Abstract

We observe the self-development of a light beam propagating along the bisector of two waves, one ordinary and the other extraordinary, that intersect inside a BaTiO3 sample at the plane that is normal to the optical axis of the crystal at the angle meeting the phase-matching condition for forward vectorial parametric mixing of three coplanar light beams. When the angle of intersection exceeds the phase-matching angle, a cone of extraordinary waves appears instead of only one beam. Both the geometry and the intensity behavior were shown to conform well to the predictions of the model in which the process is considered as forward parametric four-wave mixing.

© 1992 Optical Society of America

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  1. R. Rupp, F. Drees, Appl. Phys. B 39, 223–229 (1986).
    [CrossRef]
  2. D. Temple, C. Warde, J. Opt. Soc. Am. B 3, 337–341 (1986).
    [CrossRef]
  3. M. Ewbank, P. Yeh, J. Feinberg, Opt. Commun. 59, 423–428 (1986).
    [CrossRef]
  4. N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
    [CrossRef]
  5. R. Magnussen, T. K. Gaylord, Appl. Opt. 13, 1546–1548 (1974).
  6. E. Avakian, K. Belabaev, S. Odoulov, Sov. Phys. Solid State 25, 1887–1890 (1983).
  7. S. Odoulov, K. Belabaev, I. Kiseleva, Opt. Lett. 10, 31–33 (1985).
    [CrossRef] [PubMed]
  8. A. Khyznyak, B. Soloviev, Sov. J. Opt. Spectros. 53, 723–726 (1982).
  9. S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
    [CrossRef]
  10. K. Ringhofer, L. Solymar, Appl. Phys. Lett. 53, 1039–1040 (1988).
    [CrossRef]
  11. K. Ringhofer, L. Solymar, Appl. Phys. B 48, 395–400 (1989).
    [CrossRef]
  12. A. Yariv, D. Pepper, Opt. Lett. 1, 16–19 (1977).
    [CrossRef]
  13. A. Novikov, S. Odoulov, M. Soskin, Sov. Phys. Dokl. 295, 596–600 (1987).
  14. A. Novikov, S. Odoulov, Ukr. Phys. J. 32, 1514–1520 (1987).
  15. D. C. Jones, L. Solymar, Electron. Lett. 25, 844–845 (1989).
    [CrossRef]
  16. D. C. Jones, L. Solymar, Appl. Phys. B 50, 355–359 (1990).
    [CrossRef]
  17. D. Eberschloe, L. Solymar, Appl. Phys. Lett. 53, 1135–1137 (1988).
    [CrossRef]
  18. D. Eberschloe, L. Solymar, Electron. Lett. 24, 683–684 (1988).
    [CrossRef]
  19. J. H. Webb, K. H. Ringhofer, L. Solymar, Electron. Lett. 27, 889–890 (1991).
    [CrossRef]
  20. D. Temple, C. Warde, J. Opt. Soc. Am. B 5, 1800–1805 (1988).
    [CrossRef]
  21. A. Blendowski, B. Sturman, J. Otten, K. H. Ringhofer, Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439;J. Opt. Soc. Am. B 9, 672–681 (1992).
  22. S. Odoulov, Ukr. Phys. J. 35, 1657 (1990).
  23. S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
    [CrossRef]

1991

J. H. Webb, K. H. Ringhofer, L. Solymar, Electron. Lett. 27, 889–890 (1991).
[CrossRef]

S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

1990

S. Odoulov, Ukr. Phys. J. 35, 1657 (1990).

D. C. Jones, L. Solymar, Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

1989

D. C. Jones, L. Solymar, Electron. Lett. 25, 844–845 (1989).
[CrossRef]

K. Ringhofer, L. Solymar, Appl. Phys. B 48, 395–400 (1989).
[CrossRef]

1988

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

K. Ringhofer, L. Solymar, Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

D. Eberschloe, L. Solymar, Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

D. Eberschloe, L. Solymar, Electron. Lett. 24, 683–684 (1988).
[CrossRef]

D. Temple, C. Warde, J. Opt. Soc. Am. B 5, 1800–1805 (1988).
[CrossRef]

1987

A. Novikov, S. Odoulov, M. Soskin, Sov. Phys. Dokl. 295, 596–600 (1987).

A. Novikov, S. Odoulov, Ukr. Phys. J. 32, 1514–1520 (1987).

1986

M. Ewbank, P. Yeh, J. Feinberg, Opt. Commun. 59, 423–428 (1986).
[CrossRef]

R. Rupp, F. Drees, Appl. Phys. B 39, 223–229 (1986).
[CrossRef]

D. Temple, C. Warde, J. Opt. Soc. Am. B 3, 337–341 (1986).
[CrossRef]

1985

1984

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

1983

E. Avakian, K. Belabaev, S. Odoulov, Sov. Phys. Solid State 25, 1887–1890 (1983).

1982

A. Khyznyak, B. Soloviev, Sov. J. Opt. Spectros. 53, 723–726 (1982).

1977

1974

R. Magnussen, T. K. Gaylord, Appl. Opt. 13, 1546–1548 (1974).

Albers, N.

