Abstract

The analogy between electromagnetic wave propagation in multidimensionally periodic structures and electron-wave propagation in real crystals has proven to be a fruitful one. Initial efforts were motivated by the prospect of a photonic band gap, a frequency band in three-dimensional dielectric structures in which electromagnetic waves are forbidden irrespective of the propagation direction in space. Today many new ideas and applications are being pursued in two and three dimensions and in metallic, dielectric, and acoustic structures. We review the early motivations for this research, which were derived from the need for a photonic band gap in quantum optics. This need led to a series of experimental and theoretical searches for the elusive photonic band-gap structures, those three-dimensionally periodic dielectric structures that are to photon waves as semiconductor crystals are to electron waves. We describe how the photonic semiconductor can be doped, producing tiny electromagnetic cavities. Finally, we summarize some of the anticipated implications of photonic band structure for quantum electronics and for other areas of physics and electrical engineering.

© 1993 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. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef] [PubMed]
  3. Y. Yamamoto, S. Machida, W. H. Richardson, Science 255, 1219 (1992).
    [CrossRef] [PubMed]
  4. E. M. Purcell, Phys. Rev. 69, 681 (1946).
    [CrossRef]
  5. K. H. Drexhage, in Progress in Optics 12, E. Wolf, ed. (North-Holland, Amsterdam, 1974), p. 165.
  6. V. P. Bykov, Sov. J. Quantum Elec. 4, 861 (1975).
    [CrossRef]
  7. R. G. Hulet, E. S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
    [CrossRef] [PubMed]
  8. A. Van der Ziel, Noise(Prentice-Hall, New York, 1954).
  9. There have been some reports recently of a photonic band gap in a simple cubic geometry. H. S. Sozuer, J. W. Haus, R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
    [CrossRef]
  10. C. G. Darwin, Philos. Mag. 27, 675 (1914).
  11. E. Yablonovitch, T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
    [CrossRef] [PubMed]
  12. S. Satpathy, Z. Zhang, M. R. Salehpour, Phys. Rev. Lett. 64, 1239 (1990).
    [CrossRef] [PubMed]
  13. K. M. Leung, Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
    [CrossRef]
  14. S. John, R. Rangarajan, Phys. Rev. B 38, 10101 (1988).
    [CrossRef]
  15. E. N. Economu, A. Zdetsis, Phys. Rev. B 40, 1334 (1989).
    [CrossRef]
  16. K. M. Leung, Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
    [CrossRef] [PubMed]
  17. Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
    [CrossRef] [PubMed]
  18. K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
    [CrossRef] [PubMed]
  19. J. Maddox, Nature (London) 348, 481 (1990).
    [CrossRef]
  20. C. T. Chan, K. M. Ho, C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
    [CrossRef]
  21. E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
    [CrossRef] [PubMed]
  22. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
    [CrossRef] [PubMed]
  23. H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976); S. L. McCall, P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
    [CrossRef]
  24. E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).
  25. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, Appl. Phys. Lett. (to be published).

1992 (2)

Y. Yamamoto, S. Machida, W. H. Richardson, Science 255, 1219 (1992).
[CrossRef] [PubMed]

There have been some reports recently of a photonic band gap in a simple cubic geometry. H. S. Sozuer, J. W. Haus, R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[CrossRef]

1991 (3)

C. T. Chan, K. M. Ho, C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

1990 (6)

S. Satpathy, Z. Zhang, M. R. Salehpour, Phys. Rev. Lett. 64, 1239 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[CrossRef]

K. M. Leung, Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

J. Maddox, Nature (London) 348, 481 (1990).
[CrossRef]

1989 (2)

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

E. N. Economu, A. Zdetsis, Phys. Rev. B 40, 1334 (1989).
[CrossRef]

1988 (1)

S. John, R. Rangarajan, Phys. Rev. B 38, 10101 (1988).
[CrossRef]

1987 (3)

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

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).

1985 (1)

R. G. Hulet, E. S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

1976 (1)

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976); S. L. McCall, P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

1975 (1)

V. P. Bykov, Sov. J. Quantum Elec. 4, 861 (1975).
[CrossRef]

1946 (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[CrossRef]

1914 (1)

C. G. Darwin, Philos. Mag. 27, 675 (1914).

Bhat, R.

E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).

Brommer, K. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Bykov, V. P.

V. P. Bykov, Sov. J. Quantum Elec. 4, 861 (1975).
[CrossRef]

Caneau, C.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, Appl. Phys. Lett. (to be published).

Chan, C. T.

C. T. Chan, K. M. Ho, C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Darwin, C. G.

C. G. Darwin, Philos. Mag. 27, 675 (1914).

Drexhage, K. H.

