Abstract

We present a versatile illumination system where white light emitting diodes are coupled through a planar waveguide to periodically patterned extraction features at the focal plane of a two dimensional lenslet array. Adjusting the position of the lenslet array allows control over both the directionality and divergence of the emitted beam. We describe an analytic design process, and show optimal designs can achieve high luminous emittance (1.3x104 lux) over a 2x2 foot aperture with over 75% optical efficiency while simultaneously allowing beam steering over ± 60° and divergence control from ± 5° to fully hemispherical output. Finally, we present experimental results of a prototype system which validate the design model.

© 2014 Optical Society of America

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References

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  1. J.-G. Chang and Y.-B. Fang, “Dot-pattern design of a light guide in an edge-lit backlight using a regional partition approach,” Opt. Eng. 46(4), 10984–10995 (2012).
  2. D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
    [Crossref]
  3. T. C. Teng and J. C. Ke, “A novel optical film to provide a highly collimated planar light source,” Opt. Express 21(18), 21444–21455 (2013).
    [Crossref] [PubMed]
  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]
  5. J. M. Hallas, K. A. Baker, J. H. Karp, E. J. Tremblay, and J. E. Ford, “Two-axis solar tracking accomplished through small lateral translations,” Appl. Opt. 51(25), 6117–6124 (2012).
    [Crossref] [PubMed]
  6. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28(23), 4996–4998 (1989).
    [Crossref] [PubMed]
  7. D. T. Moore, G. R. Schmidt, and B. L. Unger, “Concentrated photovoltaic stepped planar light guide,” in International Optical Design Conference, Technical Digest (Optical Society of America, 2010), paper JMB46P.
    [Crossref]
  8. J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
    [Crossref]
  9. H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
    [Crossref]
  10. H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
    [Crossref]
  11. T.-C. Teng, W.-S. Sun, L.-W. Tseng, and W.-C. Chang, “A slim apparatus of transferring discrete LEDs’ light into an ultra-collimated planar light source,” Opt. Express 21(22), 26972–26982 (2013).
    [Crossref] [PubMed]
  12. R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Academic, 2005).
  13. F. Fournier, W. J. Cassarly, and J. P. Rolland, “Method to improve spatial uniformity with lightpipes,” Opt. Lett. 33(11), 1165–1167 (2008).
    [Crossref] [PubMed]
  14. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
    [Crossref]
  15. S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
    [Crossref]
  16. K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
    [Crossref]
  17. A. O. Marcano, C. Loper, and N. Melikechi, “High-sensitivity absorption measurement in water and glass samples using a mode-mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78(22), 3415–3417 (2001).
    [Crossref]
  18. P. S. Chechurov and G. E. Romanova, “Using the ZEMAX software complex to form photometric models of LED illuminator devices,” J. Opt. Technol. 79(5), 302–304 (2012).
    [Crossref]

2013 (3)

2012 (3)

2011 (1)

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

2010 (1)

2008 (1)

2005 (2)

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

2004 (1)

J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
[Crossref]

2001 (1)

A. O. Marcano, C. Loper, and N. Melikechi, “High-sensitivity absorption measurement in water and glass samples using a mode-mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78(22), 3415–3417 (2001).
[Crossref]

1989 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[Crossref]

Baker, K. A.

Cai, S.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Cai, Y.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Cantore, M.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
[Crossref]

Cassarly, W. J.

Chang, J.-G.

J.-G. Chang and Y.-B. Fang, “Dot-pattern design of a light guide in an edge-lit backlight using a regional partition approach,” Opt. Eng. 46(4), 10984–10995 (2012).

Chang, W.-C.

Chechurov, P. S.

Cho, J.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

Cimo, P.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Cornelissen, H. J.

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

Denault, K. A.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
[Crossref]

DenBaars, S. P.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
[Crossref]

Fan, S.

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

Fang, Y.-B.

J.-G. Chang and Y.-B. Fang, “Dot-pattern design of a light guide in an edge-lit backlight using a regional partition approach,” Opt. Eng. 46(4), 10984–10995 (2012).

Feng, D.

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

Fong, H.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Ford, J. E.

Fournier, F.

Garner, S.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Gessmann, T.

J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
[Crossref]

Hallas, J. M.

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[Crossref]

He, M.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[Crossref]

Ho, C.

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

Jin, G.

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

Karp, J. H.

Ke, J. C.

Kim, J. K.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
[Crossref]

Li, M.

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

Li, X.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Lohmann, A. W.

Loper, C.

A. O. Marcano, C. Loper, and N. Melikechi, “High-sensitivity absorption measurement in water and glass samples using a mode-mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78(22), 3415–3417 (2001).
[Crossref]

Luo, H.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
[Crossref]

Ma, H.

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

Marcano, A. O.

