Abstract

Waveguiding phenomena are investigated in an inverted opal photonic crystal made of interpenetrating air spheres, coated with amorphous Ge. Here we focus on the complete gap between the 8th and the 9th band, since a projected band analysis reveals that it is difficult to use the large lower incomplete gap for guiding purposes. Two kinds of line defects are analyzed within this photonic structure, with the plane-wave expansion method. The first one consists of an air cylinder in the Γ – K direction. It gives rise to a large number of defect modes in the bandgap. Most of these modes have large field components at the surface. The second defect is an array of air spheres, also along the Γ – K direction. This is shown to avoid the surface-like modes and sustain only two modes associated with different polarizations, in the frequency range of interest. The air mode waveguiding bandwidth reaches up to 113 nm centered at a wavelength of 1.5μm.

© 2006 Optical Society of America

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Adv. Mater. (4)

Y.A. Vlasov, N. Yao, and D.J. Norris, "Synthesis of photonic crystals for optical wavelengths from semiconductor quantum dots." Adv. Mater. 11, 165 (1999).
[CrossRef]

M. Muller, R. Zentel, T. Maka, S.G. Romanov, and C.M. Sotomayor-Torres, "Photonic crystal films with high refractive index contrast," Adv. Mater. 12, 1499 (2000).
[CrossRef]

F. Garcia-Santamaria, M. Ibisate, I. Rodriguez, F. Meseguer, and C. Lopez, "Photonic band engineering in opals by growth of Si/Ge multilayer shells," Adv. Mater. 15, 788 (2003).
[CrossRef]

W. Lee, S.A. Pruzinsky, and P.V. Braun, "Multi-photon polymerization of waveguide structures within three-dimensional photonic crystals," Adv. Mater. 14, 271 (2002).
[CrossRef]

Appl. Phys. Lett. (3)

A. Chutinan, and S. Noda, "Highly confined waveguides and waveguide bends in three-dimensional photonic crystal," Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

C. Sell, C. Christensen, J. Muehlmeier, G. Tuttle, Z.Y. Li, and K.M. Ho, "Waveguide networks in three-dimensional layer-by-layer photonic crystals," Appl. Phys. Lett. 84, 4605 (2004).
[CrossRef]

W.T. Lau, and S. Fan, "Creating large bandwidth line defects by embedding dielectric waveguides into photonic crystal slabs," Appl. Phys. Lett. 81, 3915 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

H.K. Kim, J. Shin, S. Fan, M.J.F. Digonnet, and G.S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Electron. 40, 551 (2004).
[CrossRef]

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

Nature (3)

A. Blanco, et al, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437 (2000).
[CrossRef] [PubMed]

Y.A. Vlasov, X.-Z. Bo, J.C. Sturm, and D.J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289 (2001).
[CrossRef] [PubMed]

P.V. Braun, and P.Wiltzius, "Electrochemically grown photonic crystals," Nature 402, 603 (1999).
[CrossRef]

Nuovo Cimento D (1)

V.N. Astratov, et al, "Optical spectroscopy of opal matrices with CdS embedded in its pores: quantum confinement and photonic bandgap effects." Nuovo Cimento D 17, 1349 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (4)

V. Yannopapas, A. Modinos, and N. Stefanou, "Waveguides of defect chains in photonic crystals," Phys. Rev. B 65, 235201 (2002).
[CrossRef]

Z.Y. Li, and Z.Q. Zhang, "Fragility of photonic band gaps in inverse-opal photonic crystals," Phys. Rev. B 62, 1516 (2000).
[CrossRef]

M.L. Povinelli, S.G. Johnson, S. Fan, and J.D. Joannopoulos, "Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap," Phys. Rev. B 64, 075,313 (2001).
[CrossRef]

K. Ohtaka, and M. Inoue, "Light scattering from macroscopic spherical bodies," Phys. Rev. B 25, 677 (1982).
[CrossRef]

Phys. Rev. E (1)

A. Chutinan, and S. John, "Diffractionless flow of light in two- and three-dimensional photonic band gap heterostructures: Theory, design rules, and simulations," Phys. Rev. E 71, 026,605 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

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

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, "High transmission through sharp bends in photonic crystals waveguides," Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Science (3)

S.-Y. Lin, E. Chow, V. Hietala, P.R. Villeneuve, and J.D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274 (1998).
[CrossRef] [PubMed]

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604 (2000).
[CrossRef] [PubMed]

J. Wijnhoven, and W. Vos, "Preparation of photonic crystals made of air spheres in titania," Science 281, 802 (1998).
[CrossRef]

Other (1)

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals, Molding the Flow of Light. (Princeton University Press, Princeton, New Jersey, 1995).

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Figures (13)

Fig. 1.
Fig. 1.

Band structure of interpenetrating air spheres coated with amorphous Ge (n=4.1) in a fcc lattice. The radius of the air spheres is 0.3645a and the external radius of the semiconductor shell is 0.409a. A complete bandgap of 12.8%, marked by a yellow region, exists between the 8 th and 9 th bands.

