Abstract

We consider mode coupling in multimode optical fibers using either two Bragg gratings or a Bragg grating and a long-period grating. We show that the magnitude of the band edge curvature can be controlled leading to a flat, quartic band-edge or to two band edges at distinct, nonequivalent k-values, allowing precise control of slow light propagation.

© 2007 Optical Society of America

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  1. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nature Physics 2,775-780 (2006).
    [CrossRef]
  2. X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
    [CrossRef]
  3. A. Figotin and I. Vitebskiy, "Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers," Phys. Rev. E 72,036619-12 (2005).
    [CrossRef]
  4. M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
    [CrossRef] [PubMed]
  5. M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30,552-554 (2005).
    [CrossRef] [PubMed]
  6. Y. Cao, R. Hudgins, T. J. Suleski, M. A. Fiddy, J. Raquet, K. Burbank, M. Graham, and P. Sanger, "1-D Photonic Crystal Exhibiting Degenerate Band Edge to Slow Light," In Slow and Fast Light Technical Digest, p. ME16 (Optical Society of America, Washington DC, 2006).
  7. A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85,4866-4868 (2004).
    [CrossRef]
  8. D. Mori and T. Baba, "Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide," Opt. Express 13,9398-9408 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-23-9398.
    [CrossRef] [PubMed]
  9. J. U. Kang, X. B. Xie, and J. Khurgin, "Group velocity modified Ti-diffused LiNbO3 waveguides with dual Bragg gratings," Electron. Lett. 38,1049-1051 (2002).
    [CrossRef]
  10. T. Erdogan and J. E. Sipe, "Tilted fiber phase gratings," J. Opt. Soc. Am. A 13,296-313 (1996).
    [CrossRef]
  11. S. K. Case, "Coupled-wave theory for multiply exposed thick holographic gratings," J. Opt. Soc. Am. 65,724-729 (1975).
    [CrossRef]
  12. V. Mizrahi and J. E. Sipe, "Optical-properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11,1513-1517 (1993).
    [CrossRef]
  13. A. Yariv, "Frustration of Bragg reflection by cooperative dual-mode interference: a new mode of optical propagation," Opt. Lett. 23,1835-1836 (1998).
    [CrossRef]
  14. K. S. Lee and T. Erdogan, "Fiber mode conversion with tilted gratings in an optical fiber," J. Opt. Soc. Am. A 18,1176-1185 (2001).
    [CrossRef]
  15. R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. Hulst, van, K. Asakawa, and L. Kuipers, "The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides," Opt. Express 14,1658-1672 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1658.
    [CrossRef] [PubMed]

2006 (2)

2005 (3)

2004 (2)

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85,4866-4868 (2004).
[CrossRef]

2002 (1)

J. U. Kang, X. B. Xie, and J. Khurgin, "Group velocity modified Ti-diffused LiNbO3 waveguides with dual Bragg gratings," Electron. Lett. 38,1049-1051 (2002).
[CrossRef]

2001 (2)

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

K. S. Lee and T. Erdogan, "Fiber mode conversion with tilted gratings in an optical fiber," J. Opt. Soc. Am. A 18,1176-1185 (2001).
[CrossRef]

1998 (1)

1996 (1)

1993 (1)

V. Mizrahi and J. E. Sipe, "Optical-properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11,1513-1517 (1993).
[CrossRef]

1975 (1)

Baba, T.

Case, S. K.

Cassagne, D.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

d’Yerville, M. L.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nature Physics 2,775-780 (2006).
[CrossRef]

Eggleton, B. J.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nature Physics 2,775-780 (2006).
[CrossRef]

Eich, M.

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85,4866-4868 (2004).
[CrossRef]

Engelen, R. J. P.

Erdogan, T.

Figotin, A.

A. Figotin and I. Vitebskiy, "Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers," Phys. Rev. E 72,036619-12 (2005).
[CrossRef]

Fink, Y.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30,552-554 (2005).
[CrossRef] [PubMed]

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Grillet, C.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

Hulst, N. F.

Ibanescu, M.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30,552-554 (2005).
[CrossRef] [PubMed]

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Ikeda, N.

Joannopoulos, J. D.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30,552-554 (2005).
[CrossRef] [PubMed]

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Johnson, S. G.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30,552-554 (2005).
[CrossRef] [PubMed]

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Jouanin, C.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

Kang, J. U.

J. U. Kang, X. B. Xie, and J. Khurgin, "Group velocity modified Ti-diffused LiNbO3 waveguides with dual Bragg gratings," Electron. Lett. 38,1049-1051 (2002).
[CrossRef]

Khurgin, J.

J. U. Kang, X. B. Xie, and J. Khurgin, "Group velocity modified Ti-diffused LiNbO3 waveguides with dual Bragg gratings," Electron. Lett. 38,1049-1051 (2002).
[CrossRef]

Korterik, J. P.

Lee, K. S.

Letartre, X.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

Littler, I. C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nature Physics 2,775-780 (2006).
[CrossRef]

Luo, C.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Mizrahi, V.

V. Mizrahi and J. E. Sipe, "Optical-properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11,1513-1517 (1993).
[CrossRef]

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nature Physics 2,775-780 (2006).
[CrossRef]

Mori, D.

Petrov, A. Yu.

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85,4866-4868 (2004).
[CrossRef]

Rojo Romeo, P.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

Roundy, D.

M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30,552-554 (2005).
[CrossRef] [PubMed]

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Seassal, C.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

Sipe, J. E.

T. Erdogan and J. E. Sipe, "Tilted fiber phase gratings," J. Opt. Soc. Am. A 13,296-313 (1996).
[CrossRef]

V. Mizrahi and J. E. Sipe, "Optical-properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11,1513-1517 (1993).
[CrossRef]

Sugimoto, Y.

