Abstract

Aplanats are imaging optics that completely eliminate both spherical aberration and coma. They can fulfill the practical virtues of permitting sizable gaps between the absorber and the optic, as well as compactness. However, the ability of aplanats to efficiently approach the thermodynamic limit to flux concentration and optical tolerance had remained unrecognized. Both fundamental and applied aspects of dual-mirror aplanats are reviewed and elaborated, motivated by the exigencies of tenable, maximum-performance solar concentrators, including examples from commercial concentrator photovoltaics (CPV). Promising designs for future photovoltaic concentrators are also identified, illustrating how pragmatic constraints translate into devising fundamentally new optics.

© 2010 OSA

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References

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  1. K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis. Gottingen Math-phys. Kl. 4, Nos. 1–3 (1905–1906).
  2. A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
    [CrossRef]
  3. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
    [CrossRef]
  4. R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
    [CrossRef]
  5. J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44(12), 2327–2331 (2005).
    [CrossRef] [PubMed]
  6. R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30(19), 2617–2619 (2005).
    [CrossRef] [PubMed]
  7. J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33(10), 1114–1116 (2008).
    [CrossRef] [PubMed]
  8. N. Ostroumov, J. M. Gordon, and D. Feuermann, “Panorama of dual-mirror aplanats for maximum concentration,” Appl. Opt. 48(26), 4926–4931 (2009).
    [CrossRef] [PubMed]
  9. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).
  10. D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy 72(6), 459–472 (2002).
    [CrossRef]
  11. G. D. Conley, and S. J. Horne, SolFocus Inc., 510 Logue Ave., Mountain View, CA 94043 (personal communications and company technical reports, 2008).
  12. J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 3642–3644 (2004).
    [CrossRef]
  13. E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100(4), 044514 (2006).
    [CrossRef]
  14. 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(6), 064101 (2007).
    [CrossRef]
  15. J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
    [CrossRef]
  16. H. C. Hottel, “Radiation heat transmission”, in Heat Transmission, ed. W.H. McAdams, 3rd ed., Ch. 4, McGraw-Hill (1954).
  17. E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9(1), 413–418 (1873).
    [CrossRef]
  18. R. Clausius, “Ueber die Concentration von Wärme- und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie, Series 5 197(1), 1–44 (1864).
    [CrossRef]
  19. H. Helmholtz, “Die theoretische grenze für die leistungsfähigkeit der mikroskope,” Annalen der Physik und Chemie, Series 6, 557–584 (1874).
  20. D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45(3), 030502 (2006).
    [CrossRef]
  21. D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88(11), 114104 (2006).
    [CrossRef]
  22. D. Feuermann and J. M. Gordon, “High-irradiance reactors with unfolded aplanatic optics,” Appl. Opt. 47(31), 5722–5727 (2008).
    [CrossRef]

2009 (1)

2008 (2)

2007 (1)

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(6), 064101 (2007).
[CrossRef]

2006 (3)

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100(4), 044514 (2006).
[CrossRef]

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45(3), 030502 (2006).
[CrossRef]

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88(11), 114104 (2006).
[CrossRef]

2005 (3)

J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44(12), 2327–2331 (2005).
[CrossRef] [PubMed]

R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30(19), 2617–2619 (2005).
[CrossRef] [PubMed]

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

2004 (1)

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 3642–3644 (2004).
[CrossRef]

2003 (1)

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[CrossRef]

2002 (2)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[CrossRef]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy 72(6), 459–472 (2002).
[CrossRef]

1957 (1)

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
[CrossRef]

1874 (1)

H. Helmholtz, “Die theoretische grenze für die leistungsfähigkeit der mikroskope,” Annalen der Physik und Chemie, Series 6, 557–584 (1874).

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9(1), 413–418 (1873).
[CrossRef]

1864 (1)

R. Clausius, “Ueber die Concentration von Wärme- und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie, Series 5 197(1), 1–44 (1864).
[CrossRef]

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9(1), 413–418 (1873).
[CrossRef]

Clausius, R.

R. Clausius, “Ueber die Concentration von Wärme- und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie, Series 5 197(1), 1–44 (1864).
[CrossRef]

Feuermann, D.

