Abstract

The Maxwellian-view display can provide visual information to people with low vision because retinal images can be formed independently of the refractive power of an eye by using rays converging on its pupil. This study presents the holographic Maxwellian-view display, which generates a wavefront converging on the pupil and forming images on the retina. The beam convergent point can be moved electrically in accordance with the pupil movement, and the beam width in the pupil can be changed electrically to control the depth of field of the eye. A compact optical system configuration for the holographic Maxwellian-view display is also proposed. The prototype system was constructed and experimentally tested. Because this holographic technique allows the phase modulation in the pupil, eye aberrations can be corrected; thus, retinal images can be formed for eyes with astigmatism.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  4. R. D. Beer, D. I. MacLeod, and T. P. Miller, “The extended Maxwellian view (BIGMAX): a high-intensity, high-saturation color display for clinical diagnosis and vision research,” Behav. Res. Methods 37(3), 513–521 (2005).
    [Crossref] [PubMed]
  5. M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).
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  10. M. Sugawara, M. Suzuki, and N. Miyauchi, “Retinal imaging laser eyewear with focus-free and augmented reality,” in SID Symposium Digest of Technical Papers (2016), paper 14–5L.
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    [Crossref]

2017 (2)

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(4), 85 (2017).
[Crossref]

2014 (2)

F.-C. Huang, G. Wetzstein, B. A. Barsky, and R. Raskar, “Eyeglasses-free display: towards correcting visual aberrations with computational light field displays,” ACM Trans. Graph. 33(4), 59 (2014).
[Crossref]

H. Takahashi, Y. Ito, S. Nakata, and K. Yamada, “Retinal projection type super multi-view head-mounted display,” Proc. SPIE 9012, 90120L (2014).
[Crossref]

2012 (3)

A. Yuuki, K. Itoga, and T. Satake, “A new Maxwellian view display for accommodation trouble free,” J. Int. SID 20, 581–588 (2012).

V. F. Pamplona, M. M. Oliveira, D. G. Aliaga, and R. Raskar, “Tailored displays to compensate for visual aberrations,” ACM Trans. Graph. 31(4), 81 (2012).
[Crossref]

F.-C. Huang, D. Lanman, B. A. Barsky, and R. Raskar, “Correcting for optical aberrations using multilayer displays,” ACM Trans. Graph. 31(6), 185 (2012).
[Crossref]

2010 (1)

2009 (1)

2008 (1)

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

2006 (2)

2005 (1)

R. D. Beer, D. I. MacLeod, and T. P. Miller, “The extended Maxwellian view (BIGMAX): a high-intensity, high-saturation color display for clinical diagnosis and vision research,” Behav. Res. Methods 37(3), 513–521 (2005).
[Crossref] [PubMed]

1999 (2)

1998 (1)

T. Ando, K. Yamasaki, M. Okamoto, and E. Shimizu, “Head mounted display using holographic optical element,” Proc. SPIE 3293, 183–189 (1998).
[Crossref]

1980 (1)

D. G. Green, M. K. Powers, and M. S. Banks, “Depth of focus, eye size and visual acuity,” Vision Res. 20(10), 827–835 (1980).
[Crossref] [PubMed]

1966 (1)

G. Westheimer, “The Maxwellian view,” Vision Res. 6(12), 669–682 (1966).
[Crossref] [PubMed]

Akeley, K.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

Aliaga, D. G.

V. F. Pamplona, M. M. Oliveira, D. G. Aliaga, and R. Raskar, “Tailored displays to compensate for visual aberrations,” ACM Trans. Graph. 31(4), 81 (2012).
[Crossref]

Ando, T.

T. Ando, K. Yamasaki, M. Okamoto, T. Matsumoto, and E. Shimizu, “Evaluation of HOE for head mounted display,” Proc. SPIE 3637, 110–118 (1999).
[Crossref]

T. Ando, K. Yamasaki, M. Okamoto, and E. Shimizu, “Head mounted display using holographic optical element,” Proc. SPIE 3293, 183–189 (1998).
[Crossref]

Bang, K.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

Banks, M. S.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

D. G. Green, M. K. Powers, and M. S. Banks, “Depth of focus, eye size and visual acuity,” Vision Res. 20(10), 827–835 (1980).
[Crossref] [PubMed]

Barsky, B. A.

