Abstract

We study the propagation and switching of slow-light pulses in nonlinear directional couplers composed of two parallel waveguides, where each waveguide contains a Bragg grating. We show that by optimizing the phase shift between the Bragg gratings, one can obtain specific dispersion characteristics enabling all-optical pulse manipulation in space and in time. We demonstrate that the power-controlled nonlinear self-action of light can be used to compensate dispersion-induced broadening of pulses through the formation of gap solitons, to control pulse switching in the coupler, and to tune the propagation velocity. We also confirm that the switching is tolerant to deviations of the phase shift from the optimal value, which can occur in the fabrication process.

© 2008 Optical Society of America

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2007 (5)

2006 (7)

2005 (7)

2004 (3)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[CrossRef]

A. Gubeskys and B. A. Malomed, “Solitons in a system of three linearly coupled fiber gratings,” Eur. Phys. J. D 28, 283-299 (2004).
[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]

2003 (1)

2002 (1)

2001 (1)

1999 (2)

1998 (1)

1997 (1)

1987 (2)

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

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

1982 (2)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982).
[CrossRef]

A. A. Maier, “Optical transistors and bistable elements on the basis of non-linear transmission of light by the systems with unidirectional coupled waves,” Kvantovaya Elektron. (Moscow) 9, 2296-2302 (1982) A. A. Maier,(in Russian) [IEEE J. Quantum Electron. 12, 1490-1494 (1982)].

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1988).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Aitchison, J. S.

Andrejco, M. J.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

Asakawa, K.

Aslund, M.

Baba, T.

Baker, N. J.

Binder, R.

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Borel, P. I.

Broderick, N. G. R.

Canning, J.

Castro, J. M.

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]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[CrossRef]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685-1692 (1998).
[CrossRef]

De la Rue, R. M.

de Sterke, C. M.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418-1419 (2007).
[CrossRef]

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

M. Aslund, J. Canning, L. Poladian, C. M. de Sterke, and A. Judge, “Antisymmetric grating coupler: experimental results,” Appl. Opt. 42, 6578-6583 (2003).
[CrossRef] [PubMed]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrodinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587-599 (1999).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1994), Vol. 33, pp. 203-260.

Eggleton, B. J.

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.

Fage Pedersen, J.

Fan, S. H.

Figotin, A.

A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293-382 (2006).
[CrossRef]

Frandsen, L. H.

Friberg, S. R.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

Geraghty, D. F.

Gersen, H.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Greiner, C. M.

Gubeskys, A.

A. Gubeskys and B. A. Malomed, “Solitons in a system of three linearly coupled fiber gratings,” Eur. Phys. J. D 28, 283-299 (2004).
[CrossRef]

Ha, S.

Hamann, H. F.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Honkanen, S.

Huang, S. C.

Iazikov, D.

Ibanescu, M.

Ibsen, M.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418-1419 (2007).
[CrossRef]

Ikeda, N.

Imai, M.

M. Imai and S. Sato, “Optical switching devices using nonlinear fiber-optic grating coupler,” in Photonics Based on Wavelength Integration and Manipulation, Vol. 2 of IPAP Books, K.Tada, T.Suhara, K.Kikuchi, Y.Kokubun, K.Utaka, M.Asada, F.Koyama, and T.Arakawa, eds. (Institute of Pure and Applied Physics, 2005), pp. 293-302.

Ippen, E.

Jacobsen, R. S.

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Judge, A.

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Kato, M.

Khurgin, J. B.

Kivshar, Yu. S.

S. Ha, A. A. Sukhorukov, and Yu. S. Kivshar, “Slow-light switching in nonlinear Bragg-grating couplers,” Opt. Lett. 32, 1429-1431 (2007).
[CrossRef] [PubMed]

A. A. Sukhorukov and Yu. S. Kivshar, “Slow-light optical bullets in arrays of nonlinear Bragg-grating waveguides,” Phys. Rev. Lett. 97, 233901 (2006).
[CrossRef]

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Korterik, J. P.

R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, 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).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Krauss, T. F.

Kuipers, L.

Kuramochi, E.

Kwong, N. H.

Laurenzano, M.

Lavrinenko, A. V.

Lee, C. P.

Littler, I. C. M.

Love, J. D.

Luther-Davies, B.

Maier, A. A.

