Abstract

Transmission of coherent light through photonic gratings with varying Kerr nonlinearity is modeled within a coupled-mode system derived from the Maxwell equations. The incident light waves are uniformly stable in time-dependent dynamics if the photonic grating has zero net-average Kerr nonlinearity. When the average nonlinearity is weak but nonzero, light waves exhibit oscillatory instabilities and long-term high-amplitude oscillations in the out-of-phase linear gratings. We show that a two-step transmission map between lower-transmissive and higher-transmissive states has a narrow stability domain, which limits its applicability for logic and switching functions. Light waves exhibit cascades of real and complex instabilities in the multistable gratings with strong net-average Kerr nonlinearity. Only the first lower-transmissive stationary state can be stimulated by the incident light of small intensities. Light waves of moderate and large intensities are essentially nonstationary in the multistable gratings, and they exhibit periodic generation of Bragg solitons and blowup.

© 2002 Optical Society of America

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References

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  1. D. Pelinovsky, J. Sears, L. Brzozowski, and E. H. Sargent, “Stable all-optical limiting in nonlinear periodic structures. I. Analysis,” J. Opt. Soc. Am. B 19, 43–53 (2002).
    [CrossRef]
  2. L. Brzozowski and E. H. Sargent, “Nonlinear distributed feedback structures as passive optical limiters,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
    [CrossRef]
  3. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
    [CrossRef]
  4. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” Appl. Phys. Lett. 35, 379–381 (1979).
    [CrossRef]
  5. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).
  6. W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
    [CrossRef]
  7. J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
    [CrossRef]
  8. C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
    [CrossRef] [PubMed]
  9. Yu. N. Ovchinnikov, “Stability problem in nonlinear wave propagation,” JETP 87, 807–813 (1998).
    [CrossRef]
  10. 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]
  11. C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
    [CrossRef] [PubMed]
  12. G. P. Agrawal, Nonlinear Fiber Optic (Academic, San Diego, 1989), Chap. 7.
  13. C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
    [CrossRef]

2002 (1)

2000 (1)

1998 (1)

Yu. N. Ovchinnikov, “Stability problem in nonlinear wave propagation,” JETP 87, 807–813 (1998).
[CrossRef]

1994 (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

1992 (1)

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

1991 (2)

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
[CrossRef]

1990 (1)

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

1987 (1)

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[CrossRef]

1982 (1)

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 structure,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Brzozowski, L.

Cada, M.

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[CrossRef]

Cooperman, G. D.

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]

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

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

C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Garmire, E.

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

He, J.

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

Jackson, K. R.

Marburger, J. H.

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

Mills, D. L.

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[CrossRef]

Ovchinnikov, Yu. N.

Yu. N. Ovchinnikov, “Stability problem in nonlinear wave propagation,” JETP 87, 807–813 (1998).
[CrossRef]

Pelinovsky, D.

Robert, B. D.

Sargent, E. H.

Sears, J.

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Winful, H. G.

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]

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

Appl. Phys. Lett. (2)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” 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]

IEEE J. Quantum Electron. (1)

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

J. Opt. Soc. Am. B (3)

JETP (1)

Yu. N. Ovchinnikov, “Stability problem in nonlinear wave propagation,” JETP 87, 807–813 (1998).
[CrossRef]

Phys. Rev. A (2)

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

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Phys. Rev. B (1)

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[CrossRef]

Prog. Opt. (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Other (2)

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, New York, 1985).

G. P. Agrawal, Nonlinear Fiber Optic (Academic, San Diego, 1989), Chap. 7.

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Figures (14)

Fig. 1
Fig. 1

Input–output transmission curve for the nonlinear periodic optical structure with parameters (Ia) nnl=1, n0k=0, (Ib) nnl=0, n0k=-0.15, (II) nnl=1, n0k=-0.15, and (III) nnl=1.4, n0k=0. The other material parameters are standardized as L=20 and n2k=1. The photonic structure is in the stable all-optical limiting regime for curves Ia and Ib, the locally multistable limiting regime for curve II, and the multistable regime for curve III.

