Abstract

A method of head-up display is presented that uses an in-line surface relief grating attached to a waveguide propagation head-up display to achieve a large field of view without the need for large-projection optics. Horizontal pupil expansion is achieved using an extraction hologram that is multiple times the size of the injection hologram and is recorded with modulated diffraction efficiency. Vertical pupil expansion is achieved by coupling the surface relief grating to the waveguide surface between the injection and extraction holograms. The grating replicates the beam along the propagation direction, which allows for a larger field of view at the extraction. Using this technique, both a Zemax OpticStudio computer model and a physical system demonstrator achieve a field of view of $ 16^\circ \times 14.25^\circ $.

© 2019 Optical Society of America

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Holographic waveguide heads-up display for longitudinal image magnification and pupil expansion

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References

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  1. Y.-C. Liu and M.-H. Wen, “Comparison of head-up display (HUD) vs. head-down display (HDD): driving performance of commercial vehicle operators in Taiwan,” Int. J. Hum.-Comput. Stud. 61, 679–697 (2004).
    [Crossref]
  2. M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.
  3. C. Wickens and P. Ververs, “Allocation of attention with head-up displays,” (Aviation Research Laboratory, Institute of Aviation, 1998).
  4. S. Fadden, C. D. Wickens, and P. Ververs, “Costs and benefits of head up displays: an attention perspective and a meta analysis,” (SAE International, 2000).
  5. W. J. Horrey, C. D. Wickens, and A. L. Alexander, “The effects of head-up display clutter and in-vehicle display separation on concurrent driving performance,” Proc. Hum. Factors Ergon. Soc. Annu. Meet. 47, 1880–1884 (2003).
    [Crossref]
  6. J. Upatnieks, “Compact head-up display,” U.S. patent4, 711, 512 (8December1987).
  7. R. B. Wood and M. J. Hayford, “Holographic and classical head up display technology for commercial and fighter aircraft,” Proc. SPIE 0883, 36–52 (1988).
    [Crossref]
  8. H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).
  9. M. H. Kalmanash, “Digital HUDs for tactical aircraft,” Proc. SPIE 6225, 62250L (2006).
    [Crossref]
  10. H. Peng, D. Cheng, J. Han, C. Xu, W. Song, L. Ha, J. Yang, Q. Hu, and Y. Wang, “Design and fabrication of a holographic head-up display with asymmetric field of view,” Appl. Opt. 53, H177–H185 (2014).
    [Crossref]
  11. J. Han, J. Liu, X. Yao, and Y. Wang, “Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms,” Opt. Express 23, 3534–3549 (2015).
    [Crossref]
  12. P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
    [Crossref]
  13. P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).
  14. C. H. Vallance, “The approach to optical system designs for aircraft head up displays,” Proc. SPIE 0399, 15–25 (1983).
    [Crossref]
  15. C. T. Bartlett, M. L. Busbridge, and O. T. Horton, “Considerations of a head-up display field of view,” Proc. SPIE 4712, 468–479 (2002).
    [Crossref]
  16. P. L. Wisely, “Head up and head mounted display performance improvements through advanced techniques in the manipulation of light,” Proc. SPIE 7327, 732706 (2009).
    [Crossref]
  17. J. A. Cox, T. A. Fritz, and T. R. Werner, “Application and demonstration of diffractive optics for head-mounted displays,” Proc. SPIE 2218, 32–40 (1994).
    [Crossref]
  18. A. A. Cameron, “Optical waveguide technology and its application in head-mounted displays,” Proc. SPIE 8383, 83830E (2012).
    [Crossref]
  19. S. D. Harbour, “Three-dimensional system integration for HUD placement on a new tactical airlift platform: design eye point vs. HUD eye box with accommodation and perceptual implications,” Proc. SPIE 8383, 83830V (2012).
    [Crossref]
  20. M. Homan, “The use of optical waveguides in head up display (HUD) applications,” Proc. SPIE 8736, 87360E (2013).
    [Crossref]
  21. I. K. Wilmington and M. S. Valera, “Paper no 18.2: waveguide-based display technology,” SID Symp. Digest Tech. Pap. 44, 278–280 (2013).
    [Crossref]
  22. C. M. Bigler, P.-A. Blanche, and K. Sarma, “Holographic waveguide heads-up display for longitudinal image magnification and pupil expansion,” Appl. Opt. 57, 2007–2013 (2018).
    [Crossref]
  23. C. T. Draper, C. M. Bigler, M. S. Mann, K. Sarma, and P.-A. Blanche, “Holographic waveguide head-up display with 2-D pupil expansion and longitudinal image magnification,” Appl. Opt. 58, A251–A257 (2019).
    [Crossref]
  24. “Reflective holographic gratings,” 2019, https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=25 .
  25. P.-A. Blanche, Field Guide to Holography (SPIE, 2014).

