Abstract

Under suitable conditions polystyrene microsphere colloids form photonic crystals capable of diffracting visible light, analogous to x-ray diffraction from atomic crystal planes. The lattice spacings of these crystals can be tailored to satisfy the Bragg condition along a certain direction for a particular desired wavelength. A modified Mach–Zehnder interferometer has been developed for accurately measuring relative phase shifts of light propagating in photonic crystals to determine the dispersion resulting from photonic band structure near the band edges.

© 1995 Optical Society of America

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
    [CrossRef]
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    [PubMed]
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    [CrossRef]
  7. Duke Scientific Corporation, 2463 Faber Place, Palo Alto, Calif. 94303.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. P. Hariharan, Appl. Opt. 8, 1925 (1969).
    [CrossRef] [PubMed]

1994 (2)

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

E. Özbay, E. Michel, G. Tuttel, R. Biswas, K. M. Ho, J. Bostak, D. M. Bloom, Opt. Lett. 19, 1155 (1994).
[PubMed]

1991 (1)

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

1987 (2)

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

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

1984 (1)

T. Yoshiyama, I. Sogami, N. Ise, Phys. Rev. Lett. 53, 22 (1984).
[CrossRef]

1983 (1)

P. Pieranski, Contemp. Phys. 24, 25 (1983).
[CrossRef]

1982 (1)

A. J. Hurd, N. A. Clark, R. C. Mockler, W. J. O’Sullivan, Phys. Rev. A 26, 2869 (1982).
[CrossRef]

1981 (1)

P. Pieranski, E. Dubois-Violette, F. Rothen, L. Strzelecki, J. Phys. (Paris) 42, 53 (1981).
[CrossRef]

1979 (1)

N. A. Clark, A. J. Hurd, B. J. Ackerson, Nature (London) 281, 57 (1979).
[CrossRef]

1969 (1)

Ackerson, B. J.

N. A. Clark, A. J. Hurd, B. J. Ackerson, Nature (London) 281, 57 (1979).
[CrossRef]

Biswas, R.

E. Özbay, E. Michel, G. Tuttel, R. Biswas, K. M. Ho, J. Bostak, D. M. Bloom, Opt. Lett. 19, 1155 (1994).
[PubMed]

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

Bloom, D. M.

Bostak, J.

Clark, N. A.

A. J. Hurd, N. A. Clark, R. C. Mockler, W. J. O’Sullivan, Phys. Rev. A 26, 2869 (1982).
[CrossRef]

N. A. Clark, A. J. Hurd, B. J. Ackerson, Nature (London) 281, 57 (1979).
[CrossRef]

Dubois-Violette, E.

P. Pieranski, E. Dubois-Violette, F. Rothen, L. Strzelecki, J. Phys. (Paris) 42, 53 (1981).
[CrossRef]

Gmitter, T. J.

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

Hariharan, P.

Ho, K. M.

E. Özbay, E. Michel, G. Tuttel, R. Biswas, K. M. Ho, J. Bostak, D. M. Bloom, Opt. Lett. 19, 1155 (1994).
[PubMed]

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

Hurd, A. J.

A. J. Hurd, N. A. Clark, R. C. Mockler, W. J. O’Sullivan, Phys. Rev. A 26, 2869 (1982).
[CrossRef]

N. A. Clark, A. J. Hurd, B. J. Ackerson, Nature (London) 281, 57 (1979).
[CrossRef]

Ise, N.

T. Yoshiyama, I. Sogami, N. Ise, Phys. Rev. Lett. 53, 22 (1984).
[CrossRef]

John, S.

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

Leung, K. M.

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

Michel, E

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

Michel, E.

Mockler, R. C.

A. J. Hurd, N. A. Clark, R. C. Mockler, W. J. O’Sullivan, Phys. Rev. A 26, 2869 (1982).
[CrossRef]

O’Sullivan, W. J.

A. J. Hurd, N. A. Clark, R. C. Mockler, W. J. O’Sullivan, Phys. Rev. A 26, 2869 (1982).
[CrossRef]

Özbay, E.

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

E. Özbay, E. Michel, G. Tuttel, R. Biswas, K. M. Ho, J. Bostak, D. M. Bloom, Opt. Lett. 19, 1155 (1994).
[PubMed]

Pieranski, P.

