Abstract

The superposition of a long-period grating and a fiber Bragg grating, which we call an optical superlattice, causes high-efficiency narrow-band reflections to be induced on either side of the Bragg wavelength. This effect was recently observed experimentally in a fiber-based acousto-optic superlattice modulator. We develop in detail the theory of optical superlattices in fiber Bragg gratings, treating both the acousto-optic and the fixed-grating cases. Applications include reconfigurable wavelength division multiplexers, fiber lasers and sensors, tunable filters, modulators, and frequency shifters.

© 2000 Optical Society of America

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References

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  1. P. St. J. Russell, J.-L. Archambault, “Fibre gratings,” in Optical Fibre Sensors Vol. III: Components & Sub-Systems, B. Culshaw, J. Dakin, eds. (Artech House, Norwood, Mass., 1996), pp. 9–67.
  2. R. Kashyap, Fibre Bragg Gratings (Academic, New York, 1999).
  3. P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
    [CrossRef] [PubMed]
  4. P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344–3355 (1986).
    [CrossRef]
  5. W. F. Liu, P. St. J. Russell, L. Dong, “Acousto-optic superlattice modulator using fibre Bragg grating,” Opt. Lett. 22, 1515–1517 (1997).
    [CrossRef]
  6. W. F. Liu, P. St. J. Russell, L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fibre Bragg gratings,” J. Lightwave Technol. 16, 2006–2009 (1998).
    [CrossRef]
  7. B. Y. Kim, J. N. Blake, H. E. Engan, H. J. Shaw, “All-fibre acousto-optic frequency shifter,” Opt. Lett. 11, 389–391 (1986).
    [CrossRef] [PubMed]
  8. H. Sabert, L. Dong, P. St. J. Russell, “Versatile acoustooptical flexural wave modulator, filter and frequency shifter in dual-core fibre,” Int. J. Optoelectron. 7, 189–194 (1992).
  9. T. A. Birks, D. O. Culverhouse, P. St. J. Russell, “The acousto-optic effect in single mode fibre tapers and couplers,” J. Lightwave Technol. 14, 2519–2529 (1996).
    [CrossRef]
  10. P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991); erratum, 41, 163–164 (1994).
    [CrossRef]
  11. B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
    [CrossRef]
  12. M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
    [CrossRef]

1998

1997

1996

T. A. Birks, D. O. Culverhouse, P. St. J. Russell, “The acousto-optic effect in single mode fibre tapers and couplers,” J. Lightwave Technol. 14, 2519–2529 (1996).
[CrossRef]

1995

M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
[CrossRef]

1994

B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
[CrossRef]

1992

H. Sabert, L. Dong, P. St. J. Russell, “Versatile acoustooptical flexural wave modulator, filter and frequency shifter in dual-core fibre,” Int. J. Optoelectron. 7, 189–194 (1992).

1991

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991); erratum, 41, 163–164 (1994).
[CrossRef]

1986

B. Y. Kim, J. N. Blake, H. E. Engan, H. J. Shaw, “All-fibre acousto-optic frequency shifter,” Opt. Lett. 11, 389–391 (1986).
[CrossRef] [PubMed]

P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
[CrossRef] [PubMed]

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344–3355 (1986).
[CrossRef]

Archambault, J.-L.

P. St. J. Russell, J.-L. Archambault, “Fibre gratings,” in Optical Fibre Sensors Vol. III: Components & Sub-Systems, B. Culshaw, J. Dakin, eds. (Artech House, Norwood, Mass., 1996), pp. 9–67.

Birks, T. A.

T. A. Birks, D. O. Culverhouse, P. St. J. Russell, “The acousto-optic effect in single mode fibre tapers and couplers,” J. Lightwave Technol. 14, 2519–2529 (1996).
[CrossRef]

Blake, J. N.

Culverhouse, D. O.

T. A. Birks, D. O. Culverhouse, P. St. J. Russell, “The acousto-optic effect in single mode fibre tapers and couplers,” J. Lightwave Technol. 14, 2519–2529 (1996).
[CrossRef]

Dong, L.

Eggleton, B. J.

M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
[CrossRef]

B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
[CrossRef]

Engan, H. E.

Ibsen, M.

M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
[CrossRef]

Kashyap, R.

R. Kashyap, Fibre Bragg Gratings (Academic, New York, 1999).

Kim, B. Y.

Krug, P. A.

B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
[CrossRef]

Liu, W. F.

Ouellette, F.

M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
[CrossRef]

B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
[CrossRef]

Poladian, L.

B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
[CrossRef]

Russell, P. St. J.