2019 (1)

2018 (1)

2016 (1)

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (2)

M. Homan, “The use of optical waveguides in head up display (HUD) applications,” Proc. SPIE 8736, 87360E (2013).
[Crossref]

I. K. Wilmington and M. S. Valera, “Paper no 18.2: waveguide-based display technology,” SID Symp. Digest Tech. Pap. 44, 278–280 (2013).
[Crossref]

2012 (2)

A. A. Cameron, “Optical waveguide technology and its application in head-mounted displays,” Proc. SPIE 8383, 83830E (2012).
[Crossref]

S. D. Harbour, “Three-dimensional system integration for HUD placement on a new tactical airlift platform: design eye point vs. HUD eye box with accommodation and perceptual implications,” Proc. SPIE 8383, 83830V (2012).
[Crossref]

2009 (1)

P. L. Wisely, “Head up and head mounted display performance improvements through advanced techniques in the manipulation of light,” Proc. SPIE 7327, 732706 (2009).
[Crossref]

2006 (1)

M. H. Kalmanash, “Digital HUDs for tactical aircraft,” Proc. SPIE 6225, 62250L (2006).
[Crossref]

2004 (1)

Y.-C. Liu and M.-H. Wen, “Comparison of head-up display (HUD) vs. head-down display (HDD): driving performance of commercial vehicle operators in Taiwan,” Int. J. Hum.-Comput. Stud. 61, 679–697 (2004).
[Crossref]

2003 (1)

W. J. Horrey, C. D. Wickens, and A. L. Alexander, “The effects of head-up display clutter and in-vehicle display separation on concurrent driving performance,” Proc. Hum. Factors Ergon. Soc. Annu. Meet. 47, 1880–1884 (2003).
[Crossref]

2002 (1)

C. T. Bartlett, M. L. Busbridge, and O. T. Horton, “Considerations of a head-up display field of view,” Proc. SPIE 4712, 468–479 (2002).
[Crossref]

1994 (1)

J. A. Cox, T. A. Fritz, and T. R. Werner, “Application and demonstration of diffractive optics for head-mounted displays,” Proc. SPIE 2218, 32–40 (1994).
[Crossref]

1988 (1)

R. B. Wood and M. J. Hayford, “Holographic and classical head up display technology for commercial and fighter aircraft,” Proc. SPIE 0883, 36–52 (1988).
[Crossref]

1983 (1)

C. H. Vallance, “The approach to optical system designs for aircraft head up displays,” Proc. SPIE 0399, 15–25 (1983).
[Crossref]

Ablaßmeier, M.

M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.

Alexander, A. L.

W. J. Horrey, C. D. Wickens, and A. L. Alexander, “The effects of head-up display clutter and in-vehicle display separation on concurrent driving performance,” Proc. Hum. Factors Ergon. Soc. Annu. Meet. 47, 1880–1884 (2003).
[Crossref]

Bartlett, C. T.

C. T. Bartlett, M. L. Busbridge, and O. T. Horton, “Considerations of a head-up display field of view,” Proc. SPIE 4712, 468–479 (2002).
[Crossref]

Bengler, K.

M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.

Bigler, C. M.

Blanche, P.-A.

Busbridge, M. L.

C. T. Bartlett, M. L. Busbridge, and O. T. Horton, “Considerations of a head-up display field of view,” Proc. SPIE 4712, 468–479 (2002).
[Crossref]

Cameron, A. A.

