Abstract

The diffraction efficiency of bismuth silicate photorefractive gratings recorded in a nonlinear regime (light modulation depth, ≃1) was numerically calculated; phase coupling and off-Bragg reading conditions were taken into account. We considered gratings recorded with different applied electric dc fields under strong light modulation. We calculated the amplitude and the phase of the refractive-index variation of the gratings by numerically solving first the material rate equations to obtain the phase and the amplitude of the space-charge field and then the wave coupling equations for writing. We also obtained numerical solutions of the wave coupling equations for reading, considering angular deviations from the Bragg condition and bending of the fringes along the thickness of the sample. The diffraction efficiency for a 2-cm bismuth silicate sample recorded with an applied electric field of 2.5 kV/cm and a light modulation depth of 0.9 can be enhanced from 14% to 70% by use of an angular deviation of 3×10-4 rad from the Bragg condition during reading.

© 2003 Optical Society of America

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References

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  1. N. V. Kukhtarev, V. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949-960 (1979).
    [CrossRef]
  2. J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
    [CrossRef]
  3. S. Tao, Z. H. Song, and D. R. Selviah, “Bragg-shift of holographic gratings in photorefractive Fe:LiNbO3 crystals,” Opt. Commun. 108, 144-152 (1994).
    [CrossRef]
  4. R. De Vre, M. Jeganathan, J. Wilde, and L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910-912 (1994).
    [CrossRef] [PubMed]
  5. K. Buse, J. Frejlich, and K. H. Ringhofer, “Tilting of holograms in photorefractive SR0.61Ba0.39Nb2O6 crystals by self-diffraction,” Opt. Lett. 20, 21, 2249-2251 (1995).
    [CrossRef]
  6. J. G. Murillo, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Effects of strong modulations on beam coupling gain in photorefractive materials: application to BSO,” J. Opt. Soc. Am. B 15, 2092-2098 (1998).
    [CrossRef]
  7. I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
    [CrossRef]
  8. E. Garcia, I. Casar, and L. F. Magan˜a, “Optimization of the diffraction efficiency for BSO under strong modulation and applied electric fields,” J. Opt. Soc. Am. B 17, 1961-1966 (2000).
    [CrossRef]
  9. E. Garcia, I. Casar, and L. F. Magan˜a, “On the optimization of the diffraction efficiency by angular selectivity for BSO under strong modulation and applied electric fields,” Opt. Commun. 204, 363–369 (2002).
    [CrossRef]
  10. E. Agullo Lopez, J. M. Cabrera, and F. Agullo Rueda, Electrooptics (Academic, Cambridge, 1994), Chap. 10, pp. 295–297.
  11. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 2, pp. 42–55.

2002 (1)

E. Garcia, I. Casar, and L. F. Magan˜a, “On the optimization of the diffraction efficiency by angular selectivity for BSO under strong modulation and applied electric fields,” Opt. Commun. 204, 363–369 (2002).
[CrossRef]

2000 (1)

1998 (2)

J. G. Murillo, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Effects of strong modulations on beam coupling gain in photorefractive materials: application to BSO,” J. Opt. Soc. Am. B 15, 2092-2098 (1998).
[CrossRef]

I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
[CrossRef]

1995 (1)

K. Buse, J. Frejlich, and K. H. Ringhofer, “Tilting of holograms in photorefractive SR0.61Ba0.39Nb2O6 crystals by self-diffraction,” Opt. Lett. 20, 21, 2249-2251 (1995).
[CrossRef]

1994 (2)

S. Tao, Z. H. Song, and D. R. Selviah, “Bragg-shift of holographic gratings in photorefractive Fe:LiNbO3 crystals,” Opt. Commun. 108, 144-152 (1994).
[CrossRef]

R. De Vre, M. Jeganathan, J. Wilde, and L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910-912 (1994).
[CrossRef] [PubMed]

1984 (1)

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

1979 (1)

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

Agulló-López, F.

J. G. Murillo, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Effects of strong modulations on beam coupling gain in photorefractive materials: application to BSO,” J. Opt. Soc. Am. B 15, 2092-2098 (1998).
[CrossRef]

I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
[CrossRef]

Buse, K.

