Abstract

This paper describes a compound lens for solar photovoltaic system applications, which is composed of a front aspheric or Fresnel surface and a back array of concave surfaces. In contrast to earlier designs, the proposed method can simultaneously focus and shape sun light into a uniform-square pattern on the solar cell. For a square solar cell, this approach can maximize the solar cell’s opto-electric conversion efficiency by enhancing the concentrated pattern’s uniformity. In this article, the theoretical models of the beam shaping focused lens is derived and then compared with experimental data. The tolerance in assembling the component of the concentrator is also analyzed and the corresponding simulation and experimental results are discussed in detail.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Kurtz and J. Geisz, “Multijunction solar cells for conversion of concentrated sunlight to electricity,” Opt. Express 18, A73–A78 (2010).
    [CrossRef]
  2. R. Winston, “Principles of solar concentrators of a novel design,” Sol. Eng. 16, 89–95 (1974).
    [CrossRef]
  3. K. Araki, T. Yano, and Y. Kuroda, “30 kW concentrator photovoltaic system using dome-shaped Fresnel lenses,” Opt. Express 18, A53–A63 (2010).
    [CrossRef]
  4. M. Park, K. Oh, J. Kim, H. W. Shin, and B. D. Oh, “A tapered dielectric waveguide solar concentrator for a compound semiconductor photovoltaic cell,” Opt. Express 18, 1777–1787 (2010).
    [CrossRef]
  5. J. M. Gordon, “Aplanatic optics for solar concentration,” Opt. Express 18, A41–A52 (2010).
    [CrossRef]
  6. J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18, 1122–1133 (2010).
    [CrossRef]
  7. C. M. Wang, H. I. Huang, J. W. Pan, H. Z. Kuo, H. F. Hong, H. Y. Shin, and J. Y. Chang, “Single stage transmission type broadband solar concentrator,” Opt. Express 18, A118–A125 (2010).
    [CrossRef]
  8. N. Shatz, J. Bortz, and R. Winston, “Thermodynamic efficiency of solar concentrators,” Opt. Express 18, A5–A16 (2010).
    [CrossRef]
  9. C. G. Young, “A sun-pumped cw one-watt laser,” Appl. Opt. 5, 993–998 (1966).
    [CrossRef]
  10. A. Segal, M. Epstein, and A. Yogev, “Hybrid concentrated photovoltaic and thermal power conversion at different spectral bands,” Sol. Eng. 76, 591–601 (2004).
    [CrossRef]
  11. R. Winston and J. M. Gordon, “Planar concentrators near the étendue limit,” Opt. Lett. 30, 2617–2619 (2005).
    [CrossRef]
  12. K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
    [CrossRef]
  13. C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
    [CrossRef]
  14. D. Chemisana, “Building integrated concentrating photovoltaics : a review,” Renew. Sustain. Energ. Rev. 15, 603–611(2011).
    [CrossRef]
  15. P. A. Davies, “Light-trapping lenses for solar cells,” Appl. Opt. 31, 6021–6026 (1992).
    [CrossRef]
  16. M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17, 6487–6492 (2009).
    [CrossRef]
  17. 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]
  18. M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
    [CrossRef]
  19. D. Esparza and I. Moreno, “Solar concentrator with diffuser segments,” Proc. SPIE 8011, 80117B (2011).
    [CrossRef]
  20. E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Eng. 22, 119–123 (1979).
    [CrossRef]
  21. N. C. Yeh, “Analysis of spectrum distribution and optical losses under Fresnel lenses,” Renew. Sustain. Energ. Rev. 14, 2926–2935 (2010).
    [CrossRef]
  22. H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
    [CrossRef]
  23. D. Chemisana and M. Ibáñez, “Linear Fresnel concentrators for building integrated applications,” Energy Convers. Manage. 51, 1476–1480 (2010).
    [CrossRef]
  24. V. D. Rumyantsev, “Solar concentrator modules with silicone-on-glass Fresnel lens panels and multijunction cells,” Opt. Express 18, A17–A24 (2010).
    [CrossRef]
  25. X. Deng, X. Liang, Z. Chen, W. Yu, and R. Ma, “Uniform illumination of large targets using a lens array,” Appl. Opt. 25, 377–381 (1986).
    [CrossRef]
  26. Y. Lin, G. N. Lawrence, and J. Buck, “Characterization of excimer lasers for application to lenslet array homogenizers,” Appl. Opt. 40, 1931–1941 (2001).
    [CrossRef]
  27. A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
    [CrossRef]
  28. J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
    [CrossRef]
  29. T. R. M. Sales, “Structured microlens arrays for beam shaping,” Proc. SPIE 5175, 109–120 (2003).
    [CrossRef]
  30. T. R. M. Sales, “Structured microlens arrays for beam shaping,” Opt. Eng. 42, 3084–3085 (2003).
    [CrossRef]
  31. S. I. Chang, J. B. Yoon, H. Kim, J. J. Kim, B. K. Lee, and D. H. Shin, “Microlens array diffuser for a light-emitting diode backlight system,” Opt. Lett. 31, 3016–3018 (2006).
    [CrossRef]
  32. J. J. Yang, Y. S. Liao, and C. F. Chen, “Fabrication of long hexagonal micro-lens array by applying gray-scale lithography in micro-replication process,” Opt. Commun. 270, 433–440 (2007).
    [CrossRef]
  33. O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
    [CrossRef]
  34. T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
    [CrossRef]
  35. D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
    [CrossRef]
  36. K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
    [CrossRef]
  37. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1976).
  38. A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
    [CrossRef]
  39. O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
    [CrossRef]
  40. T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
    [CrossRef]
  41. D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
    [CrossRef]