Fig. 2
Fig. 2

Existence domains I, II, and III of the three stationary regimes of photonic gratings: (i) stable all-optical limiting, (ii) locally multistable limiting, and (iii) multistability on the plane (n0k, nnl). The other material parameters are standardized as L=20 and n2k=1. The particular parameter values from Fig. 1 are marked by stars.

Fig. 3
Fig. 3

Intensities of (a) the forward and (b) backward waves in different time instances: t=12.9 and t=64.4. The stationary distribution is shown by a dotted curve for Iout=0.04. The standardized photonic structure has the material parameters nnl=1 and n0k=0 (see curve Ia in Fig. 1). The perturbation of the stationary transmission disappears by means of radiation through the ends of the structure.

Fig. 4
Fig. 4

Intensities of (a) the forward and (b) backward waves in different time instances: t=11.1, t=44.5, and t=88.9. The initial (almost stationary) distribution at t=0 is shown by a dotted curve for Iout=0.12. The standardized photonic structure has the material parameters nnl=1 and n0k=-0.15 (see curve II in Fig. 1). The stationary regime is unstable, and the light transmission exhibits high-amplitude oscillations. The transmitted intensity Iout as a function of time t is shown in (c).

Fig. 5
Fig. 5

Intensities of (a) the forward and (b) backward waves in different time instances: t=10.1 and t=50.5. The initial unstable stationary regime at t=0 is shown by a dashed curve for Iout=0.09. The final stable stationary regime is shown by a dotted curve for Iout=0.062. Both the regimes are stimulated by the same incident intensity, Iin=0.135. The standardized photonic structure has the material parameters nnl=1.4 and n0k=0 (see curve III in Fig. 1). The time-dependent dynamics of the periodic structure results in switching from an unstable highly transmissive state to a stable lower-transmissive state.

Fig. 6
Fig. 6

Spectrum of eigenvalues for Iout=0, L=20, and n0k=-0.15. (a) The eigenvalues are found from the analytical solution (30) and (31). (b) The eigenvalues are found from the analytical solution (32) and (33). (c) The eigenvalues are found numerically from the linear matrix problem (28).

Fig. 7
Fig. 7

Unstable (a) real and (b) complex eigenvalues versus the transmitted intensity Iout of the linear matrix problem (28) for the standardized photonic structure with material parameters nnl=1 and n0k=-0.1. The input–output transmission characteristics Iout(Iin) is shown in (c).

Fig. 8
Fig. 8

Unstable (a) real and (b) complex eigenvalues versus the transmitted intensity Iout of the linear matrix problem (28) for the standardized photonic structure with material parameters nnl=1 and n0k=-0.15. The input–output transmission characteristics Iout(Iin) is shown in (c), and it corresponds to curve II on Fig. 1.

Fig. 9
Fig. 9

Unstable (a) real and (b) complex eigenvalues versus the transmitted intensity Iout of the linear matrix problem (28) for the standardized photonic structure with material parameters nnl=1 and n0k=-0.3. The input–output transmission characteristics Iout(Iin) is shown in (c).

Fig. 10
Fig. 10

Unstable (a) real and (b) complex eigenvalues versus the transmitted intensity Iout of the linear matrix problem (28) for the standardized photonic structure with material parameters nnl=1.4 and n0k=0. The input–output transmission characteristics Iout(Iin) is shown in (c), and it corresponds to curve III in Fig. 1.

Fig. 11
Fig. 11

Unstable (a) real and (b) complex eigenvalues versus the transmitted intensity Iout of the linear matrix problem (28) for the standardized photonic structure with material parameters nnl=1.4 and n0k=-0.1. The input–output transmission characteristics Iout(Iin) is shown in (c).

Fig. 12
Fig. 12

Transmitted [Iout(t)] and reflected [Iref (t)] intensities as functions of time t. The incident intensity Iin(t) is given by Eqs. (34) for (a) I0=0.02 and (b) I0=0.2. The standardized photonic structure has the material parameters nnl=1 and n0k=0. The dotted curves display constant values of Iout and Iref for stationary transmission that correspond to the incident intensity Iin=I0.

