Abstract

A complete theoretical analysis of a modified grating-frustrated coupler involving a second grating in an input channel is presented. We show that this second grating allows for improved performance of the device, especially in terms of backreflected light. The condition for zero backreflection is found analytically, and a physical interpretation of the results is given.

© 2003 Optical Society of America

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References

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  1. F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
    [Crossref]
  2. A. S. Kewitch, G. A. Rakuljic, P. A. Willems, and A. Yariv, “All-fiber zero-insertion-loss add–drop filter for wavelength-division multiplexing,” Opt. Lett. 23, 106–108 (1998).
    [Crossref]
  3. F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
    [Crossref]
  4. L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
    [Crossref]
  5. J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
    [Crossref]
  6. A.-C. Jacob-Poulin, R. Vallée, S. LaRochelle, D. Faucher, and G. R. Atkins, “Channel-dropping filter based on a grating-frustrated two-core fiber,” J. Lightwave Technol. 18, 715–720 (2000).
    [Crossref]
  7. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 27.
  8. R. R. A. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24, 717–726 (1985).
    [Crossref] [PubMed]
  9. S. S. Orlov, A. Yariv, and S. V. Essen, “Coupled-mode analysis of fiber-optic add–drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22, 688–690 (1997).
    [Crossref] [PubMed]
  10. A. Yesayan and R. Vallée, “Optimized grating-frustrated coupler,” Opt. Lett. 26, 1329–1331 (2001).
    [Crossref]
  11. A. W. Snyder and A. Ankiewicz, “Optical fiber couplers—optimum solution for unequal cores,” J. Lightwave Technol. 6, 463–474 (1988).
    [Crossref]

2001 (1)

2000 (1)

1998 (1)

1997 (2)

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

S. S. Orlov, A. Yariv, and S. V. Essen, “Coupled-mode analysis of fiber-optic add–drop filters for dense wavelength-division multiplexing,” Opt. Lett. 22, 688–690 (1997).
[Crossref] [PubMed]

1996 (1)

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

1995 (1)

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

1994 (1)

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
[Crossref]

1988 (1)

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers—optimum solution for unequal cores,” J. Lightwave Technol. 6, 463–474 (1988).
[Crossref]

1985 (1)

Albert, J.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

Ankiewicz, A.

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers—optimum solution for unequal cores,” J. Lightwave Technol. 6, 463–474 (1988).
[Crossref]

Archambault, J.-L.

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
[Crossref]

Atkins, G. R.

Bakhti, F.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Barcelos, S.

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
[Crossref]

Bilodeau, F.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

Birks, T. A.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

Daxhelet, X.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Dong, L.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

Essen, S. V.

Faucher, D.

Gasca, L.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Gonthier, F.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Hill, K. O.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

Hua, P.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
[Crossref]

Jacob-Poulin, A.-C.

Johnson, D. C.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

Kewitch, A. S.

Lacroix, S.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

LaRochelle, S.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 27.

Malo, B.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

Martineau, L.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Orlov, S. S.

Rakuljic, G. A.

Reekie, L.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
[Crossref]

Sansonetti, P.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Sinet, C.

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

Snyder, A. W.

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers—optimum solution for unequal cores,” J. Lightwave Technol. 6, 463–474 (1988).
[Crossref]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 27.

St. J. Russell, P.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 182–180 (1994).
[Crossref]

Syms, R. R. A.

Thériault, S.

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

Vallée, R.

Willems, P. A.

Yariv, A.

Yesayan, A.

Appl. Opt. (1)

Electron. Lett. (1)

F. Bakhti, P. Sansonetti, C. Sinet, L. Gasca, L. Martineau, S. Lacroix, X. Daxhelet, and F. Gonthier, “Optical add/drop multiplexer based on UV-written Bragg grating in a fused 100% coupler,” Electron. Lett. 33, 803–804 (1997).
[Crossref]

IEEE Photonics Technol. Lett. (2)

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photonics Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

F. Bilodeau, D. C. Johnson, S. Thériault, B. Malo, J. Albert, and K. O. Hill, “An all-fiber dense-wavelength-division multiplexer/demultiplexer using photoimprinted Bragg gratings,” IEEE Photonics Technol. Lett. 7, 388–390 (1995).
[Crossref]

J. Lightwave Technol. (2)

Opt. Lett. (4)

Other (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 27.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the modified GFC. A second grating is added over the coupling zone.