2011

D. Chemisana, “Building integrated concentrating photovoltaics : a review,” Renew. Sustain. Energ. Rev. 15, 603–611(2011).
[CrossRef]

D. Esparza and I. Moreno, “Solar concentrator with diffuser segments,” Proc. SPIE 8011, 80117B (2011).
[CrossRef]

2010

N. C. Yeh, “Analysis of spectrum distribution and optical losses under Fresnel lenses,” Renew. Sustain. Energ. Rev. 14, 2926–2935 (2010).
[CrossRef]

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

D. Chemisana and M. Ibáñez, “Linear Fresnel concentrators for building integrated applications,” Energy Convers. Manage. 51, 1476–1480 (2010).
[CrossRef]

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

M. Park, K. Oh, J. Kim, H. W. Shin, and B. D. Oh, “A tapered dielectric waveguide solar concentrator for a compound semiconductor photovoltaic cell,” Opt. Express 18, 1777–1787 (2010).
[CrossRef]

V. D. Rumyantsev, “Solar concentrator modules with silicone-on-glass Fresnel lens panels and multijunction cells,” Opt. Express 18, A17–A24 (2010).
[CrossRef]

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]

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

N. Shatz, J. Bortz, and R. Winston, “Thermodynamic efficiency of solar concentrators,” Opt. Express 18, A5–A16 (2010).
[CrossRef]

K. Araki, T. Yano, and Y. Kuroda, “30 kW concentrator photovoltaic system using dome-shaped Fresnel lenses,” Opt. Express 18, A53–A63 (2010).
[CrossRef]

S. Kurtz and J. Geisz, “Multijunction solar cells for conversion of concentrated sunlight to electricity,” Opt. Express 18, A73–A78 (2010).
[CrossRef]

C. M. Wang, H. I. Huang, J. W. Pan, H. Z. Kuo, H. F. Hong, H. Y. Shin, and J. Y. Chang, “Single stage transmission type broadband solar concentrator,” Opt. Express 18, A118–A125 (2010).
[CrossRef]

2009

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
[CrossRef]

C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
[CrossRef]

M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17, 6487–6492 (2009).
[CrossRef]

2008

M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

2007

J. J. Yang, Y. S. Liao, and C. F. Chen, “Fabrication of long hexagonal micro-lens array by applying gray-scale lithography in micro-replication process,” Opt. Commun. 270, 433–440 (2007).
[CrossRef]

2006

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
[CrossRef]

S. I. Chang, J. B. Yoon, H. Kim, J. J. Kim, B. K. Lee, and D. H. Shin, “Microlens array diffuser for a light-emitting diode backlight system,” Opt. Lett. 31, 3016–3018 (2006).
[CrossRef]

2005

2004

A. Segal, M. Epstein, and A. Yogev, “Hybrid concentrated photovoltaic and thermal power conversion at different spectral bands,” Sol. Eng. 76, 591–601 (2004).
[CrossRef]

2003

J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Opt. Eng. 42, 3084–3085 (2003).
[CrossRef]

2002

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

2001

1992

1986

1979

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Eng. 22, 119–123 (1979).
[CrossRef]

1974

R. Winston, “Principles of solar concentrators of a novel design,” Sol. Eng. 16, 89–95 (1974).
[CrossRef]

1966

Antón, I.

Araki, K.

Aschke, L.

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

Bayer, A.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

Benitez, P.

M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
[CrossRef]

Benítez, P.

Bizjak, T.

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

Bortz, J.

Buck, J.

Buljan, M.

Büttner, A.