F.-C. Huang, G. Wetzstein, B. A. Barsky, and R. Raskar, “Eyeglasses-free display: towards correcting visual aberrations with computational light field displays,” ACM Trans. Graph. 33(4), 59 (2014).
[Crossref]

F.-C. Huang, D. Lanman, B. A. Barsky, and R. Raskar, “Correcting for optical aberrations using multilayer displays,” ACM Trans. Graph. 31(6), 185 (2012).
[Crossref]

Beer, R. D.

R. D. Beer, D. I. MacLeod, and T. P. Miller, “The extended Maxwellian view (BIGMAX): a high-intensity, high-saturation color display for clinical diagnosis and vision research,” Behav. Res. Methods 37(3), 513–521 (2005).
[Crossref] [PubMed]

Clark, R. L.

Cole, D. G.

Cooper, J.

Dileonardo, R.

Georgiou, A.

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(4), 85 (2017).
[Crossref]

Gibson, G.

Girshick, A. R.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

Green, D. G.

D. G. Green, M. K. Powers, and M. S. Banks, “Depth of focus, eye size and visual acuity,” Vision Res. 20(10), 827–835 (1980).
[Crossref] [PubMed]

Hoffman, D. M.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

Huang, F.-C.

F.-C. Huang, G. Wetzstein, B. A. Barsky, and R. Raskar, “Eyeglasses-free display: towards correcting visual aberrations with computational light field displays,” ACM Trans. Graph. 33(4), 59 (2014).
[Crossref]

F.-C. Huang, D. Lanman, B. A. Barsky, and R. Raskar, “Correcting for optical aberrations using multilayer displays,” ACM Trans. Graph. 31(6), 185 (2012).
[Crossref]

Inami, M.

M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).

Ito, Y.

H. Takahashi, Y. Ito, S. Nakata, and K. Yamada, “Retinal projection type super multi-view head-mounted display,” Proc. SPIE 9012, 90120L (2014).
[Crossref]

Itoga, K.

A. Yuuki, K. Itoga, and T. Satake, “A new Maxwellian view display for accommodation trouble free,” J. Int. SID 20, 581–588 (2012).

Jang, C.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

Kawakami, N.

M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).

Kim, J.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

Kollin, J. S.

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(4), 85 (2017).
[Crossref]

Lanman, D.

F.-C. Huang, D. Lanman, B. A. Barsky, and R. Raskar, “Correcting for optical aberrations using multilayer displays,” ACM Trans. Graph. 31(6), 185 (2012).
[Crossref]

Leach, J.

Lee, B.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

Lee, S.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

MacLeod, D. I.

R. D. Beer, D. I. MacLeod, and T. P. Miller, “The extended Maxwellian view (BIGMAX): a high-intensity, high-saturation color display for clinical diagnosis and vision research,” Behav. Res. Methods 37(3), 513–521 (2005).
[Crossref] [PubMed]

Maeda, T.

M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).

Maimone, A.

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(4), 85 (2017).
[Crossref]

Matsumoto, T.

T. Ando, K. Yamasaki, M. Okamoto, T. Matsumoto, and E. Shimizu, “Evaluation of HOE for head mounted display,” Proc. SPIE 3637, 110–118 (1999).
[Crossref]

Miller, T. P.

R. D. Beer, D. I. MacLeod, and T. P. Miller, “The extended Maxwellian view (BIGMAX): a high-intensity, high-saturation color display for clinical diagnosis and vision research,” Behav. Res. Methods 37(3), 513–521 (2005).
[Crossref] [PubMed]

Mishina, T.

Miyauchi, N.

M. Sugawara, M. Suzuki, and N. Miyauchi, “Retinal imaging laser eyewear with focus-free and augmented reality,” in SID Symposium Digest of Technical Papers (2016), paper 14–5L.

Moon, S.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

Nago, N.

Nakata, S.

H. Takahashi, Y. Ito, S. Nakata, and K. Yamada, “Retinal projection type super multi-view head-mounted display,” Proc. SPIE 9012, 90120L (2014).
[Crossref]

Okamoto, M.