Fig. 2
Fig. 2

Plots of the four coupler outputs as a function of the grating’s relative phase shift φ at λB. Other parameters are κ1=5 cm-1, κ2=1 cm-1, l=1 cm, lo=0, and cl=π/2.

Fig. 3
Fig. 3

Schematic of the three gratings, F1F3, that are responsible for the backreflection.

Fig. 4
Fig. 4

Plots of the four coupler outputs as a function of the grating strength κ2 at λB for κ1=5 cm-1, l=1 cm, and cl=π/2. The external grating segment length is (a) lo=0 and (b) lo=0.5 cm; for (b) R1 and T1 are negligible on the scale used.

Fig. 5
Fig. 5

Dependence of κ2 on κ1, leading to zero backreflection conditions at λB for l=1 cm and cl=π/2.

Fig. 6
Fig. 6

Output powers as a function of the strength of similar gratings κ at λB. Other parameters are l=1 cm, lo=2 cm, and cl=π/2.

Fig. 7
Fig. 7

Dependence of the strength of similar gratings κ versus external grating segment length lo, leading to zero backreflection at λB for l=1 cm and cl=π/2.

Fig. 8
Fig. 8

Backreflection R2 as a function of grating strength κ2 at λB, with (solid curve) and without (dashed curve) cross coupling. Other parameters are κ1=5 cm-1, l=1 cm, cl=π/2, and lo=0.

Fig. 9
Fig. 9

Transmission and reflection of (a) core 1 and (b) core 2 versus wavelength for κ1=1.5 cm-1, κ2=0.75 cm-1, lo=0, l=1 cm, and cl=π/2.

Fig. 10
Fig. 10

Same dependences as in Fig. 9 for κ1=1.5 cm-1, κ2=0.8 cm-1, lo=2.3 cm, l=1 cm, and cl=π/2.

Fig. 11
Fig. 11

Schematic of the energy exchange between reflected beams F1 and F2.

Equations (45)