A. A. Maier, “Optical transistors and bistable elements on the basis of non-linear transmission of light by the systems with unidirectional coupled waves,” Kvantovaya Elektron. (Moscow) 9, 2296-2302 (1982) A. A. Maier,(in Russian) [IEEE J. Quantum Electron. 12, 1490-1494 (1982)].

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[CrossRef]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685-1692 (1998).
[CrossRef]

Malomed, B. A.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[CrossRef]

A. Gubeskys and B. A. Malomed, “Solitons in a system of three linearly coupled fiber gratings,” Eur. Phys. J. D 28, 283-299 (2004).
[CrossRef]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685-1692 (1998).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Michaeli, A.

Millar, P.

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]

Mok, J. T.

J. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418-1419 (2007).
[CrossRef]

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

Montrosset, I.

Mori, D.

Moss, D. J.

Mossberg, T. W.

Moulin, G.

Notomi, M.

O'Boyle, M.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Oliver, M. K.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

Orlov, S. S.

Pedersen, J. Fage

Perrone, G.

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]

Peucheret, C.

Poladian, L.

Richardson, D. J.

Ruan, Y. L.

Saifi, M. A.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

Salib, M.

Saremi, M. Shokooh

Sato, S.

M. Imai and S. Sato, “Optical switching devices using nonlinear fiber-optic grating coupler,” in Photonics Based on Wavelength Integration and Manipulation, Vol. 2 of IPAP Books, K.Tada, T.Suhara, K.Kikuchi, Y.Kokubun, K.Utaka, M.Asada, F.Koyama, and T.Arakawa, eds. (Institute of Pure and Applied Physics, 2005), pp. 293-302.

Settle, M. D.

Silberberg, Y.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1994), Vol. 33, pp. 203-260.

Slusher, R. E.

Smirl, A. L.

Smith, P. W.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[CrossRef]

Soljacic, M.

Sugimoto, Y.

Sukhorukov, A. A.

S. Ha, A. A. Sukhorukov, and Yu. S. Kivshar, “Slow-light switching in nonlinear Bragg-grating couplers,” Opt. Lett. 32, 1429-1431 (2007).
[CrossRef] [PubMed]

A. A. Sukhorukov and Yu. S. Kivshar, “Slow-light optical bullets in arrays of nonlinear Bragg-grating waveguides,” Phys. Rev. Lett. 97, 233901 (2006).
[CrossRef]

Ta'eed, V. G.

Tomljenovic Hanic, S.

van Hulst, N. F.

R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, 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).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

VanEssen, S.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293-382 (2006).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Watanabe, Y.

Yang, Z. S.

Yariv, A.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Zsigri, B.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135-1137 (1987).
[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. T. Mok, M. Ibsen, C. M. de Sterke, and B. J. Eggleton, “Dispersionless slow light with 5-pulse-width delay in fibre Bragg grating,” Electron. Lett. 43, 1418-1419 (2007).
[CrossRef]

Eur. Phys. J. D (1)

A. Gubeskys and B. A. Malomed, “Solitons in a system of three linearly coupled fiber gratings,” Eur. Phys. J. D 28, 283-299 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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Figures (9)

Fig. 1
Fig. 1

Schematic of a Bragg-grating coupler.

Fig. 2
Fig. 2

Dependence of characteristic dispersion (top), intensity inside semi-infinite coupler ( z 0 ) at z = L c = 7.5 mm (middle), and coupling length L c (bottom) on wavelength detuning δ λ for (a) φ = 0 , (b) φ = 0.9 π , and (c) φ = π . For all the plots parameters correspond to Fig. 3, and light and dark shaded regions indicate the bandgap and single-mode region, respectively. Top row, solid and dotted curves correspond to even and odd modes, respectively. Middle row, intensities in the first (solid curve) and second (dashed curve) arms when the incident cw light is coupled to the first arm.

Fig. 3
Fig. 3

Dependence of slow-light regimes and the threshold condition cos ( φ 2 ) = C ρ on the structural parameters C, ρ, and φ. The shaded and unshaded regions represent one slow-mode and two slow-mode regimes, respectively. For points A, B, and C, φ = 0 , 0.9 π , and π, respectively, and C 0.38 and ρ = 0.5 for all points. The points correspond to each column in Figs. 2, 4, 5.

