Abstract

We describe a method in which holograms can be produced by calculation from images captured by integral photography (IP). We present a basic algorithm obtained by simulating IP reconstruction, in which conditions are set so as not to cause aliasing in the holograms after the calculations. To reduce the calculation load, we also propose a way to limit the range of calculation considering the distribution of light and a way to shift the optical field on the exit plane of microlenses in a lens array. Finally, by optical experiments, we confirm that three-dimensional images can be reconstructed from holograms calculated from an integral photograph of a real object captured with an IP camera.

© 2006 Optical Society of America

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  1. P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
    [CrossRef]
  2. N. Hashimoto, S. Morokawa, and K. Kitamura, "Real-time holography using the high-resolution LCTV-SLM," in Practical Holography V, S. A. Benton, ed., Proc. SPIE 1461, 291-302 (1991).
    [CrossRef]
  3. T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
    [CrossRef]
  4. O. Bryngdahl and A. Lohmann, "Single-sideband holography," J. Opt. Soc. Am. 58, 620-624 (1968).
    [CrossRef]
  5. T. Takemori, "3-dimentional display using liquid crystal devices--fast computation of hologram," ITE Tech. Rep. 21, No. 46 (Institute of Image Information and Television Engineers, 1997), pp. 13-19.
  6. T. Mishina, F. Okano, and I. Yuyama, "Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones," Appl. Opt. 38, 3703-3713 (1999).
    [CrossRef]
  7. T. Mishina, M. Okui, and F. Okano, "Viewing-zone enlargement method for sampled hologram that uses high-order diffraction," Appl. Opt. 41, 1489-1499 (2002).
    [CrossRef] [PubMed]
  8. M. G. Lippmann, "La photographie integrals," Compt. Rend. 146, 446-451 (1908).
  9. F. Okano, J. Arai, H. Hoshino, and I. Yuyama, "Three-dimensional video system based on integral photography," Opt. Eng. 38, 1072-1077 (1999).
    [CrossRef]
  10. S.-H. Shin and B. Javidi, "Speckle-reduced three-dimensional volume holographic display by use of integral imaging," Appl. Opt. 41, 2644-2649 (2002).
    [CrossRef] [PubMed]
  11. K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).
  12. O. Matoba, E. Tajahuerce, and B. Javidi, "Real-time three-dimensional object recognition with multiple perspective imaging," Appl. Opt. 40, 3318-3325 (2001).
    [CrossRef]
  13. T. Naemura, T. Yoshida, and H. Harashima, "3-D computer graphics based on integral photography," Opt. Express 8, 255-262 (2001).
    [CrossRef] [PubMed]
  14. H. Arimoto and B. Javidi, "Integral three-dimensional imaging with digital reconstruction," Opt. Lett. 26, 157-159 (2001).
    [CrossRef]
  15. R. V. Pole, "3-D imagery and holograms of objects illuminated in white light," Appl. Phys. Lett. 10, 20-22 (1967).
    [CrossRef]
  16. J. Arai, H. Hoshino, M. Okui, and F. Okano, "Effect of focusing on the resolution characteristics of integral photography," J. Opt. Soc. Am. A 20, 996-1004 (2003).
    [CrossRef]
  17. J. Arai, M. Okui, M. Kobayashi, and F. Okano, "Geometrical effects of positional errors in integral photography," J. Opt. Soc. Am. A 21, 951-958 (2004).
    [CrossRef]
  18. J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
    [CrossRef]

2005 (2)

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

2001 (3)

1999 (3)

T. Mishina, F. Okano, and I. Yuyama, "Time-alternating method based on single-sideband holography with half-zone-plate processing for the enlargement of viewing zones," Appl. Opt. 38, 3703-3713 (1999).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, "Three-dimensional video system based on integral photography," Opt. Eng. 38, 1072-1077 (1999).
[CrossRef]

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

1991 (1)

