Abstract

The economic use of high-efficiency solar cells in photovoltaics requires high concentration of sunlight and therefore precise dual-axis tracking of the sun. Due to their size and bulkiness, these trackers are less adequate for small- to mid-scale installations like flat rooftops. Our approach to combine concentrating and tracking of sunlight utilizes two laterally moving lens arrays. The presented analytic optics design method allows direct calculation of the free-form lens surfaces while incorporating the lateral movement. The obtained concentration performance exceeds a factor of 500. This demonstrates that one can benefit from high-efficiency solar cells and more compact and flexible single-axis trackers at the same time.

© 2013 OSA

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References

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  1. S. Kurtz, “Opportunities and challenges for development of a mature concentrating photovoltaic power industry (revision),” Tech. Rep., National Renewable Energy Laboratory (NREL), Golden, CO (2012).
  2. A. Mohr, T. Roth, S. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14, 663–674 (2006).
    [CrossRef]
  3. F. Duerr, Y. Meuret, H. Thienpont, “Tracking integration in concentrating photovoltaics using laterally moving optics,” Opt. Express 19, A207–A218 (2011).
    [CrossRef] [PubMed]
  4. F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
    [CrossRef] [PubMed]
  5. F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces,” Opt. Express 20, 10839–10846 (2012).
    [CrossRef] [PubMed]
  6. SunCycle. Retrieved Dec. 16, 2012, from http://www.suncycle.nl/
  7. Zettasun, Inc. Retrieved Dec. 16, 2012, from http://www.zettasun.com/zettasun/technology.html
  8. Sunfish Energy Ltd. Retrieved Dec. 16, 2012, from http://www.sunfishsolar.com/
  9. J. Karp, E. Tremblay, J. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18, 1122–1133 (2010).
    [CrossRef] [PubMed]
  10. J. Hallas, J. Karp, E. Tremblay, J. Ford, “Lateral translation micro-tracking of planar micro-optic solar concentrator,” Proc. SPIE 7769, 776904 (2010).
    [CrossRef]
  11. K. Baker, J. Karp, E. Tremblay, J. Hallas, J. Ford, “Reactive self-tracking solar concentrators: concept, design, and initial materials characterization,” Appl. Opt. 51, 1086–1094 (2012).
    [CrossRef] [PubMed]
  12. J. Hallas, K. Baker, J. Karp, E. Tremblay, J. Ford, “Two-axis solar tracking accomplished through small lateral translations,” Appl. Opt. 51, 6117–6124 (2012).
    [CrossRef] [PubMed]
  13. V. Zagolla, E. Tremblay, C. Moser, “Light induced fluidic waveguide coupling,” Opt. Express 20, A924–A931 (2012).
    [CrossRef] [PubMed]
  14. E. Tremblay, D. Loterie, C. Moser, “Thermal phase change actuator for self-tracking solar concentration,” Opt. Express 20, A964–A976 (2012).
    [CrossRef] [PubMed]
  15. W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
    [CrossRef]
  16. T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
    [CrossRef]
  17. F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Integrating tracking in concentrating photo-voltaics using non-rotational symmetric laterally moving optics,” Proc. SPIE 8124, 81240M (2011).
    [CrossRef]
  18. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
    [CrossRef]
  19. D. Grabovičkić, P. Benítez, J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
    [CrossRef]
  20. D. Grabovičkić, P. Benítez, J. C. Miñano, “Free-form V-groove reflector design with the SMS method in three dimensions,” Opt. Express 19, A747–A756 (2011).
    [CrossRef]
  21. F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Perfect imaging of three object points with only two analytic lens surfaces in two dimensions,” Proc. SPIE 8429, 842908 (2012).
    [CrossRef]
  22. H. Baig, K. Heasman, T. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012).
    [CrossRef]
  23. W. Lin, P. Benítez, J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
    [CrossRef]

2012 (9)

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Perfect imaging of three object points with only two analytic lens surfaces in two dimensions,” Proc. SPIE 8429, 842908 (2012).
[CrossRef]

H. Baig, K. Heasman, T. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012).
[CrossRef]

W. Lin, P. Benítez, J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[CrossRef]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
[CrossRef] [PubMed]

