Abstract

We present a theoretical analysis of light propagation in one-dimensional resonant photonic bandgap structures (RPBGs). The analysis is aimed at evaluating the feasibility of controlled stopping, storing, and releasing of light pulses by parametric manipulation of the RPBG’s bandstructure. First we lay the conceptual foundation of light-pulse delay by means of band structure control in infinite RPBGs, and then we contrast the idealized concepts with numerical results for realistic, finite-sized RPBGs. For a physical model for RPBGs, we use semiconductor quantum-well Bragg structures, but the general analysis is valid for a wider class of RPBG. We show that the usefulness of RPBGs for optical delay lines depends critically on the number of quantum wells and the dephasing and loss mechanisms in each unit cell of the RPBG, and we also outline optimization strategies in terms of spectral light characteristics as well as quasi-antireflection coating of the RPBGs.

© 2005 Optical Society of America

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  4. M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
    [CrossRef]
  5. J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
    [CrossRef]
  6. T. Ikawa and K. Cho, "Fate of superradiant mode in a resonant Bragg reflector," Phys. Rev. B 66, 085338 (2002).
    [CrossRef]
  7. S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, "Gap solitons in a two-channel microresonator structure," Opt. Lett. 27, 536-538 (2002).
    [CrossRef]
  8. N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."
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  10. M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2003).
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    [CrossRef] [PubMed]
  13. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
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  14. W. N. Xiao, J. Y. Zhou, and J. P. Prineas, "Storage of ultrashort optial pulses in a resonantly absorbing Bragg reflector," Opt. Express 11, 3277-3283 (2003).
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    [CrossRef] [PubMed]
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  21. A. E. Kozhekin, G. Kurizki, and B. Malomed, "Standing and moving gap solitons in resonantly absorbing gratings," Phys. Rev. Lett. 81, 3647-3650 (1998).
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    [CrossRef]
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    [CrossRef]
  29. I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
    [CrossRef] [PubMed]
  30. M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
    [CrossRef]
  31. V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
    [CrossRef]
  32. P. Chak, J. E. Sipe, and S. Pereira, "Lorentzian model for nonlinear switching in a microresonator structure," Opt. Commun. 213, 163-171 (2002).
    [CrossRef]
  33. We are grateful to J. Sipe, University of Toronto, for suggesting this approach to the bandstructure to us.
  34. L. I. Deych and A. A. Lisyansky, "Polariton dispersion law in periodic-Bragg and near-Bragg multiple quantum well structures," Phys. Rev. B 62, 4242-4244 (2000).
    [CrossRef]
  35. L. C. Andreani, "Exciton-polaritons in superlattices," Phys. Lett. A 192, 99-109 (1994).
    [CrossRef]
  36. D. Citrin, "Material and optical approaches to exciton polaritons in multiple quantum wells: formal results," Phys. Rev. B 50, 5497-5505 (1994).
    [CrossRef]
  37. N. Ashcroft and N. Mermin, Solid State Physics (Sounders College Publishing, 1976).
  38. G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
    [CrossRef]
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    [CrossRef]

2005 (2)

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, "Toward the slowing and storage of light," Opt. Photonics News 16(2), 36-40 (February 2005).
[CrossRef]

J. Khurgin, "Expanding the bandwidth of slow-light photonic devices based on coupled resonators," Opt. Lett. 30, 513-515 (2005).
[CrossRef] [PubMed]

2004 (3)

P. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. L. Wang, S. W. Chang, and S. L. Chuang, "Slow light in semiconductor quantum wells," Opt. Lett. 29, 2291-2293 (2004).
[CrossRef] [PubMed]

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

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

2003 (4)

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

W. N. Xiao, J. Y. Zhou, and J. P. Prineas, "Storage of ultrashort optial pulses in a resonantly absorbing Bragg reflector," Opt. Express 11, 3277-3283 (2003).
[CrossRef] [PubMed]

M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

2002 (4)

A. Andre and M. Lukin, "Manipulating light pulses via dynamically controlled photonic band gap," Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

