Abstract

Metallo-dielectric photonic crystals with cubic symmetries have been studied here both experimentally and theoretically in the millimeter wavelength region (15–60 mm). In a direct analogy to linear systems, we considered the three-dimensional lattices as a stack of two-dimensional resonating screens. The overall three-dimensional structure was introduced in the calculation through a structural phase. Such an approach proved useful in understanding the related mode propagation and guided us in a study of the transition between cubic and centered body cubic symmetries.

© 2005 Optical Society of America

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References

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  1. J. M. Tobias, M. Ajgaonkar, H. Grebel, “Morphology-dependent transmission through photonic crystals,” J. Opt. Soc. Am. B 19, 385–391 (2002).
    [CrossRef]
  2. H. Grebel, J. Tobias, “Study of hybrid metal-dielectric photonic crystals,” presented at the Quantum Electronics and Laser Science Conference, San Francisco, Calif., May 2002, paper QTuF1.
  3. H. Grebel, Z. Iqbal, A. Lan, “Detection of C60 using surface enhanced Raman scattering from metal coated periodic structures,” Appl. Phys. Lett. 79, 3194–3196 (2001).
    [CrossRef]
  4. J. M. Tobias, H. Grebel, “Self-imaging in photonic crystals in a subwavelength range,” Opt. Lett. 24, 1660–1662 (1999).
    [CrossRef]
  5. S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
    [CrossRef]
  6. A. Serpenguzel, “Transmission characteristics of metallodielectric photonic crystals and resonantors,” IEEE Microwave Wireless Compon. Lett. 134, 134–136 (2002).
    [CrossRef]
  7. J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
    [CrossRef]
  8. J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
    [CrossRef] [PubMed]
  9. K. D. Möller, O. Sternberg, H. Grebel, K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942–1948 (2002).
    [CrossRef] [PubMed]
  10. J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  17. R. Ulrich, “Mode of propagation on an open periodic waveguide for far infrared,” Microwave Symposia Series, Vol. XXIII (Polytechnique Press, Brooklyn, N.Y., 1974).
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    [CrossRef]
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    [CrossRef] [PubMed]

2003 (1)

J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
[CrossRef]

2002 (5)

K. D. Möller, O. Sternberg, H. Grebel, P. Lalanne, “Thick inductive cross shaped metal meshes,” J. Appl. Phys. 91, 9461–1965 (2002).
[CrossRef]

J. M. Tobias, M. Ajgaonkar, H. Grebel, “Morphology-dependent transmission through photonic crystals,” J. Opt. Soc. Am. B 19, 385–391 (2002).
[CrossRef]

A. Serpenguzel, “Transmission characteristics of metallodielectric photonic crystals and resonantors,” IEEE Microwave Wireless Compon. Lett. 134, 134–136 (2002).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

K. D. Möller, O. Sternberg, H. Grebel, K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942–1948 (2002).
[CrossRef] [PubMed]

2001 (3)

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

H. Grebel, Z. Iqbal, A. Lan, “Detection of C60 using surface enhanced Raman scattering from metal coated periodic structures,” Appl. Phys. Lett. 79, 3194–3196 (2001).
[CrossRef]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

1999 (1)

1996 (1)

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

1985 (1)

1981 (1)

1967 (1)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

Ajgaonkar, M.

Biswas, R.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, UK, 1964).

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
[CrossRef]

Compton, R. C.

Dorsinville, R.

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

Ebbesen, T. W.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

El-Kady, I.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Fleming, J. G.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Garcia-Vidal, F. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Grebel, H.