K. Baker, J. Karp, E. Tremblay, J. Hallas, J. Ford, “Reactive self-tracking solar concentrators: concept, design, and initial materials characterization,” Appl. Opt. 51, 1086–1094 (2012).
[CrossRef] [PubMed]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces,” Opt. Express 20, 10839–10846 (2012).
[CrossRef] [PubMed]

J. Hallas, K. Baker, J. Karp, E. Tremblay, J. Ford, “Two-axis solar tracking accomplished through small lateral translations,” Appl. Opt. 51, 6117–6124 (2012).
[CrossRef] [PubMed]

V. Zagolla, E. Tremblay, C. Moser, “Light induced fluidic waveguide coupling,” Opt. Express 20, A924–A931 (2012).
[CrossRef] [PubMed]

E. Tremblay, D. Loterie, C. Moser, “Thermal phase change actuator for self-tracking solar concentration,” Opt. Express 20, A964–A976 (2012).
[CrossRef] [PubMed]

2011 (3)

2010 (5)

J. Hallas, J. Karp, E. Tremblay, J. Ford, “Lateral translation micro-tracking of planar micro-optic solar concentrator,” Proc. SPIE 7769, 776904 (2010).
[CrossRef]

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
[CrossRef]

J. Karp, E. Tremblay, J. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18, 1122–1133 (2010).
[CrossRef] [PubMed]

D. Grabovičkić, P. Benítez, J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[CrossRef]

2006 (1)

A. Mohr, T. Roth, S. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14, 663–674 (2006).
[CrossRef]

2004 (1)

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Baig, H.

H. Baig, K. Heasman, T. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012).
[CrossRef]

Baker, K.

Benítez, P.

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
[CrossRef] [PubMed]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces,” Opt. Express 20, 10839–10846 (2012).
[CrossRef] [PubMed]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Perfect imaging of three object points with only two analytic lens surfaces in two dimensions,” Proc. SPIE 8429, 842908 (2012).
[CrossRef]

W. Lin, P. Benítez, J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[CrossRef]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Integrating tracking in concentrating photo-voltaics using non-rotational symmetric laterally moving optics,” Proc. SPIE 8124, 81240M (2011).
[CrossRef]

D. Grabovičkić, P. Benítez, J. C. Miñano, “Free-form V-groove reflector design with the SMS method in three dimensions,” Opt. Express 19, A747–A756 (2011).
[CrossRef]

D. Grabovičkić, P. Benítez, J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[CrossRef]

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Blen, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Cebecauer, T.

T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
[CrossRef]

Chaves, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Cruz-Campa, J.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Dross, O.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Duerr, F.

Dunlop, E.

T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
[CrossRef]

Falicoff, W.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Filatov, A.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Ford, J.

Glunz, S.

A. Mohr, T. Roth, S. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14, 663–674 (2006).
[CrossRef]

Grabovickic, D.

Hallas, J.

Heasman, K.

H. Baig, K. Heasman, T. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012).
[CrossRef]

Hernandez, M.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Huld, T.

T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
[CrossRef]

Jared, B.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Karp, J.

Kurtz, S.

S. Kurtz, “Opportunities and challenges for development of a mature concentrating photovoltaic power industry (revision),” Tech. Rep., National Renewable Energy Laboratory (NREL), Golden, CO (2012).

Lentine, A.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Lin, W.

W. Lin, P. Benítez, J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[CrossRef]

Loterie, D.

Mallick, T.

H. Baig, K. Heasman, T. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012).
[CrossRef]

Meuret, Y.

Miñano, J. C.

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
[CrossRef] [PubMed]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces,” Opt. Express 20, 10839–10846 (2012).
[CrossRef] [PubMed]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Perfect imaging of three object points with only two analytic lens surfaces in two dimensions,” Proc. SPIE 8429, 842908 (2012).
[CrossRef]

W. Lin, P. Benítez, J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[CrossRef]

D. Grabovičkić, P. Benítez, J. C. Miñano, “Free-form V-groove reflector design with the SMS method in three dimensions,” Opt. Express 19, A747–A756 (2011).
[CrossRef]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Integrating tracking in concentrating photo-voltaics using non-rotational symmetric laterally moving optics,” Proc. SPIE 8124, 81240M (2011).
[CrossRef]

D. Grabovičkić, P. Benítez, J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[CrossRef]

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Mohedano, R.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Mohr, A.

