Abstract

We develop a strategy to reduce large reflection losses from quantum-well Bragg structures through an antireflection coating technique. It is based on generalized refractive indices, which we call “effective coupling indices” (ECIs), that can be introduced to describe the coupling of light into quantum-well Bragg structures. For the example of a spectrally narrow band, which is relevant for slow-light applications, we clarify the dependence of the ECIs on the spectral bandwidth and discuss the relation between the ECIs and the group-velocity index. Numerical simulations of reflection spectra demonstrate the effectiveness of the ECI concept.

© 2007 Optical Society of America

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  1. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  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]
  3. 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] [PubMed]
  4. 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]
  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]
  6. T. Ikawa and K. Cho, "Fate of superradiant mode in a resonant Bragg reflector," Phys. Rev. B 66, 085338 (2002).
    [CrossRef]
  7. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, "Stopping, storing and releasing light in quantum well Bragg structures," J. Opt. Soc. Am. B 22, 2144-2156 (2005).
    [CrossRef]
  8. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, "Distortionless light pulse delay in quantum-well Bragg structures," Opt. Lett. 30, 2790-2792 (2005).
    [CrossRef] [PubMed]
  9. A. Andre and M. Lukin, "Manipulating light pulses via dynamically controlled photonic band gap," Phys. Rev. Lett. 89, 143602 (2002).
    [CrossRef] [PubMed]
  10. M. F. Yanik and S. Fan, "Stopping light all optically," Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]
  11. T. Baba and D. Ohsaki, "Interface of photonic crystals for high efficiently light transmission," Jpn. J. Appl. Phys. 40, 5920-5920 (2001).
    [CrossRef]
  12. J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
    [CrossRef]
  13. B. Momeni and A. Adibi, "Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals," Appl. Phys. Lett. 87, 171104 (2005).
    [CrossRef]
  14. Note that in it is called "effective refractive index."
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  16. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1955).
  17. E. D. Palik, Handbook of Optical Constants in Solids (Academic, 1985).
  18. Note that the last sentence of Appendix in contains an ambiguous formulation. In the IB, there exists a close relation between neff and the group velocity index, not the phase index as could be understood from that sentence.
  19. L. C. Andreani, "Exciton-polaritons in superlattices," Phys. Lett. A 192, 99-109 (1994).
    [CrossRef]
  20. D. Citrin, "Material and optical approaches to exciton polaritons in multiple quantum wells: formal results," Phys. Rev. B 50, 5497-5505 (1994).
    [CrossRef]
  21. 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]
  22. 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]

2005 (3)

2004 (2)

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

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
[CrossRef]

2002 (2)

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]

2001 (1)

T. Baba and D. Ohsaki, "Interface of photonic crystals for high efficiently light transmission," Jpn. J. Appl. Phys. 40, 5920-5920 (2001).
[CrossRef]

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 (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]

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] [PubMed]

1995 (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]

1994 (2)

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]

Adibi, A.

B. Momeni and A. Adibi, "Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals," Appl. Phys. Lett. 87, 171104 (2005).
[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, "Exciton-polaritons in superlattices," Phys. Lett. A 192, 99-109 (1994).
[CrossRef]

Baba, T.

T. Baba and D. Ohsaki, "Interface of photonic crystals for high efficiently light transmission," Jpn. J. Appl. Phys. 40, 5920-5920 (2001).
[CrossRef]

Baehr-Jones, T.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
[CrossRef]

Binder, R.

Born, M.

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

Cho, K.

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

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]

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]

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]

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]

Heavens, O. S.

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

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] [PubMed]

Hochberg, M.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
[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] [PubMed]

Ikawa, T.

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

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]

Joannopoulos, J.

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

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]

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.

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] [PubMed]

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]

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]

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] [PubMed]

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] [PubMed]

Kwong, N. H.

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]

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]

Meade, R.

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

Momeni, B.

B. Momeni and A. Adibi, "Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals," Appl. Phys. Lett. 87, 171104 (2005).
[CrossRef]

Ohsaki, D.

T. Baba and D. Ohsaki, "Interface of photonic crystals for high efficiently light transmission," Jpn. J. Appl. Phys. 40, 5920-5920 (2001).
[CrossRef]

Palik, E. D.