A. A. Cameron, “Optical waveguide technology and its application in head-mounted displays,” Proc. SPIE 8383, 83830E (2012).
[Crossref]

Cheng, D.

Coni, P.

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).

Cox, J. A.

J. A. Cox, T. A. Fritz, and T. R. Werner, “Application and demonstration of diffractive optics for head-mounted displays,” Proc. SPIE 2218, 32–40 (1994).
[Crossref]

Draper, C. T.

Fadden, S.

S. Fadden, C. D. Wickens, and P. Ververs, “Costs and benefits of head up displays: an attention perspective and a meta analysis,” (SAE International, 2000).

Fritz, T. A.

J. A. Cox, T. A. Fritz, and T. R. Werner, “Application and demonstration of diffractive optics for head-mounted displays,” Proc. SPIE 2218, 32–40 (1994).
[Crossref]

Gueguen, A.

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).

Ha, L.

Han, J.

Harbour, S. D.

S. D. Harbour, “Three-dimensional system integration for HUD placement on a new tactical airlift platform: design eye point vs. HUD eye box with accommodation and perceptual implications,” Proc. SPIE 8383, 83830V (2012).
[Crossref]

Hayford, M. J.

R. B. Wood and M. J. Hayford, “Holographic and classical head up display technology for commercial and fighter aircraft,” Proc. SPIE 0883, 36–52 (1988).
[Crossref]

Homan, M.

M. Homan, “The use of optical waveguides in head up display (HUD) applications,” Proc. SPIE 8736, 87360E (2013).
[Crossref]

Horrey, W. J.

W. J. Horrey, C. D. Wickens, and A. L. Alexander, “The effects of head-up display clutter and in-vehicle display separation on concurrent driving performance,” Proc. Hum. Factors Ergon. Soc. Annu. Meet. 47, 1880–1884 (2003).
[Crossref]

Horton, O. T.

C. T. Bartlett, M. L. Busbridge, and O. T. Horton, “Considerations of a head-up display field of view,” Proc. SPIE 4712, 468–479 (2002).
[Crossref]

Hourlier, S.

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).

Hu, Q.

Imaizumi, M.

H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).

Ito, H.

H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).

Kalmanash, M. H.

M. H. Kalmanash, “Digital HUDs for tactical aircraft,” Proc. SPIE 6225, 62250L (2006).
[Crossref]

Kato, H.

H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).

Laluque, L.

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).

Liu, J.

Liu, Y.-C.

Y.-C. Liu and M.-H. Wen, “Comparison of head-up display (HUD) vs. head-down display (HDD): driving performance of commercial vehicle operators in Taiwan,” Int. J. Hum.-Comput. Stud. 61, 679–697 (2004).
[Crossref]

Mann, M. S.

Peng, H.

Poitschke, T.

M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.

Rigoll, G.

M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.

Sarma, K.

Servantie, X.

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).

Shibata, H.

H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).

Shima, J.

H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).

Song, W.

Upatnieks, J.

J. Upatnieks, “Compact head-up display,” U.S. patent4, 711, 512 (8December1987).

Valera, M. S.

I. K. Wilmington and M. S. Valera, “Paper no 18.2: waveguide-based display technology,” SID Symp. Digest Tech. Pap. 44, 278–280 (2013).
[Crossref]

Vallance, C. H.

C. H. Vallance, “The approach to optical system designs for aircraft head up displays,” Proc. SPIE 0399, 15–25 (1983).
[Crossref]

Ververs, P.

C. Wickens and P. Ververs, “Allocation of attention with head-up displays,” (Aviation Research Laboratory, Institute of Aviation, 1998).

S. Fadden, C. D. Wickens, and P. Ververs, “Costs and benefits of head up displays: an attention perspective and a meta analysis,” (SAE International, 2000).

Wallhoff, F.

M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.

Wang, Y.

Wen, M.-H.

Y.-C. Liu and M.-H. Wen, “Comparison of head-up display (HUD) vs. head-down display (HDD): driving performance of commercial vehicle operators in Taiwan,” Int. J. Hum.-Comput. Stud. 61, 679–697 (2004).
[Crossref]

Werner, T. R.

