Abstract

The effects of the donor acceptor concentration ratio in iron-doped lithium niobate (Fe:LiNbO3) crystal plates on photorefractive response time and on grating diffraction efficiency are studied both theoretically and experimentally. The results provide a useful guide for designing photorefractive plates for optical information-processing applications. Two devices for real-time image recognition have been demonstrated: a photorefractive joint-transform correlator and a VanderLugt correlator. The former emphasizes fast response time, and the latter emphasizes high diffraction efficiency. By appropriate adjustment of the dopant concentration and the ratio of the donor acceptor levels, photorefractive Fe:LiNbO3 crystals that facilitate specific applications have been designed and fabricated. Shift-invariant image correlations have been achieved.

© 1999 Optical Society of America

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References

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  1. D. Psaltis, D. Brady, and K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1759 (1988).
    [CrossRef]
  2. F. H. Mok, “Angle-multiplexed storage of 5000 holograms in lithium niobate,” Opt. Lett. 18, 915–917 (1993).
    [CrossRef] [PubMed]
  3. K. Y. Hsu, S. H. Lin, and T. C. Hsieh, “Photorefractive memories for real-time image processing,” Opt. Mem. Neural Netw. 4, 277–285 (1995).
  4. K. Itoh, O. Matoba, and Y. Ichioka, “ODINN in LiN: optical dynamic interconnections for neural networks in lithium niobate,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. Yu, ed., Proc. SPIE 2529, 71–81 (1995).
    [CrossRef]
  5. C. Sun, R. Tsou, J. Chang, and M. Chang, “Real-time photorefractive interferometer for dynamic phase perturbation by self-interference in LiNbO3,” Appl. Opt. 36, 3581–3585 (1997).
    [CrossRef] [PubMed]
  6. H. Kurz, “Lithium niobate as a material for holographic information storage,” Philips Tech. Rev. 37(5/6), 109–120 (1977).
  7. H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
    [CrossRef]
  8. S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
    [CrossRef]
  9. M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
    [CrossRef]
  10. Y. Fainman, E. Klancnik, and S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. (Bellingham) 25, 228–234 (1986).
    [CrossRef]
  11. J. J. Amodei, W. Philips, and D. S. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
    [CrossRef] [PubMed]
  12. D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
    [CrossRef]
  13. W. Phillips, J. J. Amodei, and D. S. Staebler, “Optical and holographic storage properties of transition metal doped lithium niobate,” RCA Rev. 33, 95–109 (1972).
  14. P. Gunter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer-Verlag, New York, 1988), Chap. 6.
  15. R. Orlowski and E. Kratzig, “Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystal,” Solid State Commun. 27, 1351–1354 (1978).
    [CrossRef]
  16. P. Yeh, “Fundamental limit of the speed of the photorefractive effect and its impact on device applications and material research,” Appl. Opt. 26, 602–605 (1987).
    [CrossRef] [PubMed]
  17. N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electro-optic crystal,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).
  18. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  19. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  20. For example, see P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 2.

1997

1995

K. Y. Hsu, S. H. Lin, and T. C. Hsieh, “Photorefractive memories for real-time image processing,” Opt. Mem. Neural Netw. 4, 277–285 (1995).

K. Itoh, O. Matoba, and Y. Ichioka, “ODINN in LiN: optical dynamic interconnections for neural networks in lithium niobate,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. Yu, ed., Proc. SPIE 2529, 71–81 (1995).
[CrossRef]

1993

F. H. Mok, “Angle-multiplexed storage of 5000 holograms in lithium niobate,” Opt. Lett. 18, 915–917 (1993).
[CrossRef] [PubMed]

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

1988

1987

1986

Y. Fainman, E. Klancnik, and S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. (Bellingham) 25, 228–234 (1986).
[CrossRef]

1981

H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

1979

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

1978

R. Orlowski and E. Kratzig, “Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystal,” Solid State Commun. 27, 1351–1354 (1978).
[CrossRef]

1977

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

H. Kurz, “Lithium niobate as a material for holographic information storage,” Philips Tech. Rev. 37(5/6), 109–120 (1977).

1976

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electro-optic crystal,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

1975

D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

1972

W. Phillips, J. J. Amodei, and D. S. Staebler, “Optical and holographic storage properties of transition metal doped lithium niobate,” RCA Rev. 33, 95–109 (1972).

J. J. Amodei, W. Philips, and D. S. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Amodei, J. J.

D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. J. Amodei, W. Philips, and D. S. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

W. Phillips, J. J. Amodei, and D. S. Staebler, “Optical and holographic storage properties of transition metal doped lithium niobate,” RCA Rev. 33, 95–109 (1972).

