Abstract

A fully nonlinear frequency response of a moving grating in bismuth silicon oxide, including the effects of an applied electric field, is modeled by solution of the time-dependent Kukhtarev equations for photorefractive materials. The numerical results are used to define fully the nonlinear response function Fm=a-11-exp-am, where m is the modulation index in the intensity pattern, to yield the unknown quantity a over a broad range of detuning frequencies f. For low f, the response is superlinear with a<0, and for relatively large f it is sublinear with a>0. In the midrange we predict, for the first time to our knowledge, a characteristic frequency fl at which a=0 and the response is linear, that is, Fmm, despite the presence of nonlinearly generated higher harmonics of the fundamental grating wave number. In view of this linear behavior, writing a hologram at the linear-response frequency fl might permit a more faithful reproduction of an object than that which is possible by writing at the frequency of maximum response at the resonance.

© 1999 Optical Society of America

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References

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  1. N. V. Kukhtarev, Ferroelectrics 22, 949 (1979).
    [CrossRef]
  2. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 3.
  3. H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
    [CrossRef]
  4. G. A. Brost, K. M. Magde, J. J. Larkin, and M. T. Harris, J. Opt. Soc. Am. B 11, 1764 (1994).
    [CrossRef]
  5. N. Singh, S. P. Nadar, and P. P. Banerjee, Opt. Commun. 136, 487 (1997).
    [CrossRef]
  6. M. Reiser, Comput. Methods Appl. Mech. Eng. 1, 17 (1972).
    [CrossRef]
  7. R. Richymyer and K. Morton, Difference Methods for Initial-Value Problems (Interscience, New York, 1967), Chap. 8.
  8. L. B. Au and L. Solymar, J. Opt. Soc. Am. A 7, 1554 (1990).
    [CrossRef]

1997 (1)

N. Singh, S. P. Nadar, and P. P. Banerjee, Opt. Commun. 136, 487 (1997).
[CrossRef]

1994 (1)

1990 (1)

1985 (1)

H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

1979 (1)

N. V. Kukhtarev, Ferroelectrics 22, 949 (1979).
[CrossRef]

1972 (1)

M. Reiser, Comput. Methods Appl. Mech. Eng. 1, 17 (1972).
[CrossRef]

Au, L. B.

Banerjee, P. P.

N. Singh, S. P. Nadar, and P. P. Banerjee, Opt. Commun. 136, 487 (1997).
[CrossRef]

Brost, G. A.

Harris, M. T.

Huignard, J. P.

H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, Ferroelectrics 22, 949 (1979).
[CrossRef]

Larkin, J. J.

Magde, K. M.

Morton, K.

R. Richymyer and K. Morton, Difference Methods for Initial-Value Problems (Interscience, New York, 1967), Chap. 8.

Nadar, S. P.

N. Singh, S. P. Nadar, and P. P. Banerjee, Opt. Commun. 136, 487 (1997).
[CrossRef]

Rajbenbach, H.

H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Réfrégier, H.

H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Reiser, M.

M. Reiser, Comput. Methods Appl. Mech. Eng. 1, 17 (1972).
[CrossRef]

Richymyer, R.

R. Richymyer and K. Morton, Difference Methods for Initial-Value Problems (Interscience, New York, 1967), Chap. 8.

Singh, N.

N. Singh, S. P. Nadar, and P. P. Banerjee, Opt. Commun. 136, 487 (1997).
[CrossRef]

Solymar, L.

L. B. Au and L. Solymar, J. Opt. Soc. Am. A 7, 1554 (1990).
[CrossRef]

H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 3.

Comput. Methods Appl. Mech. Eng. (1)

M. Reiser, Comput. Methods Appl. Mech. Eng. 1, 17 (1972).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, Ferroelectrics 22, 949 (1979).
[CrossRef]

J. Appl. Phys. (1)

H. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

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

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

Opt. Commun. (1)

N. Singh, S. P. Nadar, and P. P. Banerjee, Opt. Commun. 136, 487 (1997).
[CrossRef]

Other (2)

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 3.

R. Richymyer and K. Morton, Difference Methods for Initial-Value Problems (Interscience, New York, 1967), Chap. 8.

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Figures (2)

Fig. 1
Fig. 1

Frequency response of Ep/m for an applied field of 1×105 V/m and modulation indices of 0.01, 0.4, 0.7, and 0.9. For m=0.01 the solid curve is the analytical result of Réfrégier et al.3; the numerical results are shown as open circles. At f=fl, Ep/m is the same for all m, implying that Epm. Note the secondary resonances for f<fl.

Fig. 2
Fig. 2

(a) Frequency dependence of a for applied external electric fields of 0.5×105, 1×105, and 1.5×105 V/m. The grating wavelength is Λ=20 µm. Inset, the curve fit between numerically and analytically3 determined responses for E0=1×105 V/m at 100 Hz. The analytical response includes the factor Fm. (b) Frequency dependence of a for grating wavelengths of 10, 20, and 30 µm. The applied electric field is 1×105 V/m.

Equations (4)

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ρd+t=SI+βρd-ρd+-γr/eρd+ρe-,
ρe-t=ρd+t+μxρe-Esc+D2ρex2,
2Φx2=-Escx-1ρd+-ρe--ρa-,
Fm=a-11-exp-am.

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