Abstract

We characterize the Hopf bifurcations that rule self-pulsing of spatially varying fields in distributed feedback structures with Kerr nonlinear response. Our analysis allows us to distinguish, inside the stop band, between fully unstable regions and those where stable high transmission mediated by localized waves is achievable dynamically. Outside the stop band, the grating reveals a complex behavior where islands of stability are interspersed between regions of self-pulsing and bistability. Numerical integration of the time-dependent coupled-mode equations validates our linear stability analysis and illustrates the dynamics.

© 2007 Optical Society of America

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  1. R. E. Slusher and B. J. Eggleton, eds., Nonlinear Photonics Crystals (Springer-Verlag, 2003).
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    [CrossRef] [PubMed]
  5. C. J. Herbert, W. S. Capinski, and M. S. Malcuit, "Optical power limiting with nonlinear periodic structures," Opt. Lett. 17, 1037-1039 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
  7. C. M. de Sterke and J. Sipe, "Switching dynamics of finite periodic nonlinear media: a numerical study," Phys. Rev. A 422858-2869 (1990).
    [CrossRef] [PubMed]
  8. C. M. de Sterke, "Stability analysis of nonlinear periodic media," Phys. Rev. A 45, 8252-8258 (1992).
    [CrossRef] [PubMed]
  9. A. B. Aceves, S. Wabnitz, and C. De Angelis, "Generation of solitons in a nonlinear periodic medium," Opt. Lett. 17, 1566-1568 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. S. Pitois, M. Haelterman, and G. Millot, "Theoretical and experimental study of Bragg modulational instability in a dynamic fiber grating," J. Opt. Soc. Am. B 19, 782-791 (2002).
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    [CrossRef] [PubMed]
  14. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
    [CrossRef] [PubMed]
  15. C. M. de Sterke, B. Eggleton, and J. Sipe, in Spatial Solitons, S.Trillo and W.E.Torruellas, eds. (Springer, 2001).
  16. C. Conti and S. Trillo, "Bifurcation of gap solitons through catastrophe theory," Phys. Rev. E 64, 036617 (2001).
    [CrossRef]
  17. J. T. Mok, C. M. De Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Phys. 2, 775-780 (2006).
    [CrossRef]
  18. S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, "From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings," Chaos 10, 590-600 (2000).
    [CrossRef]
  19. H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of bifurcation to chaos in all-optical bistable system," J. Opt. Soc. Am. B 50, 109-112 (1983).
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    [CrossRef]
  21. M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
    [CrossRef]
  22. W. Yang, A. Joshi, and M. Xiao, "Controlling dynamic instability of three-level atoms inside an optical ring cavity," Phys. Rev. A 70, 033807 (2004).
    [CrossRef]
  23. Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
    [CrossRef]
  24. A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, "Instabilities and chaos in the polarization of counterpropagating light fields," Phys. Rev. Lett. 58, 2432-2435 (1987).
    [CrossRef] [PubMed]
  25. M. Conforti, A. Locatelli, C. De Angelis, A. Parini, G. Bellanca, and S. Trillo, "Self-pulsing instabilities in backward parametric wave mixing," J. Opt. Soc. Am. B 22, 2178-2184 (2005).
    [CrossRef]
  26. N. D. Sankey, D. F. Prelewitz, and T. G. Brown, "All-optical switching in a nonlinear periodic-waveguide structure," Appl. Phys. Lett. 60, 1427-1429 (1992).
    [CrossRef]
  27. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  28. V. I. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
    [CrossRef]
  29. A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
    [CrossRef]
  30. J. Danckaert, K. Foebelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, "Dispersive optical bistability in stratified structures," Phys. Rev. B 44, 8214-8225 (1991).
    [CrossRef]
  31. S. Dutta Gupta, "Nonlinear optics of stratified media," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
    [CrossRef]
  32. T. P. Valkering and S. A. Van Gils, "Geometrical approach to stationary waves in a shallow grating," Opt. Quantum Electron. 35959-966 (2003).
    [CrossRef]

2006 (1)

