Abstract

Planar micro-optic concentrators are passive optical structures which combine a lens array with faceted microstructures to couple sunlight into a planar slab waveguide. Guided rays propagate within the slab to edge-mounted photovoltaic cells. This paper provides analysis and preliminary experiments describing modifications and additions to the geometry which increase concentration ratios along both the vertical and orthogonal waveguide axes. We present simulated results for a 900x concentrator with 85% optical efficiency, measured results for small-scale experimental systems and briefly discuss implementations using low-cost fabrication on continuous planar waveguides.

© 2011 OSA

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References

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  1. D. Feuermann and J. M. Gordon, “High-concentration photovoltaic designs based on miniature parabolic dishes,” Sol. Energy 70(5), 423–430 (2001).
    [CrossRef]
  2. A. Bett, C. Baur, F. Dimroth, G. Lange, M. Meusel, S. van Riesen, G. Siefer, V. Andreev, V. Rumyantsev, and N. Sadchikov, “FLATCON™-modules: technology and characterisation,” WCPEC-3 634–637 (2003).
  3. J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18(2), 1122–1133 (2010).
    [CrossRef] [PubMed]
  4. J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
    [CrossRef]
  5. J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
    [CrossRef]
  6. C. Y. Chang, S. Y. Yang, and J. L. Sheh, “A roller embossing process for rapid fabrication of microlens arrays on glass substrates,” Microsyst. Technol. 12(8), 754–759 (2006).
    [CrossRef]
  7. S. H. Ahn and L. J. Guo, “High-Speed Roll-to-Roll Nanoimprint Lithography on Flexible Plastic Substrates,” Adv. Mater. (Deerfield Beach Fla.) 20(11), 2044–2049 (2008).
    [CrossRef]
  8. J. P. Morgan, “Light-guide solar panel and method of fabrication thereof,” Morgan Solar, Inc. World Intellectual Property Organization, WO 2008/131561, 11 June 2008.
  9. S. Ghosh and D. S. Schultz, “solar energy concentrator,” Banyan Energy, Inc., Patent US 7,672,549B2, 2 March 2010.
  10. B. L. Unger, G. R. Schmidt, and D. T. Moore, “Dimpled planar lightguide solar concentrators,” in OSA International Optical Design Conference, ITuE5P (2010).
  11. R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).
  12. W. T. Welford and R. Winston, High Collection Nonimaging Optics (1989).
  13. H. Ries, “Thermodynamic limitations of the concentration of electromagnetic radiation,” J. Opt. Soc. Am. 72(3), 380 (1982).
    [CrossRef]
  14. A. Rabl, Active Solar Collectors and their Applications (Oxford University Press, 1985).
  15. R. Winston and H. Hinterberger, “Principles of cylindrical concentrators for solar energy,” Sol. Energy 17(4), 255–258 (1975).
    [CrossRef]
  16. J. C. Miñano, J. C. Gonźlez, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34(34), 7850–7856 (1995).
    [CrossRef] [PubMed]
  17. R. Winston, W. T. Welford, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic Press, 2005).
  18. N. Fraidenraich, “Analytic solutions for the optical properties of V-trough concentrators,” Appl. Opt. 31(1), 131–139 (1992).
    [CrossRef] [PubMed]
  19. J. H. Karp, E. J. Tremblay, and J. E. Ford, “Radial coupling method for orthogonal concentration within planar micro-optic solar collectors,” in OSA Optics For Solar Energy, STuD2 (2010).
  20. M. Collares-Pereira, A. Rabl, and R. Winston, “Lens-mirror combinations with maximal concentration,” Appl. Opt. 16(10), 2677–2683 (1977).
    [CrossRef] [PubMed]
  21. A. T. Cannistra and T. J. Suleski, “Characterization of hybrid molding and lithography for SU-8 micro-optical components,” J. Micro/Nanolith. MEMS MOEMS 9(1), 013025 (2010).
    [CrossRef]
  22. J. H. Karp and J. E. Ford, “Planar micro-optic solar concentration using multiple imaging lenses into a common slab waveguide,” Proc. SPIE 7407, 7407–7411 (2009).
  23. E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1998).
  24. S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
    [CrossRef]
  25. L. Fu, R. Leutz, and H. P. Annen, “Secondary optics for Fresnel lens solar concentrators,” Proc. SPIE 7785, 778509, 778509-6 (2010).
    [CrossRef]

2010 (4)

