Abstract

Holographic recording methods require the use of a reference beam that is coherent with the signal beam carrying the information to be recorded. In this paper, we propose self-referential holography (SRH) for holographic recording without the use of a reference beam. SRH can realize purely one-beam holographic recording by considering the signal beam itself as the reference beam. The readout process in SRH is based on energy transfer by inter-pixel interference in holographic diffraction, which depends on the spatial phase difference between the recorded phase and the readout phase. The phase-modulated recorded signal is converted into an intensity-modulated beam that can be easily detected using a conventional image sensor. SRH can be used effectively for holographic data storage and phase-to-intensity conversion.

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2010

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

2009

J. Tanaka, A. Okamoto, and M. Kitano, “Development of image-based simulation for holographic data storage system by fast Fourier transform beam-propagation method,” Jpn. J. Appl. Phys.48(3), 03A028 (2009).
[CrossRef]

2008

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

2007

2006

2005

2004

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett.85(6), 1069–1071 (2004).
[CrossRef]

1998

D. Psaltis and G. W. Burr, “Holographic data storage,” Computer31(2), 52–60 (1998).
[CrossRef]

1997

1996

1978

1969

H. Kogelnik, “Coupled-wave theory for thick hologram grating,” Bell Syst. Tech. J.48, 2909–2947 (1969).

1963

1962

E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am.52(10), 1123–1128 (1962).
[CrossRef]

Y. N. Denisyuk, “Photographic reconstruction of the optical properties of an object in its own scattered radiation field,” Sov. Phys. Dokl.7, 543–545 (1962).

1948

D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Ahmed, S.

Aota, H.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett.85(6), 1069–1071 (2004).
[CrossRef]

Bablumian, A.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Blanche, P. A.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Blanche, P.-A.

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Burr, G. W.

D. Psaltis and G. W. Burr, “Holographic data storage,” Computer31(2), 52–60 (1998).
[CrossRef]

Christenson, C.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Denisyuk, Y. N.

Y. N. Denisyuk, “Photographic reconstruction of the optical properties of an object in its own scattered radiation field,” Sov. Phys. Dokl.7, 543–545 (1962).

Feit, M. D.

Fleck, J. A.

Flores, D.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Glytsis, E. N.

Gu, T.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Horimai, H.

Hsieh, W. Y.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Ikeda, J.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Inoue, M.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Iwasaki, Y.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Joseph, J.

Katahira, C.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Kathaperumal, M.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Kitano, M.

J. Tanaka, A. Okamoto, and M. Kitano, “Development of image-based simulation for holographic data storage system by fast Fourier transform beam-propagation method,” Jpn. J. Appl. Phys.48(3), 03A028 (2009).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled-wave theory for thick hologram grating,” Bell Syst. Tech. J.48, 2909–2947 (1969).

Koppa, P.

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett.85(6), 1069–1071 (2004).
[CrossRef]

Leith, E. N.

Li, J.

Lim, P. B.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Lin, W.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Matsumoto, A.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Morishita, N.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Norwood, R.

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Norwood, R. A.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Okamoto, A.

J. Tanaka, A. Okamoto, and M. Kitano, “Development of image-based simulation for holographic data storage system by fast Fourier transform beam-propagation method,” Jpn. J. Appl. Phys.48(3), 03A028 (2009).
[CrossRef]

Peyghambarian, N.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Psaltis, D.

D. Psaltis and G. W. Burr, “Holographic data storage,” Computer31(2), 52–60 (1998).
[CrossRef]

Rachwal, B.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett.85(6), 1069–1071 (2004).
[CrossRef]

Siddiqui, O.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Tan, X.

Tanaka, J.

J. Tanaka, A. Okamoto, and M. Kitano, “Development of image-based simulation for holographic data storage system by fast Fourier transform beam-propagation method,” Jpn. J. Appl. Phys.48(3), 03A028 (2009).
[CrossRef]

Tay, S.

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Thomas, J.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Upatnieks, J.

van Heerden, P. J.