N. Ostroumov, J. M. Gordon, and D. Feuermann, “Panorama of dual-mirror aplanats for maximum concentration,” Appl. Opt. 48(26), 4926–4931 (2009).
[CrossRef] [PubMed]

D. Feuermann and J. M. Gordon, “High-irradiance reactors with unfolded aplanatic optics,” Appl. Opt. 47(31), 5722–5727 (2008).
[CrossRef]

J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33(10), 1114–1116 (2008).
[CrossRef] [PubMed]

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100(4), 044514 (2006).
[CrossRef]

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45(3), 030502 (2006).
[CrossRef]

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88(11), 114104 (2006).
[CrossRef]

J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44(12), 2327–2331 (2005).
[CrossRef] [PubMed]

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 3642–3644 (2004).
[CrossRef]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy 72(6), 459–472 (2002).
[CrossRef]

Gordon, J. M.

N. Ostroumov, J. M. Gordon, and D. Feuermann, “Panorama of dual-mirror aplanats for maximum concentration,” Appl. Opt. 48(26), 4926–4931 (2009).
[CrossRef] [PubMed]

D. Feuermann and J. M. Gordon, “High-irradiance reactors with unfolded aplanatic optics,” Appl. Opt. 47(31), 5722–5727 (2008).
[CrossRef]

J. M. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33(10), 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(6), 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(4), 044514 (2006).
[CrossRef]

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88(11), 114104 (2006).
[CrossRef]

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45(3), 030502 (2006).
[CrossRef]

R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30(19), 2617–2619 (2005).
[CrossRef] [PubMed]

J. M. Gordon and D. Feuermann, “Optical performance at the thermodynamic limit with tailored imaging designs,” Appl. Opt. 44(12), 2327–2331 (2005).
[CrossRef] [PubMed]

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 3642–3644 (2004).
[CrossRef]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy 72(6), 459–472 (2002).
[CrossRef]

Head, A. K.

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
[CrossRef]

Helmholtz, H.

H. Helmholtz, “Die theoretische grenze für die leistungsfähigkeit der mikroskope,” Annalen der Physik und Chemie, Series 6, 557–584 (1874).

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(6), 064101 (2007).
[CrossRef]

Huleihil, M.

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 3642–3644 (2004).
[CrossRef]

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy 72(6), 459–472 (2002).
[CrossRef]

Israeli, T.

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[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(6), 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(4), 044514 (2006).
[CrossRef]

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 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(6), 064101 (2007).
[CrossRef]

Lynden-Bell, D.

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[CrossRef]

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[CrossRef]

Nakar, D.

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45(3), 030502 (2006).
[CrossRef]

Ng, T. W.

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88(11), 114104 (2006).
[CrossRef]

Ostroumov, N.

Reddy, T. A.

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis. Gottingen Math-phys. Kl. 4, Nos. 1–3 (1905–1906).

Scoles, K.

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

Sun, J.

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

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(4), 044514 (2006).
[CrossRef]

Willstrop, R. V.

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[CrossRef]

Winston, R.

Young, P.

Abh. Konigl. Ges. Wis. Gottingen Math-phys. Kl. (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I-III,” Abh. Konigl. Ges. Wis. Gottingen Math-phys. Kl. 4, Nos. 1–3 (1905–1906).

Annalen der Physik und Chemie, Series (1)

H. Helmholtz, “Die theoretische grenze für die leistungsfähigkeit der mikroskope,” Annalen der Physik und Chemie, Series 6, 557–584 (1874).

Annalen der Physik und Chemie, Series 5 (1)

R. Clausius, “Ueber die Concentration von Wärme- und Lichtstrahlen und die Grenzen ihrer Wirkung,” Annalen der Physik und Chemie, Series 5 197(1), 1–44 (1864).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (3)

D. Feuermann, J. M. Gordon, and T. W. Ng, “Photonic surgery with noncoherent light,” Appl. Phys. Lett. 88(11), 114104 (2006).
[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(6), 064101 (2007).
[CrossRef]

J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84(18), 3642–3644 (2004).
[CrossRef]

Archiv für mikroskopische Anatomie (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9(1), 413–418 (1873).
[CrossRef]

J. Appl. Phys. (1)

E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. 100(4), 044514 (2006).
[CrossRef]

J. Sol. Energy Eng. (1)

J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. 127(1), 138–145 (2005).
[CrossRef]

Mon. Not. R. Astron. Soc. (2)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[CrossRef]

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[CrossRef]

Opt. Eng. (1)

D. Nakar, D. Feuermann, and J. M. Gordon, “Aplanatic near-field optics for efficient light transfer,” Opt. Eng. 45(3), 030502 (2006).
[CrossRef]

Opt. Lett. (2)

Proc. Phys. Soc. London Sec. B (1)

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
[CrossRef]

Sol. Energy (1)

D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy 72(6), 459–472 (2002).
[CrossRef]

Other (3)

G. D. Conley, and S. J. Horne, SolFocus Inc., 510 Logue Ave., Mountain View, CA 94043 (personal communications and company technical reports, 2008).