K. Buse, J. Frejlich, and K. H. Ringhofer, “Tilting of holograms in photorefractive SR0.61Ba0.39Nb2O6 crystals by self-diffraction,” Opt. Lett. 20, 21, 2249-2251 (1995).
[CrossRef]

Carrascosa, M.

J. G. Murillo, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Effects of strong modulations on beam coupling gain in photorefractive materials: application to BSO,” J. Opt. Soc. Am. B 15, 2092-2098 (1998).
[CrossRef]

I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
[CrossRef]

Casar, I.

E. Garcia, I. Casar, and L. F. Magan˜a, “On the optimization of the diffraction efficiency by angular selectivity for BSO under strong modulation and applied electric fields,” Opt. Commun. 204, 363–369 (2002).
[CrossRef]

E. Garcia, I. Casar, and L. F. Magan˜a, “Optimization of the diffraction efficiency for BSO under strong modulation and applied electric fields,” J. Opt. Soc. Am. B 17, 1961-1966 (2000).
[CrossRef]

I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
[CrossRef]

De Vre, R.

Frejlich, J.

K. Buse, J. Frejlich, and K. H. Ringhofer, “Tilting of holograms in photorefractive SR0.61Ba0.39Nb2O6 crystals by self-diffraction,” Opt. Lett. 20, 21, 2249-2251 (1995).
[CrossRef]

Garcia, E.

E. Garcia, I. Casar, and L. F. Magan˜a, “On the optimization of the diffraction efficiency by angular selectivity for BSO under strong modulation and applied electric fields,” Opt. Commun. 204, 363–369 (2002).
[CrossRef]

E. Garcia, I. Casar, and L. F. Magan˜a, “Optimization of the diffraction efficiency for BSO under strong modulation and applied electric fields,” J. Opt. Soc. Am. B 17, 1961-1966 (2000).
[CrossRef]

Heaton, J. M.

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

Hesselink, L.

Jeganathan, M.

Kukhtarev, N. V.

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

Magan˜a, L. F.

E. Garcia, I. Casar, and L. F. Magan˜a, “On the optimization of the diffraction efficiency by angular selectivity for BSO under strong modulation and applied electric fields,” Opt. Commun. 204, 363–369 (2002).
[CrossRef]

E. Garcia, I. Casar, and L. F. Magan˜a, “Optimization of the diffraction efficiency for BSO under strong modulation and applied electric fields,” J. Opt. Soc. Am. B 17, 1961-1966 (2000).
[CrossRef]

I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
[CrossRef]

J. G. Murillo, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Effects of strong modulations on beam coupling gain in photorefractive materials: application to BSO,” J. Opt. Soc. Am. B 15, 2092-2098 (1998).
[CrossRef]

Markov, V.

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

Mills, P. A.

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

Murillo, J. G.

Odulov, S. G.

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

Paige, E. G.

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

Ringhofer, K. H.

K. Buse, J. Frejlich, and K. H. Ringhofer, “Tilting of holograms in photorefractive SR0.61Ba0.39Nb2O6 crystals by self-diffraction,” Opt. Lett. 20, 21, 2249-2251 (1995).
[CrossRef]

Selviah, D. R.

S. Tao, Z. H. Song, and D. R. Selviah, “Bragg-shift of holographic gratings in photorefractive Fe:LiNbO3 crystals,” Opt. Commun. 108, 144-152 (1994).
[CrossRef]

Solymar, L.

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

Song, Z. H.

S. Tao, Z. H. Song, and D. R. Selviah, “Bragg-shift of holographic gratings in photorefractive Fe:LiNbO3 crystals,” Opt. Commun. 108, 144-152 (1994).
[CrossRef]

Soskin, M. S.

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

Tao, S.

S. Tao, Z. H. Song, and D. R. Selviah, “Bragg-shift of holographic gratings in photorefractive Fe:LiNbO3 crystals,” Opt. Commun. 108, 144-152 (1994).
[CrossRef]

Vinetskii, V. L.

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

Wilde, J.

Wilson, T.

