Abstract

A new method to obtain elliptic ray bundles in three-dimensional geometry is presented. This type of bundle is of special interest in nonimaging optics because the only known three-dimensional ideal concentrators with homogeneous media transmit elliptic bundles. First, we reformulate the condition for a bundle to be elliptic; that is, the rays passing through any point of the space form a cone with an elliptic base. The solution found includes all the elliptic bundles obtained by other methods as a particular case. Second, we look for the concentrators defined by the new elliptic bundles with the flow-line design method of nonimaging optics. Most of these bundles do not generate new devices but are transmitted ideally through the already known three-dimensional ideal concentrators.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Miñano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1986).
    [CrossRef]
  2. G. W. Forbes, I. M. Basset, “An axially symmetric variable angle nonimaging transformer,” Opt. Acta 29, 1283–1297 (1982).
    [CrossRef]
  3. R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
    [CrossRef]
  4. 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]
  5. M. Gutiérrez, J. C. Miñano, C. Vega, P. Benı́tez, “Application of Lorentz geometry to nonimaging optics: new three-dimensional ideal concentrators,” J. Opt. Soc. Am. A 13, 532–540 (1996).
    [CrossRef]
  6. See, for instance, J. G. Semple, G. T. Kneebone, Algebraic Projective Geometry (Oxford U. Press, Oxford, UK, 1952), p. 266.
  7. J. C. Miñano, “Optical confinement in photovoltaics,” in Physical Limitations to Photovoltaic Solar Energy Conversion, A. Luque, G. L. Araújo, eds. (Hilger, Bristol, UK, 1990), pp. 50–83.

1996 (1)

1986 (1)

1982 (1)

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

1979 (2)

Basset, I. M.

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

Beni´tez, P.

Forbes, G. W.

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

Gutiérrez, M.

Kneebone, G. T.

See, for instance, J. G. Semple, G. T. Kneebone, Algebraic Projective Geometry (Oxford U. Press, Oxford, UK, 1952), p. 266.

Miñano, J. C.

Semple, J. G.

See, for instance, J. G. Semple, G. T. Kneebone, Algebraic Projective Geometry (Oxford U. Press, Oxford, UK, 1952), p. 266.

Vega, C.

Welford, W. T.

Winston, R.

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

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

Other (2)

See, for instance, J. G. Semple, G. T. Kneebone, Algebraic Projective Geometry (Oxford U. Press, Oxford, UK, 1952), p. 266.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Flow-line design method in 3D geometry applied to bundles such that the cone formed by the rays passing through any point X is symmetric with respect to the plane.

Fig. 2
Fig. 2

Vectors {J, U, V} define the planes of symmetry for the cone of edge rays of the elliptic bundle at point X.

Fig. 3
Fig. 3

The cone of edge rays that results at point X is not, in general, an elliptic cone, although the cones at x1=0 are elliptic.

Fig. 4
Fig. 4

Cone of edge rays that passes through a point X of plane x3=0 for elliptic bundles 1–4.

Fig. 5
Fig. 5

Schematic representation of elliptic bundles 5–8.

Fig. 6
Fig. 6

In elliptic bundles 9 and 10 the cones of edge rays are not symmetrical with respect to the meridian planes. The flow lines are helices with constant pitch.  

Fig. 7
Fig. 7

The CC with linear symmetry transmits not only the conventional linear bundle but also elliptic bundle 8 (which includes the former as a limit case).

Fig. 8
Fig. 8

In bundle 10 with p=f, the path shown is normal at every point to U if γ>0 and to J if γ<0, but it is not closed. This is equivalent to the nonintegrability of these vector fields.

Tables (3)

Tables Icon

Table 1 Some Characteristics of the Elliptic Bundles with γ=0, s2-ap=0, a  0, mf 0 (ρ=x22+x33)

Tables Icon

Table 2 Some Characteristics of the Elliptic Bundles with γ=0, s=0, a=0, mfp  0 (ρ=x22+x33)

Tables Icon

Table 3 Some Characteristics of the Elliptic Bundles with γ  0, s=0, a=0, m=0, pf  0 (ρ=x22+x33)

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

YtG(X)Y=0,
λ1Y12+λ2Y22+λ3Y32=0.
YtH(x2, x3)Y=0.
X2=x2-y2y1 x1,
X3=x3-y3y1 x1,
YtHx2-y2y1 x1, x3-y3y1 x1Y=0.
YtHx2-y2y1 x1, x3-y3y1 x1YYtG(X)Y
forallYandXofR.
f(y2, y3)=(1y2y3)H(x2-y2x1, x3-y3x1)1y2y3.
f(y2, y3)=(1y2y3)G(x1, x2, x3)1y2y3+o(y2, y3),
g11=h11,
g12=h12-12 h11x2 x1,
g13=h13-12 h11x3 x1,
g22=h22-2 h12x2 x1+12 2h11x22 x12,
g23=h23-h12x3+h13x2x1+12 2h11x2x3 x12,
g33=h33-2 h13x3 x1+12 2h11x32 x12
3h11x23=3h11x22x3=3h11x2x32=3h11x33=0,
2h12x22=2h13x32=2 2h12x2x3+2h13x22=2 2h13x2x3+2h12x32=0,
h22x2=h33x3=h22x3+2 h23x2=h33x2+2 h23x3=0.
h11=ax22+bx32+cx2x3+dx2+ex3+f,
h12=(rx3+s)x2-tx32+γx3+δ,
h13=(tx2+u)x3-rx22+αx2+β,
h22=mx32-2Bx3+p,
h23=-mx2x3+Bx2+Dx3+E,
h33=mx22-2Dx2+q,
g11=ax22+bx32+cx2x3+dx2+ex3+f,
g12=(rx3+s)x2-tx32+γx3+δ-12(2ax2+cx3+d)x1,
g13=(tx2+u)x3-rx22+αx2+β-12(2bx3+cx2+e)x1,
g22=mx32-2Bx3+p-2(rx3+s)x1+ax12,
g23=-mx2x3+Bx2+Dx3+E-(-tx3+γ-rx2+α)x1+12cx12,
g33=mx22-2Dx2+q-2(tx2+u)x1+bx12.
G=a(x22+x32)+f(-ax1+s)x2+γx3(-ax1+s)x3-γx2(-ax1+s)x2+γx3mx32+ax12-2sx1+p-mx2x3(-ax1+s)x3-γx2-mx2x3mx22+ax12-2sx1+p.
G|x3=0ax22+f(-ax1+s)x20(-ax1+s)x2ax12-2sx1+p000mx22+ax12-2sx1+p.
G|x3=0=ax22+f-ax1x20-ax1x2ax12000mx22+ax12.
λ1=(ax12+ax22+f )-[(ax12+ax22+f )2-4afx12]1/22,
λ2=(ax12+ax22+f )+[(ax12+ax22+f )2-4afx12]1/22,
λ3=mx22+ax12.
X22(-f/a)+X32(-fx12/λ3)=1.
G|x3=0=f000p000mx22+p,
G|x3=0=f0γx20p0γx20p,
λ1=(f+p)+[(f-p)2+4γ2x22]1/22,
λ2=p,
λ3=(f+p)-[(f-p)2+4γ2x22]1/22.
φ=arctan2γρ-(p-f )+[(p-f )2+4γ2ρ2]1/2.

Metrics