T. Ikawa and K. Cho, "Fate of superradiant mode in a resonant Bragg reflector," Phys. Rev. B 66, 085338 (2002).
[CrossRef]

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, "Gap solitons in a two-channel microresonator structure," Opt. Lett. 27, 536-538 (2002).
[CrossRef]

P. Chak, J. E. Sipe, and S. Pereira, "Lorentzian model for nonlinear switching in a microresonator structure," Opt. Commun. 213, 163-171 (2002).
[CrossRef]

2001 (3)

G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
[CrossRef]

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]

O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, "Stopping light via hot atoms," Phys. Rev. Lett. 86, 628-631 (2001).
[CrossRef] [PubMed]

2000 (2)

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

L. I. Deych and A. A. Lisyansky, "Polariton dispersion law in periodic-Bragg and near-Bragg multiple quantum well structures," Phys. Rev. B 62, 4242-4244 (2000).
[CrossRef]

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metrespersecond in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

1998 (1)

A. E. Kozhekin, G. Kurizki, and B. Malomed, "Standing and moving gap solitons in resonantly absorbing gratings," Phys. Rev. Lett. 81, 3647-3650 (1998).
[CrossRef]

1996 (2)

T. Stroucken, A. Knorr, P. Thomas, and S. W. Koch, "Coherent dynamics of radiatively coupled quantum-well excitons," Phys. Rev. B 53, 2026-2033 (1996).
[CrossRef]

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

1995 (3)

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, "Electromagnetically induced transparency: propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

L. C. Andreani, "Polaritons in multiple quantum wells," Phys. Status Solidi B 188, 29-42 (1995).
[CrossRef]

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
[CrossRef] [PubMed]

1994 (3)

L. C. Andreani, "Exciton-polaritons in superlattices," Phys. Lett. A 192, 99-109 (1994).
[CrossRef]

D. Citrin, "Material and optical approaches to exciton polaritons in multiple quantum wells: formal results," Phys. Rev. B 50, 5497-5505 (1994).
[CrossRef]

E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum wells," Phys. Solid State 36, 1156-1161 (1994).

1986 (1)

B. I. Mantsyzov and R. N. Kuzmin, "Coherent interaction of light with a discrete periodic resonant medium," Sov. Phys. JETP 64, 37-44 (1986).

Agarwal, V.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Andre, A.

A. Andre and M. Lukin, "Manipulating light pulses via dynamically controlled photonic band gap," Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

Andreani, L. C.

L. C. Andreani, "Polaritons in multiple quantum wells," Phys. Status Solidi B 188, 29-42 (1995).
[CrossRef]

L. C. Andreani, "Exciton-polaritons in superlattices," Phys. Lett. A 192, 99-109 (1994).
[CrossRef]

Ashcroft, N.

N. Ashcroft and N. Mermin, Solid State Physics (Sounders College Publishing, 1976).

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metrespersecond in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Binder, R.

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

Bonetti, G.

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, "Gap solitons in a two-channel microresonator structure," Opt. Lett. 27, 536-538 (2002).
[CrossRef]

Chak, P.

P. Chak, J. E. Sipe, and S. Pereira, "Lorentzian model for nonlinear switching in a microresonator structure," Opt. Commun. 213, 163-171 (2002).
[CrossRef]

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

Chang, S. W.

Chang-Hasnain, C. J.

Cho, K.

T. Ikawa and K. Cho, "Fate of superradiant mode in a resonant Bragg reflector," Phys. Rev. B 66, 085338 (2002).
[CrossRef]

Chuang, S. L.

Citrin, D.

D. Citrin, "Material and optical approaches to exciton polaritons in multiple quantum wells: formal results," Phys. Rev. B 50, 5497-5505 (1994).
[CrossRef]

Coquillat, D.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

del Rio, J. A.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Deutsch, I. H.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
[CrossRef] [PubMed]

Deych, L. I.