W. F. Liu, P. St. J. Russell, L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fibre Bragg gratings,” J. Lightwave Technol. 16, 2006–2009 (1998).
[CrossRef]

W. F. Liu, P. St. J. Russell, L. Dong, “Acousto-optic superlattice modulator using fibre Bragg grating,” Opt. Lett. 22, 1515–1517 (1997).
[CrossRef]

T. A. Birks, D. O. Culverhouse, P. St. J. Russell, “The acousto-optic effect in single mode fibre tapers and couplers,” J. Lightwave Technol. 14, 2519–2529 (1996).
[CrossRef]

H. Sabert, L. Dong, P. St. J. Russell, “Versatile acoustooptical flexural wave modulator, filter and frequency shifter in dual-core fibre,” Int. J. Optoelectron. 7, 189–194 (1992).

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991); erratum, 41, 163–164 (1994).
[CrossRef]

P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
[CrossRef] [PubMed]

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344–3355 (1986).
[CrossRef]

P. St. J. Russell, J.-L. Archambault, “Fibre gratings,” in Optical Fibre Sensors Vol. III: Components & Sub-Systems, B. Culshaw, J. Dakin, eds. (Artech House, Norwood, Mass., 1996), pp. 9–67.

Sabert, H.

H. Sabert, L. Dong, P. St. J. Russell, “Versatile acoustooptical flexural wave modulator, filter and frequency shifter in dual-core fibre,” Int. J. Optoelectron. 7, 189–194 (1992).

Sceats, M. G.

M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
[CrossRef]

Shaw, H. J.

Electron. Lett.

B. J. Eggleton, P. A. Krug, L. Poladian, F. Ouellette, “Long superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1621–1623 (1994).
[CrossRef]

M. Ibsen, B. J. Eggleton, M. G. Sceats, F. Ouellette, “Broadly tunable DBR laser using sampled fibre Bragg gratings,” Electron. Lett. 31, 37–38 (1995).
[CrossRef]

Int. J. Optoelectron.

H. Sabert, L. Dong, P. St. J. Russell, “Versatile acoustooptical flexural wave modulator, filter and frequency shifter in dual-core fibre,” Int. J. Optoelectron. 7, 189–194 (1992).

J. Appl. Phys.

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344–3355 (1986).
[CrossRef]

J. Lightwave Technol.

T. A. Birks, D. O. Culverhouse, P. St. J. Russell, “The acousto-optic effect in single mode fibre tapers and couplers,” J. Lightwave Technol. 14, 2519–2529 (1996).
[CrossRef]

W. F. Liu, P. St. J. Russell, L. Dong, “100% efficient narrow-band acoustooptic tunable reflector using fibre Bragg gratings,” J. Lightwave Technol. 16, 2006–2009 (1998).
[CrossRef]

J. Mod. Opt.

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991); erratum, 41, 163–164 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

P. St. J. Russell, “Bragg resonance of light in optical superlattices,” Phys. Rev. Lett. 56, 596–599 (1986).
[CrossRef] [PubMed]

Other

P. St. J. Russell, J.-L. Archambault, “Fibre gratings,” in Optical Fibre Sensors Vol. III: Components & Sub-Systems, B. Culshaw, J. Dakin, eds. (Artech House, Norwood, Mass., 1996), pp. 9–67.

R. Kashyap, Fibre Bragg Gratings (Academic, New York, 1999).

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Figures (8)

Fig. 1
Fig. 1

(a) Forward (0) and backward (1) Bloch waves of unperturbed fiber Bragg grating are uncoupled (the vertical lines represent planes of constant index in the Bragg grating); (b) if the fiber is periodically stretched at the correct pitch, the Bloch waves become coupled together—a forward (0) Bloch wave incident from the left-hand side is gradually converted into a backward (1) Bloch wave traveling from the right-hand side.

Fig. 2
Fig. 2

(a) Wave-vector frequency diagram for the primary Bragg grating, (b) reflectivity spectrum as a function of frequency. The three values of δ=ϑ/2κ are the superlattice modulator operating points used in subsequent figures.

Fig. 3
Fig. 3

Sketch of an AOSLM, illustrating the labeling convention for wave amplitudes at the fundamental frequency and in the first sidebands (n=±1) for unit input amplitude. The first subscript indicates the direction (0, forward; 1, backward), and the second subscript is the sideband order. Barred amplitudes (e.g., V¯0,0) are plane-wave amplitudes outside the Bragg grating region. Unbarred amplitudes are those of the Bloch waves in the fiber Bragg grating.