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

Avakian, E.

E. Avakian, K. Belabaev, S. Odoulov, Sov. Phys. Solid State 25, 1887–1890 (1983).

Belabaev, K.

S. Odoulov, K. Belabaev, I. Kiseleva, Opt. Lett. 10, 31–33 (1985).
[CrossRef] [PubMed]

E. Avakian, K. Belabaev, S. Odoulov, Sov. Phys. Solid State 25, 1887–1890 (1983).

Blendowski, A.

A. Blendowski, B. Sturman, J. Otten, K. H. Ringhofer, Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439;J. Opt. Soc. Am. B 9, 672–681 (1992).

Drees, F.

R. Rupp, F. Drees, Appl. Phys. B 39, 223–229 (1986).
[CrossRef]

Ducollet, H.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

Eberschloe, D.

D. Eberschloe, L. Solymar, Electron. Lett. 24, 683–684 (1988).
[CrossRef]

D. Eberschloe, L. Solymar, Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

Ewbank, M.

M. Ewbank, P. Yeh, J. Feinberg, Opt. Commun. 59, 423–428 (1986).
[CrossRef]

Feinberg, J.

M. Ewbank, P. Yeh, J. Feinberg, Opt. Commun. 59, 423–428 (1986).
[CrossRef]

Gaylord, T. K.

R. Magnussen, T. K. Gaylord, Appl. Opt. 13, 1546–1548 (1974).

Herriau, J. P.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

Holtman, L.

S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

Huignard, J.-P.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

Imbert, B.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

Jones, D. C.

D. C. Jones, L. Solymar, Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

D. C. Jones, L. Solymar, Electron. Lett. 25, 844–845 (1989).
[CrossRef]

Khyznyak, A.

A. Khyznyak, B. Soloviev, Sov. J. Opt. Spectros. 53, 723–726 (1982).

Kiseleva, I.

Kraetzig, E.

S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

Kuelich, K.

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

Kukhtarev, N.

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

Magnussen, R.

R. Magnussen, T. K. Gaylord, Appl. Opt. 13, 1546–1548 (1974).

Mallick, S.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

Novikov, A.

A. Novikov, S. Odoulov, Ukr. Phys. J. 32, 1514–1520 (1987).

A. Novikov, S. Odoulov, M. Soskin, Sov. Phys. Dokl. 295, 596–600 (1987).

Odoulov, S.

S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

S. Odoulov, Ukr. Phys. J. 35, 1657 (1990).

A. Novikov, S. Odoulov, M. Soskin, Sov. Phys. Dokl. 295, 596–600 (1987).

A. Novikov, S. Odoulov, Ukr. Phys. J. 32, 1514–1520 (1987).

S. Odoulov, K. Belabaev, I. Kiseleva, Opt. Lett. 10, 31–33 (1985).
[CrossRef] [PubMed]

E. Avakian, K. Belabaev, S. Odoulov, Sov. Phys. Solid State 25, 1887–1890 (1983).

Otten, J.

A. Blendowski, B. Sturman, J. Otten, K. H. Ringhofer, Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439;J. Opt. Soc. Am. B 9, 672–681 (1992).

Pepper, D.

Ringhofer, K.

K. Ringhofer, L. Solymar, Appl. Phys. B 48, 395–400 (1989).
[CrossRef]

K. Ringhofer, L. Solymar, Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

Ringhofer, K. H.

J. H. Webb, K. H. Ringhofer, L. Solymar, Electron. Lett. 27, 889–890 (1991).
[CrossRef]

A. Blendowski, B. Sturman, J. Otten, K. H. Ringhofer, Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439;J. Opt. Soc. Am. B 9, 672–681 (1992).

Rupp, R.

R. Rupp, F. Drees, Appl. Phys. B 39, 223–229 (1986).
[CrossRef]

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

Soloviev, B.

A. Khyznyak, B. Soloviev, Sov. J. Opt. Spectros. 53, 723–726 (1982).

Solymar, L.

J. H. Webb, K. H. Ringhofer, L. Solymar, Electron. Lett. 27, 889–890 (1991).
[CrossRef]

D. C. Jones, L. Solymar, Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

K. Ringhofer, L. Solymar, Appl. Phys. B 48, 395–400 (1989).
[CrossRef]

D. C. Jones, L. Solymar, Electron. Lett. 25, 844–845 (1989).
[CrossRef]

K. Ringhofer, L. Solymar, Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

D. Eberschloe, L. Solymar, Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

D. Eberschloe, L. Solymar, Electron. Lett. 24, 683–684 (1988).
[CrossRef]

Soskin, M.