K. H. Drexhage, in Progress in Optics 12, E. Wolf, ed. (North-Holland, Amsterdam, 1974), p. 165.

Economu, E. N.

E. N. Economu, A. Zdetsis, Phys. Rev. B 40, 1334 (1989).
[CrossRef]

Gmitter, T.

E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

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

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, Appl. Phys. Lett. (to be published).

Harbison, J. P.

E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).

Haus, H. A.

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976); S. L. McCall, P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

Haus, J. W.

There have been some reports recently of a photonic band gap in a simple cubic geometry. H. S. Sozuer, J. W. Haus, R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[CrossRef]

Hilfer, E. S.

R. G. Hulet, E. S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

Ho, K. M.

C. T. Chan, K. M. Ho, C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Hulet, R. G.

R. G. Hulet, E. S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

Inguva, R.

There have been some reports recently of a photonic band gap in a simple cubic geometry. H. S. Sozuer, J. W. Haus, R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[CrossRef]

Joannopoulos, J. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

John, S.

S. John, R. Rangarajan, Phys. Rev. B 38, 10101 (1988).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Kleppner, D.

R. G. Hulet, E. S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[CrossRef]

K. M. Leung, Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

Liu, Y. F.

K. M. Leung, Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[CrossRef]

Machida, S.

Y. Yamamoto, S. Machida, W. H. Richardson, Science 255, 1219 (1992).
[CrossRef] [PubMed]

Maddox, J.

J. Maddox, Nature (London) 348, 481 (1990).
[CrossRef]

Meade, R. D.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[CrossRef]

Rangarajan, R.

S. John, R. Rangarajan, Phys. Rev. B 38, 10101 (1988).
[CrossRef]

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

Richardson, W. H.

Y. Yamamoto, S. Machida, W. H. Richardson, Science 255, 1219 (1992).
[CrossRef] [PubMed]

Salehpour, M. R.

S. Satpathy, Z. Zhang, M. R. Salehpour, Phys. Rev. Lett. 64, 1239 (1990).
[CrossRef] [PubMed]

Satpathy, S.

Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[CrossRef] [PubMed]

S. Satpathy, Z. Zhang, M. R. Salehpour, Phys. Rev. Lett. 64, 1239 (1990).
[CrossRef] [PubMed]

Schnitzer, I.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, Appl. Phys. Lett. (to be published).

Shank, C. V.

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976); S. L. McCall, P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

Soukoulis, C. M.

C. T. Chan, K. M. Ho, C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

Sozuer, H. S.

There have been some reports recently of a photonic band gap in a simple cubic geometry. H. S. Sozuer, J. W. Haus, R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[CrossRef]

Van der Ziel, A.

A. Van der Ziel, Noise(Prentice-Hall, New York, 1954).

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

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

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

E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, Appl. Phys. Lett. (to be published).

Yamamoto, Y.

Y. Yamamoto, S. Machida, W. H. Richardson, Science 255, 1219 (1992).
[CrossRef] [PubMed]

Zdetsis, A.

E. N. Economu, A. Zdetsis, Phys. Rev. B 40, 1334 (1989).
[CrossRef]

Zhang, Z.

S. Satpathy, Z. Zhang, M. R. Salehpour, Phys. Rev. Lett. 64, 1239 (1990).
[CrossRef] [PubMed]

Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

E. Yablonovitch, T. Gmitter, J. P. Harbison, R. Bhat, Appl. Phys. Lett. 51, 222 (1987).

Europhys. Lett. (1)

C. T. Chan, K. M. Ho, C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, C. V. Shank, IEEE J. Quantum Electron. QE-12, 532 (1976); S. L. McCall, P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

Nature (London) (1)

J. Maddox, Nature (London) 348, 481 (1990).
[CrossRef]

Philos. Mag. (1)

C. G. Darwin, Philos. Mag. 27, 675 (1914).

Phys. Rev. (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[CrossRef]

Phys. Rev. B (4)

There have been some reports recently of a photonic band gap in a simple cubic geometry. H. S. Sozuer, J. W. Haus, R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[CrossRef]

K. M. Leung, Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[CrossRef]

S. John, R. Rangarajan, Phys. Rev. B 38, 10101 (1988).
[CrossRef]

E. N. Economu, A. Zdetsis, Phys. Rev. B 40, 1334 (1989).
[CrossRef]

Phys. Rev. Lett. (10)

K. M. Leung, Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[CrossRef] [PubMed]

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

S. Satpathy, Z. Zhang, M. R. Salehpour, Phys. Rev. Lett. 64, 1239 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[CrossRef] [PubMed]

R. G. Hulet, E. S. Hilfer, D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

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

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

Science (1)

Y. Yamamoto, S. Machida, W. H. Richardson, Science 255, 1219 (1992).
[CrossRef] [PubMed]

Sov. J. Quantum Elec. (1)

V. P. Bykov, Sov. J. Quantum Elec. 4, 861 (1975).
[CrossRef]

Other (3)

A. Van der Ziel, Noise(Prentice-Hall, New York, 1954).

K. H. Drexhage, in Progress in Optics 12, E. Wolf, ed. (North-Holland, Amsterdam, 1974), p. 165.

I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, Appl. Phys. Lett. (to be published).

