Abstract

An upper limit on concentration for any optical device has previously been derived from the conservation of étendue. In this contribution we derive more stringent upper limits for the efficiency and the concentration of rotationally symmetric optical devices that are a consequence of skewness conservation. If the desired source and target have different skewness distributions, then losses or dilution or both will limit the performance of the optical system. We calculate the limiting curve of efficiency versus concentration and provide a design example that is virtually at this limit. We conjecture that even rotationally symmetric problems may benefit from asymmetric optical systems.

© 1997 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989).
  2. H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in NonimagingOptics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. SPIE1528, 7–14 (1991).
    [CrossRef]
  3. H. Ries, “Thermodynamic limitations of the concentration of electromagnetic radiation,” J. Opt. Soc. Am. 72, 380–385 (1982),also included in Selected Papers in Nonimaging Optics, SPIE Milestone Series Vol. MS 106, Roland Winston, ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1995).
    [CrossRef]
  4. M. E. Moorhead, N. W. Tanner, “Light-reflecting concentrators for photomultipliers with curved photocathodes,” Appl. Opt. 35, 3478–3487 (1996).
    [CrossRef] [PubMed]
  5. J. Karni, H. Ries, “Concepts for high concentration primary reflectors in central receiver system,” in Proceedings of the 7th International Symposium on Solar Thermal Concentrating Technologies (Institute for High Temperatures of the Russian Academy of Science (IVTAN), Moscow, Russia, 1994), Vol. 4, pp. 796–801.
  6. N. Shatz, J. Bortz, “An inverse engineering perspective on nonimaging optical design,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 136–156 (1995).
    [CrossRef]
  7. A. Rabl, “Solar concentrators with maximal concentration for cylindrical absorbers,” Appl. Opt. 15, 1871–1873 (1976); note also Ref. 11 below.
    [CrossRef] [PubMed]
  8. J. M. Gordon, H. Ries, “Tailored edge-ray concentrators (TERC's) as ideal second stages forFresnel reflectors,” Appl. Opt. 32, 2243–2251 (1993).
    [CrossRef] [PubMed]
  9. H. Ries, R. Winston, “Tailored edge ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994), also included in Selected Papers in Nonimaging Optics, SPIE Milestone Series Vol. MS 106, Roland Winston, ed. (SPIE Optical Engineering Press, Bellingham, Wash.1995).
    [CrossRef]
  10. H. Ries, A. Segal, J. Karni, “Extracting concentrated guided light,” Appl. Opt. 36, 2869–2874 (1997).
    [CrossRef] [PubMed]
  11. A. Rabl, “Solar concentrators with maximal concentration for cylindrical absorbers:erratum,” Appl. Opt. 16, 15 (1977).
    [CrossRef]

1997 (1)

1996 (1)

1994 (1)

1993 (1)

1982 (1)

1977 (1)

1976 (1)

Bortz, J.

N. Shatz, J. Bortz, “An inverse engineering perspective on nonimaging optical design,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 136–156 (1995).
[CrossRef]

Gordon, J. M.

Karni, J.

H. Ries, A. Segal, J. Karni, “Extracting concentrated guided light,” Appl. Opt. 36, 2869–2874 (1997).
[CrossRef] [PubMed]

J. Karni, H. Ries, “Concepts for high concentration primary reflectors in central receiver system,” in Proceedings of the 7th International Symposium on Solar Thermal Concentrating Technologies (Institute for High Temperatures of the Russian Academy of Science (IVTAN), Moscow, Russia, 1994), Vol. 4, pp. 796–801.

Moorhead, M. E.

Rabl, A.

Ries, H.

Segal, A.

Shatz, N.

N. Shatz, J. Bortz, “An inverse engineering perspective on nonimaging optical design,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 136–156 (1995).
[CrossRef]

Smestad, G.

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in NonimagingOptics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. SPIE1528, 7–14 (1991).
[CrossRef]

Tanner, N. W.

Welford, W. T.

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989).

Winston, R.

H. Ries, R. Winston, “Tailored edge ray reflectors for illumination,” J. Opt. Soc. Am. A 11, 1260–1264 (1994), also included in Selected Papers in Nonimaging Optics, SPIE Milestone Series Vol. MS 106, Roland Winston, ed. (SPIE Optical Engineering Press, Bellingham, Wash.1995).
[CrossRef]

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989).

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in NonimagingOptics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. SPIE1528, 7–14 (1991).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

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

Other (4)

J. Karni, H. Ries, “Concepts for high concentration primary reflectors in central receiver system,” in Proceedings of the 7th International Symposium on Solar Thermal Concentrating Technologies (Institute for High Temperatures of the Russian Academy of Science (IVTAN), Moscow, Russia, 1994), Vol. 4, pp. 796–801.