L. I. Deych and A. A. Lisyansky, "Polariton dispersion law in periodic-Bragg and near-Bragg multiple quantum well structures," Phys. Rev. B 62, 4242-4244 (2000).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metrespersecond in an ultracold atomic gas," Nature 397, 594-598 (1999).
[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]

Ell, C.

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

Fan, S.

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

Gaburro, Z.

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

Ghulinyan, M.

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

Gibbs, H. M.

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

Gil, B.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metrespersecond in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, "Electromagnetically induced transparency: propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metrespersecond in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, 2004).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1955).

Heebner, J. E.

Hey, R.

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

Hübner, M.

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

Ikawa, T.

T. Ikawa and K. Cho, "Fate of superradiant mode in a resonant Bragg reflector," Phys. Rev. B 66, 085338 (2002).
[CrossRef]

Imamoglu, A.

M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2003).
[CrossRef]

Ivchenko, E. L.

E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum wells," Phys. Solid State 36, 1156-1161 (1994).

Jahnke, F.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

Jain, M.

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, "Electromagnetically induced transparency: propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

Joannopoulos, J.

J. Joannopoulos, R. Meade and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Jorda, S.

E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum wells," Phys. Solid State 36, 1156-1161 (1994).

Kasapi, A.

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, "Electromagnetically induced transparency: propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

Kavokin, A.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Khitrova, G.

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

Khurgin, J.

Kira, M.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

Knorr, A.

T. Stroucken, A. Knorr, P. Thomas, and S. W. Koch, "Coherent dynamics of radiatively coupled quantum-well excitons," Phys. Rev. B 53, 2026-2033 (1996).
[CrossRef]

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

Koch, S. W.

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

T. Stroucken, A. Knorr, P. Thomas, and S. W. Koch, "Coherent dynamics of radiatively coupled quantum-well excitons," Phys. Rev. B 53, 2026-2033 (1996).
[CrossRef]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, 2004).
[CrossRef]

Kocharovskaya, O.

O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, "Stopping light via hot atoms," Phys. Rev. Lett. 86, 628-631 (2001).
[CrossRef] [PubMed]

Kozhekin, A. E.

G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
[CrossRef]

A. E. Kozhekin, G. Kurizki, and B. Malomed, "Standing and moving gap solitons in resonantly absorbing gratings," Phys. Rev. Lett. 81, 3647-3650 (1998).
[CrossRef]

Ku, P.

Kuhl, J.

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

Kurizki, G.

G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
[CrossRef]

A. E. Kozhekin, G. Kurizki, and B. Malomed, "Standing and moving gap solitons in resonantly absorbing gratings," Phys. Rev. Lett. 81, 3647-3650 (1998).
[CrossRef]

Kuzmin, R. N.

B. I. Mantsyzov and R. N. Kuzmin, "Coherent interaction of light with a discrete periodic resonant medium," Sov. Phys. JETP 64, 37-44 (1986).

Kwong, N. H.

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

Lee, E.

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[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 Electron. 37, 525-532 (2001).
[CrossRef]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Li, T.

Lisyansky, A. A.

L. I. Deych and A. A. Lisyansky, "Polariton dispersion law in periodic-Bragg and near-Bragg multiple quantum well structures," Phys. Rev. B 62, 4242-4244 (2000).
[CrossRef]

Lukin, M.

A. Andre and M. Lukin, "Manipulating light pulses via dynamically controlled photonic band gap," Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

Lukin, M. D.

M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2003).
[CrossRef]

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 Electron. 37, 525-532 (2001).
[CrossRef]

Malomed, B.

A. E. Kozhekin, G. Kurizki, and B. Malomed, "Standing and moving gap solitons in resonantly absorbing gratings," Phys. Rev. Lett. 81, 3647-3650 (1998).
[CrossRef]

Malomed, B. A.

G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
[CrossRef]

Malpuech, G.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Mantsyzov, B. I.

B. I. Mantsyzov and R. N. Kuzmin, "Coherent interaction of light with a discrete periodic resonant medium," Sov. Phys. JETP 64, 37-44 (1986).

