical confinement in photovoltaic,” in Physical Limitations to Photovoltaic Solar Energy Conversion, A. Luque, G. L. Araújo, eds. (Hilger, Bristol, UK, 1990), pp. 50–83.
  12. H. Flanders, Differential Forms (Academic, New York, 1963), pp. 82–111.

1994 (1)

1993 (1)

1992 (1)

1986 (1)

1982 (2)

G. W. Forbes, I. M. Bassett, “An axially symmetric variable-angle non-imaging transformer,” Opt. Acta 29, 1283–1297 (1982).
[CrossRef]

I. M. Bassett, G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

1979 (2)

Bassett, I. M.

G. W. Forbes, I. M. Bassett, “An axially symmetric variable-angle non-imaging transformer,” Opt. Acta 29, 1283–1297 (1982).
[CrossRef]

I. M. Bassett, G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

Flanders, H.

H. Flanders, Differential Forms (Academic, New York, 1963), pp. 82–111.

Forbes, G. W.

I. M. Bassett, G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

G. W. Forbes, I. M. Bassett, “An axially symmetric variable-angle non-imaging transformer,” Opt. Acta 29, 1283–1297 (1982).
[CrossRef]

Gordon, J. M.

Kashin, P.

Minano, J. C.

J. C. Minano, “Optical confinement in photovoltaic,” in Physical Limitations to Photovoltaic Solar Energy Conversion, A. Luque, G. L. Araújo, eds. (Hilger, Bristol, UK, 1990), pp. 50–83.

Minano, Juan C.

O’Neill, B.

B. O’Neill, Semi-Reimannian Geometry with Applications to Relativity (Academic, New York, 1983), pp. 1–96.

Rabl, A.

J. M. Gordon, P. Kashin, A. Rabl, “Nonimaging reflectors for efficient uniform illumination,” Appl. Opt. 31, 6027–6035 (1992).
[CrossRef] [PubMed]

A. Rabl, “Reflector design for illumination with extended sources: the basic solution,” in Nonimaging Optics: Maximum Efficiency Light Transfer II, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2016, 66–77 (1993).
[CrossRef]

Ries, H.

Welford, W. T.

Winston, R.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (2)

G. W. Forbes, I. M. Bassett, “An axially symmetric variable-angle non-imaging transformer,” Opt. Acta 29, 1283–1297 (1982).
[CrossRef]

I. M. Bassett, G. W. Forbes, “A new class of ideal non-imaging transformers,” Opt. Acta 29, 1271–1282 (1982).
[CrossRef]

Other (4)

B. O’Neill, Semi-Reimannian Geometry with Applications to Relativity (Academic, New York, 1983), pp. 1–96.

J. C. Minano, “Optical confinement in photovoltaic,” in Physical Limitations to Photovoltaic Solar Energy Conversion, A. Luque, G. L. Araújo, eds. (Hilger, Bristol, UK, 1990), pp. 50–83.

H. Flanders, Differential Forms (Academic, New York, 1963), pp. 82–111.

A. Rabl, “Reflector design for illumination with extended sources: the basic solution,” in Nonimaging Optics: Maximum Efficiency Light Transfer II, R. Winston, R. L. Holman, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2016, 66–77 (1993).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Cone of edge rays at the point X and two flow lines of the FLC.

Fig. 2
Fig. 2

The light cone at X is formed by the set of lightlike vectors of TxM.

Fig. 3
Fig. 3

The basis Bx′ = {J, U, V} defines the three planes P1, P2, and P3 of symmetry of the elliptic light cone.

Fig. 4
Fig. 4

Elliptic bundles 1–6. The points of the shaded regions do not belong to M.

Fig. 5
Fig. 5

Elliptic bundle 7. The 3D representation is needed because there are no meridian rays in this bundle. M is the region bounded by the hyperboloid not containing the x1 axis.

Fig. 6
Fig. 6

Cross section of the concentrators derived from elliptic bundles 1–6. Case 1 corresponds to the FLC, and the intermediate case between cases 4 and 5 (in which |G−1| = 0 is a sphere and the flow lines are straight lines) corresponds to the CC.

Fig. 7
Fig. 7

Flow lines of elliptic bundle 7. These are circumferences contained in planes parallel to the plane x1 = 0 and centered on the x1 axis.

Tables (2)

Tables Icon

Table 1 Classification of the Elliptic Bundles

Tables Icon

Table 2 Flow Lines of the Elliptic Bundles

Equations (34)