A. O. Marcano, C. Loper, and N. Melikechi, “High-sensitivity absorption measurement in water and glass samples using a mode-mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78(22), 3415–3417 (2001).
[Crossref]

Melikechi, N.

A. O. Marcano, C. Loper, and N. Melikechi, “High-sensitivity absorption measurement in water and glass samples using a mode-mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78(22), 3415–3417 (2001).
[Crossref]

Mu, C.

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

Nakamura, S.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
[Crossref]

Ouyang, S.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Park, Y.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

Rolland, J. P.

Romanova, G. E.

Schubert, E. F.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
[Crossref]

Seshadri, R.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
[Crossref]

Shi, Q.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Sone, C.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

Sun, W.-S.

Teng, T. C.

Teng, T.-C.

Tremblay, E. J.

Tseng, L.-W.

Xie, Y.

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

Yan, Y.

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

Yang, X.

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

AIP Adv. (1)

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3(7), 072107 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

A. O. Marcano, C. Loper, and N. Melikechi, “High-sensitivity absorption measurement in water and glass samples using a mode-mismatched pump-probe thermal lens method,” Appl. Phys. Lett. 78(22), 3415–3417 (2001).
[Crossref]

J. K. Kim, T. Gessmann, H. Luo, and E. F. Schubert, “GaInN light emitting diodes with RuO2/SiO2/Ag omni-directional reflector,” Appl. Phys. Lett. 84(22), 4508–4510 (2004).
[Crossref]

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 243505 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[Crossref]

J. Opt. A Pure Appl. Opt. (1)

D. Feng, Y. Yan, X. Yang, G. Jin, and S. Fan, “Novel integrated light-guide plates for liquid crystal display backlight,” J. Opt. A Pure Appl. Opt. 7(3), 111–117 (2005).
[Crossref]

J. Opt. Technol. (1)

Opt. Eng. (1)

J.-G. Chang and Y.-B. Fang, “Dot-pattern design of a light guide in an edge-lit backlight using a regional partition approach,” Opt. Eng. 46(4), 10984–10995 (2012).

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (1)

H. J. Cornelissen, H. Ma, C. Ho, M. Li, and C. Mu, “Compact collimators for high brightness blue LEDs using dielectric multilayers,” Proc. SPIE 8123, 81230J (2011).
[Crossref]

Other (3)

D. T. Moore, G. R. Schmidt, and B. L. Unger, “Concentrated photovoltaic stepped planar light guide,” in International Optical Design Conference, Technical Digest (Optical Society of America, 2010), paper JMB46P.
[Crossref]

S. Garner, H. Fong, M. He, P. Cimo, X. Li, Y. Cai, S. Ouyang, Y. Xie, Q. Shi, and S. Cai, “Flexible glass substrates for display and lighting applications,” in IEEE Photonics Conference (IPC) (2013), pp. 176–177.
[Crossref]

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Academic, 2005).

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Figures (20)

Fig. 1
Fig. 1 Conceptual illustration of the planar illumination system. The components have been exploded for clarity.

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Fig. 2
Fig. 2 Section of the array showing a collimated beam when the arrays are aligned (a), a redirected beam when the arrays are translated (b), and a diverging beam when the arrays are rotated (c).

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Fig. 3
Fig. 3 Lens geometry examples: (a) fully filled refractive Fresnel lens showing crosstalk with lateral translation and (b) partially filled reflective spherical lens showing zero crosstalk with equivalent translation and F/#.

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Fig. 4
Fig. 4 Constant (a) and stepped (b) mode volume waveguide illustrations for N = 5 extraction sites. Each section as drawn supplies light to one row of lenses above the waveguide (not shown).

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Fig. 5
Fig. 5 Angular and spatial output distributions for a conventional CPC and a Bezier collimator both with a uniform Lambertian input.

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Fig. 6
Fig. 6 Wireframe models of faceted coupler with M = 3 segments (a) and corresponding optical efficiency for M = 3, 6, and 9 segments (b).

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Fig. 7
Fig. 7 Wireframe models of curled coupler showing 3 segments (a) and corresponding optical efficiency for a few ratios of t / R (b). The efficiency is independent of aspect ratio.

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Fig. 8
Fig. 8 Top down views of the CMV (top) and SMV (bottom) waveguides with N = 5 extraction sites. The grey squares indicate the position and size of a single lens.

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Fig. 9
Fig. 9 CMV-F design space for 25% of target emittance. (a) Optimization metric for N = 60 , F / # = 0.75 . (b) Maximum beam steering angle in { F / # , N } space. (c) Optical efficiency in { F / # , N } space. Note that the axes are rotated 90° counterclockwise from (b) to (c) to clearly illustrate the data.

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Fig. 10
Fig. 10 Single section wireframe model of optimal CMV-F design.

