Abstract

Analytic expressions are derived for second- and third-order space-charge fields for a moving intensity pattern and arbitrary field strengths. The dependence on the spatial frequency, the applied field strength, and the velocity of the moving grating is examined. The magnitude of the term of correction to the fundamental harmonic is strongly dependent on experimental conditions and the relative strengths of characteristic fields. For diffusion-dominated charge transport and optimum fringe spacing the cubic correction term is only 5% of the fundamental amplitude, even for a large modulation depth of unity. Limiting cases yield the same results as previous studies.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
    [Crossref]
  2. G. C. Valley, J. Opt. Soc. Am. B 1, 868 (1984).
    [Crossref]
  3. M. Peltier and F. Micheron, J. Appl. Phys. 48, 1821 (1977).
    [Crossref]
  4. M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
    [Crossref]
  5. E. Ochoa, F. Vachss, and L. Hesselink, J. Opt. Soc. Am. A 3, 181 (1986).
    [Crossref]
  6. T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
    [Crossref]
  7. F. Vachss and L. Hesselink, J. Opt. Soc. Am. A 5, 690 (1988).
    [Crossref]
  8. F. Vachss and L. Hesselink, J. Opt. Soc. Am. B 5, 1814 (1988).
    [Crossref]
  9. L. B. Au and L. Solymar, Opt. Lett. 13, 660 (1988);J. Opt. Soc. Am. A 7, 1554 (1990).
    [Crossref]
  10. A. Bledowski, J. Otten, and K. H. Ringhofer, Opt. Lett. 16, 672 (1991).
    [Crossref] [PubMed]
  11. M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
    [Crossref]
  12. J. P. Huignard and B. Ledu, Opt. Lett. 7, 310 (1982).
    [Crossref] [PubMed]
  13. T. Y. Chang and P. Yeh, J. Opt. Soc. Am. A 3, 33 (1986).
  14. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
    [Crossref]
  15. S. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
    [Crossref]
  16. S. Sternklar, S. Weiss, and B. Fischer, Appl. Opt. 24, 3121 (1985).
    [Crossref]
  17. J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
    [Crossref]

1991 (1)

1988 (3)

1986 (3)

E. Ochoa, F. Vachss, and L. Hesselink, J. Opt. Soc. Am. A 3, 181 (1986).
[Crossref]

T. Y. Chang and P. Yeh, J. Opt. Soc. Am. A 3, 33 (1986).

S. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

1985 (3)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
[Crossref]

S. Sternklar, S. Weiss, and B. Fischer, Appl. Opt. 24, 3121 (1985).
[Crossref]

1984 (1)

1982 (1)

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

1979 (3)

M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
[Crossref]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
[Crossref]

1977 (1)

M. Peltier and F. Micheron, J. Appl. Phys. 48, 1821 (1977).
[Crossref]

Au, L. B.

Bledowski, A.

Chang, T. Y.

T. Y. Chang and P. Yeh, J. Opt. Soc. Am. A 3, 33 (1986).

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
[Crossref]

Cronin-Golomb, M.

S. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

Feinberg, J.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Fischer, B.

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
[Crossref]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
[Crossref]

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
[Crossref]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Hellwarth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Hesselink, L.

Huignard, J. P.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

J. P. Huignard and B. Ledu, Opt. Lett. 7, 310 (1982).
[Crossref] [PubMed]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Kulikov, V. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
[Crossref]

Kwong, S.

S. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

Ledu, B.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
[Crossref]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Micheron, F.

M. Peltier and F. Micheron, J. Appl. Phys. 48, 1821 (1977).
[Crossref]

Miridonov, S. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
[Crossref]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
[Crossref]

Ochoa, E.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Otten, J.

Peltier, M.

M. Peltier and F. Micheron, J. Appl. Phys. 48, 1821 (1977).
[Crossref]

Petrov, M. P.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
[Crossref]

Rajbenbach, H.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Ringhofer, K. H.

Solymar, L.

L. B. Au and L. Solymar, Opt. Lett. 13, 660 (1988);J. Opt. Soc. Am. A 7, 1554 (1990).
[Crossref]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Stepanov, S. I.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
[Crossref]

Sternklar, S.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Vachss, F.

Valley, G. C.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

Weiss, S.

Yariv, A.

S. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

Yeh, P.

T. Y. Chang and P. Yeh, J. Opt. Soc. Am. A 3, 33 (1986).

Young, L.