Brady, D.

Burke, W.

D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

Chang, J.

Chang, M.

Fainman, Y.

Y. Fainman, E. Klancnik, and S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. (Bellingham) 25, 228–234 (1986).
[CrossRef]

Hsieh, T. C.

K. Y. Hsu, S. H. Lin, and T. C. Hsieh, “Photorefractive memories for real-time image processing,” Opt. Mem. Neural Netw. 4, 277–285 (1995).

Hsu, K. Y.

K. Y. Hsu, S. H. Lin, and T. C. Hsieh, “Photorefractive memories for real-time image processing,” Opt. Mem. Neural Netw. 4, 277–285 (1995).

Ichioka, Y.

K. Itoh, O. Matoba, and Y. Ichioka, “ODINN in LiN: optical dynamic interconnections for neural networks in lithium niobate,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. Yu, ed., Proc. SPIE 2529, 71–81 (1995).
[CrossRef]

Itoh, K.

K. Itoh, O. Matoba, and Y. Ichioka, “ODINN in LiN: optical dynamic interconnections for neural networks in lithium niobate,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. Yu, ed., Proc. SPIE 2529, 71–81 (1995).
[CrossRef]

Kapphan, S.

H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Klancnik, E.

Y. Fainman, E. Klancnik, and S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. (Bellingham) 25, 228–234 (1986).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kratzig, E.

H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

R. Orlowski and E. Kratzig, “Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystal,” Solid State Commun. 27, 1351–1354 (1978).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electro-optic crystal,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Kurz, H.

H. Kurz, “Lithium niobate as a material for holographic information storage,” Philips Tech. Rev. 37(5/6), 109–120 (1977).

Lee, S. H.

Y. Fainman, E. Klancnik, and S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. (Bellingham) 25, 228–234 (1986).
[CrossRef]

Lin, S. H.

K. Y. Hsu, S. H. Lin, and T. C. Hsieh, “Photorefractive memories for real-time image processing,” Opt. Mem. Neural Netw. 4, 277–285 (1995).

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Matoba, O.

K. Itoh, O. Matoba, and Y. Ichioka, “ODINN in LiN: optical dynamic interconnections for neural networks in lithium niobate,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. Yu, ed., Proc. SPIE 2529, 71–81 (1995).
[CrossRef]

Micheron, F.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Mok, F. H.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Orlowski, R.

R. Orlowski and E. Kratzig, “Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystal,” Solid State Commun. 27, 1351–1354 (1978).
[CrossRef]

Peltier, M.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Philips, W.

D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. J. Amodei, W. Philips, and D. S. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

Phillips, W.

W. Phillips, J. J. Amodei, and D. S. Staebler, “Optical and holographic storage properties of transition metal doped lithium niobate,” RCA Rev. 33, 95–109 (1972).

Psaltis, D.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Staebler, D. L.

D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

Staebler, D. S.

J. J. Amodei, W. Philips, and D. S. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

W. Phillips, J. J. Amodei, and D. S. Staebler, “Optical and holographic storage properties of transition metal doped lithium niobate,” RCA Rev. 33, 95–109 (1972).

Sun, C.

Tsou, R.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Vormann, H.

H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Wagner, K.

Weber, G.

H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Wen, M.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Yeh, P.

Yin, S.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Yu, F. T. S.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Zhang, J.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Zhao, F.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Zhou, H.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. L. Staebler, W. Burke, W. Philips, and J. J. Amodei, “Multiple storage and erasure of fixing holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Ferroelectrics

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

J. Appl. Phys.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Opt. Commun.

S. Yin, F. Zhao, H. Zhou, M. Wen, J. Zhang, and F. T. S. Yu, “Wavelength-multiplexed holographic construction using a Ce:Fe:LiNbO3 crystal with a tunable visible-light diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Opt. Eng. (Bellingham)

Y. Fainman, E. Klancnik, and S. H. Lee, “Optimal coherent image amplification by two-wave coupling in photorefractive BaTiO3,” Opt. Eng. (Bellingham) 25, 228–234 (1986).
[CrossRef]

Opt. Lett.

Opt. Mem. Neural Netw.

K. Y. Hsu, S. H. Lin, and T. C. Hsieh, “Photorefractive memories for real-time image processing,” Opt. Mem. Neural Netw. 4, 277–285 (1995).

Philips Tech. Rev.