J. T. Mok, C. M. De Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Phys. 2, 775-780 (2006).
[CrossRef]

2005 (1)

2004 (2)

W. Yang, A. Joshi, and M. Xiao, "Controlling dynamic instability of three-level atoms inside an optical ring cavity," Phys. Rev. A 70, 033807 (2004).
[CrossRef]

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

2003 (1)

T. P. Valkering and S. A. Van Gils, "Geometrical approach to stationary waves in a shallow grating," Opt. Quantum Electron. 35959-966 (2003).
[CrossRef]

2002 (3)

2001 (1)

C. Conti and S. Trillo, "Bifurcation of gap solitons through catastrophe theory," Phys. Rev. E 64, 036617 (2001).
[CrossRef]

2000 (1)

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, "From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings," Chaos 10, 590-600 (2000).
[CrossRef]

1998 (3)

V. I. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

C. M. de Sterke, "Theory of modulational instability in fiber Bragg gratings," J. Opt. Soc. Am. B 15, 2660-2667 (1998).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1995 (1)

1992 (4)

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, "All-optical switching in a nonlinear periodic-waveguide structure," Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

C. J. Herbert, W. S. Capinski, and M. S. Malcuit, "Optical power limiting with nonlinear periodic structures," Opt. Lett. 17, 1037-1039 (1992).
[CrossRef] [PubMed]

A. B. Aceves, S. Wabnitz, and C. De Angelis, "Generation of solitons in a nonlinear periodic medium," Opt. Lett. 17, 1566-1568 (1992).
[CrossRef] [PubMed]

C. M. de Sterke, "Stability analysis of nonlinear periodic media," Phys. Rev. A 45, 8252-8258 (1992).
[CrossRef] [PubMed]

1991 (1)

J. Danckaert, K. Foebelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, "Dispersive optical bistability in stratified structures," Phys. Rev. B 44, 8214-8225 (1991).
[CrossRef]

1990 (1)

C. M. de Sterke and J. Sipe, "Switching dynamics of finite periodic nonlinear media: a numerical study," Phys. Rev. A 422858-2869 (1990).
[CrossRef] [PubMed]

1987 (3)

W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, "Instabilities and chaos in the polarization of counterpropagating light fields," Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

A. Mecozzi, S. Trillo, and S. Wabnitz, "Spatial instability, all-optical limiting and thresholding in nonlinear distributed feedback devices," Opt. Lett. 12, 1008-1010 (1987).
[CrossRef] [PubMed]

1983 (1)

H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of bifurcation to chaos in all-optical bistable system," J. Opt. Soc. Am. B 50, 109-112 (1983).

1982 (2)

Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
[CrossRef]

H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
[CrossRef]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

Appl. Phys. Lett. (3)

H. G. Winful, J. H. Marburger, and E. Garmire, "Theory of bistability in nonlinear distributed feedback structures," Appl. Phys. Lett. 35, 379-381 (1979).
[CrossRef]

H. G. Winful and G. D. Cooperman, "Self-pulsing and chaos in distributed feedback bistable optical devices," Appl. Phys. Lett. 40, 298-300 (1982).
[CrossRef]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, "All-optical switching in a nonlinear periodic-waveguide structure," Appl. Phys. Lett. 60, 1427-1429 (1992).
[CrossRef]

Chaos (1)

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, "From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings," Chaos 10, 590-600 (2000).
[CrossRef]

J. Opt. Soc. Am. B (6)

Nat. Phys. (1)

J. T. Mok, C. M. De Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nat. Phys. 2, 775-780 (2006).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

T. P. Valkering and S. A. Van Gils, "Geometrical approach to stationary waves in a shallow grating," Opt. Quantum Electron. 35959-966 (2003).
[CrossRef]

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Phys. Rev. A (4)

M. Bache, P. Lodhal, A. V. Mamaev, M. Marcus, and M. Saffman, "Observation of self-pulsing in singly resonant optical second-harmonic generation with competing nonlinearities," Phys. Rev. A 65, 033811 (2002).
[CrossRef]