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Figures (27)

Fig. 1
Fig. 1

Spontaneous-emission event from a filled upper level to an empty lower level. The density of final states is the available mode density for photons.

Fig. 2
Fig. 2

Electromagnetic wave dispersion between a pair of metal plates. The waveguide dispersion for one of the two polarizations has a cutoff frequency below which no electromagnetic modes and no spontaneous emission are allowed.

Fig. 3
Fig. 3

Right-hand side, the electromagnetic dispersion, with a forbidden gap at the wave vector of the periodicity. Left-hand side, the electron wave dispersion typical of a direct-gap semiconductor; the dots represent electrons and holes. Since the photonic band gap straddles the electronic band edge, electron–hole recombination into photons is inhibited. The photons have no place to go.

Fig. 4
Fig. 4

In a good-quality metallic resistor the current flow is quite regular, producing negligible amounts of shot noise.

Fig. 5
Fig. 5

High-quantum-efficiency laser diode, which converts the correlated flow of electrons from a low-shot-noise resistor into photon-number-state squeezed light. Random spontaneous emission outside the desired cavity mode limits the attainable noise reduction.

Fig. 6
Fig. 6

Fcc BZ in reciprocal space.

Fig. 7
Fig. 7

Forbidden gap (shaded) at the L point, which is centered at a frequency ~14% lower than the X-point forbidden gap. Therefore it is difficult to create a forbidden frequency band that overlaps all points along the surface of the BZ.

Fig. 8
Fig. 8

Two common BZ’s for bcc and fcc. The fcc case deviates least from a sphere, favoring a common overlapping band in all directions of space.

Fig. 9
Fig. 9

Homodyne detection system for measuring phase and amplitude in transmission through the photonic crystal under test. A sweep oscillator feeds a 10-dB splitter. Part of the signal is modulated (MOD) and then propagated as a plane wave through the test crystal. The other part of the signal is used as local oscillator for the mixer (MXR) to measure the amplitude change and phase shift in the crystal. Between the mixer and the XY recorder is a lock-in amplifier (not shown).

Fig. 10
Fig. 10

WS real-space unit cell of the fcc lattice, a rhombic dodecahedron. (a) Slightly oversized spherical voids are inscribed into the unit cell, breaking through the faces as illustrated. This is the WS cell, corresponding to the photograph in Plate II. (b) WS cell structure with a photonic band gap. Cylindrical holes are drilled through the top three facets of the rhombic dodecahedron and pass through the bottom three facets. The resulting atoms are roughly cylindrical and have a preferred axis in the vertical direction. This WS cell corresponds to the photograph in Plate III.

Fig. 11
Fig. 11

Fcc crystal, in which the individual WS cells are inscribed with cubes stacked in a three-dimensional checkerboard.

Fig. 12
Fig. 12

Construction of fcc crystals, consisting of spherical voids. Hemispherical holes are drilled on both faces of a dielectric sheet. When the sheets are stacked up, the hemispheres meet, producing a fcc crystal.

Fig. 13
Fig. 13

Typical semimetallic band structure for a photonic crystal with no photonic band gap. An overlap exists between the conduction band at L and the valence band at W.

Fig. 14
Fig. 14

Purported PBS of the spherical-void structure shown in Figs. 10(a) and Plate II. The rightward-sloping lines represent polarization parallel to the X plane, while the leftward-sloping lines represent the orthogonal polarization, which has a partial component out of the X plane. The cross-hatched region is the reported photonic band gap. This figure fails to show the crossing of the valence and conduction bands at the W point, which was first discovered by theory.16

Fig. 15
Fig. 15

Method for constructing a fcc lattice of the WS cells shown in Fig. 10(b). A slab of material is covered by a mask that consists of a triangular array of holes. Each hole is drilled through three times at an angle 35.26° away from normal and spread 120° on the azimuth. The resulting crisscross of holes below the surface of the slab, suggested by the cross-hatching shown here, produces a fully three-dimensionally periodic fcc structure with unit cells as given by Fig. 10(b). The drilling can be done by a real drill bit for microwave work or by reactive ion etching to create fcc structure at optical wavelengths.