Meade, R.

J. Joannopoulos, R. Meade and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mermin, N.

N. Ashcroft and N. Mermin, Solid State Physics (Sounders College Publishing, 1976).

Nesvizhskii, A. I.

E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum wells," Phys. Solid State 36, 1156-1161 (1994).

Opatrny, T.

G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
[CrossRef]

Oton, C. J.

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants in Solids (Academic, 1985).

Palinginis, P.

Paloczi, G. T.

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, "Toward the slowing and storage of light," Opt. Photonics News 16(2), 36-40 (February 2005).
[CrossRef]

Pavesi, L.

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

Pereira, S.

P. Chak, J. E. Sipe, and S. Pereira, "Lorentzian model for nonlinear switching in a microresonator structure," Opt. Commun. 213, 163-171 (2002).
[CrossRef]

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, "Gap solitons in a two-channel microresonator structure," Opt. Lett. 27, 536-538 (2002).
[CrossRef]

Phillips, W. D.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
[CrossRef] [PubMed]

Ploog, K.

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

Poon, J. K. S.

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, "Toward the slowing and storage of light," Opt. Photonics News 16(2), 36-40 (February 2005).
[CrossRef]

Prineas, J. P.

W. N. Xiao, J. Y. Zhou, and J. P. Prineas, "Storage of ultrashort optial pulses in a resonantly absorbing Bragg reflector," Opt. Express 11, 3277-3283 (2003).
[CrossRef] [PubMed]

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

Rolston, S. L.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
[CrossRef] [PubMed]

Rostovtsev, Y.

O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, "Stopping light via hot atoms," Phys. Rev. Lett. 86, 628-631 (2001).
[CrossRef] [PubMed]

Scalbert, D.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Scheuer, J.

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, "Toward the slowing and storage of light," Opt. Photonics News 16(2), 36-40 (February 2005).
[CrossRef]

Scully, M. O.

O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, "Stopping light via hot atoms," Phys. Rev. Lett. 86, 628-631 (2001).
[CrossRef] [PubMed]

Sedgwick, F.

Shah, J.

J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures, Vol. 115 of Springer Series in Solid-State Sciences (Springer-Verlag, 1996).
[CrossRef]

Sipe, J.

We are grateful to J. Sipe, University of Toronto, for suggesting this approach to the bandstructure to us.

Sipe, J. E.

P. Chak, J. E. Sipe, and S. Pereira, "Lorentzian model for nonlinear switching in a microresonator structure," Opt. Commun. 213, 163-171 (2002).
[CrossRef]

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, "Gap solitons in a two-channel microresonator structure," Opt. Lett. 27, 536-538 (2002).
[CrossRef]

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

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 Electron. 37, 525-532 (2001).
[CrossRef]

Smirl, A. L.

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

Spreeuw, R. J. C.

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
[CrossRef] [PubMed]

Stroucken, T.

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

T. Stroucken, A. Knorr, P. Thomas, and S. W. Koch, "Coherent dynamics of radiatively coupled quantum-well excitons," Phys. Rev. B 53, 2026-2033 (1996).
[CrossRef]

Thomas, P.

T. Stroucken, A. Knorr, P. Thomas, and S. W. Koch, "Coherent dynamics of radiatively coupled quantum-well excitons," Phys. Rev. B 53, 2026-2033 (1996).
[CrossRef]

Vladimirova, M.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Wang, H. L.

Winn, J.

J. Joannopoulos, R. Meade and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Xiao, W. N.

Yang, Z. S.

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

Yanik, M. F.

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

Yariv, A.

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, "Toward the slowing and storage of light," Opt. Photonics News 16(2), 36-40 (February 2005).
[CrossRef]

Yin, G. Y.

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, "Electromagnetically induced transparency: propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

Zamfirescu, M.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

Zhou, J. Y.

