Abstract

The recently developed generalized functional method provides a means of designing nonimaging concentrators and luminaires for use with extended sources and receivers. We explore the mathematical relationships between optical designs produced using the generalized functional method and edge-ray, aplanatic, and simultaneous multiple surface (SMS) designs. Edge-ray and dual-surface aplanatic designs are shown to be special cases of generalized functional designs. In addition, it is shown that dual-surface SMS designs are closely related to generalized functional designs and that certain computational advantages accrue when the two design methods are combined. A number of examples are provided.

© 2011 Optical Society of America

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References

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  1. L. Jolley, J. Waldram, and G. Wilson, Theory and Design of Illuminating Engineering Equipment (Chapman & Hall, 1930).
  2. W. Elmer, The Optical Design of Reflectors (Wiley, 1980).
  3. R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 160–161.
  4. R. Winston and H. Ries, “Nonimaging reflectors as functionals of the desired irradiance,” J. Opt. Soc. Am. A 10, 1902–1908(1993).
    [CrossRef]
  5. N. Boldyrev, “About calculation of asymmetrical specular reflectors,” Svetotekhnika 7, 7–8 (1932).
  6. V. Komissarov, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).
  7. J. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. 62, 1498–1501 (1972).
    [CrossRef]
  8. B. Westcott, Shaped Reflector Antenna Design (Wiley, 1983).
  9. I. Galindo, W. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antennas Propag. 35, 887–896 (1987).
    [CrossRef]
  10. B. Westcott and A. Zaporozhets, “Single reflector synthesis using an analytical gradient procedure,” Electron. Lett. 30, 1462–1463 (1994).
    [CrossRef]
  11. R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 173–180.
  12. V. Galindo, “Design of dual-reflector antennas with proprietary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12, 403–408 (1964).
    [CrossRef]
  13. B. Westcott, F. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc. H 128, 11–18 (1981).
    [CrossRef]
  14. B. Westcott, R. Graham, and I. Wolton, “Synthesis of dual-offset shaped reflectors for arbitrary aperture shapes using continuous domain deformation,” IEE Proc. H 133, 57–64 (1986).
    [CrossRef]
  15. B. Westcott and A. Zaporozhets, “Dual-reflector synthesis based on analytical gradient-iteration procedures,” IEE Proc. H 142, 129–135 (1995).
    [CrossRef]
  16. J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 32–47 (2006).
    [CrossRef]
  17. H. Ries and A. Rabl, “Edge-ray principle of nonimaging optics,” J. Opt. Soc. Am. A 11, 2627–2632 (1994).
    [CrossRef]
  18. R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 47–49.
  19. K. Schwarzschild, “Untersuchungen zur geometrischen Optik I–III,” Abh. Konigl. Ges. Wis. Gottingen Math-phys. Kl. 4 (1905–1906).
  20. A. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70, 945–949 (1957).
    [CrossRef]
  21. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334, 787–796 (2002).
    [CrossRef]
  22. R. Willstrop and D. Lynden-Bell, “Exact optics—II. exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342, 33–49 (2003).
    [CrossRef]
  23. J. Miñano and J. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
    [CrossRef] [PubMed]
  24. J. Miñano, P. Benítez, and J. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
    [CrossRef] [PubMed]
  25. J. Miñano, J. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34, 7850–7856(1995).
    [CrossRef] [PubMed]
  26. O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
    [CrossRef]
  27. R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 181–234.
  28. J. Miñano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826–1831 (1985).
    [CrossRef]
  29. J. Miñano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1986).
    [CrossRef]
  30. P. Davies, “Edge-ray principle of nonimaging optics,” J. Opt. Soc. Am. A 11, 1256–1259 (1994).
    [CrossRef]
  31. R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 50–63.
  32. R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 89–92.
  33. J. Gordon, D. Feuermann, and P. Young, “Unfolded aplanats for high-concentration photovoltaics,” Opt. Lett. 33, 1114–1116 (2008).
    [CrossRef] [PubMed]