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

Chang, J. Y.

Chang, S. I.

Chaves, J.

Chemisana, D.

D. Chemisana, “Building integrated concentrating photovoltaics : a review,” Renew. Sustain. Energ. Rev. 15, 603–611(2011).
[CrossRef]

D. Chemisana and M. Ibáñez, “Linear Fresnel concentrators for building integrated applications,” Energy Convers. Manage. 51, 1476–1480 (2010).
[CrossRef]

Chen, C. F.

C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
[CrossRef]

J. J. Yang, Y. S. Liao, and C. F. Chen, “Fabrication of long hexagonal micro-lens array by applying gray-scale lithography in micro-replication process,” Opt. Commun. 270, 433–440 (2007).
[CrossRef]

Chen, Z.

Chong, K. K.

K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
[CrossRef]

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]

M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
[CrossRef]

Dai, Y. J.

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

Davies, P. A.

Deng, X.

Domínguez, C.

Epstein, M.

A. Segal, M. Epstein, and A. Yogev, “Hybrid concentrated photovoltaic and thermal power conversion at different spectral bands,” Sol. Eng. 76, 591–601 (2004).
[CrossRef]

Esparza, D.

D. Esparza and I. Moreno, “Solar concentrator with diffuser segments,” Proc. SPIE 8011, 80117B (2011).
[CrossRef]

Ford, J. E.

Friesem, A. A.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Eng. 22, 119–123 (1979).
[CrossRef]

Geisz, J.

Gordon, J. M.

Hauschild, D.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

Hernandez, M.

M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
[CrossRef]

Hernández, M.

Hoffnagle, J. A.

J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

Homburg, O.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

Hong, H. F.

Huang, H. I.

Ibáñez, M.

D. Chemisana and M. Ibáñez, “Linear Fresnel concentrators for building integrated applications,” Energy Convers. Manage. 51, 1476–1480 (2010).
[CrossRef]

Ivanenko, M.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

Jan, H. T.

C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
[CrossRef]

Jarczynski, M.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

Jefferson, C. M.

J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

Karp, J. H.

Kim, H.

Kim, J.

M. Park, K. Oh, J. Kim, H. W. Shin, and B. D. Oh, “A tapered dielectric waveguide solar concentrator for a compound semiconductor photovoltaic cell,” Opt. Express 18, 1777–1787 (2010).
[CrossRef]

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
[CrossRef]

Kim, J. J.

Kritchman, E. M.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Eng. 22, 119–123 (1979).
[CrossRef]

Kuo, H. Z.

Kuroda, Y.

Kurtz, S.

Lawrence, G. N.

Lee, B. K.

Liang, X.

Liao, Y. S.

J. J. Yang, Y. S. Liao, and C. F. Chen, “Fabrication of long hexagonal micro-lens array by applying gray-scale lithography in micro-replication process,” Opt. Commun. 270, 433–440 (2007).
[CrossRef]

Lin, C. H.

C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
[CrossRef]

Lin, Y.

Lissotschenko, V.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

Ma, R.

Meinschien, J.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

Miñano, J. C.

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]

M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
[CrossRef]

Mitra, T.

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

Mohedano, R.

Moreno, I.

D. Esparza and I. Moreno, “Solar concentrator with diffuser segments,” Proc. SPIE 8011, 80117B (2011).
[CrossRef]

Oh, B. D.

Oh, K.

Pan, J. W.

Park, K. M.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
[CrossRef]

Park, M.

Rhee, J. G.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
[CrossRef]

Rumyantsev, V. D.

Ryu, K.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
[CrossRef]

Sala, G.

Sales, T. R. M.

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Opt. Eng. 42, 3084–3085 (2003).
[CrossRef]

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

Segal, A.

A. Segal, M. Epstein, and A. Yogev, “Hybrid concentrated photovoltaic and thermal power conversion at different spectral bands,” Sol. Eng. 76, 591–601 (2004).
[CrossRef]

Shatz, N.

Shin, D. H.

Shin, H. W.

Shin, H. Y.

Siaw, F. L.

K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1976).

Tremblay, E. J.

Victoria, M.

Wang, C. M.

Wang, R. Z.

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

Winston, R.

Wong, C. W.

K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
[CrossRef]

Wong, G. S.

K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
[CrossRef]

Wu, J. Y.

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

Yang, J. J.

J. J. Yang, Y. S. Liao, and C. F. Chen, “Fabrication of long hexagonal micro-lens array by applying gray-scale lithography in micro-replication process,” Opt. Commun. 270, 433–440 (2007).
[CrossRef]

Yang, Y. L.

C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
[CrossRef]

Yano, T.