Fig. 4
Fig. 4

Mode field distributions across arbitrarily shaped coupler arms when (a),(d) φ = 0 , (b),(e) φ = 0.9 π , and (c),(f) φ = π and when wavelength detuning is (a),(b),(c) large ( δ λ = 0.5 nm and v g c n 0 ) and (d),(e),(f) small ( v g = 0.1 c n 0 ) . These are visual representations of eigenvectors U n and W n in Eqs. (8, 9). Real and imaginary parts for the forward waves are plotted in solid and dotted curves and in dashed and dashed-dotted curves for the backward waves, respectively.

Fig. 5
Fig. 5

Intensity distribution (averaged over grating period) inside the first (solid curve) and second (dashed curve) arms of a semi-infinite ( z 0 ) coupler with (a) φ = 0 , (b) φ = 0.9 π , and (c) φ = π when the incident cw light is coupled to the first arm. Tops and bottoms correspond to large wavelength detuning from the resonance ( v g c n 0 ) and wavelength tuned close to the bandgap with slow group velocity v g = 0.1 c n 0 , respectively. Parameters correspond to Fig. 3, and the intensities are normalized to the input intensity.

Fig. 6
Fig. 6

Transmission coefficients as functions of input intensity for a coupler with out-of-phase Bragg gratings with (a) large wavelength detuning ( δ λ = 0.2 nm , L = 10 cm ), (b) medium detuning ( δ λ = 0.1 nm , L = 10 cm ), (c) small detuning ( δ λ = 0.05 nm , L = 1 cm ), and (d) negative detuning ( δ λ = 0.12 nm , L = 0.5 cm ). In all the plots, L = L c and solid and dashed curves correspond to transmissions from the first and second waveguides, respectively.

Fig. 7
Fig. 7

Transmitted intensities as functions of time for the case of small frequency detuning [Fig. 6c] at the input intensities of (a) 40, (b) 75, and (c) 120. In all the plots, and solid and dashed curves correspond to transmitted intensities from the first and second waveguides, respectively.

Fig. 8
Fig. 8

Dependence of output pulse characteristics on the input peak intensity for the coupler with (a) φ = π and (b)–(d) φ = 0.9 π ; (a),(b) output power normalized to the input power; (c) pulse full width at half-maximum of intensity, dotted curve marks the input pulse width; (d) pulse delay relative to propagation without the Bragg grating normalized to the input pulse width. In all the plots, solid and dashed curves correspond to the outputs at the first and second waveguides, respectively.

Fig. 9
Fig. 9

(a)–(d) Pulse dynamics inside the nonlinear coupler with φ = 0.9 π for different values of the normalized peak input intensities I 0 = 10 4 , 3.28, 3.35, and 4. Shown are the density plots of intensity in the first (left column) and second (middle column) waveguides. Output intensity profiles normalized to I 0 at the first (solid curve) and second (dashed curve) waveguides are shown in the third column, and the corresponding output pulse spectra are presented in the last column.

Equations (10)

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E ( x , y , z , t ) = { g 1 ( x , y ) [ u 1 ( z , t ) exp ( i K ̃ z ) + w 1 ( z , t ) exp ( i K ̃ z ) ] + g 2 ( x , y ) [ u 2 ( z , t ) exp ( i K ̃ z ) + w 2 ( z , t ) exp ( i K ̃ z ) ] } exp ( i Ω ̃ t ) + c.c. ,
i u 1 t + i u 1 z + ρ 1 w 1 + C u 2 + γ ( u 1 2 + 2 w 1 2 ) u 1 = 0 ,
i w 1 t i w 1 z + ρ 1 * u 1 + C w 2 + γ ( w 1 2 + 2 u 1 2 ) w 1 = 0 ,
i u 2 t + i u 2 z + ρ 2 w 2 + C u 1 + γ ( u 2 2 + 2 w 2 2 ) u 2 = 0 ,
i w 2 t i w 2 z + ρ 2 * u 2 + C w 1 + γ ( w 2 2 + 2 u 2 2 ) w 2 = 0 ,
t s = λ 0 2 ρ 1 ( π c Δ λ 0 ) ,
z s = t s c n 0 ,
u n = U n exp ( i β z i ω t ) ,
w n = W n exp ( i β z i ω t ) ,
ω 2 ( β ) = β 2 + C 2 + ρ 2 ± 2 C [ β 2 + ρ 2 cos 2 ( φ 2 ) ] 1 2 ,

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