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).
[CrossRef]

J. Appl. Phys. (1)

M. Ghulinyan, C. J. Oton, G. Bonetti, Z. Gaburro, and L. Pavesi, "Free-standing porous silicon single and multiple optical cavities," J. Appl. Phys. 93, 9724-9729 (2003).
[CrossRef]

Nature (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metrespersecond in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature 413, 273-276 (2003).
[CrossRef]

Opt. Commun. (1)

P. Chak, J. E. Sipe, and S. Pereira, "Lorentzian model for nonlinear switching in a microresonator structure," Opt. Commun. 213, 163-171 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Opt. Photonics News (1)

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, "Toward the slowing and storage of light," Opt. Photonics News 16(2), 36-40 (February 2005).
[CrossRef]

Phys. Lett. A (1)

L. C. Andreani, "Exciton-polaritons in superlattices," Phys. Lett. A 192, 99-109 (1994).
[CrossRef]

Phys. Rev. A (1)

I. H. Deutsch, R. J. C. Spreeuw, S. L. Rolston, and W. D. Phillips, "Photonic band gaps in optical lattices," Phys. Rev. A 52, 1394-1410 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (5)

L. I. Deych and A. A. Lisyansky, "Polariton dispersion law in periodic-Bragg and near-Bragg multiple quantum well structures," Phys. Rev. B 62, 4242-4244 (2000).
[CrossRef]

J. P. Prineas, C. Ell, E. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, "Exciton polariton eigenmodes in light-coupled InGaAs/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000).
[CrossRef]

T. Ikawa and K. Cho, "Fate of superradiant mode in a resonant Bragg reflector," Phys. Rev. B 66, 085338 (2002).
[CrossRef]

T. Stroucken, A. Knorr, P. Thomas, and S. W. Koch, "Coherent dynamics of radiatively coupled quantum-well excitons," Phys. Rev. B 53, 2026-2033 (1996).
[CrossRef]

D. Citrin, "Material and optical approaches to exciton polaritons in multiple quantum wells: formal results," Phys. Rev. B 50, 5497-5505 (1994).
[CrossRef]

Phys. Rev. Lett. (7)

M. Hübner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, "Collective effects of excitons in multiple quantum well Bragg and anti-Bragg structures," Phys. Rev. Lett. 76, 4199-4202 (1996).
[CrossRef]

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, "Electromagnetically induced transparency: propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

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

O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, "Stopping light via hot atoms," Phys. Rev. Lett. 86, 628-631 (2001).
[CrossRef] [PubMed]

A. Andre and M. Lukin, "Manipulating light pulses via dynamically controlled photonic band gap," Phys. Rev. Lett. 89, 143602 (2002).
[CrossRef] [PubMed]

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch oscillations in porous silicon optical superlattices," Phys. Rev. Lett. 92, 097, 401 (2004).
[CrossRef]

A. E. Kozhekin, G. Kurizki, and B. Malomed, "Standing and moving gap solitons in resonantly absorbing gratings," Phys. Rev. Lett. 81, 3647-3650 (1998).
[CrossRef]

Phys. Solid State (1)

E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, "Bragg reflection of light from quantum wells," Phys. Solid State 36, 1156-1161 (1994).

Phys. Status Solidi B (1)

L. C. Andreani, "Polaritons in multiple quantum wells," Phys. Status Solidi B 188, 29-42 (1995).
[CrossRef]

Prog. Opt. (1)

G. Kurizki, A. E. Kozhekin, T. Opatrny, and B. A. Malomed, "Optical solitons in periodic media with resonant and off-resonant nonlinearities," Prog. Opt. 42, 93-146 (2001).
[CrossRef]

Rev. Mod. Phys. (1)

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

Science (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

B. I. Mantsyzov and R. N. Kuzmin, "Coherent interaction of light with a discrete periodic resonant medium," Sov. Phys. JETP 64, 37-44 (1986).

Other (9)

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, 2004).
[CrossRef]

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1955).