W. Yang, A. Joshi, and M. Xiao, "Controlling dynamic instability of three-level atoms inside an optical ring cavity," Phys. Rev. A 70, 033807 (2004).
[CrossRef]

C. M. de Sterke and J. Sipe, "Switching dynamics of finite periodic nonlinear media: a numerical study," Phys. Rev. A 422858-2869 (1990).
[CrossRef] [PubMed]

C. M. de Sterke, "Stability analysis of nonlinear periodic media," Phys. Rev. A 45, 8252-8258 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (1)

J. Danckaert, K. Foebelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, "Dispersive optical bistability in stratified structures," Phys. Rev. B 44, 8214-8225 (1991).
[CrossRef]

Phys. Rev. E (1)

C. Conti and S. Trillo, "Bifurcation of gap solitons through catastrophe theory," Phys. Rev. E 64, 036617 (2001).
[CrossRef]

Phys. Rev. Lett. (6)

V. I. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical response of superlattices," Phys. Rev. Lett. 58, 160-163 (1987).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Y. Silberberg and I. Bar-Joseph, "Instabilities, self-oscillation and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541-1544 (1982).
[CrossRef]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, and P. W. Milonni, "Instabilities and chaos in the polarization of counterpropagating light fields," Phys. Rev. Lett. 58, 2432-2435 (1987).
[CrossRef] [PubMed]

Other (4)

C. M. de Sterke, B. Eggleton, and J. Sipe, in Spatial Solitons, S.Trillo and W.E.Torruellas, eds. (Springer, 2001).

S. Dutta Gupta, "Nonlinear optics of stratified media," in Progress in Optics, Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
[CrossRef]

R. E. Slusher and B. J. Eggleton, eds., Nonlinear Photonics Crystals (Springer-Verlag, 2003).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

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Figures (11)

Fig. 1
Fig. 1

Hopf bifurcation: the leading pair of complex-conjugate eigenvalues λ ± = λ r ± i λ i crosses the plane λ r = 0 as the control parameter P is increased. Here δ β = 1 and κ = 2.5 .

Fig. 2
Fig. 2

(a) Temporal evolution of transmitted ( P t ) and reflected ( P r ) powers of a SP-unstable solution, for δ β = 1 , κ = 2.5 , P = 0.7 ; (b) phase-space portrait.

Fig. 3
Fig. 3

(a) Phase-space evolution for initial conditions sitting on the negative-slope branch of the bistable response (here δ β = 1 , κ = 2.5 , P = 0.25 ). (b) Corresponding temporal evolutions of transmitted and reflected powers.

Fig. 4
Fig. 4

Stability map in the parameter plane δ β P (detuning–transmitted power), for κ = 2.5 (gap edges δ β = ± 2.5 shown as vertical lines). The white and hatched (cyan) areas correspond to temporally stable and unstable regions, respectively. Bistable and SP domains merge at critical detuning δ β c highlighted by a solid square. P = P c ( δ β ) [solid (red) curve] gives the level of the transmission plateau for δ β < κ .

Fig. 5
Fig. 5

(a) Steady-state response P versus P i n for grating parameters δ β = 1.75 and κ = 2.5 . The solid curve indicates stable portions, whereas unstable portions are shown as dashed (SP, complex-conjugate eigenvalues), or dotted–dashed (bistability, real eigenvalues) curves. (b) Real (solid curve) and imaginary (dashed curve) part of the leading eigenvalues versus P. Points A, B, and C denote bifurcation points where eigenvalues with positive real parts appear (A,C) or disappear (B). P i n ( 1 ) and P i n ( 2 ) (transparency point) are power levels employed in the dynamical simulation of Fig. 6.

Fig. 6
Fig. 6

(a) Reflected power P t ( t ) versus time t (solid curve) in response to the excitation P i n ( t ) (dashed curve); (b) transmitted power P r ( t ) , with inset showing a detail of the damped oscillations, (c), (d) spatial profiles ( L Λ = 50 ) of total power u ( z ) 2 after reaching steady state: (c) t = 600 , transparency point; (d) t = 1000 , knee point B. The parameters are as in Fig. 5.