K. D. Möller, O. Sternberg, H. Grebel, P. Lalanne, “Thick inductive cross shaped metal meshes,” J. Appl. Phys. 91, 9461–1965 (2002).
[CrossRef]

J. M. Tobias, M. Ajgaonkar, H. Grebel, “Morphology-dependent transmission through photonic crystals,” J. Opt. Soc. Am. B 19, 385–391 (2002).
[CrossRef]

K. D. Möller, O. Sternberg, H. Grebel, K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942–1948 (2002).
[CrossRef] [PubMed]

H. Grebel, Z. Iqbal, A. Lan, “Detection of C60 using surface enhanced Raman scattering from metal coated periodic structures,” Appl. Phys. Lett. 79, 3194–3196 (2001).
[CrossRef]

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

J. M. Tobias, H. Grebel, “Self-imaging in photonic crystals in a subwavelength range,” Opt. Lett. 24, 1660–1662 (1999).
[CrossRef]

H. Grebel, J. Tobias, “Study of hybrid metal-dielectric photonic crystals,” presented at the Quantum Electronics and Laser Science Conference, San Francisco, Calif., May 2002, paper QTuF1.

Ho, K. M.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Ho, K.-M.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

Iqbal, Z.

H. Grebel, Z. Iqbal, A. Lan, “Detection of C60 using surface enhanced Raman scattering from metal coated periodic structures,” Appl. Phys. Lett. 79, 3194–3196 (2001).
[CrossRef]

Lalanne, P.

K. D. Möller, O. Sternberg, H. Grebel, P. Lalanne, “Thick inductive cross shaped metal meshes,” J. Appl. Phys. 91, 9461–1965 (2002).
[CrossRef]

Lan, A.

H. Grebel, Z. Iqbal, A. Lan, “Detection of C60 using surface enhanced Raman scattering from metal coated periodic structures,” Appl. Phys. Lett. 79, 3194–3196 (2001).
[CrossRef]

Lezec, H. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Li, J.

J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
[CrossRef]

Lin, S. Y.

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Martin-Moreno, L.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

McCalmont, J. S.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

Möller, K. D.

K. D. Möller, O. Sternberg, H. Grebel, P. Lalanne, “Thick inductive cross shaped metal meshes,” J. Appl. Phys. 91, 9461–1965 (2002).
[CrossRef]

K. D. Möller, O. Sternberg, H. Grebel, K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942–1948 (2002).
[CrossRef] [PubMed]

K. D. Möller, W. G. Rothschild, Far Infrared Spectroscopy (Wiley, New York, 1971).

Pellerin, K. M.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Pendry, J. B.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Richards, P. L.

Rothschild, W. G.

K. D. Möller, W. G. Rothschild, Far Infrared Spectroscopy (Wiley, New York, 1971).

Serpenguzel, A.

A. Serpenguzel, “Transmission characteristics of metallodielectric photonic crystals and resonantors,” IEEE Microwave Wireless Compon. Lett. 134, 134–136 (2002).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
[CrossRef]

Sigalas, M. M.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

Soukolis, C. M.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

Sternberg, O.

K. D. Möller, O. Sternberg, H. Grebel, P. Lalanne, “Thick inductive cross shaped metal meshes,” J. Appl. Phys. 91, 9461–1965 (2002).
[CrossRef]

K. D. Möller, O. Sternberg, H. Grebel, K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942–1948 (2002).
[CrossRef] [PubMed]

O. Sternberg, “Resonances of periodic metal-dielectric meshes in the infrared wavelength region,” Ph.D. thesis (New Jersey Institute of Technology, Newark N.J. 07102, 2002<).

Stewart, K. P.

Thio, T.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

Timusk, T.

Tobias, J.

H. Grebel, J. Tobias, “Study of hybrid metal-dielectric photonic crystals,” presented at the Quantum Electronics and Laser Science Conference, San Francisco, Calif., May 2002, paper QTuF1.

Tobias, J. M.

Tuttle, G.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

Ulrich, R.

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

R. Ulrich, “Mode of propagation on an open periodic waveguide for far infrared,” Microwave Symposia Series, Vol. XXIII (Polytechnique Press, Brooklyn, N.Y., 1974).

Vijayalakshmi, S.

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

Whitbourn, L. B.

White, C. W.

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, UK, 1964).

Yaglioglu, G.

