Abstract

We present the first experimental demonstration of quasi-permanent holographic recordings in Bi12TiO20 crystals through what is to our knowledge a new electrophotochromic effect. This arises from spatial redistribution of some centers that are different from photorefractive centers. Light diffracted from this photochromic grating records a secondary photorefractive hologram. Interaction between both gratings results in asymmetric polarization properties and reversible electric field enhancement of diffracted beams.

© 1992 Optical Society of America

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References

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  1. P. Günter, J. P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1988), Vol. 1.
    [Crossref]
  2. L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
    [Crossref]
  3. S. W. McCahon, D. Rytz, G. C. Valley, M. B. Klein, B. A. Wechsler, “Hologram fixing in Bi12TiO20 using heating and an ac electric field,” Appl. Opt. 28, 1967–1969 (1989).
    [Crossref] [PubMed]
  4. J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
    [Crossref]
  5. A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
    [Crossref]
  6. G. S. Trofimov, S. I. Stepanov, “Electrical development of a hologram in a Bi12SiO20 crystal,” Sov. Tech. Phys. Lett. 10, 282–283 (1984).
  7. M. Miteva, L. Nikolova, “Oscillating behaviour of diffracted light on uniform illumination of holograms in photorefractive Bi12TiO20 crystals,” Opt. Commun. 67, 192–194 (1988).
    [Crossref]
  8. N. A. Vainos, S. L. Clapham, R. W. Eason, “Multiplexed permanent and real time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
    [Crossref] [PubMed]
  9. J. Frejlich, “Fringe-locked running hologram and multiple photoactive species in Bi12TiO20,” J. Appl. Phys. 68, 3104–3109 (1990).
    [Crossref]
  10. A. L. Khromov, A. A. Kamshilin, M. P. Petrov, “Photochromic and photorefractive gratings induced by pulsed excitation in BSO crystals,” Opt. Commun. 77, 139–143 (1990).
    [Crossref]
  11. M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
    [Crossref]
  12. A. A. Kamshilin, J. Frejlich, L. Cescato, “Photorefractive crystals for the stabilization of the holographic setup,” Appl. Opt. 25, 2376–2381 (1986).
    [Crossref]
  13. P. M. Garcia, L. Cescato, L. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
    [Crossref]
  14. P. A. M. Dos Santos, L. Cescato, J. Frejlich, “Interference-term real-time measurements for self-stabilized two-wave mixing in photorefractive crystals,” Opt. Lett. 13, 1014–1016 (1988).
    [Crossref]
  15. A. Marrakchi, R. V. Johnson, A. R. Tanguay, “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
    [Crossref]
  16. P. A. M. Santos, P. M. Garcia, J. Frejlich, “Transport length, quantum efficiency and trap density measurement in Bi12SiO20,” J. Appl. Phys. 66, 247–251 (1989).
    [Crossref]
  17. W. Wardzynski, T. Lukasiewicz, J. Zmija, “Reversible photochromic effects in doped single crystal of bismuth germanium (B12GeO20) and bismuth silicon oxide (Bi12SiO20),” Opt. Commun. 30, 203–205 (1979).
    [Crossref]
  18. R. M. Pierce, R. S. Cudney, G. D. Bacher, J. Feinberg, “Measuring photorefractive trap density without the electrooptic effect,” Opt. Lett. 15, 414–416 (1990).
    [Crossref] [PubMed]
  19. K. Walsh, T. J. Hall, R. E. Burge, “Influence of polarization state and absorption gratings on photorefractive two-wave mixing in GaAs,” Opt. Lett. 12, 1026–1028 (1987).
    [Crossref] [PubMed]
  20. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  21. S. L. Clapham, R. W. Eason, N. A. Vainos, “Spatial light modulation via enhanced diffraction efficiency of photochromic gratings in photorefractive BSO,” Opt. Commun. 74, 290–294 (1990).
    [Crossref]

1990 (4)

J. Frejlich, “Fringe-locked running hologram and multiple photoactive species in Bi12TiO20,” J. Appl. Phys. 68, 3104–3109 (1990).
[Crossref]

A. L. Khromov, A. A. Kamshilin, M. P. Petrov, “Photochromic and photorefractive gratings induced by pulsed excitation in BSO crystals,” Opt. Commun. 77, 139–143 (1990).
[Crossref]

R. M. Pierce, R. S. Cudney, G. D. Bacher, J. Feinberg, “Measuring photorefractive trap density without the electrooptic effect,” Opt. Lett. 15, 414–416 (1990).
[Crossref] [PubMed]

