Abstract

A nonimaging strategy that tailors two mirror contours for concentration near the étendue limit is explored, prompted by solar applications where a sizable gap between the optic and absorber is required. Subtle limitations of this simultaneous multiple surface method approach are derived, rooted in the manner in which phase space boundaries can be tailored according to the edge-ray principle. The fundamental categories of double-tailored reflective optics are identified, only a minority of which can pragmatically offer maximum concentration at high collection efficiency. Illustrative examples confirm that acceptance half-angles as large as 30 mrad can be realized at a flux concentration of 1000.

© 2010 Optical Society of America

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  1. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), Chaps. 8, 10, and 13.
  2. J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44, 2327–2331 (2005).
    [CrossRef] [PubMed]
  3. J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33, 1114–1116 (2008).
    [CrossRef] [PubMed]
  4. N. Ostroumov, J. M. Gordon, and D. Feuermann, “Panorama of dual-mirror aplanats for maximum concentration,” Appl. Opt. 48, 4926–4931 (2009).
    [CrossRef] [PubMed]
  5. J. M. Gordon, “Aplanatic optics for solar concentration,” Opt. Express 18, A41–A52 (2010).
    [CrossRef] [PubMed]
  6. J. C. Miñano and J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [CrossRef] [PubMed]
  7. P. Benítez and J. C. Miñano, “Ultrahigh-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A 14, 1988–1997 (1997).
    [CrossRef]
  8. P. Benítez, R. Garcia, and J. C. Miñano, “Contactless efficient two-stage solar concentrators for tubular absorber,” Appl. Opt. 36, 7119–7124 (1997).
    [CrossRef]
  9. The notation formulated in the original SMS papers refers to dual-mirror optics as “XX.” We demur from using this term in order to avoid prurient misconstrual.
  10. H. Ries and A. Rabl, “Edge-ray principle of nonimaging optics,” J. Opt. Soc. Am. A 11, 2627–2632 (1994).
    [CrossRef]
  11. A. Goldstein and J. M. Gordon, “New classes of maximum-concentration practical and robust photovoltaic concentrators with liberal optical tolerance based on dual-mirror optics,” Proc. SPIE 7769, 7769-4 (to be published).
  12. In practice, concentrator construction is terminated when the étendue transfer between the secondary mirror and the absorber becomes equal to the étendue of an absorber irradiated uniformly both spatially and in projected solid angle up to a cutoff at NAout.
  13. J. Chaves, Introduction to Nonimaging Optics (CRC, 2008), Chap. 8.
    [CrossRef]
  14. J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
    [CrossRef]
  15. E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100, 044514 (2006).
    [CrossRef]
  16. O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
    [CrossRef]
  17. R. Winston, P. Benítez, and A. Cvetkovic, “High-concentration mirror-based Köhler integrating system for tandem solar cells,” Proc. SPIE 6342, 634213 (2006).
    [CrossRef]
  18. P. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express 18, A25–A40 (2010).
    [CrossRef] [PubMed]

2010

2009

2008

2007

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[CrossRef]

2006

R. Winston, P. Benítez, and A. Cvetkovic, “High-concentration mirror-based Köhler integrating system for tandem solar cells,” Proc. SPIE 6342, 634213 (2006).
[CrossRef]

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100, 044514 (2006).
[CrossRef]

2005

2004

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

1997

1994

1992

Benítez, P.

Buljan, M.

Chaves, J.

Cvetkovic, A.

P. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express 18, A25–A40 (2010).
[CrossRef] [PubMed]

R. Winston, P. Benítez, and A. Cvetkovic, “High-concentration mirror-based Köhler integrating system for tandem solar cells,” Proc. SPIE 6342, 634213 (2006).
[CrossRef]

Feuermann, D.

Garcia, R.

Goldstein, A.

A. Goldstein and J. M. Gordon, “New classes of maximum-concentration practical and robust photovoltaic concentrators with liberal optical tolerance based on dual-mirror optics,” Proc. SPIE 7769, 7769-4 (to be published).

González, J. C.

Gordon, J. M.

J. M. Gordon, “Aplanatic optics for solar concentration,” Opt. Express 18, A41–A52 (2010).
[CrossRef] [PubMed]

N. Ostroumov, J. M. Gordon, and D. Feuermann, “Panorama of dual-mirror aplanats for maximum concentration,” Appl. Opt. 48, 4926–4931 (2009).
[CrossRef] [PubMed]

J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33, 1114–1116 (2008).
[CrossRef] [PubMed]

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[CrossRef]

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100, 044514 (2006).
[CrossRef]

J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44, 2327–2331 (2005).
[CrossRef] [PubMed]

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

A. Goldstein and J. M. Gordon, “New classes of maximum-concentration practical and robust photovoltaic concentrators with liberal optical tolerance based on dual-mirror optics,” Proc. SPIE 7769, 7769-4 (to be published).

Hernández, M.

Hirsch, B.

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[CrossRef]

Huleihil, M.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Katz, E. A.

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[CrossRef]

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100, 044514 (2006).
[CrossRef]

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Korech, O.

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[CrossRef]

Miñano, J. C.

Mohedano, R.

Ostroumov, N.

Rabl, A.

Ries, H.

Tassew, W.

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100, 044514 (2006).
[CrossRef]

Winston, R.