T. Ando, K. Yamasaki, M. Okamoto, T. Matsumoto, and E. Shimizu, “Evaluation of HOE for head mounted display,” Proc. SPIE 3637, 110–118 (1999).
[Crossref]

T. Ando, K. Yamasaki, M. Okamoto, and E. Shimizu, “Head mounted display using holographic optical element,” Proc. SPIE 3293, 183–189 (1998).
[Crossref]

Okano, F.

Oliveira, M. M.

V. F. Pamplona, M. M. Oliveira, D. G. Aliaga, and R. Raskar, “Tailored displays to compensate for visual aberrations,” ACM Trans. Graph. 31(4), 81 (2012).
[Crossref]

Padgett, M. J.

Pamplona, V. F.

V. F. Pamplona, M. M. Oliveira, D. G. Aliaga, and R. Raskar, “Tailored displays to compensate for visual aberrations,” ACM Trans. Graph. 31(4), 81 (2012).
[Crossref]

Powers, M. K.

D. G. Green, M. K. Powers, and M. S. Banks, “Depth of focus, eye size and visual acuity,” Vision Res. 20(10), 827–835 (1980).
[Crossref] [PubMed]

Raskar, R.

F.-C. Huang, G. Wetzstein, B. A. Barsky, and R. Raskar, “Eyeglasses-free display: towards correcting visual aberrations with computational light field displays,” ACM Trans. Graph. 33(4), 59 (2014).
[Crossref]

F.-C. Huang, D. Lanman, B. A. Barsky, and R. Raskar, “Correcting for optical aberrations using multilayer displays,” ACM Trans. Graph. 31(6), 185 (2012).
[Crossref]

V. F. Pamplona, M. M. Oliveira, D. G. Aliaga, and R. Raskar, “Tailored displays to compensate for visual aberrations,” ACM Trans. Graph. 31(4), 81 (2012).
[Crossref]

Satake, T.

A. Yuuki, K. Itoga, and T. Satake, “A new Maxwellian view display for accommodation trouble free,” J. Int. SID 20, 581–588 (2012).

Shimizu, E.

T. Ando, K. Yamasaki, M. Okamoto, T. Matsumoto, and E. Shimizu, “Evaluation of HOE for head mounted display,” Proc. SPIE 3637, 110–118 (1999).
[Crossref]

T. Ando, K. Yamasaki, M. Okamoto, and E. Shimizu, “Head mounted display using holographic optical element,” Proc. SPIE 3293, 183–189 (1998).
[Crossref]

Sugawara, M.

M. Sugawara, M. Suzuki, and N. Miyauchi, “Retinal imaging laser eyewear with focus-free and augmented reality,” in SID Symposium Digest of Technical Papers (2016), paper 14–5L.

Suzuki, M.

M. Sugawara, M. Suzuki, and N. Miyauchi, “Retinal imaging laser eyewear with focus-free and augmented reality,” in SID Symposium Digest of Technical Papers (2016), paper 14–5L.

Tachi, S.

M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).

Takahashi, H.

H. Takahashi, Y. Ito, S. Nakata, and K. Yamada, “Retinal projection type super multi-view head-mounted display,” Proc. SPIE 9012, 90120L (2014).
[Crossref]

Takaki, Y.

Tanemoto, Y.

Westheimer, G.

G. Westheimer, “The Maxwellian view,” Vision Res. 6(12), 669–682 (1966).
[Crossref] [PubMed]

Wetzstein, G.

F.-C. Huang, G. Wetzstein, B. A. Barsky, and R. Raskar, “Eyeglasses-free display: towards correcting visual aberrations with computational light field displays,” ACM Trans. Graph. 33(4), 59 (2014).
[Crossref]

Wulff, K. D.

Yamada, K.

H. Takahashi, Y. Ito, S. Nakata, and K. Yamada, “Retinal projection type super multi-view head-mounted display,” Proc. SPIE 9012, 90120L (2014).
[Crossref]

Yamasaki, K.

T. Ando, K. Yamasaki, M. Okamoto, T. Matsumoto, and E. Shimizu, “Evaluation of HOE for head mounted display,” Proc. SPIE 3637, 110–118 (1999).
[Crossref]

T. Ando, K. Yamasaki, M. Okamoto, and E. Shimizu, “Head mounted display using holographic optical element,” Proc. SPIE 3293, 183–189 (1998).
[Crossref]

Yanagida, Y.