N. Shatz, J. Bortz, “An inverse engineering perspective on nonimaging optical design,” in Nonimaging Optics: Maximum Efficiency Light Transfer III, R. Winston, ed., Proc. SPIE2538, 136–156 (1995).
[CrossRef]

W. T. Welford, R. Winston, High Collection Non-Imaging Optics (Academic, New York, 1989).

H. Ries, G. Smestad, R. Winston, “Thermodynamics of light concentrators,” in NonimagingOptics: Maximum Efficiency Light Transfer, R. Winston, R. L. Holman, eds., Proc. SPIE1528, 7–14 (1991).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Skewness distributions of a disk with R sin (θ)=1, a sphere of radius R=0.5 a cylinder of radius R=0.25, and height H=2. For each distribution, the area under the curve is the same.

Fig. 2
Fig. 2

If the skewness distribution of the source does not match that of the target, then the minimum of the two constitutes the upper limit for the transferred étendue. Where the distribution of the source exceeds that of the target, losses occur. Where the distribution of the target exceeds that of the source, the radiation is diluted.

Fig. 3
Fig. 3

Upper-limit efficiency as a function of concentration for a disk target and a cylindrical source of length-to-radius ratios H/R=5, H/R=10, and H/R=20. The curve is derived by varying the ratio of the radii of source and target.

Fig. 4
Fig. 4

Upper-limit efficiency as a function of concentration for a spherical source and a disk target.

Fig. 5
Fig. 5

Reflector for transferring radiation from a sphere of diameter 10 units to a disk with equal étendue within 30° found by optimization.

Fig. 6
Fig. 6

Efficiency versus concentration.

Fig. 7
Fig. 7

Efficiency versus projected half-angle.

Fig. 8
Fig. 8

Skewness distributions for sphere, disk, and all rays output by 30° optimized spline reflector.

Fig. 9
Fig. 9

Skewness distributions for sphere, disk, and subset of rays output into 30° half-angle by 30° optimized spline reflector.

Tables (1)

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Table 1 Lower and Upper Ranges of 3D OSR Optimization Parameters

Equations (27)

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s=r(k×aˆ)=rminkt
dE(s)ds=r sin(θ)>|s| 2 sin(θ)r 1-s2[r sin(θ)]2 da,
dEdds=4πR sin(θ)[1-x2-x arccos(x)],
wherex=|s|R sin(θ).
dEsds=8πR sin(θ)0arccos(x)1-x2cos2(ϕ) dϕ=4πR sin(θ)x21u-x21-uduu=4πR sin(θ)arctan2u-1-x22[(1-u)(u-x2)]1/2+x arctan2x2-u-x2u2x[(1-u)(u-x2)]1/2u=x2u=1=4π2R sin(θ)(1-|x|).
dEcds=4πH sin(θ)1-x2.
η= min(dE1/ds, dE2/ds)dsdE1/dsds.
C= min(dE1/ds, dE2/ds)dsdE2/dsds.
dPsourceds (L˜, s˜)=xS;L(x)>L˜δ(s(x)-s˜)L(x)dE(x).
dEsourceds (L˜, s˜)=xS;L(x)>L˜δ(s(x)-s˜)dE(x).
dEsourceds [L*(s), s]=!mindEtargetds (s), dEsourceds.
P= dPsourceds (L*, s)ds.
η=P dPsourceds (0, s)ds,
C=PmaxxS L(x) dEtargetds (L*, s)ds.
rCPC=rs sin(θ)-t(θ)cos(θ)
zCPC=-rs cos(θ)-t(θ)sin(θ)
0=θminθθmax=(3π/2-θi)(1-τ)
t(θ)=rsθforθθi+π/2rs θ+θi+π/2-cos(θ-θi)1+sin(θ-θi)forθi+π/2<θ.
SCPC*(θ)[SCPC(θ)]ρ
Sn*=(n-1)δS,n=1, , Ndev,
Wn=a+b(Sn*/SNdev*)1/ρ,n=1, , Ndev,
Tn=(Zn2+Rn2-rs2)1/2.
Θn=arctanRnrs-ZnTnZnrs+RnTn.
δTn=Tn-tn,
δΘn=Θn-θn.
rOSR=rs sin(θ+δθ)-(t+δt)cos(θ+δθ),
zOSR=-rs cos(θ+δθ)-(t+δt)sin(θ+δθ).

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