Abstract

We propose the use of slow-light, band-edge waveguides for compact, integrated, tunable optical time delays. We show that slow group velocities at the photonic band edge give rise to large changes in time delay for small changes in refractive index, thereby shrinking device size. Figures of merit are introduced to quantify the sensitivity, as well as the accompanying signal degradation due to dispersion. It is shown that exact calculations of the figures of merit for a realistic, three-dimensional grating structure are well predicted by a simple quadratic-band model, simplifying device design. We present adiabatic taper designs that attain <0.1% reflection in short lengths of 10 to 20 times the grating period. We show further that cascading two gratings compensates for signal dispersion and gives rise to a constant tunable time delay across bandwidths greater than 100 GHz. Given typical loss values for silicon-on-insulator waveguides, we estimate that gratings can be designed to exhibit tunable delays in the picosecond range using current fabrication technology.

© 2005 Optical Society of America

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2005 (7)

H. Altug and J. Vucković, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111,102(2005).
[Crossref]

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
[Crossref]

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” J. Appl. Phys. B 81(2–3), 283–293 (2005).
[Crossref]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033,903 (2005).
[Crossref]

M. Soljačić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E 71, 026602 (2005).
[Crossref]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13, 1202–1214 (2005).
[Crossref] [PubMed]

M. Skorobogatiy, G. Bégin, and A. Talneau, “Statistical analysis of geometrical imperfections from the images of 2D photonic crystals,” Opt. Express 13, 2487–2502 (2005).
[Crossref] [PubMed]

2004 (10)

F. Grillot, L. Vivien, S. Laval, S. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16, 1661–1663 (2004).
[Crossref]

S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett. 27, 2079–2081 (2004).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, “Waveguides, resonators, and their coupled elements in photonic crystal slabs,” Opt. Express 12, 1551–1561 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551.
[Crossref] [PubMed]

J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. 29, 2861–2863 (2004).
[Crossref]

H. Altug and J. Vučkovik, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
[Crossref]

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic bandgap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
[Crossref]

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083,901 (2004).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping Light in a Waveguide with an All-Optical Analog of Electromagnetically Induced Transparency,” Phys. Rev. Lett. 93, 233,903 (2004).
[Crossref]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063,804 (2004).
[Crossref]

D. Mori and T. Baba, “Dispersion-contolled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004).
[Crossref]

2003 (2)

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: Semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd Eur opean Symp. on Photonic Crystals, vol.  1, pp. 103–109 (2003).

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066,616 (2003).
[Crossref]

2002 (4)

S. Mookherjea and A. Yariv, “Coupled Resonator Optical Waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 448–456 (2002).
[Crossref]

J. Liu, B. Shi, D. Zhao, and X. Wang, “Optical delay in defective photonic bandgap structures,” J. Opt. A.: Pure Appl. Opt. 4, 636–639 (2002).
[Crossref]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066,608 (2002).
[Crossref]

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase locked electron-beam lithography,” J. Vac. Sci. Technol. B 20, 2753–2757 (2002).
[Crossref]

2001 (3)

1999 (1)

1998 (1)

1997 (1)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[Crossref]

1996 (1)

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54(2), R1078–1081 (1996).
[Crossref]

1986 (1)

G. E. Jellison and H. H. Burke, “The temperature depdendence of the refractive index of silicon at elevated temperatures at several laser wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
[Crossref]

Agrawal, G. P.

Akahane, Y.

Altug, H.

H. Altug and J. Vucković, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111,102(2005).
[Crossref]

H. Altug and J. Vučkovik, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
[Crossref]

Asakawa, K.

Asano, T.

Baba, T.

D. Mori and T. Baba, “Dispersion-contolled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004).
[Crossref]

Bégin, G.

Bendickson, J. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54(2), R1078–1081 (1996).
[Crossref]

Bienstman, P.

S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: Semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd Eur opean Symp. on Photonic Crystals, vol.  1, pp. 103–109 (2003).

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066,608 (2002).
[Crossref]

Bloemer, M. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54(2), R1078–1081 (1996).
[Crossref]

Bolivar, P. H.

Bowden, C. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54(2), R1078–1081 (1996).
[Crossref]

Boyd, R. W.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063,804 (2004).
[Crossref]

Burke, H. H.