Yeh, N. C.

N. C. Yeh, “Analysis of spectrum distribution and optical losses under Fresnel lenses,” Renew. Sustain. Energ. Rev. 14, 2926–2935 (2010).
[CrossRef]

Yekutieli, G.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Eng. 22, 119–123 (1979).
[CrossRef]

Yogev, A.

A. Segal, M. Epstein, and A. Yogev, “Hybrid concentrated photovoltaic and thermal power conversion at different spectral bands,” Sol. Eng. 76, 591–601 (2004).
[CrossRef]

Yoon, J. B.

Young, C. G.

Yu, W.

Zamora, P.

Zeitner, U. D.

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

Zhai, H.

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

Zhang, L. Y.

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

Appl. Opt.

Energy Convers. Manage.

H. Zhai, Y. J. Dai, J. Y. Wu, R. Z. Wang, and L. Y. Zhang, “Experimental investigation and analysis on a concentrating solar collector using linear Fresnel lens,” Energy Convers. Manage. 51, 48–55 (2010).
[CrossRef]

D. Chemisana and M. Ibáñez, “Linear Fresnel concentrators for building integrated applications,” Energy Convers. Manage. 51, 1476–1480 (2010).
[CrossRef]

Opt. Commun.

J. J. Yang, Y. S. Liao, and C. F. Chen, “Fabrication of long hexagonal micro-lens array by applying gray-scale lithography in micro-replication process,” Opt. Commun. 270, 433–440 (2007).
[CrossRef]

C. F. Chen, C. H. Lin, H. T. Jan, and Y. L. Yang, “Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method,” Opt. Commun. 282, 360–366 (2009).
[CrossRef]

Opt. Eng.

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Opt. Eng. 42, 3084–3085 (2003).
[CrossRef]

A. Büttner and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

J. A. Hoffnagle and C. M. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. 42, 3090–3099 (2003).
[CrossRef]

Opt. Express

V. D. Rumyantsev, “Solar concentrator modules with silicone-on-glass Fresnel lens panels and multijunction cells,” Opt. Express 18, A17–A24 (2010).
[CrossRef]

M. Victoria, C. Domínguez, I. Antón, and G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express 17, 6487–6492 (2009).
[CrossRef]

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]

S. Kurtz and J. Geisz, “Multijunction solar cells for conversion of concentrated sunlight to electricity,” Opt. Express 18, A73–A78 (2010).
[CrossRef]

K. Araki, T. Yano, and Y. Kuroda, “30 kW concentrator photovoltaic system using dome-shaped Fresnel lenses,” Opt. Express 18, A53–A63 (2010).
[CrossRef]

M. Park, K. Oh, J. Kim, H. W. Shin, and B. D. Oh, “A tapered dielectric waveguide solar concentrator for a compound semiconductor photovoltaic cell,” Opt. Express 18, 1777–1787 (2010).
[CrossRef]

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

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

C. M. Wang, H. I. Huang, J. W. Pan, H. Z. Kuo, H. F. Hong, H. Y. Shin, and J. Y. Chang, “Single stage transmission type broadband solar concentrator,” Opt. Express 18, A118–A125 (2010).
[CrossRef]

N. Shatz, J. Bortz, and R. Winston, “Thermodynamic efficiency of solar concentrators,” Opt. Express 18, A5–A16 (2010).
[CrossRef]

Opt. Lett.

Proc. SPIE

T. R. M. Sales, “Structured microlens arrays for beam shaping,” Proc. SPIE 5175, 109–120 (2003).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760B (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

O. Homburg, A. Bayer, T. Mitra, J. Meinschien, and L. Aschke, “Beam shaping of high power diode lasers benefits from asymmetrical refractive micro-lens arrays,” Proc. SPIE 6876, 68760 (2008).
[CrossRef]

T. Bizjak, O. Homburg, A. Bayer, T. Mitra, and L. Aschke, “Free form micro-optics enable uniform off-axis illumination and superposition of high power laser devices,” Proc. SPIE 7062, 70620T (2008).
[CrossRef]

D. Hauschild, O. Homburg, T. Mitra, M. Ivanenko, M. Jarczynski, J. Meinschien, A. Bayer, and V. Lissotschenko, “Optimizing laser beam profiles using micro-lens arrays for efficient material processing: applications to solar cells,” Proc. SPIE 7202, 72020U (2009).
[CrossRef]

M. Hernandez, A. Cvetkovic, P. Benitez, and J. C. Miñano, “High-performance Kohler concentrators with uniform irradiance on solar cell,” Proc. SPIE 7059, 705908 (2008).
[CrossRef]

D. Esparza and I. Moreno, “Solar concentrator with diffuser segments,” Proc. SPIE 8011, 80117B (2011).
[CrossRef]

Renew. Sustain. Energ. Rev.

