Abstract

An optical method for a fully parallel analog-to-digital conversion is proposed. The proposed method is carried out by means of intensity transformations of an analog input image and the thresholding for the transformed images and is suitable for two-dimensional implementation based on spatial light modulators. The intensity transformations are implemented by a liquid-crystal spatial light modulator, and thresholding is simulated by computer in consideration of the optical realization.

© 1998 Optical Society of America

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  1. H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63, 1524–1525 (1975).
    [CrossRef]
  2. H. F. Taylor, M. J. Taylor, and P. W. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32, 559–561 (1975).
    [CrossRef]
  3. H. F. Taylor, “An optical analog-to-digital converter—design and analysis,” IEEE J. Quantum Electron. QE-15, 210–216 (1979).
    [CrossRef]
  4. F. J. Leonberger, C. E. Woodward, and D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits Syst. CAS-26, 1125–1131 (1979).
    [CrossRef]
  5. F. J. Leonberger, “High-speed operation of LiNbO3 electro-optic interferometric waveguide modulators,” Opt. Lett. 5, 312–314 (1980).
    [CrossRef] [PubMed]
  6. S. Yamada, M. Minakata, and J. Noda, “Analog-to-digital conversion experiments using a LiNbO3 balanced bridge modulator,” Appl. Phys. Lett. 39, 124–126 (1981).
    [CrossRef]
  7. G. D. H. King and R. Cebulski, “Analogue-to-digital conversion using integrated electro-optic interferometers,” Electron. Lett. 18, 1099–1100 (1982).
    [CrossRef]
  8. F. J. Leonberger, C. E. Woodward, and R. A. Becker, “4-bit 828-megasample/s electro-optic guided-wave analog-to-digital converter,” Appl. Phys. Lett. 40, 565–568 (1982).
    [CrossRef]
  9. R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
    [CrossRef]
  10. P. E. Pace and D. Styer, “High-resolution encoding process for an integrated optical analog-to-digital converter,” Opt. Eng. 33, 2638–2645 (1994).
    [CrossRef]
  11. T. M. Bernard, B. Y. Zavidovique, and F. J. Devos, “A programmable artificial retina,” IEEE J. Solid-State Circuits 28, 789–798 (1993).
    [CrossRef]
  12. K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
    [CrossRef]
  13. N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
    [CrossRef]
  14. N. Mukozaka, N. Yoshida, H. Toyoda, Y. Kobayashi, and T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
    [CrossRef]
  15. G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
    [CrossRef]
  16. S. Fukushima, T. Kurokawa, S. Matsuo, and H. Kozawaguchi, “Bistable spatial light modulator using ferroelectric liquid crystals,” Opt. Lett. 15, 285–287 (1990).
    [CrossRef] [PubMed]

1994 (3)

P. E. Pace and D. Styer, “High-resolution encoding process for an integrated optical analog-to-digital converter,” Opt. Eng. 33, 2638–2645 (1994).
[CrossRef]

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

N. Mukozaka, N. Yoshida, H. Toyoda, Y. Kobayashi, and T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
[CrossRef]

1993 (1)

T. M. Bernard, B. Y. Zavidovique, and F. J. Devos, “A programmable artificial retina,” IEEE J. Solid-State Circuits 28, 789–798 (1993).
[CrossRef]

1990 (1)

1989 (1)

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

1984 (1)

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
[CrossRef]

1982 (2)

G. D. H. King and R. Cebulski, “Analogue-to-digital conversion using integrated electro-optic interferometers,” Electron. Lett. 18, 1099–1100 (1982).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and R. A. Becker, “4-bit 828-megasample/s electro-optic guided-wave analog-to-digital converter,” Appl. Phys. Lett. 40, 565–568 (1982).
[CrossRef]

1981 (1)

S. Yamada, M. Minakata, and J. Noda, “Analog-to-digital conversion experiments using a LiNbO3 balanced bridge modulator,” Appl. Phys. Lett. 39, 124–126 (1981).
[CrossRef]

1980 (1)

1979 (2)

H. F. Taylor, “An optical analog-to-digital converter—design and analysis,” IEEE J. Quantum Electron. QE-15, 210–216 (1979).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits Syst. CAS-26, 1125–1131 (1979).
[CrossRef]

1975 (2)

H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63, 1524–1525 (1975).
[CrossRef]

H. F. Taylor, M. J. Taylor, and P. W. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32, 559–561 (1975).
[CrossRef]

Banish, B.

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Bauer, P. W.

H. F. Taylor, M. J. Taylor, and P. W. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32, 559–561 (1975).
[CrossRef]

Becker, R. A.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and R. A. Becker, “4-bit 828-megasample/s electro-optic guided-wave analog-to-digital converter,” Appl. Phys. Lett. 40, 565–568 (1982).
[CrossRef]

Bernard, T. M.