H. Kurz, “Lithium niobate as a material for holographic information storage,” Philips Tech. Rev. 37(5/6), 109–120 (1977).

Proc. SPIE

K. Itoh, O. Matoba, and Y. Ichioka, “ODINN in LiN: optical dynamic interconnections for neural networks in lithium niobate,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications, F. T. Yu, ed., Proc. SPIE 2529, 71–81 (1995).
[CrossRef]

RCA Rev.

W. Phillips, J. J. Amodei, and D. S. Staebler, “Optical and holographic storage properties of transition metal doped lithium niobate,” RCA Rev. 33, 95–109 (1972).

Solid State Commun.

R. Orlowski and E. Kratzig, “Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystal,” Solid State Commun. 27, 1351–1354 (1978).
[CrossRef]

H. Vormann, G. Weber, S. Kapphan, and E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3:Fe,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Sov. Tech. Phys. Lett.

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electro-optic crystal,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Other

P. Gunter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer-Verlag, New York, 1988), Chap. 6.

For example, see P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 2.

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Figures (13)

Fig. 1
Fig. 1

Diffraction efficiency versus donor concentration for three ratios NA/ND (theoretical analysis).

Fig. 2
Fig. 2

Response-time constant versus donor concentration for three ratios NA/ND (theoretical analysis).

Fig. 3
Fig. 3

Structure of the thin crystal plate.

Fig. 4
Fig. 4

Transmission spectra of the 0.06-mol. % (dashed curve) and the 1.5-mol. % (solid curve) iron-doping LiNbO3 plates.

Fig. 5
Fig. 5

Experimental measurement of the absorption coefficients of the crystal plates versus the iron-doping concentrations.

Fig. 6
Fig. 6

Typical temporal behavior of grating formation and decay for the 1.0-mol. % iron-doped LiNbO3 plate.

Fig. 7
Fig. 7

Experimental measurement of the buildup time constant for grating formation versus the iron-doping concentrations for the crystal plates.

Fig. 8
Fig. 8

Experimental measurement of diffraction efficiency versus the iron-doping concentrations for the crystal plates.

Fig. 9
Fig. 9

Generic diagram of a photorefractive correlator: FL’s, Fourier lens; other abbreviations defined in text.

Fig. 10
Fig. 10

Schematic diagram of the PRJTC: FT, Fourier transform; JT, joint transform; B.S., beam splitter; other abbreviations defined in text.

Fig. 11
Fig. 11

(a), (b) Input images and correlation output for our PRJTC. (c) Temporal response of updating the reference image in the PRJTC.

Fig. 12
Fig. 12

Schematic diagram of our four-wave-mixing PRVLC. FT, Fourier transform; B.S., beam splitter; other abbreviations defined in text.

Fig. 13
Fig. 13

(a), (b) Input images and correlation output for our PRVLC. (c) Correlation peak versus shift-invariant distance for our PRVLC.

Tables (1)

Tables Icon

Table 1 Parameters for the Computer Simulations

Equations (24)

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

η=exp(-αd)sin2πd2λcosθn3reff ESC,
α=s(ND-NDi)hν,
ESC=ReiEd-E01+(Ed/Eq)+i(E0/Eq)I1I0,
τ=NAsI0NDEd+Eμ+iE0Ed+Eq+iE0.
ND=1dshνCξ(1-ξ)tan-1(Cξ),
1dshν(1-ξ),Cξ1,
C=eπn3I1λsI0Khνcosθreff,
{η}max=(Cξ)21+(Cξ)2exp-1Cξtan-1(Cξ)(Cξ)2,Cξ1.
ND1+C2ξ(1-ξ)ND2
×exp[-κ(1-ξ)ND]=ηmin1C1ξ(1-ξ)2,
κ=sdhν,
C1=eπdI12I0λKcosθn3reff,
C2=e2kBTK.
Edm=U3Jm2πλcosθRn1d,m=±1,±2,±3,,
n1U1*U2+U1U2*,
Ed1=U3J12πλcosθRn1d
U32πλcosθRn1d
U3(U1*U2+U1U2*).
C(x, y)=u1(x, y)u2(x, y),
Ed=-iU3 exp-αd2+iΔβ2d×n1π/λcosθR[r2+(Δβ/2)2]1/2sinrd1+Δβ2r21/2,
r=πλcosθRn1,
Δβ=-4nπλsinθΔθ-4nπλ2sin2θcosθΔλ,
C(x, y)u1(x, y)u3(x, y)sinc{rd[1+(Δβ/2r)2]1/2}.
Δ=fλrπsinθπrd2-11/2,

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