Fig. 4
Fig. 4

Coupling constants, normalized to the Bragg grating coupling constant, against power for the first, second, and third sidebands for κ=1.5 mm-1 in a fiber of diameter 30 μm at λ=1550 nm at operating points: (a) δ=ϑ/2κ=10, (b) δ=ϑ/2κ=1.1. The acoustic frequency is 10 MHz, Young’s modulus was taken as 100 kN/mm2, and the extensional acoustic-wave velocity was 5960 km/s. In (b) for n=1 note that the average index superlattice grating is comparable in strength with the stretch grating, which dominates far from the stop-band edges, i.e., for |ϑ/2κ|1.

Fig. 5
Fig. 5

Left-hand side, four-wave solutions at the operating point δ=1.95 (see Fig. 2) where the fiber Bragg grating exhibits a transmission resonance. The labeling convention in this and in subsequent plots is (k, n)[z], indicating the amplitude of a wave, propagating in the k direction, at position z in the nth sideband. Right-hand side, errors in the amplitudes estimated by substracting the four-wave solutions from the same amplitudes in the six-wave solutions. The worst-case errors are ±15%.

Fig. 6
Fig. 6

Six-wave solutions at the operating point δ=1.76 (see Fig. 2) where the fiber Bragg grating exhibits a reflection maximum of ∼30%. Note that the Bragg grating reflection (1, 0)[0] is suppressed at the superlattice resonance.

Fig. 7
Fig. 7

Four-wave solutions at the operating point δ=4.1 (see Fig. 2) where the fiber Bragg grating exhibits a reflection maximum. The accuracy of the truncation to only one sideband is good in this case.

Fig. 8
Fig. 8

Top, conversion efficiencies versus dephasing ϑs for a stationary superlattice of period 2.5 mm, phase ϕ=0, and peak strain sO=162×10-6 in a fiber of diameter 30 μm and Young’s modulus 90 kN/mm2. Bottom, conversion plotted versus superlattice phase at two fixed values of dephasing ϑs. For an acoustic wave, sensitivity to ϕ would produce amplitude modulation caused by mixing of n=0,n=±1, and higher sidebands.

Tables (1)

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Table 1 Signs of Important Quantities under Various Conditions and for Exact Phase Matching a

Equations (64)