2008 (1)

2006 (1)

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 32–47 (2006).
[CrossRef]

2004 (1)

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

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

1995 (3)

1994 (3)

1993 (1)

1992 (1)

1987 (1)

I. Galindo, W. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antennas Propag. 35, 887–896 (1987).
[CrossRef]

1986 (2)

B. Westcott, R. Graham, and I. Wolton, “Synthesis of dual-offset shaped reflectors for arbitrary aperture shapes using continuous domain deformation,” IEE Proc. H 133, 57–64 (1986).
[CrossRef]

J. Miñano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1986).
[CrossRef]

1985 (1)

1981 (1)

B. Westcott, F. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc. H 128, 11–18 (1981).
[CrossRef]

1972 (1)

1964 (1)

V. Galindo, “Design of dual-reflector antennas with proprietary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12, 403–408 (1964).
[CrossRef]

1957 (1)

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

1941 (1)

V. Komissarov, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

1932 (1)

N. Boldyrev, “About calculation of asymmetrical specular reflectors,” Svetotekhnika 7, 7–8 (1932).

Benítez, P.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

J. Miñano, P. Benítez, and J. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
[CrossRef] [PubMed]

J. Miñano, J. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34, 7850–7856(1995).
[CrossRef] [PubMed]

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 181–234.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 50–63.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 89–92.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 173–180.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 160–161.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

Blen, J.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

Boldyrev, N.

N. Boldyrev, “About calculation of asymmetrical specular reflectors,” Svetotekhnika 7, 7–8 (1932).

Bortz, J.

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 32–47 (2006).
[CrossRef]

Brickell, F.

B. Westcott, F. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc. H 128, 11–18 (1981).
[CrossRef]

Chavez, J.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

Davies, P.

Dross, O.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

Elmer, W.

W. Elmer, The Optical Design of Reflectors (Wiley, 1980).

Feuermann, D.

Galindo, I.

I. Galindo, W. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antennas Propag. 35, 887–896 (1987).
[CrossRef]

Galindo, V.

V. Galindo, “Design of dual-reflector antennas with proprietary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12, 403–408 (1964).
[CrossRef]

González, J.

Gordon, J.

Graham, R.

B. Westcott, R. Graham, and I. Wolton, “Synthesis of dual-offset shaped reflectors for arbitrary aperture shapes using continuous domain deformation,” IEE Proc. H 133, 57–64 (1986).
[CrossRef]

Head, A.

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

Hernández, M.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

Imbriale, W.

I. Galindo, W. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antennas Propag. 35, 887–896 (1987).
[CrossRef]

Jolley, L.

L. Jolley, J. Waldram, and G. Wilson, Theory and Design of Illuminating Engineering Equipment (Chapman & Hall, 1930).

Komissarov, V.

V. Komissarov, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

Lynden-Bell, D.

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

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

Miñano, J.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

J. Miñano, P. Benítez, and J. González, “RX: a nonimaging concentrator,” Appl. Opt. 34, 2226–2235 (1995).
[CrossRef] [PubMed]

J. Miñano, J. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34, 7850–7856(1995).
[CrossRef] [PubMed]

J. Miñano and J. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31, 3051–3060 (1992).
[CrossRef] [PubMed]

J. Miñano, “Design of three-dimensional nonimaging concentrators with inhomogeneous media,” J. Opt. Soc. Am. A 3, 1345–1353 (1986).
[CrossRef]

J. Miñano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look,” J. Opt. Soc. Am. A 2, 1826–1831 (1985).
[CrossRef]

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 89–92.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 181–234.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 50–63.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 160–161.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 173–180.

Mittra, R.

I. Galindo, W. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antennas Propag. 35, 887–896 (1987).
[CrossRef]

Mohedano, R.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

Muñoz, F.

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

Rabl, A.

Ries, H.

Schruben, J.

Schwarzschild, K.

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

Shatz, N.