Voorakaranam, R.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Waldman, D. A.

Wang, P.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Yamaguchi, I.

Yamamoto, M.

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Zhang, T.

Appl. Opt.

Appl. Phys. Lett.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett.85(6), 1069–1071 (2004).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled-wave theory for thick hologram grating,” Bell Syst. Tech. J.48, 2909–2947 (1969).

Computer

D. Psaltis and G. W. Burr, “Holographic data storage,” Computer31(2), 52–60 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Polym. Sci. A Polym. Chem.

C. Katahira, N. Morishita, J. Ikeda, P. B. Lim, M. Inoue, Y. Iwasaki, H. Aota, and A. Matsumoto, “Mechanistic discussion of cationic crosslinking copolymerizations of 1,2-epoxycyclohexane with diepoxide crosslinkers accompanied by intramolecular and intermolecular chain transfer reactions,” J. Polym. Sci. A Polym. Chem.48, 4445–4455 (2010).

Jpn. J. Appl. Phys.

J. Tanaka, A. Okamoto, and M. Kitano, “Development of image-based simulation for holographic data storage system by fast Fourier transform beam-propagation method,” Jpn. J. Appl. Phys.48(3), 03A028 (2009).
[CrossRef]

Nature

D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
[CrossRef] [PubMed]

P. A. Blanche, A. Bablumian, R. Voorakaranam, C. Christenson, W. Lin, T. Gu, D. Flores, P. Wang, W. Y. Hsieh, M. Kathaperumal, B. Rachwal, O. Siddiqui, J. Thomas, R. A. Norwood, M. Yamamoto, and N. Peyghambarian, “Holographic three-dimensional telepresence using large-area photorefractive polymer,” Nature468(7320), 80–83 (2010).
[CrossRef] [PubMed]

Opt. Lett.

Opt. Photon. News

N. Peyghambarian, S. Tay, P.-A. Blanche, R. Norwood, and M. Yamamoto, “Rewritable holographic 3D displays,” Opt. Photon. News19(7), 22–27 (2008).
[CrossRef]

Sov. Phys. Dokl.

Y. N. Denisyuk, “Photographic reconstruction of the optical properties of an object in its own scattered radiation field,” Sov. Phys. Dokl.7, 543–545 (1962).

Other

H. Kato, H. Horimai, P. B. Lim, K. Watanabe, M. Inoue, R. Arai, N. Morishita, and J. Ikeda, “Multi-level phase recording by collinear phase-lock holography,” Proceedings of International Workshop on Holographic Memories & Display, 79–80 (2009).

K. Curtis, L. Dhar, A. J. Hill, W. L. Wilson, and M. R. Ayres, Holographic Data Storage: From Theory to Practical Systems (John Wiley & Sons, 2010).

D. Malacara, Optical Shop Testing – Third Edition (Wiley-Interscience, 2007).

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Figures (8)

Fig. 1
Fig. 1

Conceptual diagram of self-referential holography. (a) Optical geometry for recording. The writing beam with a complex amplitude of EwSLM(x, y) is focused by an objective lens, and holograms are generated by self-interference between recording pixels. (b) The readout beam with a complex amplitude of ErSLM(x, y) is focused by an objective lens. The uniform readout intensity is changed to the signal intensity pattern Is(x, y) via the energy interchanges in the holograms. (c) Complex amplitudes of writing and readout beams. The patterns ϕw(x, y) and ϕr(x, y) are added to the writing and readout beams, respectively. If ϕw(x, y) - ϕr(x, y) ≡ ± ϕs(x, y), the intensity pattern that corresponds to the distribution of ± ϕs(x, y) can be observed.

Fig. 2
Fig. 2

Models for theoretical description of SRH principle. (a) Overview of the model. Light from the SLM pixels is converted to plane waves by a lens. These plane waves interfere with each other, and holograms are generated. (b) Layout of SLM. In the theoretical description, the pixels of the SLM are regarded as a set of many point sources.