S. L. Clapham, R. W. Eason, N. A. Vainos, “Spatial light modulation via enhanced diffraction efficiency of photochromic gratings in photorefractive BSO,” Opt. Commun. 74, 290–294 (1990).
[Crossref]

1989 (6)

P. M. Garcia, L. Cescato, L. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[Crossref]

P. A. M. Santos, P. M. Garcia, J. Frejlich, “Transport length, quantum efficiency and trap density measurement in Bi12SiO20,” J. Appl. Phys. 66, 247–251 (1989).
[Crossref]

N. A. Vainos, S. L. Clapham, R. W. Eason, “Multiplexed permanent and real time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
[Crossref] [PubMed]

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
[Crossref]

S. W. McCahon, D. Rytz, G. C. Valley, M. B. Klein, B. A. Wechsler, “Hologram fixing in Bi12TiO20 using heating and an ac electric field,” Appl. Opt. 28, 1967–1969 (1989).
[Crossref] [PubMed]

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

1988 (2)

M. Miteva, L. Nikolova, “Oscillating behaviour of diffracted light on uniform illumination of holograms in photorefractive Bi12TiO20 crystals,” Opt. Commun. 67, 192–194 (1988).
[Crossref]

P. A. M. Dos Santos, L. Cescato, J. Frejlich, “Interference-term real-time measurements for self-stabilized two-wave mixing in photorefractive crystals,” Opt. Lett. 13, 1014–1016 (1988).
[Crossref]

1987 (1)

1986 (3)

A. A. Kamshilin, J. Frejlich, L. Cescato, “Photorefractive crystals for the stabilization of the holographic setup,” Appl. Opt. 25, 2376–2381 (1986).
[Crossref]

A. Marrakchi, R. V. Johnson, A. R. Tanguay, “Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
[Crossref]

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[Crossref]

1984 (1)

G. S. Trofimov, S. I. Stepanov, “Electrical development of a hologram in a Bi12SiO20 crystal,” Sov. Tech. Phys. Lett. 10, 282–283 (1984).

1979 (2)

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
[Crossref]

W. Wardzynski, T. Lukasiewicz, J. Zmija, “Reversible photochromic effects in doped single crystal of bismuth germanium (B12GeO20) and bismuth silicon oxide (Bi12SiO20),” Opt. Commun. 30, 203–205 (1979).
[Crossref]

1969 (1)

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

Arizmendi, L.

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
[Crossref]

Bacher, G. D.

Burge, R. E.

Cescato, L.

P. M. Garcia, L. Cescato, L. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[Crossref]

P. A. M. Dos Santos, L. Cescato, J. Frejlich, “Interference-term real-time measurements for self-stabilized two-wave mixing in photorefractive crystals,” Opt. Lett. 13, 1014–1016 (1988).
[Crossref]

A. A. Kamshilin, J. Frejlich, L. Cescato, “Photorefractive crystals for the stabilization of the holographic setup,” Appl. Opt. 25, 2376–2381 (1986).
[Crossref]

Clapham, S. L.

S. L. Clapham, R. W. Eason, N. A. Vainos, “Spatial light modulation via enhanced diffraction efficiency of photochromic gratings in photorefractive BSO,” Opt. Commun. 74, 290–294 (1990).
[Crossref]

N. A. Vainos, S. L. Clapham, R. W. Eason, “Multiplexed permanent and real time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
[Crossref] [PubMed]

Cudney, R. S.

Delboulbe, A.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Dos Santos, P. A. M.

Eason, R. W.

S. L. Clapham, R. W. Eason, N. A. Vainos, “Spatial light modulation via enhanced diffraction efficiency of photochromic gratings in photorefractive BSO,” Opt. Commun. 74, 290–294 (1990).
[Crossref]

N. A. Vainos, S. L. Clapham, R. W. Eason, “Multiplexed permanent and real time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
[Crossref] [PubMed]

Feinberg, J.

Frejlich, J.

J. Frejlich, “Fringe-locked running hologram and multiple photoactive species in Bi12TiO20,” J. Appl. Phys. 68, 3104–3109 (1990).
[Crossref]

P. A. M. Santos, P. M. Garcia, J. Frejlich, “Transport length, quantum efficiency and trap density measurement in Bi12SiO20,” J. Appl. Phys. 66, 247–251 (1989).
[Crossref]

P. A. M. Dos Santos, L. Cescato, J. Frejlich, “Interference-term real-time measurements for self-stabilized two-wave mixing in photorefractive crystals,” Opt. Lett. 13, 1014–1016 (1988).
[Crossref]

A. A. Kamshilin, J. Frejlich, L. Cescato, “Photorefractive crystals for the stabilization of the holographic setup,” Appl. Opt. 25, 2376–2381 (1986).
[Crossref]

Frejlich, L.