N. Hashimoto, S. Morokawa, and K. Kitamura, "Real-time holography using the high-resolution LCTV-SLM," in Practical Holography V, S. A. Benton, ed., Proc. SPIE 1461, 291-302 (1991).
[CrossRef]

1990 (1)

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

1968 (1)

1967 (1)

R. V. Pole, "3-D imagery and holograms of objects illuminated in white light," Appl. Phys. Lett. 10, 20-22 (1967).
[CrossRef]

1908 (1)

M. G. Lippmann, "La photographie integrals," Compt. Rend. 146, 446-451 (1908).

Akimono, T.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Arai, J.

Arimoto, H.

Benton, S. A.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Bryngdahl, O.

Choi, H.

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

Choi, K.

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

Hahn, J.

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

Harashima, H.

Hashimoto, N.

N. Hashimoto, S. Morokawa, and K. Kitamura, "Real-time holography using the high-resolution LCTV-SLM," in Practical Holography V, S. A. Benton, ed., Proc. SPIE 1461, 291-302 (1991).
[CrossRef]

Higuchi, K.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Hilaire, P. S.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Horikoshi, T.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Hoshino, H.

J. Arai, H. Hoshino, M. Okui, and F. Okano, "Effect of focusing on the resolution characteristics of integral photography," J. Opt. Soc. Am. A 20, 996-1004 (2003).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, "Three-dimensional video system based on integral photography," Opt. Eng. 38, 1072-1077 (1999).
[CrossRef]

Imai, T.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Javidi, B.

Jepsen, M. L.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Kim, H.

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

Kim, J.

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

Kim, Y.

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

Kitamura, K.

N. Hashimoto, S. Morokawa, and K. Kitamura, "Real-time holography using the high-resolution LCTV-SLM," in Practical Holography V, S. A. Benton, ed., Proc. SPIE 1461, 291-302 (1991).
[CrossRef]

Kobayashi, M.

Kollin, J.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Lee, B.

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

Lim, Y.

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

Lippmann, M. G.

M. G. Lippmann, "La photographie integrals," Compt. Rend. 146, 446-451 (1908).

Lohmann, A.

Lucente, M.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Matoba, O.

Mishina, T.

Morokawa, S.

N. Hashimoto, S. Morokawa, and K. Kitamura, "Real-time holography using the high-resolution LCTV-SLM," in Practical Holography V, S. A. Benton, ed., Proc. SPIE 1461, 291-302 (1991).
[CrossRef]

Naemura, T.

Okano, F.

Okui, M.

Park, J.-H.

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

Pole, R. V.

R. V. Pole, "3-D imagery and holograms of objects illuminated in white light," Appl. Phys. Lett. 10, 20-22 (1967).
[CrossRef]

Sasaura, M.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Shin, S.-H.

Sonehara, N.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Suzuki, S.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Tajahuerce, E.

Takemori, T.

T. Takemori, "3-dimentional display using liquid crystal devices--fast computation of hologram," ITE Tech. Rep. 21, No. 46 (Institute of Image Information and Television Engineers, 1997), pp. 13-19.

Underkoffler, J.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Yagi, S.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Yamazaki, H.

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Yoshida, T.

Yoshikawa, H.

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

Yuyama, I.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

R. V. Pole, "3-D imagery and holograms of objects illuminated in white light," Appl. Phys. Lett. 10, 20-22 (1967).
[CrossRef]

Compt. Rend. (1)

M. G. Lippmann, "La photographie integrals," Compt. Rend. 146, 446-451 (1908).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys. (1)

J.-H. Park, H. Choi, Y. Kim, J. Kim, and B. Lee, "Scaling of three-dimensional integral imaging," Jpn. J. Appl. Phys. 44, 216-224 (2005).
[CrossRef]

Opt. Eng. (1)

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, "Three-dimensional video system based on integral photography," Opt. Eng. 38, 1072-1077 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (4)

K. Choi, H. Choi, H. Kim, J. Hahn, Y. Lim, J. Kim, and B. Lee, "Viewing-angle enhanced computer-generated holographic display system combined with integral imaging," in Three-Dimensional TV, Video, and Display IV, B. Javidi, F. Okano, and J.-Y. Son, eds., Proc. SPIE 6016, 601-612 (2005).