A. Novikov, S. Odoulov, M. Soskin, Sov. Phys. Dokl. 295, 596–600 (1987).

Sturman, B.

S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

A. Blendowski, B. Sturman, J. Otten, K. H. Ringhofer, Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439;J. Opt. Soc. Am. B 9, 672–681 (1992).

Temple, D.

Warde, C.

Webb, J. H.

J. H. Webb, K. H. Ringhofer, L. Solymar, Electron. Lett. 27, 889–890 (1991).
[CrossRef]

Yariv, A.

Yeh, P.

M. Ewbank, P. Yeh, J. Feinberg, Opt. Commun. 59, 423–428 (1986).
[CrossRef]

Appl. Opt.

R. Magnussen, T. K. Gaylord, Appl. Opt. 13, 1546–1548 (1974).

Appl. Phys. B

N. Kukhtarev, E. Kraetzig, K. Kuelich, R. Rupp, N. Albers, Appl. Phys. B 35, 17–21 (1984).
[CrossRef]

K. Ringhofer, L. Solymar, Appl. Phys. B 48, 395–400 (1989).
[CrossRef]

D. C. Jones, L. Solymar, Appl. Phys. B 50, 355–359 (1990).
[CrossRef]

R. Rupp, F. Drees, Appl. Phys. B 39, 223–229 (1986).
[CrossRef]

S. Odoulov, B. Sturman, L. Holtman, E. Kraetzig, Appl. Phys. B 52, 317–322 (1991).
[CrossRef]

Appl. Phys. Lett.

K. Ringhofer, L. Solymar, Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

D. Eberschloe, L. Solymar, Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

Electron. Lett.

D. Eberschloe, L. Solymar, Electron. Lett. 24, 683–684 (1988).
[CrossRef]

J. H. Webb, K. H. Ringhofer, L. Solymar, Electron. Lett. 27, 889–890 (1991).
[CrossRef]

D. C. Jones, L. Solymar, Electron. Lett. 25, 844–845 (1989).
[CrossRef]

J. Appl. Phys.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, J.-P. Huignard, J. Appl. Phys. 63, 5600–5663 (1988).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

M. Ewbank, P. Yeh, J. Feinberg, Opt. Commun. 59, 423–428 (1986).
[CrossRef]

Opt. Lett.

Sov. J. Opt. Spectros.

A. Khyznyak, B. Soloviev, Sov. J. Opt. Spectros. 53, 723–726 (1982).

Sov. Phys. Dokl.

A. Novikov, S. Odoulov, M. Soskin, Sov. Phys. Dokl. 295, 596–600 (1987).

Sov. Phys. Solid State

E. Avakian, K. Belabaev, S. Odoulov, Sov. Phys. Solid State 25, 1887–1890 (1983).

Ukr. Phys. J.

A. Novikov, S. Odoulov, Ukr. Phys. J. 32, 1514–1520 (1987).

S. Odoulov, Ukr. Phys. J. 35, 1657 (1990).

Other

A. Blendowski, B. Sturman, J. Otten, K. H. Ringhofer, Photorefractive Materials, Effects, and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439;J. Opt. Soc. Am. B 9, 672–681 (1992).

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Figures (10)

Fig. 1
Fig. 1

a, Wave-vector diagram for nonlinear mixing of copropagating orthogonally polarized eigenwaves of the crystal. b–c, Various experimental implementations of this mixing: b, conical light-induced scattering of one incident (extraordinary) wave; c, amplification of the signal wave (with ks) in the presence of two pump waves (with kpo and kpe); d, generation of the first subharmonic for two incident waves (with kpo and kpe). C’s, crystals’ c axes.

Fig. 2
Fig. 2

Schematic of experimental arrangement, BS, beam splitter; M, mirror; PRC, photorefractive crystal; SC, screen; C, crystal’s c axis.

Fig. 3
Fig. 3

Far-field patterns of output intensity for a, exact subharmonic phase matching, θ = 23°; b and c, slightly above critical angle: b, θ = 25°; c, θ = 27°. No seed radiation is introduced into the sample from outside.

Fig. 4
Fig. 4

Apex (half-) angle of scattering-wave cone θs as a function of pump intersection angle θ. The solid curve represents the dependence calculated from Eq. (15).

Fig. 5
Fig. 5

Output intensity of signal beam incident upon the sample in the direction of the first subharmonic versus time. The intensity ratio of the input signal to the pump waves is 1:1000; the pump waves have equal intensities.

Fig. 6
Fig. 6

Gain factor versus pump intensity ratio r. The solid curve is the calculated dependence [Eq. (7)].

