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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Appl. Phys. Lett.

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.

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

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.

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

Opt. Commun.

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

Opt. Lett.

Phys. Rev. Lett.

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

Other

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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