J. A. Cox, T. A. Fritz, and T. R. Werner, “Application and demonstration of diffractive optics for head-mounted displays,” Proc. SPIE 2218, 32–40 (1994).
[Crossref]

Wickens, C.

C. Wickens and P. Ververs, “Allocation of attention with head-up displays,” (Aviation Research Laboratory, Institute of Aviation, 1998).

Wickens, C. D.

W. J. Horrey, C. D. Wickens, and A. L. Alexander, “The effects of head-up display clutter and in-vehicle display separation on concurrent driving performance,” Proc. Hum. Factors Ergon. Soc. Annu. Meet. 47, 1880–1884 (2003).
[Crossref]

S. Fadden, C. D. Wickens, and P. Ververs, “Costs and benefits of head up displays: an attention perspective and a meta analysis,” (SAE International, 2000).

Wilmington, I. K.

I. K. Wilmington and M. S. Valera, “Paper no 18.2: waveguide-based display technology,” SID Symp. Digest Tech. Pap. 44, 278–280 (2013).
[Crossref]

Wisely, P. L.

P. L. Wisely, “Head up and head mounted display performance improvements through advanced techniques in the manipulation of light,” Proc. SPIE 7327, 732706 (2009).
[Crossref]

Wood, R. B.

R. B. Wood and M. J. Hayford, “Holographic and classical head up display technology for commercial and fighter aircraft,” Proc. SPIE 0883, 36–52 (1988).
[Crossref]

Xu, C.

Yang, J.

Yao, X.

Appl. Opt. (3)

Int. J. Hum.-Comput. Stud. (1)

Y.-C. Liu and M.-H. Wen, “Comparison of head-up display (HUD) vs. head-down display (HDD): driving performance of commercial vehicle operators in Taiwan,” Int. J. Hum.-Comput. Stud. 61, 679–697 (2004).
[Crossref]

Opt. Express (1)

Proc. Hum. Factors Ergon. Soc. Annu. Meet. (1)

W. J. Horrey, C. D. Wickens, and A. L. Alexander, “The effects of head-up display clutter and in-vehicle display separation on concurrent driving performance,” Proc. Hum. Factors Ergon. Soc. Annu. Meet. 47, 1880–1884 (2003).
[Crossref]

Proc. SPIE (9)

R. B. Wood and M. J. Hayford, “Holographic and classical head up display technology for commercial and fighter aircraft,” Proc. SPIE 0883, 36–52 (1988).
[Crossref]

C. H. Vallance, “The approach to optical system designs for aircraft head up displays,” Proc. SPIE 0399, 15–25 (1983).
[Crossref]

C. T. Bartlett, M. L. Busbridge, and O. T. Horton, “Considerations of a head-up display field of view,” Proc. SPIE 4712, 468–479 (2002).
[Crossref]

P. L. Wisely, “Head up and head mounted display performance improvements through advanced techniques in the manipulation of light,” Proc. SPIE 7327, 732706 (2009).
[Crossref]

J. A. Cox, T. A. Fritz, and T. R. Werner, “Application and demonstration of diffractive optics for head-mounted displays,” Proc. SPIE 2218, 32–40 (1994).
[Crossref]

A. A. Cameron, “Optical waveguide technology and its application in head-mounted displays,” Proc. SPIE 8383, 83830E (2012).
[Crossref]

S. D. Harbour, “Three-dimensional system integration for HUD placement on a new tactical airlift platform: design eye point vs. HUD eye box with accommodation and perceptual implications,” Proc. SPIE 8383, 83830V (2012).
[Crossref]

M. Homan, “The use of optical waveguides in head up display (HUD) applications,” Proc. SPIE 8736, 87360E (2013).
[Crossref]

M. H. Kalmanash, “Digital HUDs for tactical aircraft,” Proc. SPIE 6225, 62250L (2006).
[Crossref]

SID Symp. Digest Tech. Pap. (2)

P. Coni, S. Hourlier, A. Gueguen, X. Servantie, and L. Laluque, “50-3: a full windshield head-up display using simulated collimation,” SID Symp. Digest Tech. Pap. 47, 684–687 (2016).
[Crossref]

I. K. Wilmington and M. S. Valera, “Paper no 18.2: waveguide-based display technology,” SID Symp. Digest Tech. Pap. 44, 278–280 (2013).
[Crossref]

Other (8)

“Reflective holographic gratings,” 2019, https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=25 .

