Abstract

We study theoretically and experimentally nonlinear wave propagation in dynamic optical Bragg gratings. Such gratings are obtained through cross-phase modulation with the beating wave that results from two intense laser beams of different frequencies propagating in a highly birefringent fiber. We show that wave propagation in these gratings obeys the standard coupled-mode equations of static nonlinear Bragg gratings, which makes our study relevant to a wide class of problems related to nonlinear wave propagation in periodic media. The main advantage of the dynamic Bragg gratings is that they make it possible to study experimentally nonlinear wave dynamics close to the bandgap where static gratings cannot be investigated because of their high reflectivity. We illustrate this advantage through the study of Bragg modulational instability.

© 2002 Optical Society of America

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References

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  1. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam 1994), Vol. XXXIII, Chap. III, pp. 203–260.
  2. U. Mohideen, R. E. Slusher, V. Mizrahi, T. Erdogan, M. Kuwata-Gonokami, P. J. Lemaire, J. E. Sipe, C. M. de Sterke, and N. G. R. Broderick, “Gap soliton propagation in optical fiber gratings,” Opt. Lett. 20, 1674–1676 (1995).
    [CrossRef] [PubMed]
  3. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef] [PubMed]
  4. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  5. B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
    [CrossRef]
  6. T. G. Brown and B. J. Eggleton, “Bragg solitons and optical switching in nonlinear periodic structures: an historical perspective,” Opt. Express 3, 385–388 (1998), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  7. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
    [CrossRef]
  8. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap–soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 259–261 (1998).
    [CrossRef]
  9. N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
    [CrossRef]
  10. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  11. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
    [CrossRef]
  12. H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (2000).
    [CrossRef] [PubMed]
  13. S. Wabnitz, “Forward mode coupling in periodic nonlinear-optical fibers: modal dispersion cancellation and resonance solitons,” Opt. Lett. 14, 1071–1073 (1989).
    [CrossRef] [PubMed]
  14. C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
    [CrossRef]
  15. S. Pitois, M. Haelterman, and G. Millot, “Bragg modulational instability induced by a dynamic grating in an optical fiber,” Opt. Lett. 26, 780–782 (2001).
    [CrossRef]
  16. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
    [CrossRef]
  17. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [CrossRef] [PubMed]
  18. E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
    [CrossRef] [PubMed]
  19. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]

2001 (1)

2000 (1)

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (5)

1997 (1)

1996 (2)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

1995 (1)

1992 (1)

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

1990 (2)

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

1989 (2)

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Aceves, A. B.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Bilbault, J. M.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Broderick, N. G. R.

Brown, T. G.

T. G. Brown and B. J. Eggleton, “Bragg solitons and optical switching in nonlinear periodic structures: an historical perspective,” Opt. Express 3, 385–388 (1998), http://www.opticsexpress.org.
[CrossRef] [PubMed]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

de Sterke, C. M.

Dinda, P. T.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Eggleton, B. J.

Erdogan, T.

Glenn, W. H.

Haelterman, M.

S. Pitois, M. Haelterman, and G. Millot, “Bragg modulational instability induced by a dynamic grating in an optical fiber,” Opt. Lett. 26, 780–782 (2001).
[CrossRef]

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Harvey, J. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Ibsen, M.

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Kuwata-Gonokami, M.

Laming, R. I.

Lemaire, P. J.

Leonhardt, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Meltz, G.

Millot, G.

S. Pitois, M. Haelterman, and G. Millot, “Bragg modulational instability induced by a dynamic grating in an optical fiber,” Opt. Lett. 26, 780–782 (2001).
[CrossRef]

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mizrahi, V.

Mohideen, U.

Morey, W. W.

Perlin, V.

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (2000).
[CrossRef] [PubMed]

Pitois, S.

Prelewitz, D. F.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Remoissenet, M.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Richardson, D. J.

Rothenberg, J. E.

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

Sankey, N. D.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Seve, E.

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Sipe, J. E.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

U. Mohideen, R. E. Slusher, V. Mizrahi, T. Erdogan, M. Kuwata-Gonokami, P. J. Lemaire, J. E. Sipe, C. M. de Sterke, and N. G. R. Broderick, “Gap soliton propagation in optical fiber gratings,” Opt. Lett. 20, 1674–1676 (1995).
[CrossRef] [PubMed]

Slusher, R. E.

Strasser, T. A.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Taverner, D.

Wabnitz, S.

Winful, H. G.

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

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

Opt. Commun. (2)

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. A (2)

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

E. Seve, P. T. Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (2000).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Other (1)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam 1994), Vol. XXXIII, Chap. III, pp. 203–260.

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Figures (11)

Fig. 1
Fig. 1

Dispersion relation of the dynamic Bragg grating for the fiber and wave parameters given in the text. Solid curves, |r|1; dashed curves, |r|1.

Fig. 2
Fig. 2

Theoretical Bragg MI gain spectra for P=15 W and (a) r=-1 (ω=0 THz), r=-10 (ω=-0.1 THz), and r=-25 (ω=-0.24 THz); (b) r=1 (ω=0 THz), r=10 (ω=0.1 THz), and r=25 (ω=0.24 THz).

