Abstract

Signals encoded as highly stable spatial (or temporal) dark solitons can be used in nonlinear-optical devices such as logic gates, interconnects, multiplexers, and filters. The formation of these waves with the use of simple diffracting (dispersing) elements in prototypical devices is analyzed by using the direct-scattering method, which is also outlined. Noncollisional dark-soliton xor and and gates and dark-soliton level splitting and bands are also described.

© 1992 Optical Society of America

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References

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  1. V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [Sov. Phys. JETP 37, 823 (1973)].
  2. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
    [CrossRef]
  3. G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
    [CrossRef] [PubMed]
  4. P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
    [CrossRef]
  5. For a recent review, see M. Nakazawa, J. Satsuma, eds., Optical Solitons (Springer-Verlag, Berlin, 1992).
  6. W. Zhao, E. Bourkoff, Opt. Lett. 14, 703 (1989).W. Zhao, E. Bourkoff, Opt. Lett. 14, 703 (1989).W. Zhao, E. Bourkoff, Opt. Lett. 14, 703 (1989).W. Zhao, E. Bourkoff, Opt. Lett. 14, 1371 (1989).
    [CrossRef] [PubMed]
  7. K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
    [CrossRef]
  8. W. J. Tomlinson, R. J. Hawkins, A. M. Weiner, J. P. Heritage, R. N. Thurston, J. Opt. Soc. Am. B 6, 329 (1989).
    [CrossRef]
  9. For other interesting cases, see S. A. Gredeskul, Yu. S. Kivshar, M. V. Yanovskaya, Phys. Rev. A 41, 3994 (1990); G. A. Swartzlander, “Observation of continuous-wave self-deflection and spatial dark solitary waves in nonlinear media,” Ph.D. dissertation (The Johns Hopkins University, Baltimore, Md., 1990).
    [CrossRef]
  10. For two wells of width A and separation B (γ1,3,5 = 1, γ2,4 = 0, ϕ1–5 = 0), the expression is 2λ2 − 1 = P cos(2λA) ± (1 − P2)1/2 sin(2λA), where P = 1 − [1 − cos(2λA)]exp(−2νB).
  11. See, e.g., M. N. Islam, C. E. Soccolich, Opt. Lett. 16, 1460 (1991); E. Fredkin, T. Toffoli, Int. J. Theor. Phys. 21, 219 (1982).
    [CrossRef]

1991 (2)

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

See, e.g., M. N. Islam, C. E. Soccolich, Opt. Lett. 16, 1460 (1991); E. Fredkin, T. Toffoli, Int. J. Theor. Phys. 21, 219 (1982).
[CrossRef]

1990 (1)

For other interesting cases, see S. A. Gredeskul, Yu. S. Kivshar, M. V. Yanovskaya, Phys. Rev. A 41, 3994 (1990); G. A. Swartzlander, “Observation of continuous-wave self-deflection and spatial dark solitary waves in nonlinear media,” Ph.D. dissertation (The Johns Hopkins University, Baltimore, Md., 1990).
[CrossRef]

1989 (2)

1987 (1)

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

1982 (1)

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

1973 (2)

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [Sov. Phys. JETP 37, 823 (1973)].

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

Andersen, D. R.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Barthelemy, A.

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Blow, K. J.

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Bourkoff, E.

Doran, N. J.

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Emplit, P.

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Froehly, C.

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Gredeskul, S. A.

For other interesting cases, see S. A. Gredeskul, Yu. S. Kivshar, M. V. Yanovskaya, Phys. Rev. A 41, 3994 (1990); G. A. Swartzlander, “Observation of continuous-wave self-deflection and spatial dark solitary waves in nonlinear media,” Ph.D. dissertation (The Johns Hopkins University, Baltimore, Md., 1990).
[CrossRef]

Hamaide, J. P.

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Hasegawa, A.

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

Hawkins, R. J.

Heritage, J. P.

Islam, M. N.

Kaplan, A. E.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Kivshar, Yu. S.

