Abstract

We design compact (a few wavelength long) and efficient (>99%) injectors for coupling light into slow Bloch modes of periodic thin film stacks and of periodic slab waveguides. The study includes the derivation of closed-form expressions for the injection efficiency as a function of the group-velocity of injected light, and the proof that 100% coupling efficiencies for arbitrary small group velocities is possible with an injector length scaling as log(c/vg). The trade-off between the injector bandwidth and the group velocity of the injected light is also considered.

© 2007 Optical Society of America

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References

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  1. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (July 1997).
    [Crossref]
  2. J. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. 30, 2778–2780 (2005).
    [Crossref] [PubMed]
  3. J. Poon, L. Zhu, G. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. 31, 456–458 (2006).
    [Crossref] [PubMed]
  4. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
    [Crossref] [PubMed]
  5. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005).
    [Crossref] [PubMed]
  6. A. Yu. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004).
    [Crossref]
  7. D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005).
    [Crossref] [PubMed]
  8. T. F. Krauss, “Photonic Crystals shine on,” Phys. World32–36, (February 2006).
  9. A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
    [Crossref]
  10. G. Lenz, B. J. Eggleton, C. K. Madsen, and R .E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electronics 37, 525–532 (2001).
    [Crossref]
  11. Y. A. Vlasov and S. J. McNab, “Coupling into the slow light mode in slab-type photonic crystal waveguides,” Opt. Lett. 31, 50–52 (2006).
    [Crossref] [PubMed]
  12. M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tuneable time delays,” Opt. Express 13, 7145–7159 (2005).
    [Crossref] [PubMed]
  13. J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. 22, 1844–1849 (2005).
    [Crossref]
  14. P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).
  15. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E 53, 4107–4121 (1996).
    [Crossref]
  16. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).
  17. K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin2001) Chap. 11.
  18. B. Momeni and A. Adibi, “Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. 87, 171104 (2005).
    [Crossref]
  19. L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).
  20. C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
    [Crossref]
  21. P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
    [Crossref]
  22. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
    [Crossref]
  23. D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications 3, 120–128 (2005).
    [Crossref]
  24. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
    [Crossref]
  25. M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express 15, 219–226 (2007).
    [Crossref] [PubMed]
  26. H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London1969).
  27. R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

2007 (1)

2006 (3)

2005 (9)

M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tuneable time delays,” Opt. Express 13, 7145–7159 (2005).
[Crossref] [PubMed]

J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. 22, 1844–1849 (2005).
[Crossref]

B. Momeni and A. Adibi, “Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. 87, 171104 (2005).
[Crossref]

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005).
[Crossref] [PubMed]

J. Khurgin, “Expanding the bandwidth of slow-light photonic devices based on coupled resonators,” Opt. Lett. 30, 2778–2780 (2005).
[Crossref] [PubMed]

D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005).
[Crossref] [PubMed]

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications 3, 120–128 (2005).
[Crossref]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

2004 (2)

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (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]

2003 (2)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[Crossref]

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[Crossref]

2001 (1)

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

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
[Crossref]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (July 1997).
[Crossref]

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

Adibi, A.

B. Momeni and A. Adibi, “Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. 87, 171104 (2005).
[Crossref]

Andreani, L. C.

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications 3, 120–128 (2005).
[Crossref]

Baba, T.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

Coldren, L. A.

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).

Collin, R. E.

R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

Corzine, S. W.

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).

DeRose, G.

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E 53, 4107–4121 (1996).
[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 Electronics 37, 525–532 (2001).
[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.

Fan, S.

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[Crossref] [PubMed]

Gerace, D.

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications 3, 120–128 (2005).
[Crossref]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (July 1997).
[Crossref]

Haus, H. A.

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).

Hughes, S.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

Hugonin, J. P.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
[Crossref]

J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. 22, 1844–1849 (2005).
[Crossref]

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[Crossref]

Joannopoulos, J.

Johnson, S.

Khurgin, J.

Kira, G.

Krauss, T. F.

Kuipers, L.

Kuramochi, E.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
[Crossref]

Lalanne, P.

J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. 22, 1844–1849 (2005).
[Crossref]

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
[Crossref]

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[Crossref]

Lecamp, G.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
[Crossref]

Lenz, G.

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

Macleod, H. A.

H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London1969).

Madsen, C. K.

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

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[Crossref]

McNab, S. J.

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[Crossref]

Michaeli, A.

Mitsugi, S.

Momeni, B.

B. Momeni and A. Adibi, “Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. 87, 171104 (2005).
[Crossref]

Mori, D.

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[Crossref]

Notomi, M.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005).
[Crossref] [PubMed]

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]

Poon, J.

Povinelli, M.

Ramunno, L.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
[Crossref]

Sakoda, K.

K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin2001) Chap. 11.

Salib, M.

Sauvan, C.

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
[Crossref]

Scalora, M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

Settle, M. D.

Shinya, A.

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005).
[Crossref] [PubMed]

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

Slusher, R .E.

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

Tanabe, T.

Vlasov, Y. A.

Watanabe, T.

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
[Crossref]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
[Crossref]

Yanik, M. F.

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[Crossref] [PubMed]

Yariv, A.

Yeh, P.

P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

Zhu, L.