M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
[Crossref]

Appl. Opt. (1)

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics 22, 961 (1979).
[Crossref]

IEEE J. Quantum Electron. (1)

S. Kwong, M. Cronin-Golomb, and A. Yariv, IEEE J. Quantum Electron. QE-22, 1508 (1986).
[Crossref]

J. Appl. Phys. (4)

J. Feinberg, D. Heiman, A. R. Tanguay, and R. W. Hellwarth, J. Appl. Phys. 51, 1297 (1980).
[Crossref]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[Crossref]

M. Peltier and F. Micheron, J. Appl. Phys. 48, 1821 (1977).
[Crossref]

M. G. Moharam, T. K. Gaylord, R. Magnusson, and L. Young, J. Appl. Phys. 50, 5642 (1979).
[Crossref]

J. Opt. Soc. Am. A (3)

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

Opt. Commun. (1)

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Kulikov, Opt. Commun. 31, 301 (1979).
[Crossref]

Opt. Lett. (3)

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, Prog. Quantum Electron. 10, 77 (1985).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Normalized moduli |er| of the first three harmonics as a function of the normalized frequency difference Ωt0 between the two interfering light waves. The modulation depth m is taken to be 0.5; the normalized characteristic fields are taken to be appropriate for BSO at room temperature and a 20-μm grating spacing9: eD = 0.001, eM = 0.01. The normalized electric field is e0 = 1.0.

Fig. 2
Fig. 2

Same as in Fig. 1 but with e0 = 0.5.

Fig. 3
Fig. 3

Same as in Fig. 1 but with e0 = 0.1.

Fig. 4
Fig. 4

Same as in Fig. 1 but with normalized characteristic fields appropriate for BaTiO3 at room temperature and a 1-μm grating spacing16: eD = 0.1, and eM = 1.7. The normalized electric field is e0 = 1.0.

Fig. 5
Fig. 5

Normalized moduli |er| of the first three harmonics as a function of the normalized spatial frequency K/k0. The modulation depth m is taken to be 0.5 for a stationary intensity pattern (Ω = 0) and no applied field (E0 = 0).

Fig. 6
Fig. 6

Normalized moduli |er| of the first three harmonics as a function of the normalized dc electric field. The modulation depth m is taken to be 0.5 for a stationary intensity pattern (Ω = 0) and zero diffusion field (eD = 0).

Equations (52)