Fig. 7
Fig. 7

(a) Steady-state transmission characteristic: P versus P i n for δ β = 7 , κ = 2.5 . (b) Stable temporal evolution of P t and P r corresponding to an input power raised monotonically up to the value P i n ( 1 ) in the stable domain. (c) Onset of SP when the input power is raised up to the value P i n ( 2 ) in the unstable domain.

Fig. 8
Fig. 8

Steady-state transmission obtained by means of TMM [dashed (red) curve] and CMT [Eq. (7), solid (blue) curve]. TMM results are relative to a grating with Λ = 250 nm , n 0 = 2 , Δ n p = 0.2 , n 2 I = 10 20 m 2 W , and all other parameters such to fit the normalized parameters ( δ β = 7 , κ = 5 ) and powers used in Eq. (7).

Fig. 9
Fig. 9

Bifurcation structure ( κ = 2.5 ) : equilibrium points η e as a function of (a) detuning δ β for P = 0 (thin solid and dashed lines) and P = 1 (thick solid and dashed curves), (b) power P for a fixed detuning δ β = 8 . Solid and dashed lines (curves) stand for stable points (centers) and unstable points (saddles), respectively. Physical solutions are those contained in the semiplane η e 0 ( η e is a power).

Fig. 10
Fig. 10

Bifurcation and clamping powers P b i f , P c (see text for definitions) versus detuning δ β ( κ = 2.5 ) .

Fig. 11
Fig. 11

Steady-state spatial evolutions of powers (a), (c), (e) and corresponding phase-space pictures (b), (d), (f). Here δ β = 7 , κ = 2.5 , and P = 1 (a), (b), P = 1.22 (c), (d), P = 1.28 (e), (f). The trajectories followed in phase space are highlighted in bold (blue).

Equations (26)

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i ( u + t + u + z ) + δ β u + + κ u + ( u + 2 + 2 u 2 ) u + = 0 ,
i ( u t u z ) + δ β u + κ u + + ( u 2 + 2 u + 2 ) u = 0 ,
δ β = ( β ( ω ) m β g 2 ) L ,
κ = Γ m L = k 0 Δ n p c m L ,
u = u + ( z , t ) exp ( i β g L 2 z ) + u ( z , t ) exp ( i β g L 2 z ) .
u + ( z = 0 ) = P i n + , u ( z L = 1 ) = P i n exp ( i ϕ 0 ) .
u ± ( z , t ) = a ± ( z ) + p ± ( z , t ) ,
i a ± z = δ β a ± + κ a + ( a ± 2 + 2 a 2 ) a ± ,
p ± t ± p ± z = i δ β p ± + i κ p + i 2 ( a ± 2 + a 2 ) p ± + i ( a ± 2 p ± * + 2 a ± a p * + 2 a ± a * p ) .
p ̇ ( t ) = M p ( t ) ,
t SP = 2 π λ i ,
P = a + ( z ) 2 a ( z ) 2 ,
M 11 = A B 2 R + I + , M 22 = A B + 2 R + I + ,
M 12 = δ β I N R + 2 3 I + 2 2 R 2 2 I 2 ,
M 13 = 4 R I + , M 31 = 4 R + I ,
M 14 = κ I N 4 I I + , M 41 = κ I N 4 R + R ,
M 21 = δ β I N 2 R 2 + 2 I 2 + I + 2 + 3 R + 2 ,
M 23 = κ I N + 4 R R + , M 32 = κ I N 4 I + I ,
M 24 = 4 I R + , M 42 = 4 I + R ,
M 33 = A F 2 I R , M 44 = A F + 2 I R ,
M 34 = δ β I N R 2 3 I 2 2 R + 2 ,
M 43 = δ β I N + I 2 + 2 R + 2 + 2 I + 2 + 3 R 2 ,
i d a ± d z = H a ± * ,
d η d z = H r ϕ , d ϕ d z = H r η ,
H r = 2 δ β η + 2 κ η ( η + P ) cos ϕ + 3 η ( η + P ) ,
z z L = 0 η ( z ) d η f ( η ) ,

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