E. D. Palik, Handbook of Optical Constants in Solids (Academic, 1985).

We are grateful to J. Sipe, University of Toronto, for suggesting this approach to the bandstructure to us.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

J. Joannopoulos, R. Meade and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

N. H. Kwong, P. Chak, J. E. Sipe, Z. S. Yang, R. Binder, and A. L. Smirl are preparing a manuscript titled "A unified analysis of quantum well and microresonator resonant photonic bandgap structures."

J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures, Vol. 115 of Springer Series in Solid-State Sciences (Springer-Verlag, 1996).
[CrossRef]

N. Ashcroft and N. Mermin, Solid State Physics (Sounders College Publishing, 1976).

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Figures (13)

Fig. 1
Fig. 1

Schematic of the quantum-well RPBG structure. The vertical solid lines represent infinitely thin quantum wells, spaced by distance a; the background refractive index is n b ; E + and E are the forward- and backward-traveling field components, respectively, in the uniform medium between each pair of quantum wells.

Fig. 2
Fig. 2

(a) Photonic band structure of the model quantum-well Bragg RPBG with a single exciton resonance, ω x = 1.496 eV , which is below (but approximately equal to) Bragg resonance ω B = 1.497 eV . The presence of the exciton resonance leads to the almost horizontal IB. (b) The IB shown in (a) on an expanded scale.

Fig. 3
Fig. 3

Relation between ω and the imaginary part of K. The regions of the nonzero imaginary part of K correspond to the photonic bandgaps. The IB is clearly seen as a region with zero imaginary part of K around 1.496 eV .

Fig. 4
Fig. 4

Imaginary part of ω as a function of real K for the IB close to the zone boundary. Away from the zone boundary, the imaginary part of ω approaches γ . Inset, relation between the imaginary and the real parts of ω within the IB.

Fig. 5
Fig. 5

(a) Reflection spectrum of the quantum-well Bragg structure with 2000 wells, without AR coatings, (b) the same as (a) but only for the spectral region of the IB.

Fig. 6
Fig. 6

Reflection spectrum in the spectral region of the IB for the 200 quantum-well structure, without AR coatings. Inset, the spectrum over a wider frequency range, covering the IB and parts of the upper and lower bands.

Fig. 7
Fig. 7

Same as Fig. 5 but for the 2000 well structure with AR coating.

Fig. 8
Fig. 8

Same as Fig. 7 but for 200 quantum wells.

Fig. 9
Fig. 9

Top, time dependence of exciton resonance ω x and bottom, the transmitted pulse intensity normalized to the peak input pulse intensity E 0 2 , for 2000 quantum wells. Two cases are shown: Dotted curves, labeled (a), are for the case when the light is not stopped; the solid curves, labeled (b), are for the case when the light is stopped for 8.2 ps inside the RPBG. The input field at the front end of the RPBG versus time (dashed) is also shown.

Fig. 10
Fig. 10

Similar to Fig. 9 but for 200 quantum wells. Here the fields are plotted on a logarithmic scale. The dotted curves, labeled (a), are for a stopping time of 3.5 ps , and the solid curves, labeled (b), are for a stopping time of 6.2 ps . The first peaks of the two curves coincide. The input pulse is shown as the dashed–dotted curve.

Fig. 11
Fig. 11

Spatial distribution of the pulse inside the quantum-well Bragg structure (2000 wells) at different times for case (b) of Fig. 9. For clarity, the intensity of the forward- ( n b E + 2 ) and the negative intensity of the backward- ( n b E 2 ) traveling components are shown separately.

Fig. 12
Fig. 12

Logarithmic plot of the relation between the transmitted pulse energy (in units of incident pulse energy ε in ) and delay time T d (in units of the pulse duration T w ). The solid line is the result for 2000 quantum wells (Fig. 9). The dotted line shows the energy of the secondary transmission peak for 200 quantum wells (Fig. 10). For comparison, the dashed line shows the result for 2000 wells with all the parameters identical to those for the solid line, except that the center frequency of the incident light is shifted by 0.4 meV . For the 2000 quantum-well structure a pulse duration of T w = 6 ps was used, and for the 200 well structure, T w = 3 ps . Also, for comparison in the case of T w = 6 ps (2000 well structure), we plot a hypothetical decay curve (dashed–dotted line) with a decay rate of 2 γ . The corresponding hypothetical decay curve for T w = 3 ps (200 well structure) is not shown; it is close to the actual result shown as the dotted line.