Viktorovitch, P.

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, "Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers," Phys. Rev. E 72,036619-12 (2005).
[CrossRef]

Watanabe, Y.

Xie, X. B.

J. U. Kang, X. B. Xie, and J. Khurgin, "Group velocity modified Ti-diffused LiNbO3 waveguides with dual Bragg gratings," Electron. Lett. 38,1049-1051 (2002).
[CrossRef]

Yariv, A.

Appl. Phys. Lett. (2)

X. Letartre, C. Seassal, C. Grillet, P. Rojo Romeo, P. Viktorovitch, M. L. d’Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a singleline photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79,2312-2314 (2001).
[CrossRef]

A. Yu. Petrov and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85,4866-4868 (2004).
[CrossRef]

Electron. Lett. (1)

J. U. Kang, X. B. Xie, and J. Khurgin, "Group velocity modified Ti-diffused LiNbO3 waveguides with dual Bragg gratings," Electron. Lett. 38,1049-1051 (2002).
[CrossRef]

J. Lightwave Technol. (1)

V. Mizrahi and J. E. Sipe, "Optical-properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11,1513-1517 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature Physics (1)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, "Dispersionless slow light using gap solitons," Nature Physics 2,775-780 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. E (1)

A. Figotin and I. Vitebskiy, "Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers," Phys. Rev. E 72,036619-12 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, "Anomalous dispersion relations by symmetry breaking in axially uniform waveguides," Phys. Rev. Lett. 92,063903-4 (2004).
[CrossRef] [PubMed]

Other (1)

Y. Cao, R. Hudgins, T. J. Suleski, M. A. Fiddy, J. Raquet, K. Burbank, M. Graham, and P. Sanger, "1-D Photonic Crystal Exhibiting Degenerate Band Edge to Slow Light," In Slow and Fast Light Technical Digest, p. ME16 (Optical Society of America, Washington DC, 2006).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the coupling between the forward and backward propagating modes in a guided-wave structure using two superimposed Bragg gratings or Bragg and long-period gratings. Top: schematic of mode wavenumbers. Bottom: Solid and dashed arrows indicate mode coupling by individual gratings.

Fig. 2.
Fig. 2.

Dispersion relation for structures based on (a), (b) two Bragg gratings (σ=+1) and (c), (d) Bragg and long-period gratings (σ=-1). Shading marks the band-gap, and filled circles indicate the band edges. For all the plots, V 1=1, V 2=0.95, ρ 1=1, ρ 2=0.5, δ1=0. Grating detunings are (a) δ2=0, (b) δ2=-1.765, (c) δ2=0, (d) δ2=0.995.

Fig. 3.
Fig. 3.

Tuning characteristics of slow-light modes for (a),(b) coupled Bragg gratings (σ=1) and (c),(d) coupled Bragg and long-period gratings (σ=-1). Shown are the dependencies of the absolute values of the band-edge (a,c) wavenumber and (b,d) group-velocity dispersion on the grating strength (ρ2) and detuning (+δ2 at upper or -δ2 at lower gap edges). Marked points A-D correspond to the dispersion plots in Figs. 2(a)(d), respectively, where the subscripts u and l indicate the upper and lower gap edges (points A and C do not have subscripts as they coincide for both band-edges). Note that in (a) and (c), the band-edge wavenumbers are exactly zero throughout the dark blue region. Normalized parameters are the same as in Fig. 2.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

Δ n ( x , y , z ) = Δ n 1 cos [ κ 1 z + ϕ 1 ] R ( x , y ) + Δ n 2 cos [ κ 2 z + ϕ 2 ] R ( x , y ) ,
E ( x , y , z , t ) = { g 1 ( x , y ) [ u 1 ( z , t ) exp ( i K ˜ 1 z ) + w 1 ( z , t ) exp ( i K ˜ 1 z ) ]
+ g 2 ( x , y ) [ u 2 ( z , t ) exp ( i σ K ˜ 2 z ) + w 2 ( z , t ) exp ( i σ K ˜ 2 z ) ] } exp ( i Ω ˜ t ) + c . c . ,
i u 1 t + i V 1 u 1 z + ρ 1 w 1 exp ( i δ 1 z + i ϕ 1 ) + ρ 2 w 2 exp ( i δ 2 z + i ϕ 2 ) = 0 ,
i w 1 t i V 1 w 1 z + ρ 1 u 1 exp ( i δ 1 z i ϕ 1 ) + ρ 2 u 2 exp ( i δ 2 z i ϕ 2 ) = 0 ,
i u 2 t + i σ V 2 u 2 z + ρ 2 w 1 exp ( i δ 2 z + i ϕ 2 ) = 0 ,
i w 2 t i σ V 2 w 2 z + ρ 2 u 1 exp ( i δ 2 z i ϕ 2 ) = 0 .
u 1 = U 1 exp [ i ( k + δ 1 2 ) z i ω t + i ϕ 1 2 ] ,
w 1 = W 1 exp [ i ( k δ 1 2 ) z i ω t i ϕ 1 2 ] ,
u 2 = U 2 exp [ i ( k δ 1 2 + δ 2 ) z i ω t i ϕ 1 2 + i ϕ 2 ] ,
w 2 = W 2 exp [ i ( k + δ 1 2 + δ 2 ) z i ω t + i ϕ 1 2 i ϕ 2 ] .
M + ( F 1 + F 2 + ) = k ( F 1 F 2 ) , M ( F 1 F 2 ) = k ( F 1 + F 2 + ) ,
M ± = ( ( ω ± ρ 1 ) V 1 δ 1 2 , ± ρ 2 V 1 ± ρ 2 σ V 2 , ω σ V 2 + δ 1 2 δ 2 ) .

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