Abstract

I examine the conditions for formation of gap solitons in an infinite periodic Kerr medium by illuminating the nonlinear grating with two intense cw beams, providing a novel experimental method for studying gap solitons. New applications of nonlinear gratings are proposed.

© 1993 Optical Society of America

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References

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  1. W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
    [Crossref] [PubMed]
  2. C. M. de Sterke, J. E. Sipe, Opt. Lett. 14, 871 (1989); D. N. Christodoulides, R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
    [Crossref] [PubMed]
  3. S. Wabnitz, Opt. Lett. 14, 1071 (1989).
    [Crossref] [PubMed]
  4. N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
    [Crossref]
  5. A. B. Aceves, S. Wabnitz, Phys. Lett. A 141, 37 (1989).
    [Crossref]
  6. C. M. de Sterke, J. E. Sipe, Phys. Rev. A 42, 550 (1990).
    [Crossref]
  7. J. Feng, F. K Kneubühl, IEEE J. Quantum Electron. 29, 590 (1993).
    [Crossref]
  8. H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
    [Crossref]
  9. M. Thakur, in Polymersfor Lightwave and Integrated Optics, L. A. Hornak, ed. (Dekker, New York, 1992), p. 667.
  10. G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
    [Crossref]
  11. For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
    [Crossref]

1993 (1)

J. Feng, F. K Kneubühl, IEEE J. Quantum Electron. 29, 590 (1993).
[Crossref]

1992 (1)

N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[Crossref]

1990 (1)

C. M. de Sterke, J. E. Sipe, Phys. Rev. A 42, 550 (1990).
[Crossref]

1989 (3)

1988 (1)

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

1987 (1)

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

1984 (1)

G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[Crossref]

1976 (1)

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[Crossref]

Aceves, A. B.

A. B. Aceves, S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[Crossref]

Brown, T. G.

N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[Crossref]

Chen, W.

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

Chilwell, C.

G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[Crossref]

de Sterke, C. M.

Feng, J.

J. Feng, F. K Kneubühl, IEEE J. Quantum Electron. 29, 590 (1993).
[Crossref]

Finlayson, N.

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

Haus, H. A.

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[Crossref]

Kneubühl, F. K

J. Feng, F. K Kneubühl, IEEE J. Quantum Electron. 29, 590 (1993).
[Crossref]

Mills, D. L.

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

Prelewitz, D. F.

N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[Crossref]

Sankey, N. D.

N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[Crossref]

Seaton, C. T.

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[Crossref]

Shank, C. V.

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[Crossref]

Sipe, J. E.

Smith, S. D.

G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[Crossref]

Stegeman, G. I.

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[Crossref]

Thakur, M.

M. Thakur, in Polymersfor Lightwave and Integrated Optics, L. A. Hornak, ed. (Dekker, New York, 1992), p. 667.

Wabnitz, S.

S. Wabnitz, Opt. Lett. 14, 1071 (1989).
[Crossref] [PubMed]

A. B. Aceves, S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[Crossref]

Wright, E. M.

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

Zanoni, R.

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

Appl. Phys. Lett. (2)

N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[Crossref]

G. I. Stegeman, C. T. Seaton, C. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[Crossref]

IEEE J. Quantum Electron. (2)

J. Feng, F. K Kneubühl, IEEE J. Quantum Electron. 29, 590 (1993).
[Crossref]

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976).
[Crossref]

J. Light-wave Technol. (1)

For a review, see, e.g.,G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, C. T. Seaton, J. Light-wave Technol. 6, 953 (1988).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (1)

A. B. Aceves, S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[Crossref]

Phys. Rev. A (1)

C. M. de Sterke, J. E. Sipe, Phys. Rev. A 42, 550 (1990).
[Crossref]

Phys. Rev. Lett. (1)

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[Crossref] [PubMed]

Other (1)

M. Thakur, in Polymersfor Lightwave and Integrated Optics, L. A. Hornak, ed. (Dekker, New York, 1992), p. 667.

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Figures (3)

Fig. 1
Fig. 1

Geometry for excitation of the spatial gap soliton.

Fig. 2
Fig. 2

Formation of the SGS, or fusion of two beams. The grating is assumed to have the same average linear refractive index n0 as the bounding medium nh. The angle brackets denote the average over several wavelengths in the x and z directions and over several periods of time t. For z ≤ 0, Eq. (10) is displayed; for z > 0, Eqs. (3) are solved numerically. The parameters are α > 0, θB = π/4, θ = 0.4π, and = 0.0866.