Appl. Phys. Lett. (2)

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

B. Momeni and A. Adibi, “Adiabatic stage for coupling of light to extended Bloch modes of photonic crystal,” Appl. Phys. Lett. 87, 171104 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

P. Lalanne and J. P. Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[Crossref]

IEEE J. Quantum Electronics (1)

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

J. Opt. Soc. Am. A. (1)

J. P. Hugonin and P. Lalanne, “Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A. 22, 1844–1849 (2005).
[Crossref]

Opt. Exp. (1)

C. Sauvan, G. Lecamp, P. Lalanne, and J. P. Hugonin, “Modal-reflectivity enhancement by geometry tuning in photonic crystal microcavities,” Opt. Exp. 13, 245–255 (2005).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and Nonlinear propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365–379 (2003).
[Crossref]

Photonics and Nanostructures-fundamentals and applications (1)

D. Gerace and L. C. Andreani, “Effects of disorder on propagation losses and cavity Q-factors in photonic crystal slabs,” Photonics and Nanostructures-fundamentals and applications 3, 120–128 (2005).
[Crossref]

Phys. Rev. B (1)

E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005).
[Crossref]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structure,” Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

Phys. Rev. Lett. (1)

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[Crossref] [PubMed]

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (July 1997).
[Crossref]

Phys. World (1)

T. F. Krauss, “Photonic Crystals shine on,” Phys. World32–36, (February 2006).

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead Simplex Method in Low Dimensions,” SIAM J. Optim. 9,112–147 (1998).
[Crossref]

Other (6)

H. A. Macleod, Thin-film optical filters, (Adam Hilger LTD, London1969).

R. E. Collin, Field theory of guided waves, (London, Section 9 1960).

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall International, London, 1984).

K. Sakoda, Optical properties of photonic crystals, (Springer-Verlag, Berlin2001) Chap. 11.

L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, (J. Wiley and Sons, New York, 1995).

P. Yeh, Optical waves in layered media, (J. Wiley and Sons, New York1988).

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Figures (4)

Fig. 1.
Fig. 1.

Impedance mismatch problem. (a): Scattering at an interface between a uniform medium (refractive index n) and a semi-infinite bi-layer periodic stack (refractive indices nH and nL). (b) and (c): Injection efficiency T as a function of the group-velocity of the periodic-stack Bloch mode for different fill factors (0.1<f<0.9). Colormaps and white curves are exact numerical results obtained with the 2×2 transfer-matrix method in [14], and the superimposed solid-red curves are obtained using the approximate closed-form expressions. (b): data obtained in the vicinity of the valence band edge of a weak-modulation (nH=1.5 and nL=1.4) stack for n=nH. (c): data obtained in the vicinity of the conduction band edge of a strong-modulation stack (nH=3.495 and nL=1) for n=nL.

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Fig. 2.
Fig. 2.

A slow-mode injector at a single frequency is basically a mirror. (a) Definition of the injector parameters : length L , interface S1 between the coupler and the uniform medium and interface S2 between the coupler and the periodic stack supporting a slow Bloch mode. (b) Perfect injector at a single frequency for a plane wave incident from the uniform medium. (c) Reciprocal problem. The incident illumination is the reciprocal Bloch mode propagating towards the negative z-direction. (d) General solution of the synthesis problem : the injector is composed of a mirror and of a phase plate.

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Fig. 3.
Fig. 3.

Coupling efficiency into the slow-mode of a periodic stack (nH, nL) as a function of the group velocity (log scale) of the slow mode. Different injectors are considered. They are all composed of Bragg mirrors (nH, nL) with an increasing number of repeated pairs, m=1, …5. The results are obtained in the vicinity of the valence band edge of the periodic stack. The horizontal arrows indicate the bandwidths in GHz of the different injectors for T=0.8 (-1dB).

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Fig. 4.
Fig. 4.

Broadband injection from a planar waveguide to a periodic waveguide near the valence-band edge (λ=1.65 μm). (a) Injector geometry optimized for coupling at vg/c≈0.01. From left to right, the injector slit- and ridge-widths are 84, 143,127, 158, 174 nm and 239, 213, 173, 166, 166 nm, respectively. The superimposed red curve represents the squared modulus of the transverse electric field at optimal coupling. (b) Performance of the injector predicted by fully vectorial computational results for the radiation loss L (dashed curve), the modal reflection R (blue circles) and the transmission T (actually 1-T is shown with a solid black curve). All quantities are displayed in a log scale. Inset: Injection efficiency T as a function of vg with (red) and without (blue) injector. The maximum injection efficiency is as large as 0.999. In the absence of injector, this efficiency is only 0.25%.

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Equations (11)

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∣ B + ( z = z 0 + p a ) > = A H ∣ P + > + B H ∣ P − > ,
0 < u < 1 ,
− 1 < u < 0 ,
∣ B − ( z = z 0 + p a ) > = A ' H ∣ P + > + B ' H ∣ P − > ,
r = [ ( n − n H ) + u ( n + n H ) exp ( jϕ ) ] [ ( n + n H ) + u ( n − n H ) exp ( jϕ ) ] ,
( v g c ) [ α ( 1 + u 2 ) + 2 β u ] = 2 n H ( 1 − u 2 ) ,
u VB = 1 − ( α + β VB 2 n H ) v g c + O ( v g c ) 2 , and
u CB = − 1 + ( α − β CB 2 n H ) v g c + O ( v g c ) 2 ,
ϕ ( 0 ) ≈ πf n H [ 1 ± ε 1 ( 2 ε 0 ) ] ε 0 1 2 ,
T VB = ( α + β VB n H ) v g c , T CB = ( α − β CB n H ) v g c .
L = gλ log ( c v g ) ,

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