A. Mohr, T. Roth, S. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14, 663–674 (2006).
[CrossRef]

Moser, C.

Nielson, G.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Okandan, M.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Roth, T.

A. Mohr, T. Roth, S. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14, 663–674 (2006).
[CrossRef]

Sinclair, M.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Šúri, M.

T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
[CrossRef]

Sweatt, W.

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

Thienpont, H.

Tremblay, E.

Zagolla, V.

Appl. Opt. (2)

Opt. Eng. (1)

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489 (2004).
[CrossRef]

Opt. Express (8)

Proc. SPIE (5)

J. Hallas, J. Karp, E. Tremblay, J. Ford, “Lateral translation micro-tracking of planar micro-optic solar concentrator,” Proc. SPIE 7769, 776904 (2010).
[CrossRef]

W. Sweatt, B. Jared, G. Nielson, M. Okandan, A. Filatov, M. Sinclair, J. Cruz-Campa, A. Lentine, “Micro-optics for high-efficiency optical performance and simplified tracking for concentrated photovoltaics (CPV),” Proc. SPIE 7652, 765210 (2010).
[CrossRef]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Perfect imaging of three object points with only two analytic lens surfaces in two dimensions,” Proc. SPIE 8429, 842908 (2012).
[CrossRef]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, H. Thienpont, “Integrating tracking in concentrating photo-voltaics using non-rotational symmetric laterally moving optics,” Proc. SPIE 8124, 81240M (2011).
[CrossRef]

W. Lin, P. Benítez, J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[CrossRef]

Prog. Photovolt. Res. Appl. (2)

T. Huld, T. Cebecauer, M. Šúri, E. Dunlop, “Analysis of one-axis tracking strategies for PV systems in Europe,” Prog. Photovolt. Res. Appl. 18, 183–194 (2010).
[CrossRef]

A. Mohr, T. Roth, S. Glunz, “BICON: high concentration PV using one-axis tracking and silicon concentrator cells,” Prog. Photovolt. Res. Appl. 14, 663–674 (2006).
[CrossRef]

Renew. Sust. Energ. Rev. (1)

H. Baig, K. Heasman, T. Mallick, “Non-uniform illumination in concentrating solar cells,” Renew. Sust. Energ. Rev. 16, 5890–5909 (2012).
[CrossRef]

Other (4)

S. Kurtz, “Opportunities and challenges for development of a mature concentrating photovoltaic power industry (revision),” Tech. Rep., National Renewable Energy Laboratory (NREL), Golden, CO (2012).

SunCycle. Retrieved Dec. 16, 2012, from http://www.suncycle.nl/

Zettasun, Inc. Retrieved Dec. 16, 2012, from http://www.zettasun.com/zettasun/technology.html

Sunfish Energy Ltd. Retrieved Dec. 16, 2012, from http://www.sunfishsolar.com/

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Figures (8)

Fig. 1
Fig. 1

Today’s utility-scale concentrating photovoltaic systems are less suitable for small-to mid-scale installations due to their size and bulkiness (Photo courtesies of (a) Jeff Aubin, (b) Soitec Concentrix and (c) SolFocus)

Fig. 2
Fig. 2

Examples of different tracking integration concepts: using (a) rotary or (b) rectilinear motion; (c) a motion-free example based on liquid crystal (LC) light steering.

Fig. 3
Fig. 3

Shown are different solar trackers used in both PV and CPV; (a) dual-axis tracker, (b) polar aligned single-axis tracker, and (c) horizontal single-axis tracker.

Fig. 4
Fig. 4

(a) Schematic assembly of a conventional CPV module for pedestal-mounted dual-axis trackers. (b) Our tracking integrated CPV module for polar aligned single-axis trackers. (c) The laterally moving lens design in this paper is based on two plan-convex lenses with rectangular apertures, suitable for lens array structures.