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

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] [PubMed]

Prineas, J. P.

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]

Scherer, A.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
[CrossRef]

Smirl, A. L.

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] [PubMed]

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]

Winn, J.

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

Witzens, J.

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
[CrossRef]

Wolf, E.

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

Yang, Z. S.

Yanik, M. F.

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

Appl. Phys. Lett. (1)

B. Momeni and A. Adibi, "Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals," Appl. Phys. Lett. 87, 171104 (2005).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

T. Baba and D. Ohsaki, "Interface of photonic crystals for high efficiently light transmission," Jpn. J. Appl. Phys. 40, 5920-5920 (2001).
[CrossRef]

Opt. Lett. (1)

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)

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

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]

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

Phys. Rev. E (1)

J. Witzens, M. Hochberg, T. Baehr-Jones, and A. Scherer, "Mode matching interface for efficient coupling of light into planar photonic crystals," Phys. Rev. E 69, 046609 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

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] [PubMed]

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

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

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]

Other (6)

Note that in it is called "effective refractive index."

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

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

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

Note that the last sentence of Appendix in contains an ambiguous formulation. In the IB, there exists a close relation between neff and the group velocity index, not the phase index as could be understood from that sentence.

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

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Figures (8)

Fig. 1
Fig. 1

Schematic of the quantum-well Bragg structure. The vertical solid lines represent infinitely thin identical quantum wells located at positions z m and spaced by the distance a; the background refractive index is n b . The unit cell is defined to extend from z m τ to z m + 1 τ . E + and E are the “forward” and “backward” traveling field components in the uniform medium between adjacent quantum wells, respectively.

Fig. 2
Fig. 2

(a) Boundary at z b between two uniform media with refractive indices n L on the left and n R on the right. (b) A boundary at z τ = z 1 τ between a uniform medium with refractive index n un and a quantum-well Bragg structure as illustrated in Fig. 1; z 1 is the position of the first quantum well of the Bragg structure.

Fig. 3
Fig. 3

Photonic band structure of the quantum-well Bragg structure specified in Eq. (16) near the Bragg resonance ω B and exciton resonance ω x . (a) Large frequency scale, showing the upper and lower polariton band as well as the IB. (b) Extended view of the IB on a smaller frequency scale.

Fig. 4
Fig. 4

ECI, n eff , of the quantum-well Bragg structure as a function of ω in the (a) upper and (b) lower polariton band. The inset in (a) shows n eff in the IB.

Fig. 5
Fig. 5

(a) Reflection spectrum of the quantum-well Bragg structure without AR coating. (b) The same as (a) but restricted to the spectral region of the IB.

Fig. 6
Fig. 6

Same as Fig. 5a [and the inset is the same as Fig. 5b], but for the case with AR coatings.

Fig. 7
Fig. 7

ECI for τ = a 2 given by the rigorous formula, Eq. (15), as functions of ω within the IB, shown as solid curve. The dots correspond to the approximate formula, Eq. (22).

Fig. 8
Fig. 8

ECI for τ = a and group-velocity index as functions of ω within the IB. The lines are based on the rigorous formulas: Eqs. (25, 15) for the ECI, and n g = c d K d ω together with Eq. (A12) for the GVI. The dots are based on the approximate expressions, Eq. (26) for the ECI and Eq. (29) for the GVI.

Equations (51)