T. M. Bernard, B. Y. Zavidovique, and F. J. Devos, “A programmable artificial retina,” IEEE J. Solid-State Circuits 28, 789–798 (1993).
[CrossRef]

Cebulski, R.

G. D. H. King and R. Cebulski, “Analogue-to-digital conversion using integrated electro-optic interferometers,” Electron. Lett. 18, 1099–1100 (1982).
[CrossRef]

Devos, F. J.

T. M. Bernard, B. Y. Zavidovique, and F. J. Devos, “A programmable artificial retina,” IEEE J. Solid-State Circuits 28, 789–798 (1993).
[CrossRef]

Fukushima, S.

Handschy, M. A.

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

Hara, T.

Hermanns, A.

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Ishikawa, M.

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Johnson, K. M.

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

King, G. D. H.

G. D. H. King and R. Cebulski, “Analogue-to-digital conversion using integrated electro-optic interferometers,” Electron. Lett. 18, 1099–1100 (1982).
[CrossRef]

Kobayashi, Y.

N. Mukozaka, N. Yoshida, H. Toyoda, Y. Kobayashi, and T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
[CrossRef]

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Komuro, T.

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Kozawaguchi, H.

Kurokawa, T.

Kyuma, K.

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Lange, E.

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Leonberger, F. J.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and R. A. Becker, “4-bit 828-megasample/s electro-optic guided-wave analog-to-digital converter,” Appl. Phys. Lett. 40, 565–568 (1982).
[CrossRef]

F. J. Leonberger, “High-speed operation of LiNbO3 electro-optic interferometric waveguide modulators,” Opt. Lett. 5, 312–314 (1980).
[CrossRef] [PubMed]

F. J. Leonberger, C. E. Woodward, and D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits Syst. CAS-26, 1125–1131 (1979).
[CrossRef]

Li, W.

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

Matsuo, S.

McArdle, N.

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Minakata, M.

S. Yamada, M. Minakata, and J. Noda, “Analog-to-digital conversion experiments using a LiNbO3 balanced bridge modulator,” Appl. Phys. Lett. 39, 124–126 (1981).
[CrossRef]

Moddel, G.

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

Mukozaka, N.

Naruse, M.

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Noda, J.

S. Yamada, M. Minakata, and J. Noda, “Analog-to-digital conversion experiments using a LiNbO3 balanced bridge modulator,” Appl. Phys. Lett. 39, 124–126 (1981).
[CrossRef]

Ohta, J.

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Oita, M.

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Pace, P. E.

P. E. Pace and D. Styer, “High-resolution encoding process for an integrated optical analog-to-digital converter,” Opt. Eng. 33, 2638–2645 (1994).
[CrossRef]

Pagano Stauffer, L. A.

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

Rice, A.

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

Sakaida, H.

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Spears, D. L.

F. J. Leonberger, C. E. Woodward, and D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits Syst. CAS-26, 1125–1131 (1979).
[CrossRef]

Styer, D.

P. E. Pace and D. Styer, “High-resolution encoding process for an integrated optical analog-to-digital converter,” Opt. Eng. 33, 2638–2645 (1994).
[CrossRef]

Taylor, H. F.

H. F. Taylor, “An optical analog-to-digital converter—design and analysis,” IEEE J. Quantum Electron. QE-15, 210–216 (1979).
[CrossRef]

H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63, 1524–1525 (1975).
[CrossRef]

H. F. Taylor, M. J. Taylor, and P. W. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32, 559–561 (1975).
[CrossRef]

Taylor, M. J.

H. F. Taylor, M. J. Taylor, and P. W. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32, 559–561 (1975).
[CrossRef]

Toyoda, H.

N. Mukozaka, N. Yoshida, H. Toyoda, Y. Kobayashi, and T. Hara, “Diffraction efficiency analysis of a parallel-aligned nematic-liquid-crystal spatial light modulator,” Appl. Opt. 33, 2804–2811 (1994).
[CrossRef]

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

Williamson, R. C.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
[CrossRef]

Woodward, C. E.

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and R. A. Becker, “4-bit 828-megasample/s electro-optic guided-wave analog-to-digital converter,” Appl. Phys. Lett. 40, 565–568 (1982).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits Syst. CAS-26, 1125–1131 (1979).
[CrossRef]

Yamada, S.

S. Yamada, M. Minakata, and J. Noda, “Analog-to-digital conversion experiments using a LiNbO3 balanced bridge modulator,” Appl. Phys. Lett. 39, 124–126 (1981).
[CrossRef]

Yoshida, N.

Zavidovique, B. Y.