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

N D i t = s I ( N D N D i ) γ R n N D i ,
J = n q μ E μ k B T n x ,
J x = q t ( N D i n N A ) ,
E x = q ( n + N A N D i ) ,
I ( x , t ) = I 0 + { I 1 exp [ i ( K x Ω t ) ] + c . c . } ,
N D i t s I N D γ R n N D i ,
q μ ( n E ) x μ k B T 2 n x 2 q N D i t ,
E x q ( N A N D i ) ,
N D i = N D 0 + { N D 1 exp [ i ( K x Ω t ) ] + N D 2 exp [ 2 i ( K x Ω t ) ] + N D 3 exp [ 3 i ( K x Ω t ) ] + c . c . } ,
n = n 0 + { n 1 exp [ i ( K x Ω t ) ] + n 2 exp [ 2 i ( K x Ω t ) ] + n 3 exp [ 3 i ( K x Ω t ) ] + c . c . } ,
E = E 0 + { E 1 exp [ i ( K x Ω t ) ] + E 2 exp [ 2 i ( K x Ω t ) ] + E 3 exp [ 3 i ( K x Ω t ) ] + c . c . } ,
n 0 N D 0 + n 1 N D 1 * + n 2 N D 2 * + n 3 N D 3 * + n 1 * N D 1 + n 2 * N D 2 + n 3 * N D 3 = s I 0 N D / γ R ,
N D 0 = N A ;
n 0 N D 1 + n 1 N D 0 + n 2 N D 1 * + n 3 N D 2 * + n 1 * N D 2 + n 2 * N D 3 = s I 1 N D / γ R + i Ω N D 1 / γ R ,
n 0 E 1 + n 1 E 0 + n 2 E 1 * + n 3 E 2 * + n 1 * E 2 + n 2 * E 3 = i E D n 1 Ω N D 1 / ( K μ ) ,
E 1 = i E Q N D 1 / N A ;
n 0 N D 2 + n 1 N D 1 + n 2 N D 0 + n 3 N D 1 * + n 1 * N D 3 = 2 i Ω N D 2 / γ R ,
n 0 E 2 + n 1 E 1 + n 2 E 0 + n 3 E 1 * + n 1 * E 3 = 2 i E D n 2 Ω N D 2 / ( K μ ) ,
E 2 = i E Q N D 2 / ( 2 N A ) ;
n 0 N D 3 + n 1 N D 2 + n 2 N D 1 + n 3 N D 0 = 3 i Ω N D 3 / γ R ,
n 0 E 3 + n 1 E 2 + n 2 E 1 + n 3 E 0 = 3 i E D n 3 Ω N D 3 / ( K μ ) ,
E 3 = i E Q N D 3 / ( 3 N A ) .
e = e 0 , n 0 = s I 0 N D / γ R N A , N D 0 = N A ;
e 1 = i I 1 I 0 e D + i e 0 1 + e D + Ω t 0 e 0 + i [ e 0 Ω t 0 ( e D + e M ) ] ,
n 1 n 0 = e 1 1 i Ω t 0 e M e 0 i e D ,
N D 1 N A = i e 1 ;
e 2 = i n 1 n 0 N D 1 N A × 1 + 2 e D + i e 0 1 + 4 ( e D + Ω t 0 e 0 ) + 2 i [ e 0 Ω t 0 ( 4 e D + e M ) ] ,
n 2 n 0 = e 2 ( 1 2 i Ω t 0 e M ) + ( n 1 / n 0 ) e 1 e 0 2 i e D ,
N D 2 N A = 2 i e 2 ;
e 3 = ( n 1 / n 0 ) e 2 ( 1 + 6 e D + 2 i e 0 ) + ( n 2 / n 0 ) e 1 ( 1 + 3 e D + i e 0 ) 1 + 9 ( e D + Ω t 0 e 0 ) + 3 i [ e 0 Ω t 0 ( 9 e D + e M ) ] ,
n 3 n 0 = e 3 ( 1 3 i Ω t 0 e M ) + ( n 1 / n 0 ) e 2 + ( n 2 / n 0 ) e 1 e 0 3 i e D ,
N D 3 N A = 3 i e 3 .
( Ω t 0 ) r 1 / ( r 2 e 0 ) ,
e 1 = n 1 n 0 e 0 i e D 1 i Ω t 0 e M ( n 2 / n 0 ) e 1 * + ( n 1 * / n 0 ) e 2 1 i Ω t 0 e M .
C = e 1 | e 1 | 2 1 + 3 e D ( e 0 2 + e D 2 ) ( 1 + 4 e D + 2 i e 0 ) .
e 1 = i I 1 I 0 e D 1 + e D ,
e 2 = i I 1 2 I 0 2 e D ( 1 + e D ) 2 1 + 2 e D 1 + 4 e D ,
e 3 = i I 1 3 I 0 3 e D ( 1 + e D ) 3 1 + 7 e D + 9 e D 2 ( 1 + 4 e D ) ( 1 + 9 e D ) .
E 1 = i I 1 I 0 E D E Q E D + E Q [ 1 + | I 1 | 2 I 0 2 E Q 2 ( E D + E Q ) 2 E Q + 3 E D E Q + 4 E D ] .
e 1 = i ( I 1 / I 0 ) ( e D + i e 0 ) ,
e 2 = i ( I 1 2 / I 0 2 ) ( e D + i e 0 ) ,
e 3 = i ( I 1 3 / I 0 3 ) ( e D + i e 0 ) ,
| e r | = | I 1 | r I 0 r ( e D 2 + e 0 2 ) 1 / 2 .
| E 1 | = | I 1 | I 0 ( E 0 2 + E D 2 ) 1 / 2 [ 1 + ( | I 1 | 2 / I 0 2 ) ] .
e 1 = I 1 I 0 e 0 1 + i e 0 ,
e 2 = I 1 2 I 0 2 e 0 1 + i e 0 1 1 + 2 i e 0 ,
e 3 = I 1 3 I 0 3 e 0 ( 1 + i e 0 ) 1 ( 1 + 2 i e 0 ) ( 1 + 3 i e 0 ) .
η 1 I 1 2 I 0 2 e 0 2 1 + e 0 2 ,
η 2 I 1 4 I 0 4 e 0 2 ( 1 + e 0 2 ) ( 1 + 4 e 0 2 ) ,
η 3 I 1 6 I 0 6 e 0 2 ( 1 + e 0 2 ) ( 1 + 4 e 0 2 ) ( 1 + 9 e 0 2 ) .
e r ( i ) r I 1 r I 0 r e 0 r ! ( e 0 ) r ,
E 1 = i I 1 I 0 E 0 E Q E 0 i E Q [ 1 + | I 1 | 2 I 0 2 E Q 3 ( E 0 2 + E Q 2 ) ( E Q + 2 i E 0 ) ] .

Metrics