Fig. 3
Fig. 3

(a) Combinations of incident peak intensities I± = |ri±E+ (0,0)|2 and angles θ± for the excitation of the SGS (θ0 = θB). (b) The resulting peak intensity and deflection angle of the SGS. The parameters are nh = n0, θB = π/4, and = 0.02.

Equations (11)

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E ( x , z ) = E ( x , z ) exp ( - i ω t ) y ^ + c . c . = { E + ( x , z ) exp [ i ( k B x + k Z z ) ] + E - ( x , z ) × exp [ i ( - k B x + k Z z ) ] } exp ( - i ω t ) y ^ + c . c . ,
n = n 0 + n 1 cos ( 2 k B x ) + n 2 E ( x , z ) 2 ,
E + x + k Z k B E + z = i Δ k E + + i κ E - + i α ( E + 2 + 2 E - 2 ) E + , E - x - k Z k B E - z = - i Δ k E - - i κ E + - i α ( E - 2 + 2 E + 2 ) E - ,
E ± = A ± [ x - β z k B / k Z ) exp [ i ϕ ± ( x - β z k B / k Z ) ] ,
A ± = [ κ α 2 ( 1 ± β ) 1 - β 2 3 - β 2 cos 2 θ sin θ + cosh 2 ζ ] 1 / 2 , ϕ ± = sgn ( α ) [ ( β tan θ ) ζ + ( 4 β 3 - β 2 ± 1 ) × arctan ( cos θ 1 + sin θ tanh ζ ) ] 1 + sgn ( α ) 4 π + ψ 0 , θ = arcsin sgn ( α ) Δ k κ 1 - β 2             ( - π 2 θ < π 2 ) , ζ = κ cos θ 1 - β 2 ( x - β k B k Z z ) .
E ( x , z < 0 ) = 1 2 - + { [ E ^ ( k ) - i E ^ Z ( k ) / k h 2 - k 2 ] × exp [ i ( k x + k h 2 - k 2 z ) ] + [ E ^ ( k ) + i E ^ Z ( k ) / k h 2 - k 2 ] × exp [ i ( k x - k h 2 - k 2 z ) ] } d k ,
E ± ( x , 0 ) E ± ( 0 , 0 ) exp ( - x 2 / w x 2 + i ϕ ± x ) ,
ϕ ± = ( ϕ ± / x ) x , z = 0 = sgn ( α ) κ 1 - β 2 [ β sin θ + ( 4 β 3 - β 2 ± 1 ) ( 1 - sin θ ) ] .
w x = 2 π - + [ A + 2 ( x , z ) + A - 2 ( x , z ) ] d x max ( A + 2 + A - 2 ) = 1 - β 2 κ 2 π π 2 - θ 1 - sin θ .
E / E + ( 0 , 0 ) = r i + ( w x cos θ + / w + ) exp [ - ( r · m ^ i + ) 2 / w + 2 + i k h r · n ^ i + ] + r i + ( w x cos θ + / w + * ) exp [ - ( r · m ^ r + ) 2 / ( w + * ) 2 + i k h r · n ^ r + ] - r i - sgn ( α ) [ ( 1 - β ) / ( 1 + β ) ] 1 / 2 ( w x cos θ - / w - ) exp [ - ( r · m ^ i - ) 2 / w - 2 + i k h r · n ^ i - ] - r r - sgn ( α ) [ ( 1 - β ) / ( 1 + β ) ] 1 / 2 ( w x cos θ - / w - * ) exp [ - ( r · m ^ r - ) 2 / ( w - * ) 2 + i k h r · n ^ r - ] ,
w ± = [ ( w x cos θ ± ) 2 + i 2 z / ( k h cos θ ± ) ] 1 / 2 , r i ± = [ 1 + ( k Z - tan θ S ϕ ± ) / ( k h cos θ ± ) ] / 2 , r r ± = [ 1 - ( k Z - tan θ S ϕ ± ) / ( k h cos θ ± ) ] / 2 , k Z = k 0 cos θ B [ 1 - 2 tan θ B sin θ ( 1 - β 2 ) 1 / 2 ] 1 / 2 , = sgn ( α ) κ / ( k 0 cos θ B ) ,

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