Abstract

Based on the nonlinearly coupled mode equations (NLCME), the bistable steady characteristics and dynamic stability of linearly tapered nonlinear Bragg grating (LT-NLBG) have been investigated in detail. The results show that, when the device is tuned near an edge of “photonic band gap” (PBG), in contrast with the free-tapered grating, the negative-tapered grating enhances the switching-on threshold, but increases the on-off switching ratio, enlarges the stable regime, and strengthens the stability significantly. On the other hand, the positive-tapered grating decreases the switching-on threshold, but lower the on-off switching ratio, and worsen the stability remarkably.

© 2004 Optical Society of America

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References

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  1. D. N. Christodoulides and R. I. Joseph, “Slow bragg solitons in Nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [Crossref] [PubMed]
  2. B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber and Integrated Opt. 19, 383–421 (2000).
    [Crossref]
  3. N. G. R. Broderick, P. Millar, D. J. Richardson, J. S. Aitchson, R. De La Rue, and T. Krauss, “Spectral features associated with nonlinear pulse compression in Bragg gratings,” Opt. Lett. 25, 740–742 (2000).
    [Crossref]
  4. 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]
  5. N. G. R. Broderick, D. Taverner, and D. J. Richardson, “Nonlinear switching in fibre Bragg gratings,” Opt. Express 3, 447–453 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-11-447
    [Crossref] [PubMed]
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    [Crossref]
  7. L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
    [Crossref]
  8. G. P. Agrawal and S. Radic, “Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
    [Crossref]
  9. S. Radic, N. George, and G. P. Agrawal, “Optical switching in π/4-shifted nonlinear periodic structures,” Opt. Lett. 19, 1789–1791 (1994).
    [Crossref] [PubMed]
  10. S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
    [Crossref]
  11. S. Radic, N. George, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: Generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
    [Crossref]
  12. Y. A. Logvin and V. M. Volkov, “Phase sensitivity of a nonlinear Bragg grating response under bidirectional illumination,” J. Opt. Soc. Am. B 16, 774–780 (1999).
    [Crossref]
  13. J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
    [Crossref]
  14. H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003).
    [Crossref]
  15. F. Marquis, P. Dobiasch, P. Meystre, and E. M. Wright, “Slaved bistability and self-pulsing in a nonlinear interferometer,” J. Opt. Soc. Am. B 3, 50–59 (1986).
    [Crossref]
  16. D. Pelinovsky and E. H. Sargent, “Stable all-optical limiting in nonlinear periodic structures. II. Computations,” J. Opt. Soc. Am. B 19, 1873–1889 (2002).
    [Crossref]
  17. R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
    [Crossref]
  18. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).

2003 (1)

H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003).
[Crossref]

2002 (1)

2000 (3)

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber and Integrated Opt. 19, 383–421 (2000).
[Crossref]

N. G. R. Broderick, P. Millar, D. J. Richardson, J. S. Aitchson, R. De La Rue, and T. Krauss, “Spectral features associated with nonlinear pulse compression in Bragg gratings,” Opt. Lett. 25, 740–742 (2000).
[Crossref]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[Crossref]

1999 (1)

1998 (1)

1996 (1)

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
[Crossref]

1995 (2)

S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: Generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[Crossref]

1994 (2)

G. P. Agrawal and S. Radic, “Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Optical switching in π/4-shifted nonlinear periodic structures,” Opt. Lett. 19, 1789–1791 (1994).
[Crossref] [PubMed]

1991 (1)

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

1989 (1)

D. N. Christodoulides and R. I. Joseph, “Slow bragg solitons in Nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

1986 (1)

1985 (1)

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]

Agrawal, G. P.

H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: Generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Optical switching in π/4-shifted nonlinear periodic structures,” Opt. Lett. 19, 1789–1791 (1994).
[Crossref] [PubMed]

G. P. Agrawal and S. Radic, “Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).

Aitchson, J. S.

Betts, R. A.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Broderick, N. G. R.

Brzozowski, L.

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[Crossref]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow bragg solitons in Nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Chu, P. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

De La Rue, R.

Dobiasch, P.

Eggleton, B. J.

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber and Integrated Opt. 19, 383–421 (2000).
[Crossref]

Garmire, E.

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]

George, N.

Gibbs, H. M.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow bragg solitons in Nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Krauss, T.

Lee, H.

H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003).
[Crossref]

Lenz, G.

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber and Integrated Opt. 19, 383–421 (2000).
[Crossref]

Liao, C. J.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
[Crossref]

Litchinitser, N. M.

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber and Integrated Opt. 19, 383–421 (2000).
[Crossref]

Liu, J. M.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
[Crossref]

Liu, S. H.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
[Crossref]

Logvin, Y. A.

Marburger, J. H.

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]

Marquis, F.

Meystre, P.

Millar, P.

Pelinovsky, D.

Peyghambarian, N.

Radic, S.

S. Radic, N. George, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: Generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear, phase-shifted, periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Optical switching in π/4-shifted nonlinear periodic structures,” Opt. Lett. 19, 1789–1791 (1994).
[Crossref] [PubMed]

G. P. Agrawal and S. Radic, “Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

Richardson, D. J.

Sargent, E. H.

D. Pelinovsky and E. H. Sargent, “Stable all-optical limiting in nonlinear periodic structures. II. Computations,” J. Opt. Soc. Am. B 19, 1873–1889 (2002).
[Crossref]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[Crossref]

Taverner, D.

Tjugiarto, T.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Volkov, V. M.

Winful, H. G.

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]

Wright, E. M.

Xu, W. C.