Fig. 13
Fig. 13

Transmitted [Iout(t)] and reflected [Iref(t)] intensities as functions of time t. The incident intensity Iin(t) is given by Eqs. (34) for (a) I0=0.030 and (b) I0=0.075. The standardized photonic structure has the material parameters nnl=1 and n0k=-0.15. The dotted curves display constant values of stationary intensities Iout and Iref, respectively, that correspond to the incident intensity Iin=I0.

Fig. 14
Fig. 14

(a) Transmitted [Iout(t)] and reflected [Iref(t)] intensities as functions of time t. (b),(c) The intensities of the forward wave |A+|2 and the backward wave |A-|2 across the photonic device at the time instances t=230, t=250, and t=260. The incident intensity Iin(t) is given by Eqs. (34) for I0=0.2. The standardized photonic structure has the material parameters nnl=1.4 and n0k=0. The dotted curves display stationary intensities and distributions that correspond to the incident intensity Iin=0.2.

Equations (88)

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iA+z+A+t+n0kA-+nnl(|A+|2+2|A-|2)A+
+n2k[(2|A+|2+|A-|2)A-+A+2A¯-]=0,
-iA-z-A-t+n0kA++nnl(2|A+|2+|A-|2)A-
+n2k[(|A+|2+2|A-|2)A++A-2A¯+]=0,
Iin=|A+|2(0),Iref=|A-|2(0),Iout=|A+|2(L),
n2k34|nnl|.
n0k-πn2k3|nnl|L.
A+(0, t)=Iin exp(iθin),A-(L, t)=0,
A+=u+iw,A-=v+iy.
ut+uz+n0ky+fu=0,
-yt+yz+n0ku+fy=0,
-wt-wz+n0kv+fw=0,
vt-vz+n0kw+fv=0,
fu=f(u, w, v, y),fw=f(w, u, y, v),
fv=f(v, y, u, w),fy=f(y, v, w, u),
f(u, w, v, y)=nnl(u2+w2+2v2+2y2)w+n2k[(u2+3w2+v2+y2)y+2uwv].
2ut2-2uz2+n0k2u=0.
z=zn=nh,n=0,1 ,, N,(N+1),
uk=u1,ku2,k...uN,kuN+1,k,wk=w1,kw2,k...wN,kwN+1,k,
vk=v0,kv1,k...vN-1,kvN,k,yk=y0,ky1,k...yN-1,kyN,k.
u0,k=Iin(tk) cos θin(tk),
w0,k=Iin(tk) sin θin(tk),
vN+1,k=0,yN+1,k=0.
un,k+1-2un,k+un,k-1τ2+n0k2(un,k+1+un,k-1)2
-un+1,k+1-2un,k+1+un-1,k+12h2-un+1,k-1-2un,k-1+un-1,k-12h2=0.
A(r)κI+-κI-B(r)uk+1yk+1=A(-r)-κI+κI-B(-r)uk-1yk-1+HukHyk,
A(r)-κI+κI-B(r)wk+1vk+1=A(-r)κI+-κI-B(-r)wk-1vk-1+HwkHvk,
r=τ2h,κ=τn0k,
τ=0.5hhM<0.0020.001M-1hM>0.002,
A+(z, 0)=A+0(z),
A-(z, 0)=A-0(z)[1+α exp(-z)],
u=u0(z)+u1(z)exp(λt),
w=w0(z)+w1(z)exp(λt),