Fig. 13
Fig. 13

Group velocity as a function of K in the IB. Inset, group velocity as a function of ω .

Equations (47)

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

( 2 z 2 n b 2 c 2 2 t 2 ) E ( z , t ) = 4 π c 2 2 t 2 P res ( z , t ) ,
t p m ( t ) = i [ ω x ( t ) i γ ] p m ( t ) + i μ ϕ ̃ ( 0 ) E ( z m , t ) ,
ϕ ̃ ( 0 ) = k ϕ ( k ) = 2 2 a 0 π ,
P res ( z ) = μ ϕ ̃ ( 0 ) m = δ ( z z m ) p m .
E ( z , t ) = E + ( z , t ) E ( z , t ) = E + ( z m + ϵ , t n b z z m c ) E ( z m + 1 ϵ , t n b z m + 1 z c ) , z m < z < z m + 1 , ϵ 0 + ,
z E ( z m + ϵ , t ) z E ( z m ϵ , t ) = 4 π c 2 μ ϕ ̃ ( 0 ) 2 t 2 p m ( t ) ,
E ( z m ϵ , t ) = E ( z m + ϵ , t ) = E ( z m , t ) .
E m , R + ( t ) = E m , L + ( t ) 2 π n b c μ ϕ ̃ ( 0 ) t p m ( t ) ,
E m , R ( t ) = E m , L ( t ) 2 π n b c μ ϕ ̃ ( 0 ) t p m ( t ) ,
E m , L ± ( t ) = E ± ( z m ϵ , t ) , E m , R ± ( t ) = E ± ( z m + ϵ , t ) .
p m ( ω ) = χ ( ω ) E ( z m , ω ) , χ ( ω ) = μ ϕ ̃ ( 0 ) 1 ω ( ω x i γ ) ,
[ E m + 1 , R + ( ω ) E m + 1 , R ( ω ) ] = M ¯ ( ω ) [ E m , R + ( ω ) E m , R ( ω ) ] ,
M ¯ ( ω ) = [ α ( 1 + β ) ( 1 α ) β α β ( 1 α ) ( 1 β ) ]
β = i 2 π ω n b c μ ϕ ̃ ( 0 ) χ ( ω ) = i Γ 1 ( ω x i γ ) ω ,
Γ = 2 π n b c μ 2 ϕ ̃ ( 0 ) 2 ,
α = exp ( i q α ) , q = n b ( ω c ) .
M ¯ ( ω ) [ E 0 , R + ( ν , K ) E 0 , R ( ν , K ) ] = exp ( i K a ) [ E 0 , R + ( ν , K ) E 0 , R ( ν , K ) ] ,
exp ( i K ± a ) = [ α ( 1 + β ) + ( 1 α ) ( 1 β ) ] ± { [ α ( 1 + β ) + ( 1 α ) ( 1 β ) ] 2 4 } 1 2 2 ,
[ E m , R + ( ν , K ) E m , R ( ν , K ) ] = exp ( i m K a ) [ E 0 , R + ( ν , K ) E 0 , R ( ν , K ) ] .
E ( z , ω ) = A E ( ν , K ) B E ( ν , K ) ,
E ( ν , K ) = u ν , K ( z ) exp ( i K z ) ,
u ν , K ( z ) = E 0 , R + ( ν , K ) exp [ i ( q K ) ( z z m ) ] E 0 , R ( ν , K ) exp [ i ( q + K ) ( z z m ) ] ,
u ν , K ( z + a ) = u ν , K ( z ) , E ( ν , K + 2 π a ) = E ( ν , K ) ,
cos ( K a ) = 1 2 [ α + 1 α + β ( α 1 α ) ] .
cos ( K a ) = cos ( q a ) + Γ ( ω + i γ ) ω x 1 ( ω ω x ) sin ( q a ) .