For other interesting cases, see S. A. Gredeskul, Yu. S. Kivshar, M. V. Yanovskaya, Phys. Rev. A 41, 3994 (1990); G. A. Swartzlander, “Observation of continuous-wave self-deflection and spatial dark solitary waves in nonlinear media,” Ph.D. dissertation (The Johns Hopkins University, Baltimore, Md., 1990).
[CrossRef]

Regan, J. J.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Reynaud, F.

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [Sov. Phys. JETP 37, 823 (1973)].

Soccolich, C. E.

Swartzlander, G. A.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Tappert, F.

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

Thurston, R. N.

Tomlinson, W. J.

Weiner, A. M.

Yanovskaya, M. V.

For other interesting cases, see S. A. Gredeskul, Yu. S. Kivshar, M. V. Yanovskaya, Phys. Rev. A 41, 3994 (1990); G. A. Swartzlander, “Observation of continuous-wave self-deflection and spatial dark solitary waves in nonlinear media,” Ph.D. dissertation (The Johns Hopkins University, Baltimore, Md., 1990).
[CrossRef]

Yin, H.

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [Sov. Phys. JETP 37, 823 (1973)].

Zhao, W.

Appl. Phys. Lett. (1)

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

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

Opt. Commun. (2)

P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

K. J. Blow, N. J. Doran, Opt. Commun. 42, 403 (1982).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

For other interesting cases, see S. A. Gredeskul, Yu. S. Kivshar, M. V. Yanovskaya, Phys. Rev. A 41, 3994 (1990); G. A. Swartzlander, “Observation of continuous-wave self-deflection and spatial dark solitary waves in nonlinear media,” Ph.D. dissertation (The Johns Hopkins University, Baltimore, Md., 1990).
[CrossRef]

Phys. Rev. Lett. (1)

G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov, A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973) [Sov. Phys. JETP 37, 823 (1973)].

Other (2)

For two wells of width A and separation B (γ1,3,5 = 1, γ2,4 = 0, ϕ1–5 = 0), the expression is 2λ2 − 1 = P cos(2λA) ± (1 − P2)1/2 sin(2λA), where P = 1 − [1 − cos(2λA)]exp(−2νB).

For a recent review, see M. Nakazawa, J. Satsuma, eds., Optical Solitons (Springer-Verlag, Berlin, 1992).

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Figures (2)

Fig. 1
Fig. 1

Dark-soliton eigenvalues λ generated from a three-region mask. The central region has normalized width L. (a) Cross-hatched curves, a pure amplitude mask with an opaque region; thick solid curves, an opaque central region and ϕ3ϕ1 = π. (b) A pure phase mask in which the phase of the central region is ϕ2 = π (cross-hatched curves) and 2π/3 (thick solid curves).

Fig. 2
Fig. 2

Dark-soliton spectrum of eigenvalues showing level splitting and band formation as the number of diffracting regions is increased. (a) Pure amplitude mask with the opaque well width and separation equal to two. The data points are connected by curves to aid the eye. (b) Pure phase mask with an even (filled triangles) and odd (filled circles) number of regions.

Equations (4)

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u j ( x , z = 0 ) = γ j exp ( i ϕ j ) ,             x j < x < x j + 1 ,
A j exp ( p j x j + 1 ) + C j g j exp ( - i ϕ j - p j x j + 1 ) = A j + 1 exp ( p j + 1 x j + 1 ) + C j + 1 g j + 1 exp ( - i ϕ j + 1 - p j + 1 x j + 1 ) ,
A j g j exp ( i ϕ j + p j x j + 1 ) + C j exp ( - p j x j + 1 ) = A j + 1 g j + 1 exp ( i ϕ j + 1 + p j + 1 x j + 1 ) + C j + 1 exp ( - p j + 1 x j + 1 ) ,
[ g 1 exp ( + i ϕ 1 ) - g 2 exp ( + i ϕ 2 ) ] [ g 2 exp ( - i ϕ 2 ) - g 3 exp ( - i ϕ 3 ) ] exp ( - p 2 L ) = { g 1 g 2 exp [ i ϕ 1 - ϕ 2 ) ] - 1 } { 1 - g 2 g 3 exp [ i ( ϕ 2 - ϕ 3 ) ] } exp ( + p 2 L ) ,

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