Abstract

The nonlinear differential equations describing the response of photorefractive material to a moving sinusoidal intensity pattern that permits large modulation depth are solved by a numerical technique under steady-state condition. It is found that, at large modulation, the higher harmonics of the variables (space charge field, electron density, and ionized donor density) have appreciable amplitudes that can be further enhanced by moving the fringes. It is also shown that the optimized amplitudes of the higher harmonics of the space charge field can be strong and comparable with the fundamental and that the dependence of these higher harmonics of the space charge field on the spatial frequency is much weaker at large modulation.

© 1990 Optical Society of America

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References

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  1. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  2. Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
    [CrossRef]
  3. E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefractive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
    [CrossRef]
  4. S. Rai, Studies in Static and Dynamic Volume Holography, M.S. thesis (University of Oxford, Oxford, 1986).
  5. T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  6. F. Vachss, L. Hesselink, “Nonlinear photorefractive response at high modulation depths,” J. Opt. Soc. Am. A 5, 690–701 (1988).
    [CrossRef]
  7. F. Vachss, L. Hesselink, “Selective enhancement of spatial harmonics of a photorefractive grating,” J. Opt. Soc. Am. B 5, 1814–1821 (1988).
    [CrossRef]
  8. L. B. Au, L. Solymar, “Space-charge field in photorefractive materials at large modulation,” Opt. Lett. 13, 660–662 (1988).
    [CrossRef] [PubMed]
  9. W. H. Press, B. P. Flanery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).
  10. J. M. Heaton, L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
    [CrossRef]
  11. In our arguments leading to relation (21), the summation terms of the right-hand of Eq. (17) were assumed to be constant. This is, of course, not true and makes the prediction too rough to agree with the actual locations of the maxima when the nonlinearity is large. Nevertheless, it can still provide a good qualitative explanation of the problem.
  12. G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).
  13. J. M. Thompson, H. B. Stewart, Nonlinear Dynamics and Chaos (Wiley, New York, 1986).
  14. D. W. Jordan, P. Smith, Non-Linear Ordinary Differential Equations (Clarendon, Oxford, 1979).

1988

1986

1985

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

1979

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

1975

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Alig, R. C.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Alphonse, G. A.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Au, L. B.

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Flanery, B. P.

W. H. Press, B. P. Flanery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Heaton, J. M.

J. M. Heaton, L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

Hesselink, L.

Huignard, J. P.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Jordan, D. W.

D. W. Jordan, P. Smith, Non-Linear Ordinary Differential Equations (Clarendon, Oxford, 1979).

Kukhtarev, N. V.

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

Markov, V. B.

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

Ochoa, E.

Odulov, S. G.

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

Philips, W.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Press, W. H.

W. H. Press, B. P. Flanery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Rai, S.

S. Rai, Studies in Static and Dynamic Volume Holography, M.S. thesis (University of Oxford, Oxford, 1986).

Rajbenbach, H.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Smith, P.

D. W. Jordan, P. Smith, Non-Linear Ordinary Differential Equations (Clarendon, Oxford, 1979).

Solymar, L.

J. M. Heaton, L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

L. B. Au, L. Solymar, “Space-charge field in photorefractive materials at large modulation,” Opt. Lett. 13, 660–662 (1988).
[CrossRef] [PubMed]

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Soskin, M. S.

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

Staebler, D. L.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Stewart, H. B.

J. M. Thompson, H. B. Stewart, Nonlinear Dynamics and Chaos (Wiley, New York, 1986).

Teukolsky, S. A.

W. H. Press, B. P. Flanery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Thompson, J. M.

J. M. Thompson, H. B. Stewart, Nonlinear Dynamics and Chaos (Wiley, New York, 1986).

Vachss, F.

Vetterling, W. T.

W. H. Press, B. P. Flanery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

Vinetskii, V. L.

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

Ferroelectrics

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

IEEE J. Quantum Electron.

J. M. Heaton, L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

J. Appl. Phys.

Ph. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Prog. Quantum Electron.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

RCA Rev.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Philips, “Time-dependent characteristics and photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Other

J. M. Thompson, H. B. Stewart, Nonlinear Dynamics and Chaos (Wiley, New York, 1986).

D. W. Jordan, P. Smith, Non-Linear Ordinary Differential Equations (Clarendon, Oxford, 1979).

S. Rai, Studies in Static and Dynamic Volume Holography, M.S. thesis (University of Oxford, Oxford, 1986).

W. H. Press, B. P. Flanery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1987).

In our arguments leading to relation (21), the summation terms of the right-hand of Eq. (17) were assumed to be constant. This is, of course, not true and makes the prediction too rough to agree with the actual locations of the maxima when the nonlinearity is large. Nevertheless, it can still provide a good qualitative explanation of the problem.

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Figures (7)

Fig. 1
Fig. 1

Spectral amplitudes of the normalized space charge field w = E/EQ when the fringe is moving at a normalized velocity of y = 1.25 × 10−4 for different values of modulation m.