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

Ferroelectrics (1)

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

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

Opt. Acta (1)

J. M. Heaton, P. A. Mills, E. G. Paige, L. Solymar, and T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885-901 (1984).
[CrossRef]

Opt. Commun. (2)

S. Tao, Z. H. Song, and D. R. Selviah, “Bragg-shift of holographic gratings in photorefractive Fe:LiNbO3 crystals,” Opt. Commun. 108, 144-152 (1994).
[CrossRef]

E. Garcia, I. Casar, and L. F. Magan˜a, “On the optimization of the diffraction efficiency by angular selectivity for BSO under strong modulation and applied electric fields,” Opt. Commun. 204, 363–369 (2002).
[CrossRef]

Opt. Lett. (2)

R. De Vre, M. Jeganathan, J. Wilde, and L. Hesselink, “Effect of applied fields on the Bragg condition and the diffraction efficiency in photorefractive crystals,” Opt. Lett. 19, 910-912 (1994).
[CrossRef] [PubMed]

K. Buse, J. Frejlich, and K. H. Ringhofer, “Tilting of holograms in photorefractive SR0.61Ba0.39Nb2O6 crystals by self-diffraction,” Opt. Lett. 20, 21, 2249-2251 (1995).
[CrossRef]

Phys. Rev. B (1)

I. Casar, L. F. Magan˜a, M. Carrascosa, and F. Agulló-López, “Calculation of beam gain and fringe bending under electric fields and strong modulations,” Phys. Rev. B 58, 9591-9594 (1998).
[CrossRef]

Other (2)

E. Agullo Lopez, J. M. Cabrera, and F. Agullo Rueda, Electrooptics (Academic, Cambridge, 1994), Chap. 10, pp. 295–297.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 2, pp. 42–55.

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of the recording geometry that uses two beams. The grating period is Λ. The electric field is E0, and it is applied along the x axis. The z axis is along the thickness of the sample. (b) Schematic of the reading geometry that uses a single beam, Ii, at an incident angle θi and permits an angular deviation Δθ from the Bragg angle. The diffracted intensity is Id, and the diffracted angle is θd.

Fig. 2
Fig. 2

Phase difference of the recording beams, in radians, as a function of z for a grating recorded with m0=0.9 and several values of the applied electric field E0.

Fig. 3
Fig. 3

Variation of diffraction efficiency with z for a grating recorded with 5 kV/cm and m0=0.9 and replayed with different values of ξ.

Fig. 4
Fig. 4

Variation of diffraction efficiency with z for a grating recorded with 10 kV/cm and m0=0.9 and replayed with different values of ξ.

Fig. 5
Fig. 5

Diffraction efficiency as a function of z for a grating recorded without an electric field and replayed with different values of angular deviation Δθ.

Fig. 6
Fig. 6

Diffraction efficiency as a function of z for a grating recorded with 2.5 kV/cm and m0=0.9 and replayed with different values of angular deviation Δθ.

Fig. 7
Fig. 7

Diffraction efficiency as a function of angular variation for a sample of 2.0 cm for gratings recorded with different values of light modulation mo and for an applied dc field of 10 kV/cm.

Tables (1)

Tables Icon

Table 1 Values of the Physical Parameters of Bismuth Silicate Used in the Numerical Solution of the Material Rate Equations

Equations (14)

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A1=A10(z, t)exp{-i[k1r+Ψ1(z, t)]}exp(iωt),
A2=A20(z, t)exp{-i[k2r+Ψ2(z, t)]}exp(iωt).
m(z)=2[I1(z)I2(z)]1/2I0.
Ψ(z)dz=Γ I1-I22I0cos Φg,
Δn(x, z)=½n03rE1[x, m(z)]×exp {-i[Φg(z)+Ψ(z)+K * r]},
δ=α2-α1=2k(cos θd-cos θi)=-2kΔθ sin θB=-2KΔθ,
E=|Ai|exp(-iα1z)exp(-iωt)+|Ad|exp(-iα2z)exp(iωt),
dAidz+iκ12 Adexp(-iδz)exp[-iΨ(z)]exp[-iΦg(z)]=0,
dAddz+iκ21 Aiexp(+iδz)exp[+iΨ(z)]exp[+iΦg(z)]=0,
2Aiz2-iΨz+Φgz+δAiz+κ12κ21 Ai=0.
Ai(z)=Ai(0)exp(-iξz)iξssin sz+cos sz,
Ad(z)=-k21s Ad(0)exp(+iξz)sin(sz),
s2=κ21κ12+δ+Ψz+Φgz=κ12κ21+ξ2
η(z)=|Ad(z)|2cos θd||A1(0)|2cos θi.

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