P. M. Garcia, L. Cescato, L. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[Crossref]

Fromont, C.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Garcia, P. M.

P. M. Garcia, L. Cescato, L. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[Crossref]

P. A. M. Santos, P. M. Garcia, J. Frejlich, “Transport length, quantum efficiency and trap density measurement in Bi12SiO20,” J. Appl. Phys. 66, 247–251 (1989).
[Crossref]

Hall, T. J.

Herriau, J. P.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[Crossref]

Huignard, J. P.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[Crossref]

Johnson, R. V.

Kamshilin, A. A.

A. L. Khromov, A. A. Kamshilin, M. P. Petrov, “Photochromic and photorefractive gratings induced by pulsed excitation in BSO crystals,” Opt. Commun. 77, 139–143 (1990).
[Crossref]

A. A. Kamshilin, J. Frejlich, L. Cescato, “Photorefractive crystals for the stabilization of the holographic setup,” Appl. Opt. 25, 2376–2381 (1986).
[Crossref]

Khromov, A. L.

A. L. Khromov, A. A. Kamshilin, M. P. Petrov, “Photochromic and photorefractive gratings induced by pulsed excitation in BSO crystals,” Opt. Commun. 77, 139–143 (1990).
[Crossref]

Klein, M. B.

Kogelnik, H.

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

Kulikov, V. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
[Crossref]

Lukasiewicz, T.

W. Wardzynski, T. Lukasiewicz, J. Zmija, “Reversible photochromic effects in doped single crystal of bismuth germanium (B12GeO20) and bismuth silicon oxide (Bi12SiO20),” Opt. Commun. 30, 203–205 (1979).
[Crossref]

Mallick, S.

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Marrakchi, A.

McCahon, S. W.

Miridonov, S. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
[Crossref]

Miteva, M.

M. Miteva, L. Nikolova, “Oscillating behaviour of diffracted light on uniform illumination of holograms in photorefractive Bi12TiO20 crystals,” Opt. Commun. 67, 192–194 (1988).
[Crossref]

Nikolova, L.

M. Miteva, L. Nikolova, “Oscillating behaviour of diffracted light on uniform illumination of holograms in photorefractive Bi12TiO20 crystals,” Opt. Commun. 67, 192–194 (1988).
[Crossref]

Petrov, M. P.

A. L. Khromov, A. A. Kamshilin, M. P. Petrov, “Photochromic and photorefractive gratings induced by pulsed excitation in BSO crystals,” Opt. Commun. 77, 139–143 (1990).
[Crossref]

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
[Crossref]

Pierce, R. M.

Rytz, D.

Santos, P. A. M.

P. A. M. Santos, P. M. Garcia, J. Frejlich, “Transport length, quantum efficiency and trap density measurement in Bi12SiO20,” J. Appl. Phys. 66, 247–251 (1989).
[Crossref]

Stepanov, S. I.

G. S. Trofimov, S. I. Stepanov, “Electrical development of a hologram in a Bi12SiO20 crystal,” Sov. Tech. Phys. Lett. 10, 282–283 (1984).

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
[Crossref]

Tanguay, A. R.

Trofimov, G. S.

G. S. Trofimov, S. I. Stepanov, “Electrical development of a hologram in a Bi12SiO20 crystal,” Sov. Tech. Phys. Lett. 10, 282–283 (1984).

Vainos, N. A.

S. L. Clapham, R. W. Eason, N. A. Vainos, “Spatial light modulation via enhanced diffraction efficiency of photochromic gratings in photorefractive BSO,” Opt. Commun. 74, 290–294 (1990).
[Crossref]

N. A. Vainos, S. L. Clapham, R. W. Eason, “Multiplexed permanent and real time holographic recording in photorefractive BSO,” Appl. Opt. 28, 4381–4385 (1989).
[Crossref] [PubMed]

Valley, G. C.

Walsh, K.

Wardzynski, W.

W. Wardzynski, T. Lukasiewicz, J. Zmija, “Reversible photochromic effects in doped single crystal of bismuth germanium (B12GeO20) and bismuth silicon oxide (Bi12SiO20),” Opt. Commun. 30, 203–205 (1979).
[Crossref]

Wechsler, B. A.

Zmija, J.

