Abstract

We derive a simple analytical expression for the modulational instability of two counterpropagating waves in a periodic nonlinear structure. Numerical studies reveal that the nonlinear development of this instability leads to time-recurrent energy storage in the form of steady solitons and to soliton train emission from both ends of the filter.

© 1992 Optical Society of America

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References

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  1. C. M. de Sterke, J. E. Sipe, Phys. Rev. A 38, 5149 (1989).
  2. P.St. J. Russell, J. Mod. Opt. 38, 1599 (1991).
  3. H. G. Winful, G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
  4. D. N. Christodoulides, R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
  5. A. B. Aceves, S. Wabnitz, Phys. Lett. A 141, 37 (1989).
  6. W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987);D. L. Mills, S. E. Trullinger, Phys. Rev. B 36, 6269 (1987).
  7. C. M. de Sterke, Phys. Rev. A 45, 2012 (1992),
  8. N. D. Sankey, D. F. Prelewitz, T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
  9. C. M. de Sterke, J. Sipe, Phys. Rev. A 42, 2858 (1990).
  10. H. G. Winful, R. Zamir, S. Felman, Appl. Phys. Lett. 58, 1001 (1991).
  11. E. A. Kuznetsov, A. V. Mikhailov, Teor. Mat. Fiz. 30, 193 (1977);D. J. Kaup, A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).
  12. A stability analysis of finite nonlinear DFB structures was recently presented by C. M. de Sterke, Phys. Rev. A 45, 8252 (1992).

1992 (3)

C. M. de Sterke, Phys. Rev. A 45, 2012 (1992),

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

A stability analysis of finite nonlinear DFB structures was recently presented by C. M. de Sterke, Phys. Rev. A 45, 8252 (1992).

1991 (2)

H. G. Winful, R. Zamir, S. Felman, Appl. Phys. Lett. 58, 1001 (1991).

P.St. J. Russell, J. Mod. Opt. 38, 1599 (1991).

1990 (1)

C. M. de Sterke, J. Sipe, Phys. Rev. A 42, 2858 (1990).

1989 (3)

D. N. Christodoulides, R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).

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

C. M. de Sterke, J. E. Sipe, Phys. Rev. A 38, 5149 (1989).

1987 (1)

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987);D. L. Mills, S. E. Trullinger, Phys. Rev. B 36, 6269 (1987).

1982 (1)

H. G. Winful, G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).

1977 (1)

E. A. Kuznetsov, A. V. Mikhailov, Teor. Mat. Fiz. 30, 193 (1977);D. J. Kaup, A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).

Aceves, A. B.

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

Brown, T. G.

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

Chen, W.

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987);D. L. Mills, S. E. Trullinger, Phys. Rev. B 36, 6269 (1987).

Christodoulides, D. N.

D. N. Christodoulides, R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).

Cooperman, G. D.

H. G. Winful, G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).

de Sterke, C. M.

C. M. de Sterke, Phys. Rev. A 45, 2012 (1992),

C. M. de Sterke, J. Sipe, Phys. Rev. A 42, 2858 (1990).

C. M. de Sterke, J. E. Sipe, Phys. Rev. A 38, 5149 (1989).

Felman, S.

H. G. Winful, R. Zamir, S. Felman, Appl. Phys. Lett. 58, 1001 (1991).

Joseph, R. I.

D. N. Christodoulides, R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).

Kuznetsov, E. A.

E. A. Kuznetsov, A. V. Mikhailov, Teor. Mat. Fiz. 30, 193 (1977);D. J. Kaup, A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).

Mikhailov, A. V.

E. A. Kuznetsov, A. V. Mikhailov, Teor. Mat. Fiz. 30, 193 (1977);D. J. Kaup, A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).

Mills, D. L.

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987);D. L. Mills, S. E. Trullinger, Phys. Rev. B 36, 6269 (1987).

Prelewitz, D. F.

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

Russell, P.St. J.

P.St. J. Russell, J. Mod. Opt. 38, 1599 (1991).

Sankey, N. D.

