Abstract

We investigate the electromagnetic properties of a two-dimensional (2-D) photonic-crystal array of vertical cavities for use in nonlinear optical image processing. We determine the 2-D photonic band structure of the array, and we discuss how it is influenced by the degree of interaction between cavities. We study the properties of defects in the 2-D lattice and show that neighboring cavities interact through their overlapping wave functions. This interaction can be used to produce nearest-neighbor nonlinear Boolean functions such as and, or, and xor, which are useful for optical image processing. We demonstrate the use of 2-D photonic bandgap structures for image processing by removing noise from a sample image with a nearest-neighbor and function.

© 1998 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986).
  2. S. H. Lee, ed., Optical Information Processing Fundamentals (Springer-Verlag, New York, 1981).
    [CrossRef]
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  4. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  5. J. D. Joanopolous, R. D. Meade, Joshua N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
  6. P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
    [CrossRef]
  7. T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
    [CrossRef]
  8. D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
    [CrossRef] [PubMed]
  9. N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976).
  10. P. W. Anderson, “The concept of frustration in spin glasses,” J. Less Common Met. 63, 291–294 (1978).
    [CrossRef]
  11. T.-H. Chao, “Dynamically reconfigurable optical morphological processor and its application,” in Optical Information Processing Systems and Architectures 4, B. Javidi, ed., Proc. SPIE1772, 21–29 (1992).
    [CrossRef]

1996 (1)

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

1995 (1)

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

1992 (1)

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1978 (1)

P. W. Anderson, “The concept of frustration in spin glasses,” J. Less Common Met. 63, 291–294 (1978).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “The concept of frustration in spin glasses,” J. Less Common Met. 63, 291–294 (1978).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976).

Atkin, D. M.

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

Birks, T. A.

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

Brennan, T. M.

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Chao, T.-H.

T.-H. Chao, “Dynamically reconfigurable optical morphological processor and its application,” in Optical Information Processing Systems and Architectures 4, B. Javidi, ed., Proc. SPIE1772, 21–29 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986).

Gourley, P. L.

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Hammons, B. E.

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Joanopolous, J. D.

J. D. Joanopolous, R. D. Meade, Joshua N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Meade, R. D.

J. D. Joanopolous, R. D. Meade, Joshua N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

Mermin, N. D.

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976).

Roberts, P. J.

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

Russel, P. St. J.

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

Shepherd, T. S.

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Vawter, G. A.

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Warren, M. E.

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Winn, Joshua N.

J. D. Joanopolous, R. D. Meade, Joshua N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

Yablonovitch, E.

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

P. L. Gourley, M. E. Warren, G. A. Vawter, T. M. Brennan, B. E. Hammons, “Optical Bloch waves in a semiconductor photonic lattice,” Appl. Phys. Lett. 60, 2714–2716 (1992).
[CrossRef]

Electron Lett. (1)

T. A. Birks, P. J. Roberts, P. St. J. Russel, D. M. Atkin, T. S. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron Lett. 31, 1941–1943 (1995).
[CrossRef]

J. Less Common Met. (1)

P. W. Anderson, “The concept of frustration in spin glasses,” J. Less Common Met. 63, 291–294 (1978).
[CrossRef]

Phys. Rev. Lett. (3)

D. F. Sievenpiper, M. E. Sickmiller, E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Other (5)

J. D. Joanopolous, R. D. Meade, Joshua N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1986).

S. H. Lee, ed., Optical Information Processing Fundamentals (Springer-Verlag, New York, 1981).
[CrossRef]

N. W. Ashcroft, N. D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976).

T.-H. Chao, “Dynamically reconfigurable optical morphological processor and its application,” in Optical Information Processing Systems and Architectures 4, B. Javidi, ed., Proc. SPIE1772, 21–29 (1992).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Diagram of a single vertical cavity and a section of the array. The arrows indicate the large perpendicular component of the wave vector and the small parallel component in the plane of the array.

Fig. 2
Fig. 2

Photonic band structure of the vertical-cavity array. The solid curves represent the structure without spaces between the cavities, as in the empty lattice model. The filled and open circles show measurements taken for s and p polarizations, respectively, when the cavities are spaced by 0.3 cm, producing gaps. The solid horizontal bars indicate the energy levels of a single frequency-shifted cavity and a pair of altered cavities within the array. The Brillouin zone is included for reference.

Fig. 3
Fig. 3

(a) Highest energy level at point H on the top face of the Brillouin zone, composed entirely of p waves. The electric field produces a hedgehog pattern. (b) The lowest energy level composed entirely of s waves. The electric fields produce a swirl pattern.

Fig. 4
Fig. 4

(a) Original image of a number 7 with pepper noise. Black shading indicates which cavities have been altered with an extra layer of acrylic. (b) The nearest-neighbor and function has removed the pepper noise, leaving the image of the number 7 intact. Black pixels are those with a transmission of more than -12 dB within a narrow frequency range. (c) Measuring at a higher frequency allows one to detect the single-cavity resonance associated with pepper noise. The end points of the number 7 also have some transmission at this frequency.

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