D. Chemisana, “Building integrated concentrating photovoltaics : a review,” Renew. Sustain. Energ. Rev. 15, 603–611(2011).
[CrossRef]

N. C. Yeh, “Analysis of spectrum distribution and optical losses under Fresnel lenses,” Renew. Sustain. Energ. Rev. 14, 2926–2935 (2010).
[CrossRef]

Rew. Eng.

K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Rew. Eng. 34, 1364–1370 (2009).
[CrossRef]

Sol. Eng.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Eng. 22, 119–123 (1979).
[CrossRef]

R. Winston, “Principles of solar concentrators of a novel design,” Sol. Eng. 16, 89–95 (1974).
[CrossRef]

A. Segal, M. Epstein, and A. Yogev, “Hybrid concentrated photovoltaic and thermal power conversion at different spectral bands,” Sol. Eng. 76, 591–601 (2004).
[CrossRef]

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Eng. 80, 1580–1587 (2006).
[CrossRef]

Other

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 1976).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1.
Fig. 1.

(a) Simple schematic of a solar concentrator, which consists of a collection lens and a solar cell chip. (b) Collected sunlight is usually focused to a circular spot instead of a rectangular one.

Fig. 2.
Fig. 2.

(a) Schematic sketch of BSFL to illustrate the relationship among the aspheric front surface, the periodic concave surface, and the image plane, where the solar cell is positioned. (b) Relative locations of the refracted rays from the front aspheric surface and the concave surface normals.

Fig. 3.
Fig. 3.

(a) Initial layout, including the high-order aspheric coefficients, drawn by ZEMAX software. (b) 3D layout and rays distribution simulated by ASAP software. A square light pattern on the solar cell can be observed.

Fig. 4.
Fig. 4.

(a) Simulated light pattern on the solar cell is square. (b) Irradiance distribution of the light pattern along x axis.

Fig. 5.
Fig. 5.

Measurement points for the irradiance uniformity of light pattern.

Fig. 6.
Fig. 6.

(a) Schematic layout of F-BSFL and its optical path drawn. (b) 3D layout and rays distribution of F-BSFL drawn by ASAP software. A square light pattern on the solar cell can also be observed.

Fig. 7.
Fig. 7.

(a) Simulated light pattern from F-BSFL on the solar cell is square. (b) Irradiance distribution of the light pattern along x axis.

Fig. 8.
Fig. 8.

Light patterns on the solar cell from different concave surfaces are distorted differently. (a) Outer concave surface generates a more distorted pattern. (b) Inner concave surface generates a less distorted one.

Fig. 9.
Fig. 9.

Prototypes of (a) BSFL and (b) F-BSFL.

Fig. 10.
Fig. 10.

Transmission efficiency measurement of the BSFL. (a) Optical setup and (b) experimental setup.

Fig. 11.
Fig. 11.

Captured light patterns of (a) BSFL and (b) F-BSFL.

Fig. 12.
Fig. 12.

Angular tolerance analysis and measurement results of the incident rays’ angles into BSFL and F-BSFL.

Fig. 13.
Fig. 13.

Main degradation factors of BSFL’s optical performance, including (a) slant of incident sunlight, (b) decentering and rotation of the solar cell, and (c) rotation of the collection lens of BSFL.

Tables (3)

Tables Icon

Table 1. Specification of BSFL

Tables Icon

Table 2. Simulated and Experimental Performances of BSFL and F-BSFL

Tables Icon

Table 3. Assembly Tolerance of BSFL and F-BSFL

Equations (14)

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

Z=cra21+[1c(1+k)ra2]12+adra4+aera6+afra8+agra10+,
Upper rim ray:Um1=pdBFL·n.
Chief ray:Uc=(p1)d+d/2BFL·n.
Lower rim ray:Um2=(p1)dBFL·n.
Upper rim ray:Um1=d2r+pdBFL·n.
Chief ray:Uc=(p1)d+d/2BFL·n.
Lower rim ray:Um2=d2r+(p1)dBFL·n.
Upper rim ray:Um1=d2rn(d2r+pdBFL·n).
Chief ray:Uc=n((p1)d+d/2BFL·n).
Lower rim ray:Um2=d2r+n(d2r+(p1)dBFL·n).
Upper rim ray:ymax=pdBFL×Um1.
Chief ray:yc=(pd/2)BFL×Uc=0.
Lower rim ray:ymin=(p1)dBFL×Um2.
U=EminEavg,

Metrics