Fig. 3
Fig. 3

(a) Optimal modulation frequency Ωopt calculated from linear-stability analysis versus gap frequency detuning ω. (b) Associated MI gain.

Fig. 4
Fig. 4

Schematic diagram of the experimental setup: F’s neutral-density filters; DM, dichroic mirror; MPC, multiple-pass cell; ODL, optical delay line; P’s, polarizers; λ/2, half-wave plate; L, steering lens; BS, beam splitter; MOs, microscopic objectives; Hibi Fiber, high-birefringence fiber; PD, photodiode.

Fig. 5
Fig. 5

Pump spectra showing the forward and the backward waves for three different values ω of the gap frequency detuning and for an input pump power of 15 W. The zero frequency corresponds to the forward pump wave that has a fixed wavelength of 574.78 nm.

Fig. 6
Fig. 6

Experimental MI spectra of light emerging from the fiber for an input power of 15 W and different values of the frequency detuning: (a) ω=-0.83 THz, (b) ω=-0.23 THz, (c) ω=0.3 THz, (d) ω=0.6 THz.

Fig. 7
Fig. 7

Calculated MI spectra for the same parameters as those in Fig. 6: (a) ω=-0.83 THz, (b) ω=-0.23 THz, (c) ω=0.3 THz, (d) ω=0.6 THz.

Fig. 8
Fig. 8

Optimal modulation frequency Ωopt versus frequency detuning ω. Solid curves, calculated from linear-stability analysis; crosses, experimental measurements.

Fig. 9
Fig. 9

Calculated MI spectra for ω=0 THz and P=15 W.

Fig. 10
Fig. 10

Numerical simulation showing the evolution of the pump wave intensity profile over a propagation distance of 400 m without any grating. The parameters are Ω=0.5 THz, ρ=π/2, and κ=0.

Fig. 11
Fig. 11

Numerical simulation showing the evolution of the pump wave intensity profile over a propagation distance of 400 m when the grating is present. The parameters are Ω=0.5 THz, ρ=π/2, and κ=0.018 m-1.

Equations (29)

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Epz=δ Ept-i2β2 2Ept2+iγ|Ep|2+23|Eg|2Ep,
Egz=-i2β2 2Egt2+iγ|Eg|2+23|Ep|2Eg,
δ=B/c,
Ep(z, t)=Up(z, t)expiδβ2t-δ22β2z,
Upz=-i2β2 2Upt2+iγ|Up|2+23|Eg|2Up,
Egz=-i2β2 2Egt2+iγ|Eg|2+23|Up|2Eg.
Eg(0, t)=(a/2)[exp(iΩt)+exp(-iΩt)],
Upz=-i2β2 2Upt2+iδβ(t)Up+iγ|Up|2Up,
δβ(t)=(2γ/3)(a2/2)cos(2Ωt)=2κ cos(2Ωt),
Up(z, t)=F(z, t)exp[i(-Ωt+Kz)]+B(z, t)exp[i(Ωt+Kz)],
Fz=-i2β2 2Ft2-Ωβ2 Ft+iκB+iγ(|F|2+2|B|2)F,
Bz=-i2β2 2Bt2+Ωβ2 Bt+iκF+iγ(|B|2+2|F|2)B.
F(z, t)=A+ exp[i(βz-ωt)],
B(z, t)=A- exp[i(βz-ωt)],
ω=κ2Ωβ2 r-1r+γP2Ωβ2 r2-1r2+1,
β=12β2ω2+κ2 r+1r+3γP2,
-κ+3γP2<β<κ+3γP2.
ωB-ωF=2Ω.
kB-kF=ΔK,
δω=(β1y-β1x)/β2.
F(z, t)=[A++u(z, t)]exp[i(βz-ωt)],
B(z, t)=[A-+v(z, t)]exp[i(βz-ωt)].
u(z, t)=us(z)exp(iΩMt)+ua(z)exp(-iΩMt),
v(z, t)=vs(z)exp(iΩMt)+va(z)exp(-iΩMt),
z[Y]=i[M][Y],
[M]=β+β22(ω2+ΩM2)+β2(ωΩ+ωΩM+ΩMΩ)+2γ[(A+)2+(A)2]γ(A+)22γA+A+κ2γA+Aγ(A+)2ββ22(ω2+ΩM2)+β2(ωΩ+ωΩM+ΩMΩ)2γ[(A+)2+(A)2]2γA+A2γA+Aκ2γA+A+κ2γA+Aβ+β22(ω2+ΩM2)+β2(ωΩ+ωΩMΩMΩ)+2γ[(A+)2+(A)2]γ(A)22γA+A2γA+Aκγ(A)2ββ22(ω2+ΩM2)+β2(ωΩ+ωΩMΩMΩ2γ[(A+)2+(A)2].
ω=Bcβ2-2πcλF+2πcλ1.
F(z, t)=κ3γ sin(ρ)sechκΩβ2 sin(ρ)t-i ρ2×exp[-iκ cos(ρ)z],
B(z, t)=-κ3γ sin(ρ)sechκΩβ2 sin(ρ)t+i ρ2×exp[-iκ cos(ρ)z].

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