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
[Crossref]

Xue, Y. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Appl. Phys. Lett. (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]

Fiber and Integrated Opt. (1)

B. J. Eggleton, G. Lenz, and N. M. Litchinitser, “Optical pulse compression schemes that use nonlinear Bragg gratings,” Fiber and Integrated Opt. 19, 383–421 (2000).
[Crossref]

IEEE J. Quantum Electron. (4)

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[Crossref]

S. Radic, N. George, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: Generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[Crossref]

H. Lee and G. P. Agrawal, “Nonlinear switching of optical pulses in fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 508–515 (2003).
[Crossref]

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: Theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

IEEE Photon. Technol. Lett. (1)

G. P. Agrawal and S. Radic, “Phased-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

J. M. Liu, C. J. Liao, S. H. Liu, and W. C. Xu, “The dynamics of direction-dependent switching in nonlinear chirped gratings,” Opt. Commun. 130, 295–301 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

D. N. Christodoulides and R. I. Joseph, “Slow bragg solitons in Nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

Steady-state transmission spectrum of LT-NLBG for different Δκ in linear case.

Fig. 2.
Fig. 2.

Steady-state input-output characteristics of LT-NLBG for three different Δκ with δL=2.5.

Fig. 3.
Fig. 3.

Axial distribution of forward wave intensity with δL=2.5, where figures (a) and (b) are for Δκ=-1 and Δκ=1, respectively.

Fig. 4.
Fig. 4.

Dependence of αm on transmitted intensity for three values of taper parameter Δκ with δL=2.5.

Fig. 5.
Fig. 5.

Steady-state input-output characteristics of LT-NLBG for three different Δκ with δL=0.

Fig. 6.
Fig. 6.

Mapping of the stability boundary for three different Δκ.

Equations (33)

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n ( z ) = n 0 + n 1 ( z ) cos [ 2 π Λ z + ϕ ( z ) ] + n 2 E ( z ) 2
E = A f exp [ i ( β 0 z ϖ t ) ] + A b exp [ i ( β 0 z + ϖ t ) ]
A f z + 1 v g A f t = i [ δ A f + Γ ( A f 2 + 2 A b 2 ) A f + κ A b ]
A b z 1 v g A b t = i [ δ A b + Γ ( A b 2 + 2 A f 2 ) A b + κ * A f ]
δ = β β 0 = n 0 ϖ c β 0 , Γ = 2 π n 2 λ 0 , κ ( z ) = π n 1 ( z ) η λ 0 exp [ i ϕ ( z ) ]
κ ( z ) = κ 0 [ 1 + Δ κ ( z L 2 ) L ]
z = 0 : A f ( 0 , t ) = A i ( 0 , t ) , A r ( 0 , t ) = A b ( 0 , t )
z = L : A b ( L , t ) = 0 , A r ( L , t ) = A f ( L , t )
A f = u + iw , A b = v + iy
A f ( z , t ) = [ u 0 ( z ) + g u ( z , t ) ] + i [ w 0 ( z ) + g w ( z , t ) ]
A b ( z , t ) = [ v 0 ( z ) + g v ( z , t ) ] + i [ y 0 ( z ) + g y ( z , t ) ]
g u z + 1 v g g u t = A 1 g u + B 1 g w + C 1 g v + D 1 g y
g w z + 1 v g g w t = A 2 g u + B 2 g w + C 2 g v + D 2 g y
g v z 1 v g g v t = A 3 g u + B 3 g w + C 3 g v + D 3 g y
g y z 1 v g g y t = A 4 g u + B 4 g w + C 4 g v + D 4 g y
A 1 = f 1 ( u 0 , w 0 ) , B 1 = f 2 ( w 0 , u 0 , v 0 , y 0 ) ,
C 1 = 2 f 1 ( w 0 , v 0 ) , D 1 = 2 f 1 ( w 0 , y 0 ) κ
A 2 = f 2 ( u 0 , w 0 , v 0 , y 0 ) , B 2 = A 1 ,
C 2 = 2 f 1 ( u 0 , v 0 ) + κ , D 2 = 2 f 1 ( u 0 , y 0 )
A 3 = D 2 , B 3 = 2 f 1 ( w 0 , y 0 ) + κ * ,
C 3 = f 1 ( v 0 , y 0 ) , D 3 = f 2 ( y 0 , v 0 , u 0 , w 0 )
A 4 = 2 f 1 ( u 0 , v 0 ) κ * , B 4 = C 1 ,
C 4 = f 2 ( v 0 , y 0 , u 0 , w 0 ) , D 4 = C 3
f 1 ( ξ , η ) = 2 Γ ξ η
f 2 ( ξ , η , ζ , χ ) = δ + Γ ( 3 ξ 2 + η 2 + 2 ζ 2 + 2 χ 2 )
g u = u 1 ( z ) exp ( st ) , g w = w 1 ( z ) exp ( st ) ,
g v = v 1 ( z ) exp ( st ) , g y = y 1 ( z ) exp ( st )
d u 1 dz ( A 1 u 1 + B 1 w 1 + C 1 v 1 + D 1 y 1 ) = s v g u 1
d w 1 dz ( A 2 u 1 + B 2 w 1 + C 2 v 1 + D 2 y 1 ) = s v g w 1
d v 1 dz + ( A 3 u 1 + B 3 w 1 + C 3 v 1 + D 3 y 1 ) = s v g v 1
d y 1 dz + ( A 4 u 1 + B 4 w 1 + C 4 v 1 + D 4 y 1 ) = s v g v 1
g u ( 0 , t ) = g w ( 0 , t ) = g v ( L , t ) = g y ( L , t ) = 0
u 1 ( 0 ) = w 1 ( 0 ) = v 1 ( L ) = y 1 ( L ) = 0

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