P. Pieranski, Contemp. Phys. 24, 25 (1983).
[CrossRef]

P. Pieranski, E. Dubois-Violette, F. Rothen, L. Strzelecki, J. Phys. (Paris) 42, 53 (1981).
[CrossRef]

Rothen, F.

P. Pieranski, E. Dubois-Violette, F. Rothen, L. Strzelecki, J. Phys. (Paris) 42, 53 (1981).
[CrossRef]

Sigalas, M.

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

Sogami, I.

T. Yoshiyama, I. Sogami, N. Ise, Phys. Rev. Lett. 53, 22 (1984).
[CrossRef]

Strzelecki, L.

P. Pieranski, E. Dubois-Violette, F. Rothen, L. Strzelecki, J. Phys. (Paris) 42, 53 (1981).
[CrossRef]

Tuttel, G.

E. Özbay, E. Michel, G. Tuttel, R. Biswas, K. M. Ho, J. Bostak, D. M. Bloom, Opt. Lett. 19, 1155 (1994).
[PubMed]

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

Yablonovitch, E.

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

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

Yoshiyama, T.

T. Yoshiyama, I. Sogami, N. Ise, Phys. Rev. Lett. 53, 22 (1984).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. Özbay, E Michel, G. Tuttel, R. Biswas, M. Sigalas, K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994).
[CrossRef]

Contemp. Phys. (1)

P. Pieranski, Contemp. Phys. 24, 25 (1983).
[CrossRef]

J. Phys. (1)

P. Pieranski, E. Dubois-Violette, F. Rothen, L. Strzelecki, J. Phys. (Paris) 42, 53 (1981).
[CrossRef]

Nature (1)

N. A. Clark, A. J. Hurd, B. J. Ackerson, Nature (London) 281, 57 (1979).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

A. J. Hurd, N. A. Clark, R. C. Mockler, W. J. O’Sullivan, Phys. Rev. A 26, 2869 (1982).
[CrossRef]

Phys. Rev. Lett. (4)

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

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

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

T. Yoshiyama, I. Sogami, N. Ise, Phys. Rev. Lett. 53, 22 (1984).
[CrossRef]

Other (1)

Duke Scientific Corporation, 2463 Faber Place, Palo Alto, Calif. 94303.

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Figures (5)

Fig. 1
Fig. 1

Transmission spectrum along the [111] direction. The photonic crystal is a colloidal crystal of polystyrene microspheres, with 0.110-μm diameter and 4.8% volume fraction.

Fig. 2
Fig. 2

Modified Mach–Zehnder interferometer. The fringe pattern is positioned in front of the CCD camera by adjustment of pentaprism PP1. The optical paths are equalized by adjustment of pentaprism PP2 along a line perpendicular to PP1. The relative intensity is optimized for maximum fringe visibility around the band edge by rotation of the polarizer in the first arm. EC, empty sample cell; CC, crystal sample cell; BS’s, beam-splitter cubes; A’s, analyzers; P’s, polarizers.

Fig. 3
Fig. 3

Typical interferometer fringe pattern with optimized fringe visibility. The solid curve is the resulting nonlinear least-squares fit of Eq. (1).

Fig. 4
Fig. 4

Photon dispersion obtained from the wavelength dependence of relative phase of the fringe patterns. The broken region corresponds to the [111] stop band; the low transmitted intensity prohibits formation of a measurable fringe pattern. The dotted line represents the free photon dispersion in the crystal away from the gap along with the residual imbalance in the optical elements of the interferometer.

Fig. 5
Fig. 5

Deviation of the index of refraction near the stop band. The free photon dispersion has been linearly approximated and subtracted from the residual index of refraction obtained from Fig. 4. Note that the scanned region lies well inside the stop band, where the solid curve is the percentage transmission from Fig. 1.

Equations (3)

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I ( x ) = I 0 exp [ - ( x - μ w ) 2 ] [ 1 + V cos ( 2 π f x + Δ ϕ ) ] .
Δ Φ ( λ ) = 2 π [ n c ( λ ) - 1 ] L / λ ,
Δ n c ( λ ) = n c ( λ ) - n c ( λ 0 ) = Δ ϕ ( λ ) 2 π λ L .

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