Abstract

We derive general reciprocity relations that are applicable to a large class of one-dimensional stratified systems. These results reveal clearly the role of absorption and spatial symmetry in the nonreciprocity of reflection observed in a recent experiment by Armitage et al. [Phys. Rev. B 58, 15,376 (1998)]. We also present examples of structures for which such nonreciprocal effects can be significant.

© 2002 Optical Society of America

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References

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  1. A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
    [CrossRef]
  2. For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999), Secs. 13.3 and 13.4.
    [CrossRef]
  4. M. Nieto-Vesperinas and E. Wolf, J. Opt. Soc. Am. A 3, 2038 (1986).
    [CrossRef]
  5. An extensive review of reciprocity relations in the context of photonic bandgap structures is given by J. P. Dowling, IEE Proc. Optoelectron. 145, 420 (1998).
    [CrossRef]
  6. Vacuum field Rabi splittings9,10 are now understood also in terms of classic dispersion theory.7,8
  7. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
    [CrossRef] [PubMed]
  8. S. D. Gupta and G. S. Agarwal, Opt. Commun. 115, 597 (1995).
    [CrossRef]
  9. J. Sánchez-Mondrágon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
    [CrossRef]
  10. G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
    [CrossRef]

1999 (1)

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

1998 (2)

An extensive review of reciprocity relations in the context of photonic bandgap structures is given by J. P. Dowling, IEE Proc. Optoelectron. 145, 420 (1998).
[CrossRef]

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

1995 (1)

S. D. Gupta and G. S. Agarwal, Opt. Commun. 115, 597 (1995).
[CrossRef]

1990 (1)

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

1986 (1)

1984 (1)

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

1983 (1)

J. Sánchez-Mondrágon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Agarwal, G. S.

S. D. Gupta and G. S. Agarwal, Opt. Commun. 115, 597 (1995).
[CrossRef]

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

Armitage, A.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Astratov, V. N.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999), Secs. 13.3 and 13.4.
[CrossRef]

Carmichael, H. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Dowling, J. P.

An extensive review of reciprocity relations in the context of photonic bandgap structures is given by J. P. Dowling, IEE Proc. Optoelectron. 145, 420 (1998).
[CrossRef]

Eberly, J. H.

J. Sánchez-Mondrágon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Gauthier, D. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Gehring, G. A.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Gibbs, H. M.

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Gupta, S. D.

S. D. Gupta and G. S. Agarwal, Opt. Commun. 115, 597 (1995).
[CrossRef]

Jahnke, F.

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Kavokin, A. V.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Khitrova, G.

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Kira, M.

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Koch, S. W.

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Morin, S. E.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Mossberg, T. W.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Narozhny, N. B.

J. Sánchez-Mondrágon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Nieto-Vesperinas, M.

Roberts, J. S.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Sánchez-Mondrágon, J.

J. Sánchez-Mondrágon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

Skolnik, M. S.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Whittaker, D. M.

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Wolf, E.

M. Nieto-Vesperinas and E. Wolf, J. Opt. Soc. Am. A 3, 2038 (1986).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999), Secs. 13.3 and 13.4.
[CrossRef]

Wu, Q.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Zhu, Y.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

IEE Proc. Optoelectron. (1)

An extensive review of reciprocity relations in the context of photonic bandgap structures is given by J. P. Dowling, IEE Proc. Optoelectron. 145, 420 (1998).
[CrossRef]

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

Opt. Commun. (1)

S. D. Gupta and G. S. Agarwal, Opt. Commun. 115, 597 (1995).
[CrossRef]

Phys. Rev. B (1)

A. Armitage, M. S. Skolnik, A. V. Kavokin, D. M. Whittaker, V. N. Astratov, G. A. Gehring, and J. S. Roberts, Phys. Rev. B 58, 15,376 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

J. Sánchez-Mondrágon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. 51, 550 (1983).
[CrossRef]

G. S. Agarwal, Phys. Rev. Lett. 53, 1732 (1984).
[CrossRef]

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, Phys. Rev. Lett. 64, 2499 (1990).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

For a recent review of semiconductor microcavities, see, for example, G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999), Secs. 13.3 and 13.4.
[CrossRef]

Vacuum field Rabi splittings9,10 are now understood also in terms of classic dispersion theory.7,8

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of the medium and the solutions in regions away from the medium.

Fig. 2
Fig. 2

Schematic view of the model of the experiment of Armitage et al. consisting of coupled Fabry–Perot cavities, with the left cavity containing the resonant absorber (quantum wells). The system is illuminated by a plane wave at an angle θ from the left (the right) with incident amplitude E1i E2i leading to reflected amplitude E1R E2R.

Fig. 3
Fig. 3

Reflectivities R1=E1R2 (left) and R2=E2R2 (right) for unit incident amplitudes for two values of γ, namely, γ=5.0×10-4 µm-1 (top) and γ=1.0×10-6 µm-1 (bottom) for ωp2=1.0×10-3 µm-2. Other parameters are as follows: θ=30, ω0=1/0.85µm-1, and e0=3.9085.

Equations (17)

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d2dz2+k2zE1=0, k=ωc,
d2dz2+k2*zE2*=0.
-LLE2*d2dz2E1-E1d2dz2E2*dz+k2-LLE1E2*-*dz=0.
E1=E1i expikz+E1R exp-ikz, zl1=E1T expikz, zl2;E2=E2i exp-ikz+E2R expikz, zl2=E2T exp-ikz, zl1.
E1TE2R*+E2T*E1R+kE1zE2*zIm zdz=0.
E*dEdz-EdE*dz-L+L+2ik2E2 Imdz=0,
ET2+ER2+kE2 Imdz=Ei2.
-LLE2d2dz2E1-E1d2dz2E2dz=0.
E2iE1T=E1iE2T.
E2T=E1T,E2R*E2T*+E1RE1T+kE2T*E1T×E1zE2*zIm zdz=0.
E2T=E1T, E2R*E2T*+E1RE1T=0, E1T2+E1R2=1.
E2R2=E1R2, E2RE1R.
Ez=E-z,
Ez=expikz+E1R exp-ikz+E2T exp-ikz, z<l1=-l=exp-ikz+E2R expikz+E1T expikz, z>l2=+l.
E1R+E2T=E1T+E2R.
E2R=E1R.
eω=e0+ωp2ω02-ω2-iγω,

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