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (3)

H. Grebel, Z. Iqbal, A. Lan, “Detection of C60 using surface enhanced Raman scattering from metal coated periodic structures,” Appl. Phys. Lett. 79, 3194–3196 (2001).
[CrossRef]

S. Vijayalakshmi, H. Grebel, G. Yaglioglu, R. Dorsinville, C. W. White, “Nonlinear dispersion properties of sub-wavelength photonic crystals,” Appl. Phys. Lett. 78, 1754–1756 (2001).
[CrossRef]

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759–2761 (1996).
[CrossRef]

IEEE Microwave Wireless Compon. Lett. (1)

A. Serpenguzel, “Transmission characteristics of metallodielectric photonic crystals and resonantors,” IEEE Microwave Wireless Compon. Lett. 134, 134–136 (2002).
[CrossRef]

Infrared Phys. (1)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

J. Appl. Phys. (1)

K. D. Möller, O. Sternberg, H. Grebel, P. Lalanne, “Thick inductive cross shaped metal meshes,” J. Appl. Phys. 91, 9461–1965 (2002).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, K. M. Ho, “All metallic three-dimensional photonic crystals with a large infrared band-gap,” Nature 417, 52–55 (2002).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

J. Li, L. Zhou, C. T. Chan, P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901-1–083901-4 (2003).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, UK, 1964).

O. Sternberg, “Resonances of periodic metal-dielectric meshes in the infrared wavelength region,” Ph.D. thesis (New Jersey Institute of Technology, Newark N.J. 07102, 2002<).

“Micro-Stripes Program” by Flomerics, Inc., 275 Turnpike Road, Suite 100, Southborough, Mass., 01772.

K. D. Möller, W. G. Rothschild, Far Infrared Spectroscopy (Wiley, New York, 1971).

R. Ulrich, “Mode of propagation on an open periodic waveguide for far infrared,” Microwave Symposia Series, Vol. XXIII (Polytechnique Press, Brooklyn, N.Y., 1974).

H. Grebel, J. Tobias, “Study of hybrid metal-dielectric photonic crystals,” presented at the Quantum Electronics and Laser Science Conference, San Francisco, Calif., May 2002, paper QTuF1.

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Figures (13)

Fig. 1
Fig. 1

Schematics of simple cubic (C), body-centered cubic (BCC) and face-centered cubic (FCC) crystal structure. In the body-centered lattice, alternate cubic layers are displaced in the diagonal direction. The distance between the first and the second layer is one half of the lattice constant of the simple cubic structure. The layers of the FCC have periodicity constant of a/2 and separation of a/2; alternate layers are shifted by a/2.

Fig. 2
Fig. 2

Experimental transmission power of (a) C and (b) BCC crystals with lattice constant 20 mm and metal sphere diameter d=6.35 mm. The wavelengths in the figures are in millimeters. Successive curves are given for increasing number of layers: gray, one layer; black, five layers.

Fig. 3
Fig. 3

Simulated transmission of (a) C and (b) BCC structures with lattice constant a=20 mm and metal sphere diameter d=6.35 mm. The wavelengths in the figures are in millimeters. Successive curves are given for increasing number of layers: gray, one layer; black, five layers.

Fig. 4
Fig. 4

Simulations produced with transmission line formalism. For cubic structure (gray curve) the input data are periodicity constant g=20 mm, transverse resonance wavelength for each screen λ(R)=21 mm, resonance width Δω=0.05 mm, loss parameter α=0.2, number of layers f=6, spacing between layers s1=20 mm, and structure phase Φ(λ)cubic=tan-1[λd/πWeff2]+c1 with c1=const. For BCC (black curve), same parameters as for Figs. 2 and 3 with the exception of screen spacing s2=(0.5)s1=10 mm and a phase constant Φ(λ)BCC=tan-1[λd/πWeff2]+2c1.

Fig. 5
Fig. 5

Simulations of a cubic lattice with a=15 mm. Mode assignment: λ1(RC)=16 mm, λ1(SC)=32.5 mm, λ2(SC)=16 mm, λ3(SC)=12 mm. Note that λ1(RC) overlaps λ2(SC). Shown are layers in succession: gray, one layer; black, ten layers.