Fig. 3
Fig. 3

(a) Conditions for solving the coupled wave equations. (b) Output intensity changes obtained from the coupled wave equations. The intensities of all pixels having the same intensity before illuminating the holograms (z = 0.0) are different at the output (z = 1.0).

Fig. 4
Fig. 4

Conceptual diagram of energy interchanges between four pixels. In this example, the intensities of pixels 1 and 3 are lower than those of pixels 2 and 4 at the output.

Fig. 5
Fig. 5

Models and flows in FFT-BPM-based simulations.

Fig. 6
Fig. 6

Simulation results. (a) Signal pattern ϕs(x, y). This pattern is used as the signal to be recorded in storage-type SRH and as the target to be detected in PIC-type SRH. (b) Additional pattern ϕad(x, y). This pattern is recorded in PIC-type SRH. (c) Writing pattern ϕw(x, y) = ϕs(x, y) + ϕad(x, y) for storage-type SRH. Readout pattern for PIC-type SRH. (d) Normalized output intensity pattern of storage-type SRH. (e) Normalized output intensity pattern of PIC-type SRH.

Fig. 7
Fig. 7

Experimental setup. Light source was a diode-pumped solid-state laser with a wavelength of 532.0 nm. Focal lengths of L1, L2, L3, L4, and L5 are 150, 200, 200, 100, and 100 mm, respectively.

Fig. 8
Fig. 8

Experimental results. (a) Signal pattern ϕs(x, y). Solid and dashed lines enclose the area to be recorded and the area used for SNR calculation, respectively. Dark pixels represent π/2 pixels, whereas brighter pixels represent 0 pixels. (b) Enlarged view of the region for SNR calculation [area enclosed by dashed line in (a)]. (c) Directly captured writing beam, i.e., intensity distribution of ϕw(x, y). (d) Output intensity pattern in storage-type SRH. (e) Output intensity pattern in PIC-type SRH.

Equations (16)

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E SLM w ( x,y )= A w exp[ i ϕ w ( x,y ) ] = A w exp{ i[ ϕ s ( x,y )+ ϕ ad ( x,y ) ] }.
E SLM r ( x,y )= A r exp[ i ϕ r ( x,y ) ] A r exp[ i ϕ ad ( x,y ) ].
E SLM w ( x,y )= A w exp[ i ϕ r ( x,y ) ].
E SLM r ( x,y )= A r exp[ i ϕ w ( x,y ) ] = A r exp{ i[ ϕ s ( x,y )+ ϕ ad ( x,y ) ] }.
E w ( r,t )= p=1 N A p w exp[ i( ωt k p r+ ψ p w ) ] .
n( r )= n 0 + 1m<n<N n mn cos[ K mn rΔ ψ mn w ]
K mn = k n k m ,
Δ ψ mn w = ψ n w ψ m w .
E r ( r,t )= p=1 N A p r exp[ i( ωt k p r+ ψ p r ) ] .
{ 2 +[ 2π λ n( r ) ]iωμσ } E r ( r,t )=0,
( d dz + α cos θ 1 ) A 1 r = i cos θ 1 [ q=2 N κ 1q A q r exp( iΔ ψ 1q ) ] ( d dz + α cos θ p0 ) A p0 r = i cos θ p0 [ q=1 p01 κ qp0 A q r exp( iΔ ψ qp0 ) + q=p0+1 N κ p0q A q r exp( iΔ ψ p0q ) ] ( d dz + α cos θ N ) A N r = i cos θ N [ q=1 N1 κ qN A q r exp( iΔ ψ qN ) ].
α= μcσ 2 n 0 ,
κ mn = π n mn λ .
Δ ψ mn =Δ ψ mn r Δ ψ mn w =Δ ϕ mn r Δ ϕ mn w =Δ ϕ mn .
Δn( x,y,z )=Δ n max { 1exp[ I( x,y,z )T G sat ] }.
SNR= I ON I OFF σ ON 2 + σ OFF 2 .

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