R. Winston, P. Benítez, and A. Cvetkovic, “High-concentration mirror-based Köhler integrating system for tandem solar cells,” Proc. SPIE 6342, 634213 (2006).
[CrossRef]

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), Chaps. 8, 10, and 13.

Young, P.

Zamora, P.

Appl. Opt.

Appl. Phys. Lett.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. 91, 064101 (2007).
[CrossRef]

J. Appl. Phys.

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100, 044514 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. SPIE

R. Winston, P. Benítez, and A. Cvetkovic, “High-concentration mirror-based Köhler integrating system for tandem solar cells,” Proc. SPIE 6342, 634213 (2006).
[CrossRef]

A. Goldstein and J. M. Gordon, “New classes of maximum-concentration practical and robust photovoltaic concentrators with liberal optical tolerance based on dual-mirror optics,” Proc. SPIE 7769, 7769-4 (to be published).

Other

In practice, concentrator construction is terminated when the étendue transfer between the secondary mirror and the absorber becomes equal to the étendue of an absorber irradiated uniformly both spatially and in projected solid angle up to a cutoff at NAout.

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008), Chap. 8.
[CrossRef]

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), Chaps. 8, 10, and 13.

The notation formulated in the original SMS papers refers to dual-mirror optics as “XX.” We demur from using this term in order to avoid prurient misconstrual.

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Figures (12)

Fig. 1
Fig. 1

Illustration of the design method. Both θ in and shading by the secondary are purposely exaggerated for clarity of illustration (also in Figs. 3, 4, 7, 9, 10). Two extreme incident wavefronts represent the far-field source. Incident extreme rays r 1 i , r 2 i , and r 3 i are used to define points N, M, X, and Y. r 1 o , r 2 o , and r 3 o are the corresponding output rays to the absorber edges. Here, and in subsequent figures, incident right- and left-hand extreme rays are represented by solid and dashed lines, respectively. The origin of the ( x , z ) coordinate system is at the absorber's center. Explicit indication of the axes is omitted in subsequent drawings for clarity of absorber edge rays. z po and z so are the ordinate values for the vertices of the primary and secondary, respectively. s = the distance between the vertices of the primary and secondary and K = the distance between the absorber and the vertex of the secondary are the two dimensional input parameters (the signs of s and K are delineated in Subsection 3B).

Fig. 2
Fig. 2

2D phase space mapping from (a) the entry to (b) the absorber [12]. Extreme rays from Figs. 1, 3 are indicated. Primed rays are symmetric counterparts of unprimed rays.

Fig. 3
Fig. 3

As in Fig. 1, but showing how mirror contours are tailored to extreme rays as they move from the rim inward.

Fig. 4
Fig. 4

Design where RHI is mapped to LHA. Relative to Fig. 1, note the reversal of the design direction from X to Y.

Fig. 5
Fig. 5

Examples of the six fundamental categories of dual-mirror SMS designs with physically admissible solutions—all designed for N A out = 1 as explained in Subsection 3D. In all cases, the secondary comprises the thicker line type. Categories B and E have degenerate solutions that may seem disparate but follow from the same category, distinguished only by the magnitudes of s and K. Sample extreme rays are traced in each instance. Concentrators A, B 1 , B 2 , D, E 1 , and E 2 are designed for θ in = 30   mrad . Concentrators C and F are designed for θ in = 15   mrad because they incur excessive losses at larger θ in values. Performance figures for a few champion designs are provided in Section 5. In each case the absorber radius has unit length.

Fig. 6
Fig. 6

Illustration of a SMS solution that is valid in two dimensions but incurs large skew ray rejection in its axisymmetric 3D version. (a) Mirror contours. Two edge rays are traced. (b) The inherently disjoint input phase space.

Fig. 7
Fig. 7

Example of a self-consistent maximum-performance design, necessarily with N A out = 1 . The initial segment N M in Figs. 1, 3, 4 shrinks to zero length. In this limit, the rays r 1 i and r 2 i in Fig. 1 approach coincidence but are directed to opposite absorber edges. Except for shading, this device essentially achieves the étendue limit.

Fig. 8
Fig. 8

(a) Input and (b) output 2D phase space for the class illustrated in Fig. 7. The primed rays are the symmetric counterparts of the unprimed rays.

Fig. 9
Fig. 9

Designing from the optic axis outward (see the text).

Fig. 10
Fig. 10

Illustration of terminating the reflectors when designing from the optic axis outward—shown here for the option of oversizing the secondary (and hence oversizing the absorber). Ray r 7 i is indicative of rays excluded by this construction option.

Fig. 11
Fig. 11

A phase space interpretation of the need to oversize the secondary in the design strategy that forgoes achieving the étendue limit of Eq. (1). Only half of (a) input and (b) output phase space is graphed for clarity of illustration (the other half follows from symmetry about the origin). Ray labels correspond to those in Figs. 9, 10.

Fig. 12
Fig. 12

Acceptance angle function of the 3D axisymmetric concentrator in Fig. 5 B 1 , for θ in = 30   mrad . The function is less than unity even at small incidence angles due to the intrinsic losses of shading and blocking. The ideal limit is indicated by the step function (solid line).

Tables (1)

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Table 1 Performance of Five Concentrators that Approach the Étendue Limit a

Equations (1)

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C max = ( sin ( θ out ) sin ( θ in ) ) 2 = ( N A out N A in ) 2 ,

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