Fig. 16
Fig. 16

BZ of a fcc structure, incorporating nonspherical atoms as in Fig. 10(b). Since the space lattice is not distorted, this is simply the standard fcc BZ lying on a hexagonal face rather than the usual cubic face. Only the L points on the top and bottom hexagons are threefold symmetry axes. Therefore they are labeled L3. The L points on the other six hexagons are labeled L1. The U3 and K3 points are equivalent, since they are a reciprocal lattice vector apart. Likewise, the U1 and points are K1 equivalent.

Fig. 17
Fig. 17

Frequency versus wave vector (ω versus k) dispersion along the surface of the BZ shown in Fig. 16, where c/a is the speed of light divided by the fcc cube length. The ovals and the triangles are the experimental points for s and p polarization, respectively. The solid and dashed curves are the calculations for s and p polarization, respectively. The dark shaded band is the totally forbidden band gap. The lighter shaded stripes above and below the dark band are forbidden only for s and p polarization, respectively.

Fig. 18
Fig. 18

Construction of the nonspherical-void photonic crystal of Figs. 10(b) and 15–17 and of Plate III by reactive ion etching.

Fig. 19
Fig. 19

One-dimensional Fabry–Perot resonator, made of multilayer dielectric mirrors with a space of one half-wavelength between the left- and right-hand mirrors. The net effect is to introduce a quarter-wavelength phase slip defect into the overall periodic structure. A defect mode is introduced at midgap.

Fig. 20
Fig. 20

〈1, 1 ¯, 0〉 Cross-sectional view of our fcc photonic crystal, consisting of nonspherical air atoms centered on the large dots. Dielectric material is represented by the shaded areas. The dashed rectangle is a face-diagonal cross section of the unit cube. Donor defects consisted of a dielectric sphere centered on an atom. We selected an acceptor defect as shown centered in the unit cube. It consists of a missing horizontal slice in a single vertical rib.

Fig. 21
Fig. 21

Experimental configuration for the detection of local electromagnetic modes in the vicinity of a lattice defect. Transmission amplitude attenuation from one antenna to the other is measured. At the local mode frequency the signal hops by means of the local mode in the center of the photonic crystal, producing a local transmission peak. The signal propagates in the 〈1, 1, 1〉 direction through 8–10 atomic layers. Co-Ax, coaxial line.

Fig. 22
Fig. 22

(a) Transmission attenuation through a defect-free photonic crystal as a function of microwave frequency. The forbidden gap falls between 13 and 16 GHz. (b) Attenuation through a photonic crystal with a single acceptor in the center. The relatively large acceptor defect volume shifted its frequency to near midgap. The electromagnetic resonator Q was ~1000, limited only by the loss tangent of the dielectric material. (c) Attenuation through a photonic crystal with a single donor defect, an off-center dielectric sphere, leading to two shallow donor modes.

Fig. 23
Fig. 23

Donor and acceptor mode frequencies as a function of normalized donor and acceptor defect volume. The points are experimental, and the corresponding curves are calculated. Defect volume is normalized to (λ/2n)3, where λ is the midgap vacuum wavelength and n is the refractive index. A finite defect volume is necessary to bind a mode in the forbidden gap.

Fig. 24
Fig. 24

Properties of the SM-LED, whose cavity is represented by the small circle inside the rectangular photonic crystal at left. The words Monochromatic and Directional represent the temporal and spatial coherence of the SM-LED output, as is explained in the text. The modulation speed can be >10 GHz, and the differential quantum efficiency can be >50%, which is competitive with that of laser diodes. But there is no threshold current for the SM-LED, as indicated by the curves for light output versus the input current at the bottom. The regular stream of photoelectrons, e’s, is meant to represent photon-number-state squeezing, which can be produced by the SM-LED if the spontaneous-emission factor β of the cavity is high enough.

Plate I
Plate I

Photograph of a three-dimensional fcc crystal consisting index of Al2O3 spheres of refractive 3.06. The dielectric spheres are supported in place by the blue foam material of refractive index 1.01. These spherical-dielectric-atom structures failed to show a photonic band gap at any volume fraction.

Plate II
Plate II

Photograph of the photonic crystal corresponding to Fig. 10(a), which had only a pseudogap rather than a full photonic band gap. The spherical voids were closer than close-packed, overlapping and allowing holes to pass through as shown.

Plate III
Plate III

Top-view photograph of the nonspherical-atom structure of WS unit cells as shown in Fig. 10(a), constructed by the method of Fig. 15.

Tables (1)

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Table 1 Summary of Differences and Similarities between Photonic and Electronic Band Structures

Equations (2)

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w = 2 π V 2 ρ ( E ) ,
( Δ i ) 2 = 2 e i Δ f ,

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