Fig. 7
Fig. 7

Optical oscillators using parametric gain in BaTiO3: a, closed linear; b, closed unidirectional ring; c, unclosed unidirectional ring. M’s, mirrors; C’s, crystals’ c axes.

Fig. 8
Fig. 8

Calculated intensity of oscillating wave Iosc versus coupling strength for a closed unidirectional ring oscillator. Near each curve the value of the cavity mirror reflectivity R is shown; r = 1.

Fig. 9
Fig. 9

Time dependence of oscillation intensity for a closed unidirectional ring oscillator; r = 1.

Fig. 10
Fig. 10

Intensity of oscillating wave versus coupling strength for a closed unidirectional ring oscillator. The solid curve is a fit to the experimental points, r = 1.

Equations (27)

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

2 k s e = k p o + k p e ,
d A p o / d x = f ( A s e ) 2 ( A p e ) * , d A s e / d x = f A p o A p e ( A s e ) * , d A p e / d x = f A s o A s e ( A p e ) * , d A s o / d x = f ( A p e ) 2 ( A s e ) * ,
k s o = 2 k p e k s e
f = ( 2 π Δ n / I 0 λ ) ,
Δ n = ( r 51 n 3 E D ) / ( 1 + l D 2 / L 2 ) .
| A | 2 = | A s e | 2 = [ a s e ( 0 ) ] 2 { exp ( 2 F x ) cos 2 [ Φ ( 0 ) / 2 ] + exp ( 2 F x ) sin 2 [ Φ ( 0 ) / 2 ] } ,
F = f a p o a p e , A ( 0 ) = a s e ( 0 ) exp [ i Φ ( 0 ) / 2 ] ,
Φ = 2 ϕ s ϕ p o ϕ p e ,
[ a s ( x ) ] 2 + [ a p o ( x ) ] 2 = [ a s e ( 0 ) ] 2 + [ a p o ( 0 ) ] 2 , a s e ( x ) a p e ( x ) + a p o ( x ) a s o ( x ) = a s e ( 0 ) a p e ( 0 ) , [ a p e ( x ) ] 2 + [ a s o ( x ) ] 2 = [ a p e ( 0 ) ] 2 , a s e ( x ) a s o ( x ) a p o ( x ) a p e ( x ) = a p o ( 0 ) a p e ( 0 ) ,
d η / d x = F { η 3 + η 2 [ ( 1 η 2 ) 1 / 2 / p ] η } ,
η = a s e ( x ) / { [ a s e ( 0 ) ] 2 + [ a p o ( 0 ) ] 2 } 1 / 2 ,
p = a p o ( 0 ) / a s e ( 0 ) .
tan ( μ + ν 2 ) = tanh ( F x / 2 ) { ( 1 + p 2 ) 1 / 2 tan [ u o + ( υ / 2 ) ] 1 } p tan [ u 0 + ( υ / 2 ) ] tanh ( F x / 2 ) { tan [ u o + ( υ / 2 ) ] ( 1 + p 2 ) 1 / 2 } p ,
tan u = η ( 1 η 2 ) 1 / 2 , tan 2 υ 2 = ( 1 + p 2 ) 1 / 2 + p ( 1 + p 2 ) 1 / 2 p , tan u o = 1 p .
[ a s e ( x ) ] 2 = [ a s e ( 0 ) ] 2 [ a p o ( 0 ) ] 2 exp ( 2 F x ) [ a p o ( 0 ) ] 2 + [ a s e ( 0 ) ] 2 exp ( 2 F x ) ,
sin θ = [ n o ( n o n e ) ] 1 / 2 ,
k p e + k p o = k s 1 e + k s 2 e .
4 sin 2 θ s = [ ( n o 2 sin 2 θ ) 1 / 2 + ( n e 2 sin 2 θ ) 1 / 2 ] 2 4 n s 2 .
Γ = ( 1 / l ) In [ I s e ( ) / I s e ( 0 ) ] ,
R exp ( 2 F l ) = 1 ,
[ a s e ( 0 ) ] 2 = [ a s e ( l ) ] 2 R ,
Γ l = 2 ( 1 + r + J ) r 1 / 2 ln J 1 J [ R ( J + 1 ) J ] 1 / 2
J = [ a s e ( 0 ) / a p o ( 0 ) ] 2 ,
3 k SH 3 e = 2 k p e + k p o ,
4 k SH 4 e = 3 k p e + k p o .
θ p o = ( N 1 ) θ p e = { [ 2 ( N 1 ) n o n e ( n o n e ) ] / [ n o + ( N 1 ) n e ] } 1 / 2 .
θ p o ( SH 3 ) = 27.37 ° , θ p e ( SH 3 ) = 13.28 ° , θ p o ( SH 4 ) = 34.23 ° , θ p e ( SH 4 ) = 18.71 °

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