Abstract

It is suggested that a material with a novel spatial dielectric distribution can exhibit a bandgap that is approximately independent of propagation angle. This independence is accomplished by development of the dielectric constant from reflection vectors of equal strength that are ideally equispaced in angle.

© 1999 Optical Society of America

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References

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
    [CrossRef] [PubMed]
  3. S. John, Phys. Rev. Lett. 58, 2864 (1987).
    [CrossRef]
  4. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
    [CrossRef] [PubMed]
  5. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
    [CrossRef] [PubMed]
  6. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
    [CrossRef]
  7. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
    [CrossRef]
  8. D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
    [CrossRef]
  9. I. I. Tarhan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
    [CrossRef]

1996 (1)

I. I. Tarhan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[CrossRef]

1994 (1)

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

1993 (1)

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

1992 (2)

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
[CrossRef]

1991 (1)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

1989 (1)

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[CrossRef] [PubMed]

1987 (2)

S. John, Phys. Rev. Lett. 58, 2864 (1987).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
[CrossRef]

Dalichaouch, R.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Gmitter, T. J.

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[CrossRef] [PubMed]

Ho, K. M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

Joannopoulus, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

John, S.

S. John, Phys. Rev. Lett. 58, 2864 (1987).
[CrossRef]

Kroll, N.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

McCall, S. L.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

Platzman, P. M.

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
[CrossRef]

Robertson, W. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Sigalas, M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

Tarhan, I. I.

I. I. Tarhan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[CrossRef]

Watson, G. H.

I. I. Tarhan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[CrossRef] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Appl. Phys. Lett. 61, 495 (1992).
[CrossRef]

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Appl. Phys. Lett. 65, 645 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, J. Opt. Soc. Am. 10, 314 (1993).
[CrossRef]

Phys. Rev. B (1)

I. I. Tarhan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[CrossRef]

Phys. Rev. Lett. (5)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[CrossRef] [PubMed]

S. John, Phys. Rev. Lett. 58, 2864 (1987).
[CrossRef]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[CrossRef] [PubMed]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulus, Phys. Rev. Lett. 68, 2023 (1992).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Inverse dielectric constant within a unit cell for N=8, with kj/k0=±1,0,±12/17,±12/17,0,±1. ϵmax/ϵmin=3.5.

Fig. 2
Fig. 2

Dispersion relation for eight-reflection-vector periodic material for the scalar model discussed in the text. Dotted curves, locations of the Bragg scattering planes; solid curves, dispersion curves. In the reduced zone scheme point Γ is 0 0, point X is 1 0, and point M is 1 1. In the extended zone scheme we selected points Γ and X on the kx axis and points M on the diagonal. In Fig.  3 the corresponding extended zone scheme is illustrated, and the trajectories are illustrated. For clarity we label the bands corresponding to kx/k0=12, 14, 16. The reflection vectors correspond to a line at ω=c0k0 between Γ and X and a point at ω=122/17c0k0 at M (these points are indicated in Fig.  3). We take the coefficient for each reflection vector to be A=0.0841.

Fig. 3
Fig. 3

Extended zone scheme corresponding to the defi-nitions of the wave vectors selected for construction of the dispersion diagram in Fig.  2. For the part of the diagram including ΓX, the wave vectors that were selected lie along the kx axis. For the part of the diagram that includes XM, we swept from the on-axis X points, which include all odd numbers on the axis. For the part of the diagram that includes ΓM, we swept from the on-axis Γ points, which include the even integers on the axis. Circles, locations of the reflection vectors.

Fig. 4
Fig. 4

Schematic of the bandgap in the case of N=32 and A=0.006. This calculation assumes that the 32 reflection vectors are equispaced in angle, each with magnitude kj=k0. Although the dielectric is not periodic, one can make sensible definitions of the crystal directions to compare with the N=8 case considered above. The construction of this dispersion relation is similar to that of Fig.  2. The XM and ΓM regimes use the same directions as those used in Fig.  2. The unshaded region is the bandgap. The estimates used in the text for the upper and the lower band edges are indicated in this figure: The upper band edge at ω+ lies A/2 above the Bragg point for perfect backreflection; the lower band lies A/2 below the crossing point of two Bragg reflection planes.

Equations (12)

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ω2ur=-c2r2ur,
c2r=1μϵr.
c2r=c021+jAjexpi2kj˙r.
kjk0=icos2πjN+jsin2πjN,
kjk0=±1,0,±1217,±1217,0,±1.
ω+=c0k01+A2,
ω-=c0k01cosπ/N-A2.
Δωc0k0=1+22A-12πN2.
A11+2πN2.
ϵmaxϵmin=1+NA1-NA.
Δωc0k0=1+22Nϵmax-ϵminϵmax+ϵmin-12πN2.
ϵmaxϵminN+π21+2N-π21+2=N+4.088N-4.088.

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