M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).

Yuuki, A.

A. Yuuki, K. Itoga, and T. Satake, “A new Maxwellian view display for accommodation trouble free,” J. Int. SID 20, 581–588 (2012).

Yuyama, I.

ACM Trans. Graph. (5)

V. F. Pamplona, M. M. Oliveira, D. G. Aliaga, and R. Raskar, “Tailored displays to compensate for visual aberrations,” ACM Trans. Graph. 31(4), 81 (2012).
[Crossref]

F.-C. Huang, D. Lanman, B. A. Barsky, and R. Raskar, “Correcting for optical aberrations using multilayer displays,” ACM Trans. Graph. 31(6), 185 (2012).
[Crossref]

F.-C. Huang, G. Wetzstein, B. A. Barsky, and R. Raskar, “Eyeglasses-free display: towards correcting visual aberrations with computational light field displays,” ACM Trans. Graph. 33(4), 59 (2014).
[Crossref]

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36(6), 190 (2017).
[Crossref]

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(4), 85 (2017).
[Crossref]

Appl. Opt. (2)

Behav. Res. Methods (1)

R. D. Beer, D. I. MacLeod, and T. P. Miller, “The extended Maxwellian view (BIGMAX): a high-intensity, high-saturation color display for clinical diagnosis and vision research,” Behav. Res. Methods 37(3), 513–521 (2005).
[Crossref] [PubMed]

J. Int. SID (1)

A. Yuuki, K. Itoga, and T. Satake, “A new Maxwellian view display for accommodation trouble free,” J. Int. SID 20, 581–588 (2012).

J. Vis. (1)

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

Opt. Express (2)

Proc. IEEE (1)

Y. Takaki, “High-density directional display for generating natural three-dimensional images,” Proc. IEEE 94(3), 654–663 (2006).
[Crossref]

Proc. SPIE (3)

H. Takahashi, Y. Ito, S. Nakata, and K. Yamada, “Retinal projection type super multi-view head-mounted display,” Proc. SPIE 9012, 90120L (2014).
[Crossref]

T. Ando, K. Yamasaki, M. Okamoto, and E. Shimizu, “Head mounted display using holographic optical element,” Proc. SPIE 3293, 183–189 (1998).
[Crossref]

T. Ando, K. Yamasaki, M. Okamoto, T. Matsumoto, and E. Shimizu, “Evaluation of HOE for head mounted display,” Proc. SPIE 3637, 110–118 (1999).
[Crossref]

Vision Res. (2)

G. Westheimer, “The Maxwellian view,” Vision Res. 6(12), 669–682 (1966).
[Crossref] [PubMed]

D. G. Green, M. K. Powers, and M. S. Banks, “Depth of focus, eye size and visual acuity,” Vision Res. 20(10), 827–835 (1980).
[Crossref] [PubMed]

Other (5)

T. A. Furness III and J. S. Kollins, “Display with variable transmissive element,” US Patent 2003/0095081 A1 (2003).

N. Fujimoto and Y. Takaki, “Holographic Maxwellian-view Display System,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (online) (Optical Society of America, 2017), paper Th3A.4.

J. Park, “Waveguide-type See-through 3D Head-mounted Displays Without Accommodation Vergence Mismatch,” in Frontiers in Optics 2017, OSA Technical Digest (online) (Optical Society of America, 2017), paper FW5C.3.

M. Inami, N. Kawakami, T. Maeda, Y. Yanagida, and S. Tachi, “A stereoscopic display with large field of view using Maxwellian optics,” in Proceedings of Int. Conf. Artificial Reality and Tele-Existence‘97, pp. 71–76 (1997).

M. Sugawara, M. Suzuki, and N. Miyauchi, “Retinal imaging laser eyewear with focus-free and augmented reality,” in SID Symposium Digest of Technical Papers (2016), paper 14–5L.