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s(p)=s(ksz-Ωst)=socos(ksz-Ωst),ks=2π/λs,
so=(2Ps/EAvgs)1/2.
(p)=(1-χ)s(p)+01+M cos[K(z-s(p)dz)],
cos[Kz-a sin(p)]
=J0(a)cos Kz+n=1Jn(a)[(-1)ncos(Kz+np)+cos(Kz-np)]=J0(a)cos Kz+n=1Jn(a){(-1)ncos[(K+nks)z-nΩst]+cos[(K-nks)z+nΩst]},
a=Kso/ks,|a|=|λsso/Λ|.
Jn(a)an2nn!.
vn=±nΩs±nks+K±nΩsK,
2E+(ωn0/c)2(1+M cos Kz)E=0,
E0(z)=E1*(z)=B(z)exp-j ϑ2 (1-1/δ2)1/2z=B(z)exp(-jqz),
B(z)=exp(-jKz/2)+δ[-1+(1-1/δ2)1/2]exp(jKz/2)
=exp(-jKz/2)+η exp(jKz/2).
ϑ=2ωn0c-K=2(ω-ωB)n0c,
δ=ϑ2κ,κ=ωn0cM4ωΔn2c,
k=[±K±ϑ(1-1/δ2)1/2]/2
vg=±(c/n0)(1-1/δ2)1/2
Δ2E-n0c22t2(1-χ)s(p)n02
+1+M cosKz-s(p)dzE=0,
E(z, t)=k=01Ek(z)nVk,n(z)exp[j(ω+nΩs)t]
=k=01B[(-1)kz]exp{j[(-1)k+1qz+ωt]}×nVk,n(z)exp(jnΩst),
dV0,ndz+j(κ0-κ1)V1,n-1exp[-j(ks-2q)z]+j(κ0+κ1)V1,n+1exp[j(ks+2q)z]+jm=2κm{V1,n-m(-1)mexp[-j(mks-2q)z]+V1,n+mexp[j(mks+2q)z]}=0,
dV1,ndz-j(κ0+κ1)V0,n-1exp[-j(ks+2q)z]
 -j(κ0-κ1)V0,n+1exp[j(ks-2q)z]
 -jm=1κm{V0,n-mexp[-j(mks+2q)z]
 +V0,n+m(-1)mexp[j(mks-2q)z]}=0,
κ0=ωn0c(1-χ)soη20(1-η2),
κm=ωn0cM[1+(-1)mη2]Jm(a)4(1-η2),
η=δ[-1+(1-1/δ2)1/2],q=ϑ2 (1-1/δ2)1/2.
±mks=2q=ϑ(1-1/δ2)1/2.
dV0,0dz+j(κ0-κ1)V1,-1exp[-j(ks-2q)z]
+j(κ0+κ1)V1,1exp[j(ks+2q)z]=0.
dV1,0dz-jκn{V0,-nexp[-j(nks+2q)z]
+V0,n(-1)nexp[j(nks-2q)z]}=0.
dV0,0dz+j(κ0+κ1)V1,1exp[j(ks+2q)z]=0,
dV1,1dz-j(κ0+κ1)V0,0exp[-j(ks+2q)z]=0.
dV0dz+j((κ0-κ1)exp{-j[(ks-2q)z+ϕ]}+(κ0+κ1)exp{j[(ks+2q)z+ϕ]})V1+jm=2κm((-1)mexp{-j[(mks-2q)z+mϕ]}+exp{j[(mks+2q)z+mϕ]})V1=0,
dV1dz-j((κ0+κ1)exp{-j[(ks+2q)z+ϕ]}+(κ0-κ1)exp{j[(ks-2q)z+ϕ]})V0-jm=2κm(exp{-j[(mks+2q)z+mϕ]}+(-1)mexp{j[(mks-2q)z+mϕ]})V0=0,
E¯(z, t)=k=01exp[-j(-1)kk0z]nV¯k,n(z)×exp[j(ω+nΩs)t],
V¯0,n(0)=V0,n(0)+ηV1,n(0)=δ(n),
V¯1,n(0)=ηV0,n(0)+V1,n(0),
V¯0,n(L)=V0,n(L)exp[j(ϑ/2-q)L]+ηV1,n(L)×exp[j(ϑ/2+q)L],
V¯1,n(L)=ηV0,n(L)exp[-j(ϑ/2+q)L]+V1,n(L)×exp[-j(ϑ/2-q)L]=0,
V¯1,n(0)=V0,n(0)(η-1/η)+δ(n)/η,
V¯0,n(L)=V0,n(L)(1-η2)exp[j(ϑ/2-q)L].
dV0,ndz+jκsV1,n-1exp(-jϑsz)=0,
dV1,n-1dz-jκsV0,nexp(jϑsz)=0,
κs=κ0+κ1,ϑs=ks-2q.
V0,n(L)V1,n-1(L)=[Mn,n-1]V0,n(0)V1,n-1(0),
[Mn,n-1]=cos(qsL)+j ϑs2qssin(qsL)exp(-jϑsL/2)j κsqssin(qsL)exp(-jϑsL/2)j κsqssin(qsL)exp(jϑsL/2)cos(qsL)-j ϑs2qssin(qsL)exp(jϑsL/2),
qs=[(ϑs/2)2-κs2]1/2.
|V¯0,0(L)|2=1-2η2[1+cos(2Lq)cosh(κsL)]cosh(κsL)-4η2cos(2Lq)×sech(κsL),
|V¯1,-1(0)|2=(1-2η2)sinh(κsL)tanh(κsL)cosh(κsL)-4η2cos(2Lq),
|V¯0,-1(L)|2=sinh(κsL)tanh(κsL)cosh(κsL)-4η2cos(2Lq)η2,
|V¯1,0(0)|2=cosh(κsL)+sech(κsL)-2 cos(2Lq)cosh(κsL)-4η2cos(2Lq)η2,
|V¯1,-1(0)|2=tanh2(κsL)=1-|V¯0,0(L)|2,
|V¯0,-1(L)|2=|V¯1,0(0)|2=0
|V¯0,0(L)|2=δs2-1δs2-cos2[κsL(δs2-1)1/2]=1-|V¯1,-1(0)|2,
|V¯0,-1(L)|2=|V¯1,0(0)|2=0,δs=ϑs/(2κs)
[M]=cos(qsL)+j ϑs2qssin(qsL)exp(-jϑsL/2)j κsqssin(qsL)exp[-j(ϑsL/2+ϕ)]j κsqssin(qsL)exp[j(ϑsL/2+ϕ)]cos(qsL)-j ϑs2qssin(qsL)exp(jϑsL/2), 
V0(L)V1(L)=[M]V0(0)V1(0).
V¯0(L)exp[j(q-ϑ/2)]-ηV¯0(L)exp[j(q+ϑ/2)]=[M]1-ηV1(0)V¯1(0)-η.
|V¯1(0)|2=tanh(κsL)[tanh(κsL)-4η sin(qL)cos(ϕ-qL)]1-4η cos(ϕ-qL)sin(qL)tanh(κsL),
|V¯0(L)|2=sech2(κsL)1-4η cos(ϕ-qL)sin(qL)tanh(κsL).
max-minmax+min=4η tanh(κsL).

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