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

E ( z , ω ) = u ν , K ( z ) e i K z .
ω ( ν , K ) n eff = c κ ,
κ = ( 1 i ) d d z ln E ( z , t ) .
( E + ( z b ε , ω ) E ( z b ε , ω ) ) = [ 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 + ( z b + ε , ω ) E ( z b + ε , ω ) ) , ε 0 + .
r = n L n R n L + n R , t = 2 n L n L + n R .
n 0 × j = 1 N ( n i 2 ) ( 1 ) j × ( n f ) ( 1 ) N + 1 = 1 .
E un ( z ) = E un + ( z ) E un ( z ) = A un e in un ( ω c ) z B un e in un ( ω c ) z for z < z τ ,
E ps ( z ) = E ps + ( z ) E ps ( z ) = A ps u ν , K ( z ) e i K z B ps u ν , K ( z ) e i K z for z > z τ ,
( E un + ( z τ ε ) E un ( z τ ε ) ) = [ n ps ( 1 ) ( ν , K ; τ ) + n un 2 n un n ps ( 2 ) ( ν , K ; τ ) n un 2 n un n ps ( 1 ) ( ν , K ; τ ) n un 2 n un n ps ( 2 ) ( ν , K ; τ ) + n un 2 n un ] ( E ps + ( z τ + ε ) E ps ( z τ + ε ) ) ,
n ps ( 1 ) ( ν , K ; τ ) i K u ν , K ( z τ + ε ) + u ν , K ( z τ + ε ) i ω c u ν , K ( z τ + ε ) ,
n ps ( 2 ) ( ν , K ; τ ) i K u ν , K ( z τ + ε ) u ν , K ( z τ + ε ) i ω c u ν , K ( z τ + ε ) ,
n ps ( 2 ) ( ν , K ) = n ps ( 1 ) * ( ν , K ) ,
( E un + ( z τ ε ) E un ( z τ ε ) ) = [ n ps ( 1 ) ( ν , K ; τ ) + n un 2 n un n ps ( 1 ) * ( ν , K ; τ ) n un 2 n un n ps ( 1 ) ( ν , K ; τ ) n un 2 n un n ps ( 1 ) * ( ν , K ; τ ) + n un 2 n un ] ( E ps + ( z τ + ε ) E ps ( z τ + ε ) ) .
n ps ( 1 ) ( ν , K ; τ ) = n b E 0 + ( ν , K ; τ ) E 0 ( ν , K ; τ ) + 1 E 0 + ( ν , K ; τ ) E 0 ( ν , K ; τ ) 1 .
n eff ( ω ) = n ps ( 1 ) ( ν , K ; a 2 ) = n b E 0 + ( ν , K ; a 2 ) E 0 ( ν , K ; a 2 ) + 1 E 0 + ( ν , K ; a 2 ) E 0 ( ν , K ; a 2 ) 1 .
ω x = 1.4966 eV , ω B = 1.4970 eV , n b = 3.61 , Γ = 1.8 × 10 5 ,
δ ω x ω ω x , δ ω B ω ω B , Δ ω x B ω x ω B ,
β = i Γ ω ω ω x i Γ ω B δ ω x i Im β ,
δ ω x ω B , δ ω B ω B 1 , 1 π δ ω B ω B β π δ ω B ω B .
E 0 + ( ν , K ; 1 2 a ) E 0 ( ν , K ; 1 2 a ) = Im [ e i π ( ω ω B ) ( 1 + i Im β ) ] ± 1 { Re [ e i π ( ω ω B ) ( 1 + i Im β ) ] } 2 Im β ,
e i π ( ω ω B ) 1 i π δ ω B ω B ,
E 0 + ( ν , K ; 1 2 a ) E 0 ( ν , K ; 1 2 a ) ( Γ ω B δ ω x π δ ω B ω B ) ± 2 π Γ δ ω B δ ω x ( π Γ δ ω B δ ω x ) 2 Γ ω B δ ω x ,
1 2 π Γ δ ω B ω B δ ω x ω B ,
n eff ( ω ) n b 2 π Γ δ ω B ω B δ ω x ω B 2 + 2 π Γ δ ω B ω B δ ω x ω B 1 2 n b 2 π Γ δ ω B δ ω x δ ω x Δ ω x B Δ ω x B ω B .
n eff 1 2 n b π 2 Γ Δ ω x B ω B .