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 32–47 (2006).
[CrossRef]

Stevens, F.

B. Westcott, F. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc. H 128, 11–18 (1981).
[CrossRef]

Waldram, J.

L. Jolley, J. Waldram, and G. Wilson, Theory and Design of Illuminating Engineering Equipment (Chapman & Hall, 1930).

Westcott, B.

B. Westcott and A. Zaporozhets, “Dual-reflector synthesis based on analytical gradient-iteration procedures,” IEE Proc. H 142, 129–135 (1995).
[CrossRef]

B. Westcott and A. Zaporozhets, “Single reflector synthesis using an analytical gradient procedure,” Electron. Lett. 30, 1462–1463 (1994).
[CrossRef]

B. Westcott, R. Graham, and I. Wolton, “Synthesis of dual-offset shaped reflectors for arbitrary aperture shapes using continuous domain deformation,” IEE Proc. H 133, 57–64 (1986).
[CrossRef]

B. Westcott, F. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc. H 128, 11–18 (1981).
[CrossRef]

B. Westcott, Shaped Reflector Antenna Design (Wiley, 1983).

Willstrop, R.

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

Wilson, G.

L. Jolley, J. Waldram, and G. Wilson, Theory and Design of Illuminating Engineering Equipment (Chapman & Hall, 1930).

Winston, R.

R. Winston and H. Ries, “Nonimaging reflectors as functionals of the desired irradiance,” J. Opt. Soc. Am. A 10, 1902–1908(1993).
[CrossRef]

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 181–234.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 50–63.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 89–92.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 160–161.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 173–180.

Wolton, I.

B. Westcott, R. Graham, and I. Wolton, “Synthesis of dual-offset shaped reflectors for arbitrary aperture shapes using continuous domain deformation,” IEE Proc. H 133, 57–64 (1986).
[CrossRef]

Young, P.

Zaporozhets, A.

B. Westcott and A. Zaporozhets, “Dual-reflector synthesis based on analytical gradient-iteration procedures,” IEE Proc. H 142, 129–135 (1995).
[CrossRef]

B. Westcott and A. Zaporozhets, “Single reflector synthesis using an analytical gradient procedure,” Electron. Lett. 30, 1462–1463 (1994).
[CrossRef]

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 (1905–1906).

Appl. Opt. (3)

Electron. Lett. (1)

B. Westcott and A. Zaporozhets, “Single reflector synthesis using an analytical gradient procedure,” Electron. Lett. 30, 1462–1463 (1994).
[CrossRef]

IEE Proc. H (3)

B. Westcott, F. Stevens, and F. Brickell, “GO synthesis of offset dual reflectors,” IEE Proc. H 128, 11–18 (1981).
[CrossRef]

B. Westcott, R. Graham, and I. Wolton, “Synthesis of dual-offset shaped reflectors for arbitrary aperture shapes using continuous domain deformation,” IEE Proc. H 133, 57–64 (1986).
[CrossRef]

B. Westcott and A. Zaporozhets, “Dual-reflector synthesis based on analytical gradient-iteration procedures,” IEE Proc. H 142, 129–135 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

I. Galindo, W. Imbriale, and R. Mittra, “On the theory of the synthesis of single and dual offset shaped reflector antennas,” IEEE Trans. Antennas Propag. 35, 887–896 (1987).
[CrossRef]

V. Galindo, “Design of dual-reflector antennas with proprietary phase and amplitude distributions,” IEEE Trans. Antennas Propag. 12, 403–408 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

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

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

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

Opt. Lett. (1)

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

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

Proc. SPIE (2)

O. Dross, R. Mohedano, P. Benítez, J. Miñano, J. Chavez, J. Blen, M. Hernández, and F. Muñoz, “Review of SMS design methods and real world applications,” Proc. SPIE 5529, 35–47 (2004).
[CrossRef]

J. Bortz and N. Shatz, “Generalized functional method of nonimaging optical design,” Proc. SPIE 6338, 32–47 (2006).
[CrossRef]

Svetotekhnika (1)

N. Boldyrev, “About calculation of asymmetrical specular reflectors,” Svetotekhnika 7, 7–8 (1932).

Trudy VEI (1)

V. Komissarov, “The foundations of calculating specular prismatic fittings,” Trudy VEI 43, 6–61 (1941).

Other (9)

B. Westcott, Shaped Reflector Antenna Design (Wiley, 1983).

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 173–180.