P. S. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, "Electronic display system for computational holography," in Practical Holography IV, S. A. Benton, ed., Proc. SPIE 1212, 174-182 (1990).
[CrossRef]

N. Hashimoto, S. Morokawa, and K. Kitamura, "Real-time holography using the high-resolution LCTV-SLM," in Practical Holography V, S. A. Benton, ed., Proc. SPIE 1461, 291-302 (1991).
[CrossRef]

T. Horikoshi, M. Sasaura, T. Imai, H. Yamazaki, T. Akimono, S. Yagi, K. Higuchi, S. Suzuki, and N. Sonehara, "Time-sharing display approach using liquid crystal light valve and a photorefractive crystal for electroholography," in Practical Holography XIII, S. A. Benton, ed., Proc. SPIE 3637, 64-71 (1999).
[CrossRef]

Other (1)

T. Takemori, "3-dimentional display using liquid crystal devices--fast computation of hologram," ITE Tech. Rep. 21, No. 46 (Institute of Image Information and Television Engineers, 1997), pp. 13-19.

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Figures (16)

Fig. 1
Fig. 1

Principle of IP: (a) capturing and (b) reconstruction.

Fig. 2
Fig. 2

Generation of holograms using an IP image.

Fig. 3
Fig. 3

Configuration used to calculate the optical field of an object beam reconstructed from an elemental image on a hologram plane.

Fig. 4
Fig. 4

Hologram produced by object beams reconstructed from a single elemental image: (a) when θ > 2ϕ m , aliasing occurs in a hologram; (b) when θ ≤ 2ϕ m , aliasing does not occur in a hologram.

Fig. 5
Fig. 5

Production of holograms from an IP image. (a) Basic algorithm. The spread angle of the IP-reconstructed object beam is limited to 2ϕ m so as not to cause aliasing. (b) To reduce the calculation load, the calculation range is limited to the same size as a microlens centered on (−apl∕f, −bpl∕f), in which the IP-reconstructed beam is distributed. (c) The optical field on the exit plane of a microlens is shifted to the hologram plane.

Fig. 6
Fig. 6

Ratio of the energy of a beam from a pixel of an elemental image distributed inside a fixed range (expressed as α times aperture width w) to the energy distributed on the whole hologram plane. Distance l between the microlens and hologram plane is set at 10, 50, and 100 mm.

Fig. 7
Fig. 7

Half-zone-plate processing: (a) principle of half-zone-plate processing and (b) application of half-zone-plate processing to the optical field-shift method.

Fig. 8
Fig. 8

Experimental setup (for real IP images).

Fig. 9
Fig. 9

Objects and IP image: (a) locations of computer-generated objects and (b) computer-generated IP image.

Fig. 10
Fig. 10

Reconstructed image from a hologram with aliasing. This picture is focused on the letter P. Several images generated from aliasing and conjugate images overlap with the true image.

Fig. 11
Fig. 11

Reconstructed image from a hologram without aliasing. This picture is focused on the letter P. Images caused by aliasing and their conjugate images are reduced. A conjugate image, however, still overlaps the true image.

Fig. 12
Fig. 12

Reconstructed images from a hologram calculated using the basic algorithm. The conjugate image is reduced in these images. (a) Focus is on the letter I. (b) Focus is on the letter P.

Fig. 13
Fig. 13

Reconstructed images from a hologram calculated by limiting the calculation range to twice the size of a microlens vertically and horizontally. (a) Focus is the same position as in Fig. 12(a). (b) Focus is the same position as in Fig. 12(b).