Fig. 2.
Fig. 2.

Projected band structure of the infinite crystal along thve Γ - K and Γ - L directions in the Brillouin zone. The K point is locatevd at a distance ( 3 2 4 ) ( 2 π a ) away from Γ along [01̅1]. The L point is located at a distance ( 3 2 ) ( 2 π a ) away from Γ along [111]. The shaded regions correspond to the propagating electromagnetic modes in the inverse opal structure.

Fig. 3.
Fig. 3.

Supercell used to calculate the dispersion relations of the waveguide modes. The size of the supercell is 5 2 2 a × 5 a × 1 2 2 a , . The blue regions correspond to the Ge shells, which lie at the interstitial region between the air spheres of the inverted opal structure. A cylindrical waveguide of radius R = 0.3a is located at the center of the supercell and is oriented along the [11̅0] direction represented by the red arrow.

Fig. 4.
Fig. 4.

Projected band structure for the inverse opal photonic crystal with a linear defect created by an air cylinder of radius R. The cylinder is along the [11̅0] direction. The periodicity p in this direction is 2 2 a . The shaded blue areas indicate the propagating modes in the perfect three-dimensional photonic crystal. The thick line indicates the band that possesses a significant fraction of energy in air.

Fig. 5.
Fig. 5.

Energy in the electric field for the modes at the Brillouin zone center, in the defect bands inside the bandgap. The gray regions represent Ge. The defect consists of a cylindrical air waveguide of radius R = 0.30a along the [11̅0] direction. For each of these bands, the plot describes the electric energy distribution on the plane perpendicular to the waveguide, at the z-coordinate corresponding to the largest electric field intensity. The waveguide’s circular cross-section can be seen at the center of each map.

Fig. 6.
Fig. 6.

Energy in the electric field for a mode at the Brillouin zone center for the 392 th band. The gray regions represent Ge. The structure is the same as in Fig. 4(b). Plotted here is the energy distribution on a plane perpendicular to the waveguide, located at z = 2 4 a .

Fig. 7.
Fig. 7.

Projected band structure for the inverse opal photonic crystal with a linear defect made of an array of air spheres of radius R. The array is along the [11̅0] direction. The periodicity p in this direction is 2 2 a . The shaded blue areas indicate the propagating modes in the perfect three-dimensional photonic crystal. The brown area indicates the region covered by the band that possesses a significant fraction of energy in air.

Fig. 8.
Fig. 8.

Energy in the electric field for the modes at the Brillouin zone center, in the defect bands inside the bandgap. The gray regions represent Ge. The defect consists of an array of air spheres of radius R = 0.30a along the [11̅0] direction. For each of these bands, the plot describes the electric energy distribution on the plane perpendicular to the waveguide, at the z-coordinate corresponding to the largest electric field intensity. The center of one of these spheres can be seen at the center of the maps located at z = 2 4 a

Fig. 9.
Fig. 9.

Energy in the electric field for a mode at the Brillouin zone center for the 392 th band. The gray regions represent Ge. The structure is the same as in Fig. 7(b). Plotted here is the energy distribution on a plane perpendicular to the waveguide, located at z = 2 4 a . The center of one of the air spheres constituting the line defect is located at the center of the plot.

Fig. 10.
Fig. 10.

Energy in the electric field for the modes at the Brillouin zone center, in the defect bands inside the bandgap. The gray regions represent Ge. The defect consists of an array of air spheres of radius R = 0.45 a along the [11̅0] direction. For each of these bands, the plot describes the electric energy distribution on the plane perpendicular to the waveguide, at the z-coordinate corresponding to the largest electric field intensity. The center of one of these spheres can be seen at the center of the maps located at z = 2 4 a

Fig. 11.
Fig. 11.

Energy in the electric field for a mode at the Brillouin zone center for the 392 th band. The gray regions represent Ge. The structure is the same as in Fig. 7(c). Plotted here is the energy distribution on a plane perpendicular to the waveguide, located at z = 2 4 a . The center of one of the air spheres constituting the line defect is located at the center of the plot.

Fig. 12.
Fig. 12.

Electric field patterns of the defect modes at the Brillouin zone center. The color plots correspond to || on the same plane z = 0 as in Fig. 10, in an inverted opal structure in which an array of air spheres of radius R = 0.45a has been drilled along the [11̅0] direction. (a) Band 392. The electric fields are completely perpendicular to the plane, i.e. they lie in the direction of the waveguide. (b) Band 393. The electric fields are completely within the plane. A vector plot of the in-plane electric field is superimposed.

Fig. 13.
Fig. 13.

Waveguiding bandwidth as a function of air sphere radius, for a defect guide consisting of an array of air spheres along the [11̅0] direction in the inverse opal structure. Considered here is band 392, centered at a wavelength of 1.5μm.

Equations (1)

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r = ( 0.45 a ) 2 ( 2 a 4 ) 2 = 0.28 a ,

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