L. Jolley, J. Waldram, and G. Wilson, Theory and Design of Illuminating Engineering Equipment (Chapman & Hall, 1930).

W. Elmer, The Optical Design of Reflectors (Wiley, 1980).

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 160–161.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 47–49.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 181–234.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 50–63.

R. Winston, J. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 89–92.

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Figures (20)

Fig. 1
Fig. 1

Geometry for derivation of the basic functional method.

Fig. 2
Fig. 2

Geometry for derivation of the far-field generalized functional method.

Fig. 3
Fig. 3

Geometry for derivation of the DSF method.

Fig. 4
Fig. 4

Geometry used in the design of edge-ray concentrators.

Fig. 5
Fig. 5

Finite-conjugate aplanatic optical system depicted as a black box.

Fig. 6
Fig. 6

Geometry used in the design of dual-surface 2D SMS concentrators.

Fig. 7
Fig. 7

Source and target geometry for the first example. The source profile used in the functional design of the reflective edge-ray concentrator is the subsection of the source depicted as having a total arc length of s 1 .

Fig. 8
Fig. 8

Geometry used in deriving the output-angle function γ ( s , r ) for the first example, for source-arc-length values greater than s z .

Fig. 9
Fig. 9

Shape profile of a reflective edge-ray concentrator computed using the near-field generalized functional method, showing the path of a single edge ray emitted from a point on the circular source profile at arc-length coordinate s = 19 mm .

Fig. 10
Fig. 10

Geometry used in designing a reflective unfolded aplanat.

Fig. 11
Fig. 11

Source-profile geometry for the reflective unfolded aplanat.

Fig. 12
Fig. 12

Target-profile geometry for the reflective unfolded aplanat.

Fig. 13
Fig. 13

Reflective unfolded aplanat designed using the DSF method.

Fig. 14
Fig. 14

Shape-coordinate error as a function of θ for the reflective unfolded aplanat.

Fig. 15
Fig. 15

Sampled points produced by the discrete SMS method on the primary and secondary reflectors of the unfolded 2D SMS concentrator. Also shown are rays incident on a single sampled point of the secondary at ± 0.5 ° relative to the optical axis.

Fig. 16
Fig. 16

Wavefronts A and A used in the design of the reflective unfolded hybrid SMS-DSF concentrator.

Fig. 17
Fig. 17

Sampled target arc length u as a function of the source arc length s used in designing the reflective unfolded hybrid SMS-DSF concentrator. The solid curve connecting the sampled points was obtained by means of cubic-spline interpolation.

Fig. 18
Fig. 18

Shape profile of reflective unfolded hybrid SMS-DSF concentrator, with traced meridional rays incident on the entrance pupil of the secondary at + 0.5 ° relative to the optical axis.

Fig. 19
Fig. 19

Plots of the computed y-axis ray-position error as a function of the ray-incidence angle on the image plane for the discrete SMS and hybrid SMS-DSF designs.

Fig. 20
Fig. 20

Computed ray-position error plot as in Fig. 19, but showing only ray errors in the range of 2 to 2 μm .