Fig. 6
Fig. 6

Cubic lattice with a=15 mm for various separations values, s=7.5 mm, s=15 mm, and s=20 mm, respectively. Mode assignment: dark gray, s=7.5 mm: λ(R)=17.5 mm, λ1(S)=17.5 mm. Dotted black: s=15 mm: λ(R)=16 mm, λ1(S)=32.5 mm, λ2(S)=16 mm, λ3(S)=12 mm. Black: s=20 mm: λ(R)=16 mm, λ1(S)=43 mm, λ2(S)=21.5 mm.

Fig. 7
Fig. 7

Transmission line theory calculations for a cubic lattice with a=15 mm for various separation values. Transmission line resonance input: 16 mm. Thick black: s=7.5 mm: λ(R)=17.5 mm, λ1(S)=17.5 mm. Gray: s=15 mm: λ(R)=16 mm, λ1(S)=32.5 mm, λ2(S)=16 mm, λ3(S)=12 mm. Thin black: s=20 mm: λ(R)=16 mm, λ1(S)=43 mm, λ2(S)=21.5 mm.

Fig. 8
Fig. 8

Transition from BCC (black) to simple C (gray) structure, by decreasing the center sphere’s diameter db in BCC: db=6.35, 5, 4, 1, and 0 mm, respectively. The first-order stacking mode λ1(S) decreases in intensity with decreasing values of db. The resonance mode R shifts from λ(R)=16 mm to λ(R)=17.5 mm.

Fig. 9
Fig. 9

Cubic lattice with a=15 mm for various values of the ratio d/a of sphere diameter to lattice constant, with d=4, 6.35, 7.5, and 8 mm.

Fig. 10
Fig. 10

Simulation for ten layers of a FCC lattice. Note the resonance mode λ(RFCC)=10 mm and stacking mode λ(SFCC)=18 mm. The peak at 18 mm has been observed experimentally6 at about the same wavelength.

Fig. 11
Fig. 11

Schematic of a layered structure of C in the 〈011〉 and 〈111〉 directions. The layers in the 〈011〉 direction have two periodicity constants: a and a2; alternate layers are laterally shifted and spaced apart by a/2. In the 〈111〉 direction, the first and fourth layers have cubic structure with periodicity constant a. The second and third layers have hexagonal structure with periodicity constants of a(3/2) and a/2. The layers are shifted with respect to each other. All layers are separated by a/3.

Fig. 12
Fig. 12

Propagation in a cubic structure along the 〈011〉 direction. The strongest peak is made of two overlapping modes λ1(RC) and λ1(SC) at 30 mm. Weaker peaks are observed at λ2(RC)=20 mm and λ2(SC)=15 mm. The curve is a succession of simulations with 1, 3, 10, and 14 layers.

Fig. 13
Fig. 13

Propagation in a cubic structure along the 〈111〉 direction. The strongest peak is made of two overlapping of hexagonal, λ1(RHX) and stacking, λ1(SC) modes at λ=24 mm. The resonance λ1(RC)=22.5 mm is attributed to a cubic screen, whereas the peak at λ2(RHX)=30.5 mm is attributed to a hexagonal screen. The peak at λ=27.5 is generated by shifting the hexagonal layers with respect to each other.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

Y(λ)=1/[α+(iΩ(λ)/2ω1NΔω)].
Ω(λ)=g/λω0-λω0/g,
N=(n12+n22).
mc11=(-Y/2+1)exp[-iΦ(λ)],mc12=-Y/2,mc21=Y/2,mc22=(Y/2+1)exp[+iΦ(λ)].
ms11=exp[-i2πsn1/λ-iΦ(s)(λ)],ms12=0,
ms21=0,ms22=exp[i2πsn1/λ+iΦ(s)(λ)].
MC(λ)cubic=[MC(λ)MS1(λ)]fMC(λ),
MC(λ)BCC=[MC(λ)MS2(λ)]fMC(λ),
Φ(λ)BCC=tan-1[λd/πn2Weff2]+c1=Φ(λ)C,ΦBCC(s)(λ)=c1=(3)π/2.

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