T. M. Bernard, B. Y. Zavidovique, and F. J. Devos, “A programmable artificial retina,” IEEE J. Solid-State Circuits 28, 789–798 (1993).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

G. Moddel, K. M. Johnson, W. Li, A. Rice, L. A. Pagano Stauffer, and M. A. Handschy, “High-speed binary optically addressed spatial light modulator,” Appl. Phys. Lett. 55, 537–539 (1989).
[CrossRef]

H. F. Taylor, M. J. Taylor, and P. W. Bauer, “Electro-optic analog-to-digital conversion using channel waveguide modulators,” Appl. Phys. Lett. 32, 559–561 (1975).
[CrossRef]

S. Yamada, M. Minakata, and J. Noda, “Analog-to-digital conversion experiments using a LiNbO3 balanced bridge modulator,” Appl. Phys. Lett. 39, 124–126 (1981).
[CrossRef]

F. J. Leonberger, C. E. Woodward, and R. A. Becker, “4-bit 828-megasample/s electro-optic guided-wave analog-to-digital converter,” Appl. Phys. Lett. 40, 565–568 (1982).
[CrossRef]

Electron. Lett. (1)

G. D. H. King and R. Cebulski, “Analogue-to-digital conversion using integrated electro-optic interferometers,” Electron. Lett. 18, 1099–1100 (1982).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. F. Taylor, “An optical analog-to-digital converter—design and analysis,” IEEE J. Quantum Electron. QE-15, 210–216 (1979).
[CrossRef]

IEEE J. Solid-State Circuits (1)

T. M. Bernard, B. Y. Zavidovique, and F. J. Devos, “A programmable artificial retina,” IEEE J. Solid-State Circuits 28, 789–798 (1993).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

F. J. Leonberger, C. E. Woodward, and D. L. Spears, “Design and development of a high-speed electrooptic A/D converter,” IEEE Trans. Circuits Syst. CAS-26, 1125–1131 (1979).
[CrossRef]

Nature (1)

K. Kyuma, E. Lange, J. Ohta, A. Hermanns, B. Banish, and M. Oita, “Artificial retinas—fast, versatile image processors,” Nature 372, 197–198 (1994).
[CrossRef]

Opt. Eng. (1)

P. E. Pace and D. Styer, “High-resolution encoding process for an integrated optical analog-to-digital converter,” Opt. Eng. 33, 2638–2645 (1994).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (2)

R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wide-band electrooptic guided-wave analog-to-digital converters,” Proc. IEEE 72, 802–819 (1984).
[CrossRef]

H. F. Taylor, “An electrooptic analog-to-digital converter,” Proc. IEEE 63, 1524–1525 (1975).
[CrossRef]

Other (1)

N. McArdle, M. Naruse, T. Komuro, H. Sakaida, M. Ishikawa, Y. Kobayashi, and H. Toyoda, “A smart-pixel parallel optoelectronic computing system with free-space dynamic interconnections,” in Proceedings of the International Conference on Massively Parallel Processing Using Optical Interconnections (IEEE Computer Society, San Mateo, Calif., 1996), pp. 146–157.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic of 4-bit A/D conversion: (a) An analog input light is made of four intensity transformations, and the four transformed light beams are obtained in parallel. (b) The four transformed light beams are encoded to the 4-bit light signals after thresholding at 0.5.

Fig. 2
Fig. 2

Configuration of the fully parallel optical A/D converter. It is constructed with SLM’s and four threshold devices (TD’s). SLM1, SLM2, SLM3, and SLM4 have input–output properties of π, 2π, 4π, and 8π transformations, respectively. The 4-bit digital images (MSB, NMSB, NLSB, and LSB images) are obtained by means of thresholding on the TD’s in parallel.

Fig. 3
Fig. 3

Experimental setup: LD of 680 nm; SF, spatial-frequency filter; HM’s, half-mirrors; MR1 and MR2, mirrors; AP, aperture; AL, analyzer.

Fig. 4
Fig. 4

Duplicate input images displayed on the LCD.

Fig. 5
Fig. 5

(a) Input–output properties of the experimental setup. Their properties are measured by the placement of a photodetector instead of the CCD camera and a neutral-density filter instead of the LCD. The solid curves with the filled circles, the triangles, and the filled squares represent the π, 2π, and 4π transformations, respectively. (b) The 3-bit binary signal with the gray-code format by means of thresholding at 0.5.

Fig. 6
Fig. 6

Experimental results of the intensity transformations of the input image: (a) The π-transformed image. (b) The 2π-transformed image. (c) The 4π-transformed image.

Fig. 7
Fig. 7

Digital 3-bit gray-code images. They are obtained from the images shown in Figs. 6(a)6(c) by the threshold device simulated by computer. The threshold value is set to the half-maximum of the transformed image.

Equations (5)

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

m ij n = F x ij ,   n = 1 - cos 2 n - 1 π x ij / 2 ,   n = 1 ,   2 ,   3 ,   4 .
y ij n = T m ij n = 0 ,     if   m ij n < 0.5 , = 1 ,     otherwise ,
I out = F I in = I r 1 - β cos ϕ a I in + ϕ 0 V 0 / 2 ,
ϕ a I in = 2 π I in α / I in α + I h α ,
T a m = 1 / exp - 40 m - 0.5 + 1 ,   0.0 < m < 1.0 .

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