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

H. C. Hottel, “Radiation heat transmission”, in Heat Transmission, ed. W.H. McAdams, 3rd ed., Ch. 4, McGraw-Hill (1954).

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Figures (12)

Fig. 1
Fig. 1

Absorber flux map (for the aplanat of Fig. 6(a)) [8], for several θs values, with flux concentration C normalized to its respective thermodynamic limit (Eq. (1)). Radial position in the focal plane R is expressed relative to the minimum absorber radius commensurate with the thermodynamic limit (which, as shown below, is θs f for an aplanat of focal length f). C/Cmax does not reach unity even at R well below θs f due to inherent shading and blocking. A concentrator at the thermodynamic limit would exhibit the step function denoted by the black dashed line.

Fig. 2
Fig. 2

Schematic of off-axis orientation as displacement of the focal spot (solid red circle) relative to the absorber (larger circle).

Fig. 3
Fig. 3

Variation of concentrator optical gain with off-axis orientation for the aplanat of Figs. 6(b) and 12(a), for two θs values. The corresponding fundamental limits for θt (Eq. (2)) are also noted, indicating that this optic achieves ~80% of the basic bound.

Fig. 4
Fig. 4

Illustration of aplanat construction.

Fig. 5
Fig. 5

Upward-facing absorber, s > 0, K > 0. This class of aplanat was adopted for SolFocus Generation One (Section 5), but with a lower design NAexit .

Fig. 7
Fig. 7

Downward-facing absorber, s > 0, K < 0.

Fig. 9
Fig. 9

Upward-facing absorber, s > 0, K < 0 (of interest in X-ray and neutron optics where grazing incidence angles on the mirrors are essential).

Fig. 6
Fig. 6

Upward-facing absorber, s < 0, K < 0. The seemingly different concentrators subsumed in this class correspond to different magnitudes of s and K. The aplanat in part (b) is elaborated in Section 6.

Fig. 10
Fig. 10

Downward-facing absorber, s < 0, K < 0. Different magnitudes of s and K yield seemingly disparate shapes that are unified within this class.

Fig. 8
Fig. 8

Downward-facing absorber, s > 0, K > 0. The zoom near the focal plane confirms the gap between the absorber and the reflectors.

Fig. 11
Fig. 11

Dual-mirror aplanats comprising SolFocus photovoltaic concentrators. (a) Generation One: Air-filled design of the type in Fig. 5 but with a nominal NAexit = 0.5 at the focal plane and a glass terminal concentrator, at Cg = 625. Coplanarity (i.e., the rims of the primary and secondary lying in the same plane) is necessary to realize the fundamental compactness limit of an aspect ratio of 1/4 [6], and facilitates accurate mirror alignment because the secondary is adhered to the protective glazing placed on the primary. (b) Generation Two: All-glass, planar, externally mirrored variation of the same design (also coplanar and achieving ultra-compactness), but with a 1 mm2 cell optically bonded to the vertex of the primary mirror. (c) Prototypes of Generations One and Two. (d) Sub-field of SolFocus Generation One arrays in Spain.

Fig. 12
Fig. 12

(a) The aplanat of Fig. 6(b) for CPV. Fermat's strings are indicated as in Fig. 4. (b) A lens-enhanced version that permits lower aspect ratios. A raytrace at an incidence angle of 5° illustrates the robustness to off-axis orientation, i.e., the absence of potential damage to mirror elements from excessive irradiance.

Equations (6)

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

C max = ( N A e x i t sin ( θ s ) ) 2
θ t n sin ( θ e x i t ) C g θ s
L o + L 1 + L 2 = c o n s t .
f = r sin ( ϕ ) = c o n s t . '
r p = sin ( ϕ ) ;    x p = s cos 2 ( ϕ 2 ) + g ( ϕ ) s ( 1 K f ( ϕ ) ) cos 4 ( ϕ 2 ) r s = 2 s K f ( ϕ ) tan ( ϕ 2 ) K f ( ϕ ) tan 2 ( ϕ 2 ) + g ( ϕ ) ;    x s = r s cot ( ϕ ) where    g ( ϕ ) = s ( 1 s ) tan 2 ( ϕ 2 )    and    f ( ϕ ) = | g ( ϕ ) s | s s 1
θ s f = θ s r sin ( ϕ )    or    f = r sin ( ϕ ) = constant

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