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Figures (12)

Fig. 1
Fig. 1 Conventional Maxwellian-view display based on geometrical optics.
Fig. 2
Fig. 2 Depth of field (DOF) range where the Maxwellian view is achieved when the virtual image is produced at (a) a finite position and (b) infinity.
Fig. 3
Fig. 3 Basic idea for (a) the image projection on the retina and (b) the movement of the light convergent point in the holographic Maxwellian-view display.
Fig. 4
Fig. 4 Proposed layout for the holographic Maxwellian-view display.
Fig. 5
Fig. 5 Optical system used for explaining the hologram calculation process.
Fig. 6
Fig. 6 Calculation process to obtain the wavefront generated by the spatial light modulator (SLM).
Fig. 7
Fig. 7 Reduction of the system length of the holographic Maxwellian-view display.
Fig. 8
Fig. 8 Optical system for the holographic Maxwellian-view display with see-through function using a liquid crystal on silicon spatial light modulator (LCOS-SLM).
Fig. 9
Fig. 9 Constructed holographic Maxwellian-view display prototype: (a) system dimensions and (b) the camera mounted for monitoring the pupil movement.
Fig. 10
Fig. 10 Experimental results for the electronic movement of the light convergent point: (a) light convergence point moved on the pupil plane (simulated by a paper sheet), and (b) the corresponding retinal images capture by a video camera
Fig. 11
Fig. 11 Retinal images produced by the holographic Maxwellian-view display prototype: the focus of the video camera was changed from 300 mm to 1,000 mm.
Fig. 12
Fig. 12 Retinal images formation for astigmatic eyes: (a)–(d) normal astigmatism and (e)-(f) irregular astigmatism (combination of two cylindrical lenses); (a), (c), and (e) are without correction, and (b), (d), and (f) are with correction by the auxiliary phase. CYL: degree of astigmatism; AX: angle of astigmatism.

Equations (11)

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1/ l n =1/l+δ/ f e d,
1/ l f =1/lδ/ f e d,
g ( x i , y i )=g( x i x s , y i y s )exp{ ik[ z i + ( x i x c ) 2 + ( y i y c ) 2 2 z i ] },
u( x p , y p )= K 1 exp( ik x p 2 + y p 2 2 z i ) + + g ( x i , y i )exp( ik x i 2 + y i 2 2 z i )exp( ik x p x i + y p y i z i )d x i d y i , = K 2 exp[ ik ( x p x s ) 2 + ( y p y s ) 2 2 z i ]G( x p x c λ z i , y p y c λ z i ),
u ( x p , y p )=circ( x p x c , y p y c ;d/2 )exp[ i ϕ a ( x p x c , y p y c ) ]u( x p , y p ),
h( x o , y o )= K 3 exp( ik x o 2 + y o 2 2 z p ) + + u ( x p , y p )exp( ik x p 2 + y p 2 2 z p ) ×exp( ik x o x p + y o y p z p )d x p d y p ,
h ( x o , y o )= K 4 exp[ i k 2 ( 1 z p 1 f c )( x o 2 + y o 2 ) ] + + G( x p x c λ z i , y p y c λ z i ) ×circ( x p x c , y p y c ;d/2 )exp[ i ϕ a ( x p x c , y p y c ) ] ×exp{ i k 2 ( 1 z i 1 z p )[ ( x p x c ) 2 + ( y p y c ) 2 ] }exp( i2π x 0 x p + y 0 y p λ z p )d x p d y p ,
d= 2.44λ/( 1/ l n 1/ l f ) .
l=2/( 1/ l n +1/ l f ).
u'( x p , y p )= K 5 exp[ ik x p 2 + y p 2 2 z p /( 1g/ f c ) ] + + h ( x o , y o )exp( ik x p x o + y p y o f c )d x o d y o ,
h'( x o , y o )= 1 K 5 + + u'( x p , y p )exp[ ik x p 2 + y p 2 2 z p /( 1g/ f c ) ]exp( ik x o x p + y o y p f c )d x p d y p = K 6 + + G( x p x c λ z i , y p y c λ z i )circ( x p x c , y p y c ;d/2 )exp[ i ϕ a ( x p x c , y p y c ) ] ×exp{ i k 2 [ 1 z i 1 z p ( 1g/ f c ) ][ ( x p x c ) 2 + ( y p y c ) 2 ] }exp( i2π x 0 x p + y 0 y p λ z p )d x p d y p ,

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