Abstract

The optical performance of axisymmetric radiation concentrators and illuminators that are derived when two-dimensional edge-ray designs are rotated about their optic axis is investigated. Of particular interest are devices with spherical and cylindrical absorbers or light sources, for which the inherent ray rejection can be substantial. From the principle of etendue (phase-space) conservation, a lower bound for ray rejection can be established. With computer ray tracing, we demonstrate that this bound underestimates the actual ray rejection by only a few percent at most. Hence, to a good approximation, it can be used as an equality in analytic predictions of characteristic efficiency-concentration curves. By designing for absorbers or sources with a bald spot, the full range of efficiency and flux concentration values can be realized and the trade-off between them can be quantified. The optical performance of these edge-ray designs is also compared against fundamental upper bounds on the flux concentration and efficiency of axisymmetric devices.

© 1998 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).
  2. M. E. Moorhead, N. W. Tanner, “Light-reflecting concentrators for photomultipliers with curved photocathodes,” Appl. Opt. 35, 3479–3487 (1996).
    [CrossRef]
  3. H. Ries, N. Shatz, J. Bortz, W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14, 2855–2862 (1997).
    [CrossRef]
  4. A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
    [CrossRef]

1997 (1)

1996 (1)

M. E. Moorhead, N. W. Tanner, “Light-reflecting concentrators for photomultipliers with curved photocathodes,” Appl. Opt. 35, 3479–3487 (1996).
[CrossRef]

1976 (1)

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
[CrossRef]

Bortz, J.

Moorhead, M. E.

M. E. Moorhead, N. W. Tanner, “Light-reflecting concentrators for photomultipliers with curved photocathodes,” Appl. Opt. 35, 3479–3487 (1996).
[CrossRef]

Rabl, A.

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
[CrossRef]

Ries, H.

Shatz, N.

Spirkl, W.

Tanner, N. W.

M. E. Moorhead, N. W. Tanner, “Light-reflecting concentrators for photomultipliers with curved photocathodes,” Appl. Opt. 35, 3479–3487 (1996).
[CrossRef]

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

Winston, R.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

Appl. Opt. (1)

M. E. Moorhead, N. W. Tanner, “Light-reflecting concentrators for photomultipliers with curved photocathodes,” Appl. Opt. 35, 3479–3487 (1996).
[CrossRef]

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

Sol. Energy (1)

A. Rabl, “Comparison of solar concentrators,” Sol. Energy 18, 93–111 (1976).
[CrossRef]

Other (1)

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

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Figures (7)

Fig. 1
Fig. 1

Cross sections of edge-ray designs. θ a , acceptance half-angle. The right-hand and left-hand side reflectors are mirror images of one another. (a) With a flat one-sided slat (disk in 3D) absorber. Incident extreme rays at ±θ a are required to be reflected to the edges of the absorber. Hence reflector AE is the arc of a parabola with the focus at F and the axis parallel to AF. C 2D = AB/EF = 1/sin(θ a ). (b) With a circular (spherical in 3D) absorber of diameter d. Reflector BG is obtained from requiring that incident extreme rays be reflected as tangents to the absorber. Reflector section GE below the shadow line is the involute of a circular arc. C 2D = AB/(πd) = 1/sin(θ a ). (c) With a rectangular (cylindrical in 3D) absorber that includes the upper, but not the lower, edge (disk in 3D). r, edge half-width (cylinder radius in 3D); h, height. The reflectors are a combination of parabolic and involute sections. IJ is a circular arc (the involute of a straight line) of radius h and center at G. JK is the arc of a parabola with its focus at G and GJ as its axis. KB is the arc of a parabola with its focus at F and its axis parallel to GJ. C 2D = AB/[2(h + r) = 1/sin(θ a ).

Fig. 2
Fig. 2

Cross sections of edge-ray concentrators with a bald spot on the absorber. θ a , acceptance half-angle. Reflectors follow from the edge-ray strategy, but with reflector construction starting at the edge of the bald spot. (a) With a circular (spherical in 3D) absorber. Bald-spot size is measured by its arc half-angle φ. d, sphere diameter. (b) With a rectangular (cylindrical in 3D) absorber. Bald-spot size is measured by its height h 0 along the circumferential area. r, edge half-width (cylinder diameter in 3D); h, height.

Fig. 3
Fig. 3

Acceptance angle function (fraction of incident rays that reach the absorber as a function of incidence angle) for an axisymmetric concentrator with a full spherical absorber (zero bald spot) generated from the 2D edge-ray design. θ a = 30°.

Fig. 4
Fig. 4

Characteristic performance curve of relative flux concentration against efficiency for a concentrator with a spherical absorber. Solid curve, analytic prediction for axisymmetric devices generated from 2D edge-ray designs with varying bald-spot size; dashed curve, fundamental upper bound on concentrator performance3; solid triangles, the points at which C rel and η, respectively, reach a value of unity.

Fig. 5
Fig. 5

Plot of relative flux concentration against efficiency for axisymmetric concentrators with a cylindrical absorber generated from the 2D edge-ray designs for varying bald-spot sizes and h/ r = 4. The fundamental upper bound calculated with the formalism of Ref. 3 is also shown, along with the limiting curve for a disk absorber, i.e., the limit of h/ r → 0. Solid triangles are the points at which C rel and η, respectively, reach a value of unity.

Fig. 6
Fig. 6

Nondimensionalized power delivery (η f abs) graphed against bald-spot extent for the spherical absorber concentrator.

Fig. 7
Fig. 7

Cross section of an axisymmetric concentrator for a spherical absorber designed for a virtual disk absorber with the same diameter as the sphere. The reflectors are therefore the same as in Fig. 1(a). Although this device attains a relative concentration of only 1/4, it does so at an efficiency of unity. As such, it is an edge-ray design that realizes the extreme point in Fig. 4. of η = 1 at C rel = 1/4.

Tables (1)

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Table 1 Computer 3D Ray-Trace Simulation Results for the Efficiency of Representative Axisymmetric Edge-Ray Designsa

Equations (21)

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C 2 D = aperture   width absorber   perimeter
C 2 D max = 1 sin θ a ,
C 3 D = aperture   area absorber   area
C 3 D max = 1 sin 2 θ a
C 2 D = D d π - φ ,
C 3 D = D 2 d   cos φ / 2 2 ,
f abs = cos 2 φ / 2 .
C 2 D = D 2 h - h 0 + r ,
C 3 D = D 2 4 r 2 h - h 0 + r ,
f abs = 2 h - h 0 + r 2 h + r .
η = C 2 D 2 C 3 D ,
η = 2   cos φ / 2 π - φ 2 .
η = r 2 h - h 0 + r h - h 0 + r 2 .
η = C flux C 3 D f abs .
C flux = C 2 D 2 f abs .
C 2 D   sin θ a 1 .
C rel = C flux sin 2 θ a = C 2 D   sin θ a 2 f abs .
φ = π - D   sin θ a d ,
h 0 = 2 h + r - D   sin θ a 2 .
P = I η A ap = I η C 3 D A abs = I η C 3 D A abs full f abs .
P IC 3 D A abs full = η f abs ,

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