v=v0(z)+v1(z)exp(λt),
y=y0(z)+y1(z)exp(λt).
u1(0)=w1(0)=v1(L)=y1(L)=0.
-u1z-n0ky1-fu=λu1,y1z+n0ku1+fy=λy1,
-w1z+n0kv1+fw=λw1,v1z-n0kw1-fv=λv1,
f=nnl[2u0w0u1+(u02+3w02+2v02+2y02)w1+4w0v0v1+4w0y0y1]+n2k[2(u0y0+w0v0)u1+2(3w0y0+u0v0)w1+2(u0w0+v0y0)v1+(u02+3w02+v02+y02)y1].
A-κI+00κI-B0000AκI+00-κI-Buywv+Uuywv=γuywv,
κ=2hn0k,γ=2hλ,
u1=n0k sin kz,y1=-λ sin kz-k cos kz,
1+n0k2 sin2(kL)k2=0.
u1=n0k sin(knh),
y1=-λ sin(knh)-sin(kh)h cos knh,
1+(n0kh)2 sin2(kL)sin2(kh)=0.
Iin(t)=I0 tanh t,θin(t)=0.
un,k+1+r(un+1,k+1-un-1,k+1)+κyn,k+1
=un,k-1-r(un+1,k-1-un-1,k-1)-κyn,k-1-Fun,k,
wn,k+1+r(wn+1,k+1-wn-1,k+1)-κvn,k+1
=wn,k-1-r(wn+1,k-1-wn-1,k-1)+κvn,k-1+Fwn,k,
vn,k+1-r(vn+1,k+1-vn-1,k+1)+κwn,k+1
=vn,k-1+r(vn+1,k-1-vn-1,k-1)-κwn,k-1-Fvn,k,
yn,k+1-r(yn+1,k+1-yn-1,k+1)-κun,k+1
=yn,k-1+r(yn+1,k-1-yn-1,k-1)+κun,k-1+Fyn,k,
r=τ2h,κ=τn0k,
Fun,k=2τfu(un,k, wn,k, vn,k, yn,k).
uN+1,k+1+r(3uN+1,k+1-4uN,k+1+uN-1,k+1)
=uN+1,k-1-r(3uN+1,k-1-4uN,k-1+uN-1,k-1)
-FuN+1,k,
wN+1,k+1+r(3wN+1,k+1-4wN,k+1+wN-1,k+1)
=wN+1,k-1-r(3wN+1,k-1-4wN,k-1+wN-1,k-1)
+FwN+1,k,
v0,k+1-r(-v2,k+1+4v1,k+1-3v0,k+1)
=v0,k-1+r(-v2,k-1+4v1,k-1-3v0,k-1)-κ(w0,k+1+w0,k-1)-Fv0,k,
y0,k+1-r(-y2,k+1+4y1,k+1-3y0,k+1)
=y0,k-1+r(-y2,k-1+4y1,k-1-3y0,k-1)+κ(u0,k+1+u0,k-1)+Fy0,k,
A(r)=1r0...000-r1r...0000-r1...000.........000...-r1r000...r-4r1+3r,
B(r)=1+3r-4rr...000r1-r...0000r1...000.........000...r1-r000...0r1,
I+=010...000001...000.........000...001000...000,
I-=000...000100...000.........000...100000...010.
Huk=r(u0,k+1+u0,k-1)e1-Fuk,
Hwk=r(w0,k+1+w0,k-1)e1+Fwk,
Hvk=-κ(w0,k+1+w0,k-1)e1-Fvk,
Hyk=κ(u0,k+1+u0,k-1)e1+Fyk,
-(un+1-un-1)-κyn-Fun=γun,
-(wn+1-wn-1)+κvn+Fwn=γwn,
(vn+1-vn-1)-κwn-Fvn=γvn,
(yn+1-yn-1)+κun+Fyn=γyn,
κ=2hn0k,γ=2hλ,
Fun=2hfu(u0n, w0n, v0n, y0n;un, wn, vn, yn).
-(3uN+1-4uN+uN-1)-FuN+1=γuN+1,
-(3wN+1-4wN+wN-1)+FwN+1=γwN+1,
(-3v0+4v1-v2)-Fv0=γv0,
(-3y0+4y1-y2)+Fy0=γy0,
A=0-10...00010-1...000010...000.........000...10-1000...-14-3,
B=-34-1...000-101...0000-10...000.........000...-101000...0-10.
U=-FuFyFw-Fv,

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