Fig. 5
Fig. 5

(a) The 2D analytic design of the single thick lens is based on the convergence points construction in (b). (c) This analytic lens design can be extended for tracking integration by including separated plano-convex lenses and (d) a shifted second lens for off-axis rays.

Fig. 6
Fig. 6

Introduction of all necessary initial values in (a) and (b), and functions in (c) and (d) to derive the conditional equations from Fermat’s principle in three dimensions.

Fig. 7
Fig. 7

Ray tracing results for explicitly calculated solutions in two and three dimensions demonstrate perfect coupling of on- and off-axis ray sets, as intended by the analytic design.

Fig. 8
Fig. 8

Comparison of the concentration performance for three different designs methods: the analytic free-form solution of this paper offers superior performance over the full field of view when compared with the extended SMS3D design and the rotational symmetric SMS2D design; all designs share rectangular lens and rectangular receiver apertures.

Equations (11)

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

d 1 = v 0 ( p 1 w 0 ) , d 2 = n 2 | p 2 p 1 | , d 3 = | p 3 p 2 | , d 4 = n 3 | p 4 p 3 | , d 5 = | p 5 p 4 |
d ^ 1 = v 1 ( q 1 w 0 ) , d ^ 2 = n 2 | q 2 q 1 | , d ^ 3 = | q 3 q 2 | , d ^ 4 = n 3 | q 4 q 3 | , d ^ 5 = | q 5 q 4 |
D 1 = x ( d 1 + d 2 ) = 0 , D 2 = y ( d 1 + d 2 ) = 0 , D 3 = p 2 x ( d 2 + d 3 ) = 0 , D 4 = p 2 y ( d 2 + d 3 ) = 0 , D 5 = p 3 x ( d 3 + d 4 ) = 0 , D 6 = p 3 y ( d 3 + d 4 ) = 0 , D 7 = s ( d 4 + d 5 ) = 0 , D 8 = t ( d 4 + d 5 ) = 0
D 9 = x ( d ^ 1 + d ^ 2 ) = 0 , D 10 = y ( d ^ 1 + d ^ 2 ) = 0 , D 11 = q 2 x ( d ^ 2 + d ^ 3 ) = 0 , D 12 = q 2 y ( d ^ 2 + d ^ 3 ) = 0 , D 13 = q 3 x ( d ^ 3 + d ^ 4 ) = 0 , D 14 = q 3 y ( d ^ 3 + d ^ 4 ) = 0 , D 15 = u ( d ^ 4 + d ^ 5 ) = 0 , D 16 = v ( d ^ 4 + d ^ 5 ) = 0
f ( x , y ) = i = 0 j = 0 f i , j ( x x 0 ) i y 2 j , g ( x , y ) = i = 0 j = 0 g i , j ( x x 1 ) i y 2 j
s ( x , y ) = i = 0 j = 0 s i , j ( x x 0 ) i y 2 j , t ( x , y ) = i = 0 j = 1 t i , j ( x x 0 ) i y ( 2 j 1 )
with s ( x , y ) = ( p 2 x , p 3 x , q 2 x , q 3 x , s , y ) ( x , y ) and t ( x , y ) = ( p 2 y , p 3 y , q 2 y , q 3 y , t , v ) ( x , y )
lim x x 0 lim y 0 n x n m y m D i = 0 ( i = 1 , 3 , , 15 ) m { n 1 , m = 0 }
lim x x 0 lim y 0 n x n m y m D i = 0 ( i = 2 , 4 , 16 ) , { n = 0 , m | odd number m }
lim x x 0 lim y 0 n x n m y m D i = 0 ( i = 2 , 4 , , 16 ) , { n 1 , m | odd number m } lim x x 0 lim y 0 n 1 x n 1 m + 1 y m + 1 D i = 0 ( i = 3 , 5 , 7 , 11 , 13 , 15 ) , { n 1 , m | odd number m }
M x ( f n + 1 , 0 g n + 1 , 0 s n , 0 ) = b ( n , 0 ) , M y ( f 0 , m + 1 g 0 , m + 1 t 0 , m ) = b ( 0 , m ) , M x y ( f n , m + 1 g n , m + 1 s n 1 , m + 1 t n , m ) = b ( n , m )

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