E 0 + ( ν , K ; a 2 + Δ a ) E 0 ( ν , K ; a 2 + Δ a ) = e 2 i n b ( ω c ) Δ a E 0 + ( ν , K ; a 2 ) E 0 ( ν , K ; a 2 )
n ¯ eff ( ω ) n b = n b n eff ( ω ) ,
n ¯ eff ( ω ) 2 n b Γ 2 π δ ω x δ ω B Δ ω x B δ ω x ω B Δ ω x B ,
e i K a ( 1 π Γ δ ω B δ ω x ) ± i 2 π Γ δ ω B δ ω x ( π Γ δ ω B δ ω x ) 2 1 ± i 2 π Γ δ ω B δ ω x .
i a e i K a d K ± i 2 π Γ Δ ω x B δ ω x 2 δ ω x δ ω B 2 d ω ,
n g = c v g = c d K d ω n b 2 π Γ ( Δ ω x B δ ω x ) 2 δ ω x δ ω B 2 π ω B Δ ω x B ,
n g = n b 8 Γ π ω B Δ ω x B = n ¯ eff .
ω n ¯ eff K = c ( κ K n ¯ eff n g ) .
E ( z , ω ) = E + ( z , ω ) E ( z , ω ) = E + ( z m + ε , ω ) e in b ( ω c ) ( z z m ) E ( z m + ε , ω ) e i n b ( ω c ) ( z z m ) ,
z m < z < z m + 1 , ε 0 + ,
z E ( z m + ε , ω ) z E ( z m ε , ω ) = 4 π ω 2 c 2 μ ϕ ̃ ( 0 ) p m ( ω ) ,
E ( z m + ε , ω ) E ( z m ε , ω ) = 0 , ε 0 + ,
p m ( ω ) = μ ϕ ̃ * ( 0 ) 1 ω ( ω x i γ ) .
( E + ( z m + ε , ω ) E ( z m + ε , ω ) ) = [ 1 + β β β 1 β ] ( E + ( z m ε , ω ) E ( z m ε , ω ) ) ,
β = i Γ ω ω ω x , Γ = 2 π n b c μ 2 ϕ ̃ ( 0 ) 2 .
( E + ( z m + 1 τ , ω ) E ( z m + 1 τ , ω ) ) = M ¯ ( ω ; τ ) ( E + ( z m τ , ω ) E ( z m τ , ω ) ) ,
M ¯ ( ω ; τ ) = [ e i n b ( ω c ) ( a τ ) 0 0 e i n b ( ω c ) ( a τ ) ] [ 1 + β β β 1 β ] [ e i n b ( ω c ) τ 0 0 e i n b ( ω c ) τ ] ,
= [ e i n b ( ω c ) a ( 1 + β ) , β e i n b ( ω c ) ( a 2 τ ) β e i n b ( ω c ) ( a 2 τ ) , e i n b ( ω c ) a ( 1 β ) ] .
M ¯ ( ω ; τ ) [ E 0 + ( ν , K ; τ ) E 0 ( ν , K ; τ ) ] = exp ( i K a ) [ E 0 + ( ν , K ; τ ) E 0 ( ν , K ; τ ) ] ,
exp ( i K ± a ) = [ e i n b ( ω c ) a ( 1 + β ) + e i n b ( ω c ) a ( 1 β ) ] ± [ e i n b ( ω c ) a ( 1 + β ) + e i n b ( ω c ) a ( 1 β ) ] 2 4 2 = Re [ e i n b ( ω c ) a ( 1 + β ) ] ± { Re [ e i n b ( ω c ) a ( 1 + β ) ] } 2 1 ,
E 0 + ( ν , K ± ; τ ) E 0 ( ν , K ± ; τ ) = exp ( i K ± a ) e i n b ( ω c ) a ( 1 β ) β e i n b ( ω c ) ( a 2 τ ) .
u ν , K ( z ) = E 0 + ( ν , K ; τ ) e i ( n b ( ω c ) K ) [ z ( z m τ ) ] E 0 ( ν , K ; τ ) e i ( n b ( ω c ) + K ) [ z ( z m τ ) ] for z ( z m 1 , z m ) ,
exp ( i K ± a ) = Re [ e i n b ( ω c ) a ( 1 + β ) ] ± i 1 { Re [ e i n b ( ω c ) a ( 1 + β ) ] } 2 .
[ E 0 + ( ν , K ; τ = a 2 ) E 0 ( ν , K ; τ = a 2 ) ] * = E 0 + ( ν , K ; τ = a 2 ) E 0 ( ν , K ; τ = a 2 ) ,
( E 0 + ( ν , K ; τ ) E 0 ( ν , K ; τ ) ) = ζ ( E 0 * ( ν , K ; τ ) E 0 + * ( ν , K ; τ ) )
u ν , K ( z ) = ζ u ν , K * ( z ) ,

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