Fig. 14
Fig. 14

Reconstructed images from a hologram calculated using the field-shift method. (a) Focus is the same position as in Fig. 12(a). (b) Focus is the same position as in Fig. 12(b).

Fig. 15
Fig. 15

IP image captured by an IP camera.

Fig. 16
Fig. 16

Reconstructed images: (a) Focus is on the rear doll. (b) Focus is on the front doll.

Tables (4)

Tables Icon

Table 1 Parameters for IP Calculation of Figure 8(b)

Tables Icon

Table 2 Specifications of LCD

Tables Icon

Table 3 Parameters of an IP Image Taken by an IP Camera

Tables Icon

Table 4 Specifications of the LCD

Equations (13)

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

U L i ( x 2 , y 2 ) = a = m m b = n n ( j λ f exp ( j k f ) C a b δ ( x 1 ap , y 1 b p ) × exp { j k [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 2 f ] } × d x 1 d y 1 ) = j λ f exp ( jkf ) exp [ j k ( x 2     2 + y 2     2 2 f ) ] × a = m m b = n n { C a b exp [ j k ( a 2 p 2 + b 2 p 2 2 f ) ] ×   exp [ j k ( a p x 2 + b p y 2 f ) ] } ,
U L o ( x 2 , y 2 ) = U L i ( x 2 , y 2 ) S ( x 2 , y 2 ) exp [ j k ( x 2     2 + y 2     2 2 f ) ] = j λ f exp ( j k f ) S ( x 2 , y 2 ) a = m m b = n n { C a b × exp [ j k ( a 2 p 2 + b 2 p 2 2 f ) ] × exp [ j k ( a p f x 2 + b p f y 2 ) ] } ,
S ( x 2 , y 2 ) = { 1 ( inside of the microlens ) 0 ( outside of the microlens ) ,
U H ( x 3 , y 3 ) = j λ l exp ( j k l ) U L o ( x 2 , y 2 ) × exp { j k [ ( x 3 x 2 ) 2 + ( y 3 y 2 ) 2 2 l ] } × d x 2 d y 2 ,
ϕ m = λ 2 p ,
θ = 2 arctan ( w p / 2 f p ) ,
θ 2 ϕ m .
f p w p / [ 2 tan ( λ / 2 p ) ] .
U H ( x 3 , y 3 ) = 1 λ 2 l f exp [ j k ( l + f ) ] a = m m b = n n [ C a b exp [ j k ( a 2 p 2 + b 2 p 2 2 f ) ( 1 l f ) ] exp [ j k ( a p f x 3 + b p f y 3 ) ] × S ( x 2 , y 2 ) exp ( j k { [ x 2 ( x 3 + a p l f ) ] 2 + [ y 2 ( y 3 + b p l f ) ] 2 2 l } ) d x 2 d y 2 ] .
R H ( x 3 , y 3 ) = exp [ j k ( a p f x 3 + b p f y 3 ) ] × S ( x 2 , y 2 ) exp ( j k { [ x 2 ( x 3 + a p l f ) ] 2 + [ y 2 ( y 3 + b p l f ) ] 2 2 l } ) d x 2 d y 2 ,
E ( α ) = - b p l f α w 2 b p l f + α w 2 - a p l f α w 2 a p l f + α w 2 | R H ( x 3 , y 3 ) | 2 d x 3 d y 3 | R H ( x 3 , y 3 ) | 2 d x 3 d y 3 .
U H ( x 3 , y 3 ) = U L o ( x 3 + a p l f , y 3 + b p l f ) a = m m b = n n { C a b exp [ j k ( a p f x 3 + b p f y 3 ) S ( x 3 + a p l f , y 3 + b p l f ) ] } .
U H ( x 3 , y 3 ) a = 0 m b = n n { C a b exp [ j k ( a p f x 3 + b p f y 3 ) S ( x 3 + a p l f , y 3 + b p l f ) ] } .

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