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ε=ε10+δ1cos(qz)-lozl+lointhefirstcoreε20+δ2cos(qz+φ),0zlinthesecondcore,ε0inthecladding
E=j=12[Aj(z)exp(iβjz)+Bj(z)exp(-iβjz)]fj(x, y),
dA1dz=ic1A2exp[i(ρ1-ρ2)z]+iκ1B1exp(i2ρ1z)+i[χ11+χ21exp(iφ)]B2exp[i(ρ1+ρ2)z],
-dB1dz=ic1B2exp[-i(ρ1-ρ2)z]+iκ1A1exp(-i2ρ1z)+i[χ11+χ21exp(-iφ)]A2×exp[-i(ρ1+ρ2)z],
dA2dz=ic2A1exp[-i(ρ1-ρ2)z]+iκ2B2×exp(iφ+i2ρ2z)+i[χ12+χ22exp(iφ)]×B1exp[i(ρ1+ρ2)z],
-dB2dz=ic2B1exp[i(ρ1-ρ2)z]+iκ2A2exp×(-iφ-i2ρ2z)+i[χ12+χ22exp(-iφ)]A1×exp[-i(ρ1+ρ2)z],
cj=k022βj(j) (εjo-εo)f1f2dxdy()fj2dxdy,
κj=k024βj δj(j)fj2dxdy()fj2dxdy,χjk=k024βk δj(j)f1f2dxdy()fk2dxdy.
β1=β2(i.e.,ρ1=ρ2)
forthecodirectionalcoupling,
β1=β2=q/2(i.e.,ρ1=ρ2=0)
forthedirectBraggcoupling,
β1+β2=q(i.e.ρ1+ρ2=0)
forcontradirectionalcrosscoupling.
ddz [c2(|A1|2-|B1|2)+c1(|A2|2-|B2|2)]
=i[(c1χ12-c2χ11)+(c1χ22-c2χ21)exp(iφ)]×exp[i(ρ1+ρ2)z](A2*B1-A1*B2)+c.c.
Aj(z)=Ajoexp[i(σ+ρj)z],
Bj(z)=Bjoexp[i(σ-ρj)z],
σ+ρ1-κ1-c1-χ11-χ21exp(iφ)κ1σ-ρ1χ11+χ21exp(-iφ)c1-c2-χ12-χ22exp(iφ)σ+ρ2-κ2exp(iφ)χ12+χ22exp(-iφ)c2κ2exp(-iφ)σ-ρ2A1oB1oA2oB2o=0.
σ4+P2σ2+2iσP1sin φ+P0=0,
P2=κ12+κ22-ρ12-ρ22-2c1c2+2[χ12χ11+χ22χ21+cos φ(χ11χ22+χ21χ12)],
P1=(ρ1-ρ2)(χ11χ22-χ21χ12)+κ2(c1χ12-c2χ11)+κ1(c1χ22-c2χ21),
P0=ρ12ρ22+κ12κ22-κ22ρ12-κ12ρ22+c12c22-2c1c2ρ1ρ2-2c1c2κ1κ2cos φ-c12(χ122+χ222+2χ12χ22cos φ)-c22(χ112+χ212+2χ11χ21cos φ)+(χ112+χ212+2χ11χ21cos φ)(χ122+χ222+2χ12χ22cos φ)-2ρ1ρ2[χ11χ12+χ21χ22+cos φ(χ11χ22+χ21χ12)]-2κ1κ2[χ11χ22+χ21χ12+cos φ(χ12χ11+χ22χ21)]-2κ2ρ1[c1(χ22+χ12cos φ)+c2(χ21+χ11cos φ)]-2κ1ρ2[c1(χ12+χ22cos φ)+c2(χ11+χ21cos φ)].
A1o=Mk1D,B1o=Mk2D,
A2o=Mk3D,B2o=Mk4D,
A2(z=0)=Io,B2(z=l)=0,
A1(z=-lo)=0,B1(z=l+lo)=0,
n=14 M13(σn)Dn=Io,
n=14 M14(σn)Dnexp(iσnl)=0,
n=14[M11(σn)-iκ1sinh(μlo)M12(σn)/H]Dn=0,
n=14[M12(σn)-iκ1sinh(μlo)M11(σn)/H]Dn
×exp(iσnl)=0,
A2(z=l)=n=14M13(σn)Dnexp(iσnl),
B2(z=0)=n=14 M14(σn)Dn,
B1(z=-lo)=μn=14 M12(σn)Dn/H,
A1(z=l+lo)=μn=14 M11(σn)Dnexp(iσnl)/H.
T2|A2(z=l)|2/Io, T1|A1(z=l+lo)|2/Io,
R2|B2(z=0)|2/Io,R1|B1(z=-lo)|2/Io.
σ1=-σ2=i2 {[(κ1+κ2)2-4c2]1/2-|κ1-κ2|},
σ3=-σ4=i2 {[(κ1+κ2)2-4c2]1/2+|κ1-κ2|}.
sinh[(κ1-κ2)l]
-(κ1+κ2) sinh{[(κ1+κ2)2-4c2]1/2l}[(κ1+κ2)2-4c2]1/2×[1+tanh2(κ1lo)]+4 tanh(κ1lo)×sinh2(κ1-κ2)l2-(κ1+κ2)2-2c2(κ1+κ2)2-4c2
×sinh2[(κ1+κ2)2-4c2]1/2l2=0,
κ2=c2κ11-1κ1l+o(1/κ14),
κ2=κ1(1+2lo/l).

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