J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18(2), 1122–1133 (2010).
[CrossRef] [PubMed]

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

A. T. Cannistra and T. J. Suleski, “Characterization of hybrid molding and lithography for SU-8 micro-optical components,” J. Micro/Nanolith. MEMS MOEMS 9(1), 013025 (2010).
[CrossRef]

L. Fu, R. Leutz, and H. P. Annen, “Secondary optics for Fresnel lens solar concentrators,” Proc. SPIE 7785, 778509, 778509-6 (2010).
[CrossRef]

2009 (1)

J. H. Karp and J. E. Ford, “Planar micro-optic solar concentration using multiple imaging lenses into a common slab waveguide,” Proc. SPIE 7407, 7407–7411 (2009).

2008 (1)

S. H. Ahn and L. J. Guo, “High-Speed Roll-to-Roll Nanoimprint Lithography on Flexible Plastic Substrates,” Adv. Mater. (Deerfield Beach Fla.) 20(11), 2044–2049 (2008).
[CrossRef]

2006 (1)

C. Y. Chang, S. Y. Yang, and J. L. Sheh, “A roller embossing process for rapid fabrication of microlens arrays on glass substrates,” Microsyst. Technol. 12(8), 754–759 (2006).
[CrossRef]

2005 (2)

R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

2001 (1)

D. Feuermann and J. M. Gordon, “High-concentration photovoltaic designs based on miniature parabolic dishes,” Sol. Energy 70(5), 423–430 (2001).
[CrossRef]

2000 (1)

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[CrossRef]

1995 (1)

1992 (1)

1982 (1)

1977 (1)

1975 (1)

R. Winston and H. Hinterberger, “Principles of cylindrical concentrators for solar energy,” Sol. Energy 17(4), 255–258 (1975).
[CrossRef]

Ahn, S. H.

S. H. Ahn and L. J. Guo, “High-Speed Roll-to-Roll Nanoimprint Lithography on Flexible Plastic Substrates,” Adv. Mater. (Deerfield Beach Fla.) 20(11), 2044–2049 (2008).
[CrossRef]

Alexander, M. R.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Annen, H. P.

L. Fu, R. Leutz, and H. P. Annen, “Secondary optics for Fresnel lens solar concentrators,” Proc. SPIE 7785, 778509, 778509-6 (2010).
[CrossRef]

Benítez, P.

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

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

Buytaert, G.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Cannistra, A. T.

A. T. Cannistra and T. J. Suleski, “Characterization of hybrid molding and lithography for SU-8 micro-optical components,” J. Micro/Nanolith. MEMS MOEMS 9(1), 013025 (2010).
[CrossRef]

Chang, C. Y.

C. Y. Chang, S. Y. Yang, and J. L. Sheh, “A roller embossing process for rapid fabrication of microlens arrays on glass substrates,” Microsyst. Technol. 12(8), 754–759 (2006).
[CrossRef]

Chaves, J.

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[CrossRef]

Collares-Pereira, M.

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[CrossRef]

M. Collares-Pereira, A. Rabl, and R. Winston, “Lens-mirror combinations with maximal concentration,” Appl. Opt. 16(10), 2677–2683 (1977).
[CrossRef] [PubMed]

Dimogerontakis, T.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Feuermann, D.

D. Feuermann and J. M. Gordon, “High-concentration photovoltaic designs based on miniature parabolic dishes,” Sol. Energy 70(5), 423–430 (2001).
[CrossRef]

Ford, J. E.

J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18(2), 1122–1133 (2010).
[CrossRef] [PubMed]

J. H. Karp and J. E. Ford, “Planar micro-optic solar concentration using multiple imaging lenses into a common slab waveguide,” Proc. SPIE 7407, 7407–7411 (2009).

Fraidenraich, N.

Fu, L.

L. Fu, R. Leutz, and H. P. Annen, “Secondary optics for Fresnel lens solar concentrators,” Proc. SPIE 7785, 778509, 778509-6 (2010).
[CrossRef]

Gonzlez, J. C.

Gordon, J. M.

D. Feuermann and J. M. Gordon, “High-concentration photovoltaic designs based on miniature parabolic dishes,” Sol. Energy 70(5), 423–430 (2001).
[CrossRef]

Guo, L. J.

S. H. Ahn and L. J. Guo, “High-Speed Roll-to-Roll Nanoimprint Lithography on Flexible Plastic Substrates,” Adv. Mater. (Deerfield Beach Fla.) 20(11), 2044–2049 (2008).
[CrossRef]

Hinterberger, H.