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

Sipe, J.

C. M. de Sterke, J. Sipe, Phys. Rev. A 42, 2858 (1990).

Sipe, J. E.

C. M. de Sterke, J. E. Sipe, Phys. Rev. A 38, 5149 (1989).

Wabnitz, S.

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

Winful, H. G.

H. G. Winful, R. Zamir, S. Felman, Appl. Phys. Lett. 58, 1001 (1991).

H. G. Winful, G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).

Zamir, R.

H. G. Winful, R. Zamir, S. Felman, Appl. Phys. Lett. 58, 1001 (1991).

Appl. Phys. Lett. (3)

H. G. Winful, G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).

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

H. G. Winful, R. Zamir, S. Felman, Appl. Phys. Lett. 58, 1001 (1991).

J. Mod. Opt. (1)

P.St. J. Russell, J. Mod. Opt. 38, 1599 (1991).

Phys. Lett. A (1)

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

Phys. Rev. A (4)

C. M. de Sterke, J. Sipe, Phys. Rev. A 42, 2858 (1990).

C. M. de Sterke, J. E. Sipe, Phys. Rev. A 38, 5149 (1989).

C. M. de Sterke, Phys. Rev. A 45, 2012 (1992),

A stability analysis of finite nonlinear DFB structures was recently presented by C. M. de Sterke, Phys. Rev. A 45, 8252 (1992).

Phys. Rev. Lett. (2)

D. N. Christodoulides, R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).

W. Chen, D. L. Mills, Phys. Rev. Lett. 58, 160 (1987);D. L. Mills, S. E. Trullinger, Phys. Rev. B 36, 6269 (1987).

Teor. Mat. Fiz. (1)

E. A. Kuznetsov, A. V. Mikhailov, Teor. Mat. Fiz. 30, 193 (1977);D. J. Kaup, A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).

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Figures (3)

Fig. 1
Fig. 1

(a) Modulational instability gain G = −2 Im(Γ) and (b) fraction of intensity |bu|2 versus the wave number K = 2π/L, where L is the DFB length and A is the input cw amplitude.

Fig. 2
Fig. 2

Generation of steady breather and pulse trains for the u (top) and υ (bottom) modes. Here L = 4, σ = 0.5, and A = −B = 0.5.

Fig. 3
Fig. 3

Forward field intensities at the output end z = L for L = 4: (a) σ = 0.5, A = −B = 0.6; (b) σ = 0, A = −B = 1.4.

Equations (8)

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E 1 T + ω E 1 Z = i κ exp ( 2 ivZ ) E 2 + i R ( | E 2 | 2 + σ | E 1 | 2 ) E 1 = 0 , E 2 T ω E 2 Z = i κ exp ( 2 ivZ ) E 1 + i R ( | E 1 | 2 + σ | E 2 | 2 ) E 2 = 0 ,
E 1 = u κ / R exp [ i v ( Z ω T ) ] , E 2 = u κ / R exp [ i v ( Z + i ω T ) ] , t = T κ , z = Z κ / ω ,
u t + u z = i υ + i ( | υ | 2 + σ | u | 2 ) u , υ t υ z = i u + i ( | u | 2 + σ | υ | 2 ) υ .
u A exp ( i Ω t ) , υ = B exp ( i Ω t ) , B ± = ± A , Ω ± = 1 ( 1 + σ ) A 2 .
a ( z , t ) = f u ( t ) exp ( iKz ) + b u ( t ) exp ( iKz ) , b ( z , t ) = f υ ( t ) exp ( iKz ) + b υ ( t ) exp ( iKz ) .
Γ 4 2 { K 2 + 2 [ 1 A 2 ( 1 σ ) ] } Γ 2 + K 4 4 A 2 K 2 ( 1 + σ ) = 0.
K < 2 A 1 + σ .
u ( z , t = 0 ) = υ ( z , t = 0 ) A exp ( i Ω _ t ) , u ( z = 0 , t ) = υ ( z = L , t ) = [ A + sin ( Δ t ) ] exp ( i Ω _ t ) .

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