G. E. Jellison and H. H. Burke, “The temperature depdendence of the refractive index of silicon at elevated temperatures at several laser wavelengths,” J. Appl. Phys. 60, 841–843 (1986).
[Crossref]

Cappuzzo, M. A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
[Crossref]

Cassan, E.

F. Grillot, L. Vivien, S. Laval, S. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16, 1661–1663 (2004).
[Crossref]

Cerrina, F.

Chang, H.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063,804 (2004).
[Crossref]

Chen, E.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
[Crossref]

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[Crossref]

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, “Dispersion of Cascaded Fiber Gratings in WDM Lightwave Systems,” J. Lightwave Technol. 16, 1523–9 (1998).
[Crossref]

Fan, S.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping Light in a Waveguide with an All-Optical Analog of Electromagnetically Induced Transparency,” Phys. Rev. Lett. 93, 233,903 (2004).
[Crossref]

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083,901 (2004).
[Crossref]

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066,616 (2003).
[Crossref]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[Crossref]

Ferrera, J.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[Crossref]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54(2), R1078–1081 (1996).
[Crossref]

Foresi, J. S.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[Crossref]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54(2), R1078–1081 (1996).
[Crossref]

Fuller, K. A.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063,804 (2004).
[Crossref]

Gasparyan, A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
[Crossref]

Geis, M. W.

S. J. Spector, M. W. Geis, and T. Lyszczarz. Private communication.

Gomez, L. T.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
[Crossref]

Goodberlet, J. G.

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase locked electron-beam lithography,” J. Vac. Sci. Technol. B 20, 2753–2757 (2002).
[Crossref]

Griffin, A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
[Crossref]

Grillot, F.

F. Grillot, L. Vivien, S. Laval, S. Pascal, and E. Cassan, “Size Influence on the Propagation Loss Induced by Sidewall Roughness in Ultrasmall SOI Waveguides,” IEEE Photonics Technol. Lett. 16, 1661–1663 (2004).
[Crossref]

Hastings, J. T.

J. T. Hastings, M. H. Lim, J. G. Goodberlet, and H. I. Smith, “Optical waveguides with apodized sidewall gratings via spatial-phase locked electron-beam lithography,” J. Vac. Sci. Technol. B 20, 2753–2757 (2002).
[Crossref]

Hau, L. V.

M. Soljačić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E 71, 026602 (2005).
[Crossref]

Henschel, W.

Hughes, S.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033,903 (2005).
[Crossref]

Ibanescu, M.

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S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066,608 (2002).
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S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066,608 (2002).
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S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.
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M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. LeGrange, and S. S. Patel, “Integrated resonance-enhanced variable delay lines,” IEEE Photonics Technol. Lett. 17, 834–6 (2005).
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S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” J. Appl. Phys. B 81(2–3), 283–293 (2005).
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S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: Semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd Eur opean Symp. on Photonic Crystals, vol.  1, pp. 103–109 (2003).

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S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066,608 (2002).
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S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy, M. Soljačić, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Coupling, scattering and perturbation theory: Semi-analytical analyses of photonic-crystal waveguides,” in Proc. 2003 5th Intl. Conf. on Transparent Optical Networks and 2nd Eur opean Symp. on Photonic Crystals, vol.  1, pp. 103–109 (2003).

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Appl. Phys. Lett. (4)

H. Altug and J. Vučkovik, “Two-dimensional coupled photonic crystal resonator arrays,” Appl. Phys. Lett. 84, 161–163 (2004).
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H. Altug and J. Vucković, “Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays,” Appl. Phys. Lett. 86, 111,102(2005).
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D. Mori and T. Baba, “Dispersion-contolled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004).
[Crossref]

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljačić, “Effect of a photonic bandgap on scattering from waveguide disorder,” Appl. Phys. Lett. 84, 3639–3641 (2004).
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IEEE J. Quantum Electron. (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
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Figures (7)

Fig. 1.
Fig. 1.

(a) Types of 1-D periodic gratings. (b) Typical band structure of a 1-D periodic grating. The band gap is shaded yellow. (c) Magnified view of the band structure near the band edge, illustrating how a small shift in refractive index can lead to a large shift in group velocity. (The case shown corresponds to n′ < n, or Δn < 0.)

Fig. 2.
Fig. 2.