Equations (39)

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γ ( θ ) = θ β + β ,
β ( θ ) = arctan { n sin [ θ γ ( θ ) ] n cos [ θ γ ( θ ) ] n } ,
β ( θ ) = 1 2 [ θ γ ( θ ) π ] .
1 r d r = tan [ β ( θ ) ] d θ ,
r ( θ ) = r 0 exp { θ 0 θ tan [ β ( θ ) ] d θ }
β ( s ) = arctan { n sin [ θ ( s ) γ ( s ) ] n cos [ θ ( s ) γ ( s ) ] n } ,
β ( s ) = 1 2 [ θ ( s ) γ ( s ) π ] .
d r d s = θ ˙ ( s ) tan [ β ( s ) ] r + c g { tan [ β ( s ) ] cos [ δ ( s ) ] sin [ δ ( s ) ] } ,
r ( s ) = exp { s 0 s θ ˙ ( s ) tan [ β ( s ) ] d s } × { r 0 + c g s 0 s { tan [ β ( s ) ] cos [ δ ( s ) ] sin [ δ ( s ) ] } exp { s 0 s θ ˙ ( s ) tan [ β ( s ) ] d s } d s } .
X ( s ) = x ( s ) + r ( s ) cos [ θ ( s ) ] ,
Y ( s ) = y ( s ) + r ( s ) sin [ θ ( s ) ] .
d r d s = θ ˙ ( s ) tan [ β ( s , r ) ] r + c g { tan [ β ( s , r ) ] cos [ δ ( s ) ] sin [ δ ( s ) ] } ,
d r d s = θ ˙ ( s ) tan [ β ( s , r , ρ ) ] r + c g { tan [ β ( s , r , ρ ) ] cos [ δ ( s ) ] sin [ δ ( s ) ] } ,
γ ( s , r , ρ ) = angle { x ˜ ( s ) x ( s ) ρ cos [ Θ s ( s ) ] r cos [ θ ( s ) ] , y ˜ ( s ) y ( s ) ρ sin [ Θ s ( s ) ] r sin [ θ ( s ) ] } ,
d ρ d s = Θ ˙ s ( s ) tan [ α ( s , r , ρ ) ] ρ C g { tan [ α ( s , r , ρ ) ] cos [ Δ s ( s ) ] sin [ Δ s ( s ) ] } u ˙ s ( s ) ,
α ( s , r , ρ ) = arctan { n sin [ Θ s ( s ) γ ( s , r , ρ ) ] n cos [ Θ s ( s ) γ ( s , r , ρ ) ] n } ,
α ( s , r , ρ ) = 1 2 [ Θ s ( s ) γ ( s , r , ρ ) π ] .
X ˜ ( s ) = x ˜ ( s ) ρ ( s ) cos [ Θ s ( s ) ] ,
Y ˜ ( s ) = y ˜ ( s ) ρ ( s ) sin [ Θ s ( s ) ] .
s 1 = a [ 3 2 π arcsin ( a + A L ) ] .
s z = a [ π 2 + arcsin ( a + A L ) ] ,
γ ( s , r ) = arcsin [ A R ( s , r ) ] angle [ L X ( s , r ) , Y ( s , r ) ]
X ( s , r ) = x ( s ) + r cos [ θ ( s ) ] ,
Y ( s , r ) = y ( s ) + r sin [ θ ( s ) ]
R ( s , r ) = [ L X ( s , r ) ] 2 + Y 2 ( s , r )
γ ( s , r ) = π 2 s a .
sin ( θ ) h = constant ,
θ ( s ) = θ min + s a .
δ ( s ) = 0 ° .
x ( s ) = a cos ( θ min + s a ) ,
y ( s ) = a sin ( θ min + s a ) ,
u max = R a p [ 1 sin ( θ min ) sin ( θ max ) ] = 7.412 mm ,
x ˜ u ( u ) = L = 31 mm ,
y ˜ u ( u ) = R a p sin ( θ min ) sin ( θ max ) u = 2.588 mm u .
u s ( s ) = R a p sin ( θ min + s a ) sin ( θ min ) sin ( θ max ) ,
x ˜ ( s ) = L = 31 mm ,
y ˜ ( s ) = R a p sin ( θ min + s a ) sin ( θ max ) .
Θ s ( s ) = 0 ° ,
Δ s ( s ) = 0 ° .

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