Fig. 2
Fig. 2

Spatial variation of the normalized electron density u = n/NA for different values of normalized fringe velocity y for a modulation m = 0.6.

Fig. 3
Fig. 3

Spatial variation of the normalized space charge field w = E/EQ for different values of normalized fringe velocity y for a modulation m = 0.6.

Fig. 4
Fig. 4

Spatial variation of the normalized ionized donor density υ = ND+/NA for different values of normalized fringe velocity y for a modulation m = 0.6.

Fig. 5
Fig. 5

Spectral amplitudes of the normalized electron density u = n/NA for normalized fringe velocity (a) y = 0.05 × 10−4, (b) y = 0.15 × 10−4, and (c) y = 0.3 × 10−4, for a modulation m = 0.6.

Fig. 6
Fig. 6

Plot of the velocity optimized lth harmonic amplitude of the space charge field, |El|max, (l = 1, 2, 3, 4, 5) as a function of the inverse of the fringe spacing Λ−1, for modulations (a) m = 0.1 and (b) m = 0.6. The calculated points are indicated by crosses. Note that there is a change of scale in the Λ−1 axis in (b).

Fig. 7
Fig. 7

Plot of the velocity optimized lth harmonic amplitude of the normalized space charge field |wl| (l = 1, 2, 3, 4, 5) as a function of modulation m on a logarithmic scale. The grating spacing is 70 mm−1. The calculated points are indicated by crosses.

Equations (35)

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N D + t = s I N D γ n N D + ,
J = n e μ E μ k T n x ,
J x = e t ( N D + n ) ,
s E x = e ( n + N A N D + ) ,
I = I 0 + 1 2 { I 1 exp [ j ( K x δ ω t ) ] + I 1 * exp [ j ( K x δ ω t ) ] } ,
u υ y d υ d ξ = P I I 0 ,
E D M d 2 u d ξ 2 y d d ξ ( υ u ) = E Q M ( u d w d ξ + w d u d ξ ) ,
d w d ξ = 1 + u υ ,
u = n N A , υ = N D + N A , w = E E Q
E Q = e N A s K , E M = γ N A μ K , E D = k T K e , P = s I 0 N D γ N A 2 , y = δ ω γ N A , ξ = K x δ ω t .
η 0 + 1 2 r = 1 [ η r exp ( j r ξ ) + η r exp ( j r ξ ) ] ,
P = u 0 ( 1 + u 0 ) + 1 4 l = l 0 ( u l + j l ω l ) u l * ,
( w 0 + j r E D Q ) u r + ( u 0 + j r y E M Q ) w r = 1 2 l = l 0 , r u l w r l ,
( 1 + 2 u 0 + j r y ) u r + ( r 2 y + j r u 0 ) w r = P m r 1 2 l = l 0 , r ( u l + j l w l ) u r l
m = I 1 I 0 , m r = { m , when r = 1 0 , 0 , otherwise
υ 0 = 1 + u 0 ,
υ r = u r + j r w r .
u 0 = 1 2 { 1 + [ 1 ( 4 P l 0 ( u l + j l w l ) u l * ) ] 1 / 2 } ,
[ u r w r ] = ϒ 1 [ 1 2 l 0 , r u l w r l P m r 1 2 l 0 , r u r l ( u l + j l w l ) ] ,
ϒ = [ E 0 Q + j r E D Q u 0 + j r y E M Q 1 + 2 u 0 + j r y r 2 y + j r u 0 ] ,
F = l = 0 N f 1 2 2 ,
det ϒ | E 0 Q u 0 + j r y E M Q 1 r 2 y + j r u 0 | ,
| det ϒ | 2 ( r 2 y E 0 Q u 0 ) 2 + r 2 ( u 0 E 0 E Q + y E M E Q ) 2 ,
r 2 y u 0 E Q / E 0 ,
u υ = P ( 1 + 1 2 m e j ξ + 1 2 m * e j ξ ) ,
u w E D Q d u d ξ = const .,
d w d ξ = 1 + u υ .
η = η 0 + 1 2 η 1 e j ξ + 1 2 η 1 * e j ξ ,
d η d ξ = 1 2 ( d η 1 d ξ j η 1 ) e j ξ + c . c .
η 1 ( ξ ) = η ˜ 1 + δ η ,
u 0 δ υ + υ 0 δ u = 0 ,
u 0 δ w + w 0 δ u E D Q ( d δ u d ξ j δ u ) = 0 ,
d ξ w d ξ j δ w = δ u δ υ .
d d ξ [ δ u δ w ] = [ E 0 D + j u 0 E Q D 1 + υ 0 / u 0 j ] [ δ u δ w ] .
E 0 D ± [ E 0 D 2 + 4 E Q D 2 ( u 0 + υ 0 ) ] 1 / 2 2 + j .

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