ω x = 1.496 eV , ω B = 1.497 eV , n b = 3.61 ,
a 0 = 15.0 nm , μ = q e ( 0.243 nm ) , γ = 0 ,
Γ = 2 π n b c μ 2 ϕ ̃ ( 0 ) 2 = 8.50 × 10 6 .
ω ω B = 1 + cos ( K a ) 1 + cos ( K a ) + π Γ [ ( ω x i γ ) ω B ] .
ω ω B = 1 + cos ( K a ) 1 + cos ( K a ) + π Γ ( ω x ω B ) ,
ω ( K = ± π ) = ω B , ω ( K = 0 ) = ω B + 2 2 + π Γ ( ω x ω B ) .
R { ω } ω B = 1 + cos ( K a ) 1 + cos ( K a ) + π Γ ( ω x ω B ) ,
I { ω } = 1 + cos ( K a ) 1 + cos ( K a ) + π Γ γ .
I { ω } = R { ω } ω B ω x ω B γ .
( E L + E L ) = [ n R + n L 2 n L n R n L 2 n L n R n L 2 n L n R + n L 2 n L ] ( E R + E R ) .
n 1 2 = n n 0 ( n 2 2 n 3 2 ) .
n 1 = 1.59 , n 2 = 1.15 , n 3 = 3.61 .
ω x ( t ) = { 1.496 eV t < t 1 1.496 eV + 0.001 eV ( t t 1 ) ( t 2 t 1 ) t 1 < t < t 2 1.497 eV t 2 < t < t 3 1.497 eV 0.001 eV ( t t 3 ) ( t 4 t 3 ) t 3 < t < t 4 1.496 eV t > t 4 } .
v g = ω K = π Γ sin ( K a ) ( 1 + cos ( K a ) + π Γ ) 2 ( ω x ω B ) a .
E un ( z ) = E un + ( z ) E un ( z ) = A un exp ( i n un ω c z ) B un exp ( i n un ω c z ) , z < z x ,
E ps ( z ) = E ps + ( z ) E ps ( z ) = A ps u ν , K ( z ) exp ( i K z ) B ps u ν , K ( z ) exp ( i K z ) , z > z x ,
z E un ( z x ϵ ) = z E ps ( z x + ϵ ) ,
E un ( z x ϵ ) = E ps ( z x + ϵ ) .
[ E un + ( z x ϵ ) E un ( z x ϵ ) ] = [ n ps ( 1 ) ( ν , K ) + n un 2 n un n ps ( 2 ) ( ν , K ) n un 2 n un n ps ( 1 ) ( ν , K ) n un 2 n un n ps ( 2 ) ( ν , K ) + n un 2 n un ] [ E ps + ( z x + ϵ ) E ps ( z x + ϵ ) ] ,
n ps ( 1 ) ( ν , K ) i K u ν , K ( z x + ϵ ) + u ν , K ( z x + ϵ ) i ω c u ν , K ( z x + ϵ ) = n b [ E 0 R + ( ν , K ) E 0 R ( ν , K ) ] + 1 [ E 0 R + ( ν , K ) E 0 R ( ν , K ) ] 1 ,
n ps ( 2 ) ( ν , K ) i K u ν , K ( z x + ϵ ) u ν , K ( z x + ϵ ) i ω c u ν , K ( z x + ϵ ) = n b [ E 0 R ( ν , K ) E 0 R + ( ν , K ) ] + 1 [ E 0 R ( ν , K ) E 0 R + ( ν , K ) ] 1 .
[ E un + ( z x ϵ ) E un ( z x ϵ ) ] = [ n ps + n un 2 n un n ps n un 2 n un n ps n un 2 n un n ps + n un 2 n un ] [ E ps + ( z x + ϵ ) E ps ( z x + ϵ ) ] ,

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