Abstract

The tunneling of electromagnetic waves in the band-gap region of periodic dielectric arrays is investigated with the coherent microwave transient spectroscopy technique. Transmission probabilities at frequencies in the fundamental band gap are measured and found to depend exponentially on sample thickness. From these results the frequency dependence of the imaginary wave vector is determined. The peak imaginary wave vector, which occurs at midgap, is observed to be proportional to the width of the band gap, unlike the case for single-barrier tunneling of electrons, where the relationship is expected to vary as the square root of the barrier height.

© 1993 Optical Society of America

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987); S. John, J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
    [CrossRef] [PubMed]
  2. M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
    [CrossRef]
  3. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
    [CrossRef] [PubMed]
  4. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
    [CrossRef] [PubMed]
  5. Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
    [CrossRef]
  6. G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
    [CrossRef]
  7. See, for example, C. B. Duke, in Solid State Physics (Academic, New York, 1969), Vol. 10, pp. 30–36 and Chap. VI.
  8. See, for example, Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990); K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990); K. M. Leung, Y. F. Liu, Phys. Lett. 65, 2646 (1990).
    [CrossRef] [PubMed]
  9. In a periodic dielectric medium, Maxwell's wave equation for electric displacement vector D, when expanded in terms of plane waves, contains three terms.8 They are the kinetic energy term (k + G)2, the potential energy term ∼V(G − G′), and the total energy term ω2/c2. Here, G is a reciprocal-lattice vector and c the speed of light. When ∊(r) is properly scaled, the corresponding wave equation can be expressed in terms of dimensionless constants, i.e., kao, Gao, and ωao/c. It follows that Eg at the Brillouin zone edge scales linearly with 1/ao as does κ at the midgap. Hence, a linear relationship between Eg and κmax is expected.
  10. E. Yablonovitch, T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
    [CrossRef] [PubMed]

1992

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

1991

M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
[CrossRef]

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

1990

G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
[CrossRef]

See, for example, Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990); K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990); K. M. Leung, Y. F. Liu, Phys. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

1989

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

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
[CrossRef]

1987

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987); S. John, J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef] [PubMed]

Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
[CrossRef]

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
[CrossRef]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

Dalichaouch, R.

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

Duke, C. B.

See, for example, C. B. Duke, in Solid State Physics (Academic, New York, 1969), Vol. 10, pp. 30–36 and Chap. VI.

Gmitter, T. J.

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

Halbout, J.-M.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
[CrossRef]

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
[CrossRef]

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

Kopcsay, G. V.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
[CrossRef]

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
[CrossRef]

Maradudin, A. A.

M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
[CrossRef]

McCall, S. L.

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

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

Pastol, Y.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
[CrossRef]

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
[CrossRef]

Platzman, P. M.

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

Plihal, M.

M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
[CrossRef]

Rapp, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

Robertson, W. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

Satpathy, S.

See, for example, Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990); K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990); K. M. Leung, Y. F. Liu, Phys. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

Schultz, S.

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

Shambrook, A.

M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
[CrossRef]

Sheng, P.

M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
[CrossRef]

Smith, D.

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

Yablonovitch, E.

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

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987); S. John, J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef] [PubMed]

Zhang, Z.

See, for example, Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990); K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990); K. M. Leung, Y. F. Liu, Phys. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

Appl. Phys. Lett.

Y. Pastol, G. Arjavalingam, G. V. Kopcsay, J.-M. Halbout, Appl. Phys. Lett. 55, 2277 (1989); G. Arjavalingam, N. Theophilou, Y. Pastol, G. V. Kopcsay, M. Angelopoulos, J. Chem. Phys. 93, 6 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, G. V. Kopcsay, IEEE Trans. Microwave Theory Tech. 38, 615 (1990).
[CrossRef]

Opt. Commun.

M. Plihal, A. Shambrook, A. A. Maradudin, P. Sheng, Opt. Commun. 80, 199 (1991).
[CrossRef]

Phys. Rev. Lett.

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

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rapp, J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992); J. Opt. Soc. Am. B 10, 322 (1993).
[CrossRef] [PubMed]

See, for example, Z. Zhang, S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990); K. M. Ho, C. T. Chan, C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990); K. M. Leung, Y. F. Liu, Phys. Lett. 65, 2646 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987); S. John, J. Wang, Phys. Rev. Lett. 64, 2418 (1990).
[CrossRef] [PubMed]

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

Other

In a periodic dielectric medium, Maxwell's wave equation for electric displacement vector D, when expanded in terms of plane waves, contains three terms.8 They are the kinetic energy term (k + G)2, the potential energy term ∼V(G − G′), and the total energy term ω2/c2. Here, G is a reciprocal-lattice vector and c the speed of light. When ∊(r) is properly scaled, the corresponding wave equation can be expressed in terms of dimensionless constants, i.e., kao, Gao, and ωao/c. It follows that Eg at the Brillouin zone edge scales linearly with 1/ao as does κ at the midgap. Hence, a linear relationship between Eg and κmax is expected.

See, for example, C. B. Duke, in Solid State Physics (Academic, New York, 1969), Vol. 10, pp. 30–36 and Chap. VI.

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for COMITS measurements. The two-dimensional photonic crystal consists of arrays of 10-cm-long alumina-ceramic rods arranged in a square-lattice structure with lattice constant ao. The three samples investigated are made with rods of different diameter (0.52, 0.74, or 1.24 mm), and the ratio d/ao is kept constant at 0.40.

Fig. 2
Fig. 2

Transmission amplitude as a function of frequency f for a series of photonic crystals with the same rod diameter (d = 0.51 mm) but different thicknesses, i.e., different number of rows. The reference spectrum contains frequency components from 15 to 135 GHz. When the array thickness is increased beyond 5 rows, the fundamental band gap at 62–110 GHz becomes fully developed.

Fig. 3
Fig. 3

Thickness dependence of amplitude transmission in the fundamental gap at three different frequencies, f = 63.8, 68.2, and 85.8 GHz. The dots are experimental data, and the straight lines are least-square fits to an exponential function. From the slope of the lines the imaginary wave vector (attenuation coefficient of E field) is determined.

Fig. 4
Fig. 4

Frequency dependence of the imaginary wave vector (κ) in the band-gap region of a sample with d = 0.51 mm. The maximum κ (6.0 cm−1) occurs at midgap (f = 85.8 GHz).

Fig. 5
Fig. 5

Plot of the peak imaginary wave vector κmax versus the width of the fundamental band gap. The straight line clearly shows that κmax scales linearly with Eg.

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