(a) 3-D perspective view of a slow-light grating structure. (b) Top view. (c) Band structure.

Fig. 3.
Fig. 3.

(a) Sensitivity figure of merit for the structure of Fig. 2 as a function of fractional frequency from the band edge. (b) Required length for different amounts of tunable time delay. Symbols/solid lines are exact/quadratic-approximation calculations for a fractional index shift Δn/n = -0.01.

Fig. 4.
Fig. 4.

Dispersion figures of merit for the structure of Fig. 2 as a function of fractional frequency from the band edge for (a) fixed bandwidth Δω/2n=10 GHz and (b) fixed time delay Δτ=1ns. Symbols/solid lines are exact/quadratic-approximation calculations for a fractional index shift Δn/n = -0.01.

Fig. 5.
Fig. 5.

(a) Reflection as a function of the length of a taper (in units of a, the waveguide period) connecting a uniform waveguide to the grated waveguide of Fig. 2 for n Si = 3.45 and ω= 0.22491[2πc/a] (< 2% from the band edge). Results were obtained from 3d numerical integration of the coupling integral in Eq. 8. (b) Normalized taper profile s(z) for linear taper (dashed line) and optimized, variable-rate taper (solid line).

Fig. 6.
Fig. 6.

Device design including adiabatically-tapered waveguide segments and dispersion compensation. Insets show the band diagram corresponding to each of the two grating regions.

Fig. 7.
Fig. 7.

Effect of dispersion compensation on the tunable time delay. Using grating 1 or grating 2 alone causes the tunable delay to vary strongly across the bandwidth. Cascading the two gratings as in Fig. 7 (labeled “average”) gives a flat tunable delay in the center of the bandwidth.

Equations (22)

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s Δ τ τ Δ n n = Δ ( 1 v g ) Δ n 1 v g 1 Δ n n ,
L = Δ τ Δ ( 1 v g ) Δ n = Δ τ Δ n n v g s .
d τ ω τ ω Δ ω 1 Δ ω = L Δ ( 1 v g ) Δ ω 1 Δ ω
d = Δ τ Δ ω Δ ( 1 v g ) Δ ω Δ ( 1 v g ) Δ n .
ω ( k ) = ω be α ( π a k ) 2 ,
ω ( k ) = ω be + δω ( α + δα ) ( π a k ) 2 .
s 1 Δ n n ( 1 1 ( 1 + δα Δ ω be ) ( 1 + δα α ) ) .
L 2 Δτ αΔ ω be 1 [ ( 1 + δω Δ ω be ) ( 1 + δα α ) ] 1 2 .
d L Δ ω 2 α [ 1 Δ ω be 1 Δ ω be + Δ ω ] .
c r = 0 L dz k r e 2 πik a C ̂ z i Δ β k ( z ) × exp [ i 0 z Δ β k ( z ) dz ] ,
Δ ω FP ω v g Δ L 2 L Δ β ω .
L ~ 1 ( αΔ ω be ) 3 2 .
β ˜ 3 = d ( d 2 k 1 d ω 2 + d 2 k 2 d ω 2 ) ω o 3 4 α ( Δ 2 ) 5 2
ω 1 ( k 1 ) = ω o + Δ 2 α ( π a 1 k 1 ) 2
ω 1 ( k 1 ) = ω o + Δ 2 + δω α ( π a 1 k 1 ) 2
ω 2 ( k 2 ) = ω o Δ 2 + α ( k 2 π a 2 ) 2
ω 2 ( k 2 ) = ω o Δ 2 δω + α ( k 2 π a 2 ) 2
Δ τ = L 2 [ Δ ( 1 v g , 1 ) + Δ ( 1 v g , 2 ) ]
Δ ( 1 v g , 1 ) = 1 2 α [ 1 Δ 2 + δω ( ω ω o ) 1 Δ 2 ( ω ω o ) ] ,
Δ ( 1 v g , 2 ) = 1 2 α [ 1 Δ 2 + δω ( ω o ω ) 1 Δ 2 ( ω o ω ) ] ,
Δ τ L = 1 2 α [ 1 Δ 2 + δω 1 Δ 2 ] 0 . 8503 c 1 .
L 1 e = 10 log 10 ( 1 e ) loss in dB mm = 4.3429 dB 2.1 dB mm = 2.1 mm .

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