R. Winston and H. Hinterberger, “Principles of cylindrical concentrators for solar energy,” Sol. Energy 17(4), 255–258 (1975).
[CrossRef]

Infante, J.

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

Karam, N.

R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).

Karp, J. H.

J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18(2), 1122–1133 (2010).
[CrossRef] [PubMed]

J. H. Karp and J. E. Ford, “Planar micro-optic solar concentration using multiple imaging lenses into a common slab waveguide,” Proc. SPIE 7407, 7407–7411 (2009).

King, R.

R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).

Leutz, R.

L. Fu, R. Leutz, and H. P. Annen, “Secondary optics for Fresnel lens solar concentrators,” Proc. SPIE 7785, 778509, 778509-6 (2010).
[CrossRef]

Lillington, D.

R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).

Liu, J.

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

Miñano, J. C.

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

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

Rabl, A.

Ries, H.

Sheh, J. L.

C. Y. Chang, S. Y. Yang, and J. L. Sheh, “A roller embossing process for rapid fabrication of microlens arrays on glass substrates,” Microsyst. Technol. 12(8), 754–759 (2006).
[CrossRef]

Sherif, R.

R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).

Skeldon, P.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Stijns, E.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Suleski, T. J.

A. T. Cannistra and T. J. Suleski, “Characterization of hybrid molding and lithography for SU-8 micro-optical components,” J. Micro/Nanolith. MEMS MOEMS 9(1), 013025 (2010).
[CrossRef]

Terryn, H.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Thompson, G. E.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Tremblay, E. J.

Van Gils, S.

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

Wang, L.

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

Winston, R.

M. Collares-Pereira, A. Rabl, and R. Winston, “Lens-mirror combinations with maximal concentration,” Appl. Opt. 16(10), 2677–2683 (1977).
[CrossRef] [PubMed]

R. Winston and H. Hinterberger, “Principles of cylindrical concentrators for solar energy,” Sol. Energy 17(4), 255–258 (1975).
[CrossRef]

Yang, S. Y.

C. Y. Chang, S. Y. Yang, and J. L. Sheh, “A roller embossing process for rapid fabrication of microlens arrays on glass substrates,” Microsyst. Technol. 12(8), 754–759 (2006).
[CrossRef]

Adv. Mater. (Deerfield Beach Fla.) (1)

S. H. Ahn and L. J. Guo, “High-Speed Roll-to-Roll Nanoimprint Lithography on Flexible Plastic Substrates,” Adv. Mater. (Deerfield Beach Fla.) 20(11), 2044–2049 (2008).
[CrossRef]

Appl. Opt. (3)

IEEE PVSC (1)

R. Sherif, R. King, N. Karam, and D. Lillington, “The path to 1 GW of concentrator photovoltaics using multijunction solar cells,” IEEE PVSC 31, 17–22 (2005).

J. Appl. Phys. (1)

S. Van Gils, T. Dimogerontakis, G. Buytaert, E. Stijns, H. Terryn, P. Skeldon, G. E. Thompson, and M. R. Alexander, “Optical properties of magnetron-sputtered and rolled aluminum,” J. Appl. Phys. 98(8), 083505 (2005).
[CrossRef]

J. Micro/Nanolith. MEMS MOEMS (1)

A. T. Cannistra and T. J. Suleski, “Characterization of hybrid molding and lithography for SU-8 micro-optical components,” J. Micro/Nanolith. MEMS MOEMS 9(1), 013025 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

Microsyst. Technol. (1)

C. Y. Chang, S. Y. Yang, and J. L. Sheh, “A roller embossing process for rapid fabrication of microlens arrays on glass substrates,” Microsyst. Technol. 12(8), 754–759 (2006).
[CrossRef]

Opt. Express (1)

Proc. SPIE (3)

J. C. Miñano, P. Benítez, J. Liu, J. Infante, J. Chaves, and L. Wang, “Applications of the SMS method to the design of compact optics,” Proc. SPIE 7717, 77170I, 77170I-8 (2010).
[CrossRef]

J. H. Karp and J. E. Ford, “Planar micro-optic solar concentration using multiple imaging lenses into a common slab waveguide,” Proc. SPIE 7407, 7407–7411 (2009).