W. Wardzynski, T. Lukasiewicz, J. Zmija, “Reversible photochromic effects in doped single crystal of bismuth germanium (B12GeO20) and bismuth silicon oxide (Bi12SiO20),” Opt. Commun. 30, 203–205 (1979).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[Crossref]

A. Delboulbe, C. Fromont, J. P. Herriau, S. Mallick, J. P. Huignard, “Quasi-nondestructive readout of holographically stored information in photorefractive Bi12SiO20 crystals,” Appl. Phys. Lett. 55, 713–715 (1989).
[Crossref]

Bell Syst. Tech. J. (1)

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

J. Appl. Phys. (4)

J. Frejlich, “Fringe-locked running hologram and multiple photoactive species in Bi12TiO20,” J. Appl. Phys. 68, 3104–3109 (1990).
[Crossref]

P. M. Garcia, L. Cescato, L. Frejlich, “Phase-shift measurement in photorefractive holographic recording,” J. Appl. Phys. 66, 47–49 (1989).
[Crossref]

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
[Crossref]

P. A. M. Santos, P. M. Garcia, J. Frejlich, “Transport length, quantum efficiency and trap density measurement in Bi12SiO20,” J. Appl. Phys. 66, 247–251 (1989).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Commun. (5)

M. Miteva, L. Nikolova, “Oscillating behaviour of diffracted light on uniform illumination of holograms in photorefractive Bi12TiO20 crystals,” Opt. Commun. 67, 192–194 (1988).
[Crossref]

S. L. Clapham, R. W. Eason, N. A. Vainos, “Spatial light modulation via enhanced diffraction efficiency of photochromic gratings in photorefractive BSO,” Opt. Commun. 74, 290–294 (1990).
[Crossref]

W. Wardzynski, T. Lukasiewicz, J. Zmija, “Reversible photochromic effects in doped single crystal of bismuth germanium (B12GeO20) and bismuth silicon oxide (Bi12SiO20),” Opt. Commun. 30, 203–205 (1979).
[Crossref]

A. L. Khromov, A. A. Kamshilin, M. P. Petrov, “Photochromic and photorefractive gratings induced by pulsed excitation in BSO crystals,” Opt. Commun. 77, 139–143 (1990).
[Crossref]

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and nonlinear image processing in electrooptic Bi12SiO20 crystals,” Opt. Commun. 31, 301–305 (1979).
[Crossref]

Opt. Lett. (3)

Sov. Tech. Phys. Lett. (1)

G. S. Trofimov, S. I. Stepanov, “Electrical development of a hologram in a Bi12SiO20 crystal,” Sov. Tech. Phys. Lett. 10, 282–283 (1984).

Other (1)

P. Günter, J. P. Huignard, eds., Photorefractive Materials and Their Applications (Springer-Verlag, Berlin, 1988), Vol. 1.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

Bi12TiO20 crystal orientation for volume hologram recording.

Fig. 2
Fig. 2

Experimental configuration for two-wave mixing and hologram erasure experiments in Bi12TiO20: S1, S2, shutters; R1, R2, photodetectors; PZT, piezoelectric transducer; E, external electric field; B1, B2, recording beams; H1, H2, output beams through BTO.

Fig. 3
Fig. 3

Hologram erasure curves for holograms recorded (a) for 30 min without an external electric field (E = 0), (b) for 30 min with a dc field E = 3.8 kV/cm, (c) for 96 min with ac field of amplitude E = 5 kV/cm. In all cases, the holograms were recorded in the nonstabilized mode.

Fig. 4
Fig. 4

Polarization state for reading out and diffracted beams.

Fig. 5
Fig. 5

Influence of the external ac electric field on hologram readout. The hologram was recorded for 40 min with an ac field of amplitude E = 5 kV/cm.

Fig. 6
Fig. 6

Typical erasure curve for the slow grating.

Fig. 7
Fig. 7

Dependence of the slow grating erasure rate on spatial frequency.

Fig. 8
Fig. 8

Evolution of Ω and 2Ω terms of the intensity through BTO during hologram recording for (a) beam H1 and (b) beam H2. Recorded patterns of light were fringe locked to a permanent grating on a photographic plate placed at the side of the BTO crystal.

Equations (5)

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

η AG = s h 2 ( α 1 d 2 cos ϑ 0 ) ( α 1 d 2 cos ϑ 0 ) 2
α = α 0 + α 1 cos ( K · x ) .
η PG = ( π n 3 r 41 E sc 2 λ cos ϑ 0 ) 2 ( m d ) 2 ( sin ρ d ρ d ) 2 ,
d 1.5 mm , E sc 2 kV / cm ( for E = 0 ) , ϑ 0 20 ° , E sc 7 kV / cm ( for E = 5 kV / cm ac ) , ρ d 20 ° , r 41 5 × 10 - 12 m / V , n 2.6.
m = 2 η AG 1 + η AG α 1 d cos ϑ 0 .

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