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λ 1 Y 1 2 + λ 2 Y 2 2 + λ 3 Y 3 2 = 0 ,
λ 2 Y 2 2 + λ 3 Y 3 2 = - λ 1 .
γ ¨ k ( t ) + [ γ ˙ ( t ) ] t Γ k [ γ ( t ) ] γ ˙ ( t ) = 0 [ γ ˙ ( t ) ] t G [ γ ( t ) ] γ ˙ ( t ) = 0 } γ ¨ k ( t ) = 0.
Γ i j k = 1 / 2 g k m ( g j m , i + g m i , j - g i j , m ) ,
Γ k = f k G ,             k { 1 , 2 , 3 } ,
g i m , j = g m k f k g i j + g i k f k g j m ,             i , j , m { 1 , 2 , 3 } .
[ - a x 1 2 - 2 b 1 x 1 + d - a x 1 x 2 - b 2 x 1 - b 1 x 2 + k - a x 1 x 3 - b 3 x 1 - b 1 x 3 + l - a x 1 x 2 - b 2 x 1 - b 1 x 2 + k - a x 2 2 - 2 b 2 x 2 + m - a x 2 x 3 - b 3 x 2 - b 2 x 3 + n - a x 1 x 3 - b 3 x 1 - b 1 x 3 + l - a x 2 x 3 - b 3 x 2 - b 2 x 3 + n - a x 3 2 - 2 b 3 x 3 + p ] ,
G = 1 G - 1 [ - a m ( x 2 2 + x 3 2 ) + m 2 ( a x 1 + b ) m x 2 ( a x 1 + b ) m x 3 ( a x 1 + b ) m x 2 - ( a x 1 2 + 2 b x 1 - d ) m - ( a d + b 2 ) x 3 2 ( a d + b 2 ) x 2 x 3 ( a x 1 + b ) m x 3 ( a d + b 2 ) x 2 x 3 - ( a x 1 2 + 2 b x 1 - d ) m - ( a d + b 2 ) x 2 2 ] ,
G - 1 = - ( a x 1 2 + 2 b x 1 - d ) m 2 - ( a d + b 2 ) m ( x 2 2 + x 3 2 ) .
G - 1 x 3 = 0 = [ - a x 1 2 - 2 b x 1 + d - x 2 ( a x 1 + b ) - x 3 ( a x 1 + b ) - x 2 ( a x 1 + b ) - a x 2 2 + m - a x 2 x 3 - x 3 ( a x 1 + b ) - a x 2 x 3 - a x 3 2 + m ] x 3 = 0 = [ - a x 1 2 - 2 b x 1 + d - x 2 ( a x 1 + b ) 0 - x 2 ( a x 1 + b ) - a x 2 2 + m 0 0 0 m ] .
G - 1 = - ( a x 1 2 + 2 b x 1 - d ) m 2 - ( a d + b 2 ) m ( x 2 2 + x 3 2 ) = 0 ,
G = 1 G - 1 × [ 1 - ( x 2 2 + x 3 2 ) x 1 x 2 x 1 x 3 x 1 x 2 1 - ( x 1 2 + x 3 2 ) x 2 x 3 x 1 x 3 x 2 x 3 1 - ( x 1 2 + x 2 2 ) ] ,
G - 1 = 1 - ( x 1 2 + x 2 2 + x 3 2 ) < 0.
λ 1 = [ 1 - ( x 1 2 + x 2 2 + x 3 2 ) ] - 1 ,             λ 2 = λ 3 = 1 ,
γ ( t ) = ( c 1 e t , c 2 e t , c 3 e t ) .
1 - ( x 1 2 + x 2 2 + x 3 2 ) = 0.
G = ( g i j ) ,             f = [ f 1 f 2 f 3 ] , d x = [ d x 1 d x 2 d x 3 ] ,             H = d x f t + f d x t .
d G = GHG ,
0 = d 2 G = d G HG + G d HG - GH d G .
0 = GHG HG + G d HG - GH GHG = G d HG ,
d H = - d x d f t + d f d x t = 0 ,
f 1 = a x 1 + b 1 , f 2 = a x 2 + b 2 , f 3 = - a x 3 + b 3 ,
I = G - 1 G 0 = d I = d G - 1 G + G - 1 d G .
d G - 1 = - H .
F = [ 1 0 0 0 cos α - sin α 0 sin α cos α ] .
Y t G ( X ) Y = 0 Y t F t G ( F X ) F Y = 0.
G ( X ) = h ( X ) F G ( F X ) F ,
h ( X ) G - 1 ( X ) = F G - 1 ( F X ) F .
b 2 = b 3 = k = l = n = 0 ,             p = m .
γ ( t ) = [ x 1 x 2 x 3 ] = [ p 1 p 2 0 ] + t [ Y 1 Y 2 Y 3 ] .
[ a m 2 Y 1 2 + ( a d + b 2 ) m ( Y 2 2 + Y 3 2 ) ] t 2 + 2 [ a m 2 p 1 Y 1 + b m 2 Y 1 + ( a d + b 2 ) m p 2 Y 2 ] t + [ m 2 ( a p 1 2 + 2 b p 1 - d ) + ( a d + b 2 ) m p 2 2 ] = 0.
Δ = m 2 ( a d + b 2 ) G - 1 ( P ) [ Y t G ( P ) Y ] .
Y t G ( P ) Y = [ ( - a m p 2 2 m 2 ) Y 1 2 - ( a p 1 2 + 2 b p 1 - d ) m ( Y 2 2 + Y 3 2 ) + 2 ( a p 1 + b ) m p 2 Y 1 Y 2 - ( a d + b 2 ) p 2 2 Y 3 2 ] / G - 1 ( P ) .
x 2 2 + x 3 2 = m a + Y t G ( P ) Y m a Y 1 2 .

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