L. Fu, R. Leutz, and H. P. Annen, “Secondary optics for Fresnel lens solar concentrators,” Proc. SPIE 7785, 778509, 778509-6 (2010).
[CrossRef]

Sol. Energy (3)

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[CrossRef]

D. Feuermann and J. M. Gordon, “High-concentration photovoltaic designs based on miniature parabolic dishes,” Sol. Energy 70(5), 423–430 (2001).
[CrossRef]

R. Winston and H. Hinterberger, “Principles of cylindrical concentrators for solar energy,” Sol. Energy 17(4), 255–258 (1975).
[CrossRef]

Other (9)

R. Winston, W. T. Welford, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic Press, 2005).

A. Rabl, Active Solar Collectors and their Applications (Oxford University Press, 1985).

W. T. Welford and R. Winston, High Collection Nonimaging Optics (1989).

A. Bett, C. Baur, F. Dimroth, G. Lange, M. Meusel, S. van Riesen, G. Siefer, V. Andreev, V. Rumyantsev, and N. Sadchikov, “FLATCON™-modules: technology and characterisation,” WCPEC-3 634–637 (2003).

J. P. Morgan, “Light-guide solar panel and method of fabrication thereof,” Morgan Solar, Inc. World Intellectual Property Organization, WO 2008/131561, 11 June 2008.

S. Ghosh and D. S. Schultz, “solar energy concentrator,” Banyan Energy, Inc., Patent US 7,672,549B2, 2 March 2010.

B. L. Unger, G. R. Schmidt, and D. T. Moore, “Dimpled planar lightguide solar concentrators,” in OSA International Optical Design Conference, ITuE5P (2010).

J. H. Karp, E. J. Tremblay, and J. E. Ford, “Radial coupling method for orthogonal concentration within planar micro-optic solar collectors,” in OSA Optics For Solar Energy, STuD2 (2010).

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1998).

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Figures (13)

Fig. 1
Fig. 1

The micro-optic concentrator pairs a lens array with a planar slab waveguide (a). Localized 120° prisms placed on the waveguide surface couple light into guided modes without shadowing effects (b).

Fig. 2
Fig. 2

The lens array and coupling prisms create two bundles exiting the waveguide (a). Within the polar plot, output angles (green circles) fill only a portion of the total angular spectrum (b).

Fig. 3
Fig. 3

Orthogonal waveguides provides additional concentration along the slab width (a). Radial coupler orientation appears as a lens focusing into a v-trough (b).

Fig. 4
Fig. 4

Dimensions of the orthogonal waveguide layout.

Fig. 5
Fig. 5

Optical efficiency as a function of orthogonal concentration factor for various lens f-numbers. All systems incorporated 200mm of propagation in glass to the exit aperture.

Fig. 6
Fig. 6

Efficiency comparison between rectangular and orthogonal waveguide geometries. Both systems used an F/3 lens array and 1mm thick glass waveguides.

Fig. 7
Fig. 7

Optical layout showing 8x orthogonal concentration using F/3 lenses (note: system scale has been reduced for visualization purposes) (a). Sidewalls confine light along the slab width, increasing the angular spectrum in one dimension (b).

Fig. 8
Fig. 8

Optical layout of the prototype concentrator (a). SEM image of the waveguide surface showing coupler spacing and diameter (inset) (b). Image of the system under test (c).

Fig. 9
Fig. 9

Geometry of the experimental orthogonal concentrator (a). Image of the fabricated waveguide (b) and the concentrator under test (c).

Fig. 10
Fig. 10

Layout of the SOE positioned between two opposing waveguides.

Fig. 11
Fig. 11

Raytrace highlighting the odd and even reflection ray paths within the SOE (a) and the associated angular spectrum when capturing the output from F/3 focusing lenses (b).

Fig. 12
Fig. 12

Orthogonal waveguides can be combined with secondary optics to reach very high concentration levels (a). The associated angular spectrum increases in two dimensions (b).

Fig. 13
Fig. 13

Equivalent micro-optics increase concentration without sectioning the lens array or waveguide and enable continuous manufacturing approaches.

Equations (6)

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

C 2 D = 1 sin θ i n , C 3 D = 1 sin 2 θ i n
C g e o = l ( w 1 + w 2 ) / 2 w 2 h
tan α = 1 2 F / # + tan θ
f w i = C + 1 2 C 2 ( 3 C 2 2 C 1 ) 1 / 2   where   C = w i w 2 = 1 sin ( α / n )
tan ψ = w 1 2 f
l = f ( 1 1 C )

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