P.-A. Blanche, Field Guide to Holography (SPIE, 2014).

H. Kato, H. Ito, J. Shima, M. Imaizumi, and H. Shibata, “Development of hologram head-up display,” (SAE International, 1992).

J. Upatnieks, “Compact head-up display,” U.S. patent4, 711, 512 (8December1987).

M. Ablaßmeier, T. Poitschke, F. Wallhoff, K. Bengler, and G. Rigoll, “Eye gaze studies comparing head-up and head-down displays in vehicles,” in IEEE International Conference on Multimedia and Expo (2007), pp. 2250–2252.

C. Wickens and P. Ververs, “Allocation of attention with head-up displays,” (Aviation Research Laboratory, Institute of Aviation, 1998).

S. Fadden, C. D. Wickens, and P. Ververs, “Costs and benefits of head up displays: an attention perspective and a meta analysis,” (SAE International, 2000).

P. Coni, S. Hourlier, X. Servantie, L. Laluque, and A. Gueguen, “A 3D head up display with simulated collimation,” (SAE International, 2016).

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Figures (8)

Fig. 1.
Fig. 1. In a traditional HUD, light from the source is encoded with the desired image before it passes through a collection of relay optics that cause the image to be located at optical infinity. The light is then projected onto a partially reflective, transparent combiner so that the image is reflected to the observer while still allowing him/her to see outside the vehicle. An orthogonal view of the imaging device, the image plane, and the perceived image are shown to demonstrate how an image propagates through the system.
Fig. 2.
Fig. 2. Light from an outside source is projected through the waveguide HOE system to present an image to the observer over an expanded eyebox. Successive HOEs increase the exit pupil size, first vertically, then horizontally.
Fig. 3.
Fig. 3. Propagation angle is chosen to bisect the critical angle and the angle defined by Eq. (1).
Fig. 4.
Fig. 4. Cross-sectional splitting of incident light by SRG. After the first interaction with the grating, light is diffracted into $ + {1}$ , $ - {1}$ , and zero diffraction orders. Light in the zero order is again split into the $ + {1}$ , $ - {1}$ , and zero orders, while light diffracted into the $ \pm 1 $ orders is not diffracted or diffracted back to the zero order direction.
Fig. 5.
Fig. 5. Light incident from the source is diffracted at the design angle by the injection hologram. Light incident on the SRG is split into the $ + {1}$ , $ - {1}$ , and zero diffraction orders. Subsequent interactions with the SRG continue diffraction into one of the three propagation directions. The variable DE extraction hologram diffracts the expanded “zero order” toward the observer. The image inset provides an enlarged view of the branching diffraction that is caused by the SRG. Without the SRG, only a narrow portion of the image would be displayed.
Fig. 6.
Fig. 6. (Top) 5%, 50%, and 33% DE to $ \pm 1 $ diffraction orders. 33% DE maintains image quality without sacrificing uniform intensity. (Bottom) No SRG, $ 25\;{\rm mm} \times 25\;{\rm mm} $ SRG, and $ 50\;{\rm mm} \times 50\;{\rm mm} $ SRG. Increasing grating size allows for more of the projected image to be visible.
Fig. 7.
Fig. 7. Physical demonstrator of in-line pupil expansion HUD setup. Injection and extraction holograms are recorded in Bayfol HX200, which is attached to the waveguide surface. The SRG is coupled to the waveguide with index-matching fluid. The projector is pushed flush to the waveguide surface, minimizing the footprint of the system.
Fig. 8.
Fig. 8. Demonstrator shown in Fig. 7 is set up with an image projected in the background. When focusing on the waveguide plane, the image is not clear. Only when focus changes to the far field does the image become clear. Though the image visible in this scene is green, changing the observer’s viewing position from left to right changes the color of the displayed image from violet to red. (a) Focused at the waveguide plane; (b) focused at optical infinity.

Equations (2)

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θ m a x = tan 1 ( w / 2 t ) ,
R i = 1 N t o t N i + 1 ,

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