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Fig. 11
Fig. 11 Far field directivity (a) and divergence (b) simulations of the optimal CMV-F design, with total optical efficiency plotted on the left-hand plane (dashed blue). Part (a) shows good agreement between the Zemax (black) and analytic (red) models. Part (b) shows the Zemax model (black) on a log scale.

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Fig. 12
Fig. 12 SMV-C design space for 100% of the target emittance. (a) Maximum steering angle and (b) minimum beam divergence angle, constrained to { F / # , N } space.

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Fig. 13
Fig. 13 Single section wireframe model of optimal SMV-C design.

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Fig. 14
Fig. 14 Far field directivity (a) and divergence (b) simulations of optimal SMV-C design, with total optical efficiency plotted on the left-hand plane (dashed blue). Part (a) shows good agreement between the Zemax (black) and analytic (red) models. Part (b) shows the Zemax model (black) on a log scale.

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Fig. 15
Fig. 15 Dialux simulations of conventional 2x2 foot LED fixture (a) and optimized SMV-C design (b) - (d). The waveguide system was simulated in three configurations: [diffuse] 1° rotation, [spot 1] (Δx, Δy) = (-3, 3) mm, and [spot 2] (Δx, Δy) = (5, 0) mm.

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Fig. 16
Fig. 16 (a) Unit cell system. (b) Cut-away schematic drawn to scale and illustrative ray path. (c) Measured (top) and simulated (bottom) far field intensity patterns.

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Fig. 17
Fig. 17 Far field directivity of the unit cell system: analytic model (red), Zemax simulation (black), and lab measurement (blue). Measured drop in off-axis intensity is due to poor off-axis lens performance.

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Fig. 18
Fig. 18 (a) System components: (i) waveguide, (ii) ball-bearing extraction feature, (iii) lenses, and (iv) PCB, LEDs, and CPC coupler; (b) assembled system (shown without cover); and (c) exploded CAD model.

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Fig. 19
Fig. 19 Simulation (left column) and measurement (center column) of on-axis, off-axis, and diverged spots 3 meters from the aperture. The right column shows the corresponding view of the aperture from an angle.

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Fig. 20
Fig. 20 Near field directionality (a) and divergence (b) of the prototype system 3 meters from the aperture. Part (a) shows the analytic model (red), Zemax model (black), and measurements (blue). Part (b) shows the Zemax model (black) and measurement (blue) on a log scale.

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Tables (1)

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Table 1 System Efficiencies and Loss Mechanisms

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Equations (20)

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ψ max = sin −1 ( nsin( tan −1 ( 1 2( F/# ) −tan( θ 2 ) ) ) )
φ= sin −1 ( nsin( tan −1 ( w facet 2f ) ) )
w facet = t wg tanγ = t wg | γ= 45 ∘
χ=( 1− σ f t wg D )exp( −αD )
P ext,j = P 0 σ f t wg D ⋅ ( χ j−1 + η 2 χ 2N−j ) ( 1− η 1 η 2 χ 2N )
P ext,total = ∑ j=1 N P ext,j = P 0 σ f t wg D( χ−1 ) ⋅ ( χ N −1 )( 1+ η 2 χ N ) ( 1− η 1 η 2 χ 2N )
w facet <2 t wg
η beam = sin 2 ( θ 1 )
h 1 sin( θ 1 )= h 2 sin( θ 2 )
M = h 2 t w g
θ 0 = cos −1 ( 1− t 2R )
I o u t = η b e a m ( θ 1 ) η c o u p l e r ( M , θ 2 ) η e x t ( σ f , D , t w g , N , χ , η 1 , η 2 ) t w g cos θ N D h 2 2 P L E D
η coupler ( M, θ 2 )= f 1 ( M, A ⇀ 1 ) f 1 ( M, A ⇀ 2 ) θ 2 + f 1 ( M, A ⇀ 3 ) + f 2 ( M, A ⇀ 4 ) ( f 2 ( M, A ⇀ 5 ) θ 2 ) 2 + f 2 ( M, A ⇀ 6 )
f 1 ( M, A ⇀ i )= A i,1 M 2 + A i,2 M+ A i,3
f 2 ( M, A ⇀ i )= A i,1 A i,2 M+ A i,3
h 1 2 sin 2 θ 1 sin 2 θ 2 = η b e a m ( θ 1 ) η c o u p l e r ( M , θ 2 ) η e x t ( σ f , D , t w g , N , χ , η 1 , η 2 ) t w g P L E D I o u t N D
θ= cos −1 ( 2N( F/# )tanφ )
tanθ= N− cos 2 θ N( N−1 )+cosθsinθ
t wg = Dcosθ N
ψ max = sin − 1 ( n sin ( tan − 1 ( 1 2 ( F / # ) − tan ( sin − 1 ( h 1 N cos θ I o u t η c o u p l e r P L E D ) ) ) ) )

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