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Self-collimation of light in three-dimensional photonic crystals

R. Iliew, C. Etrich, and F. Lederer

Open Access Open Access

Abstract

We calculate three-dimensional (3D) dispersion relations of woodpile and inverse opal photonic crystals. Inspecting the iso-frequency surfaces of the four lowest-order bands at appropriate frequencies we identify regions where self-collimation of light may be expected. These predictions are verified by means of finite-difference time-domain calculations both for high- and low-index photonic crystals.

©2005 Optical Society of America

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Figures (8)
Fig. 1.
Fig. 1. Bandstructure for the fcc woodpile structure with f = 28%, n = 3.4 (left) and f = 40%, n = 1.6 (middle) and for the inverse opal with n = 3.4 (right). X′, U′, K′ and W′ are the high symmetry points with the larger z components of k, obtained by exchanging k y and k z of X, U, K and W.

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Fig. 2.
Fig. 2. IFSs of the high-index woodpile crystal for (a) band 1 at Ω = 0.44, (b) band 2 at Ω = 0.44, (c) band 3 at Ω = 0.69 and (d) band 4 at Ω = 0.73. The curvature K̅ is mapped onto the IFSs (see color bars). The values of the black isolines of lowest curvatures K̅ are 0.2, 0.3, 0.4 for (a), 0.5, 0.6, 0.7 for (b), 0.2, 0.3, 0.4 for (c) and 0.4, 0.5, 0.6 for (d). The red lines show the outline of a square and a hexagonal face of the BZ.

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Fig. 3.
Fig. 3. Rendered 3D energy density in the woodpile PC with n = 3.4 for Ω = 0.44 (left) and Ω = 0.35 (right). The wire source is located at the origin and oriented along z.

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Fig. 4.
Fig. 4. IFS (left) and rendered 3D energy density of the FDTD calculation (right) for the woodpile PC with n = 1.6 for Ω = 0.73. The curvature K̅ is mapped onto the IFS. The values of the black isolines of lowest curvatures K̅ are 0.2, 0.5, 0.6. Note that in the FDTD simulation the structure is rotated by 45° in the x-y plane compared to the conventional fcc cell used for the IFS.

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Fig. 5.
Fig. 5. IFSs of the inverse opal structure for (a) band 3 at Ω = 0.54, (b) band 4 at Ω = 0.54, (c) band 3 at Ω = 0.58 and (d) band 4 at Ω = 0.58. The curvature K̅ is mapped onto the IFSs (see color bars). The values of the black isolines of lowest curvatures K̅ are 0.2, 0.3, 0.4. The red lines show the outline of a square and a hexagonal face of the BZ.

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Fig. 6.
Fig. 6. Rendered 3D energy density in the inverse opal with n = 3.4 for Ω = 0.54 (left) and Ω = 0.58 (right). The wire source is located at the origin and oriented along z.

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Fig. 7.
Fig. 7. Movie (2.4 MByte) of H z in the plane z = 0 for the inverse opal at Ω = 0.54.

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Fig. 8.
Fig. 8. IFSs of the inverse opal structure for (a) band 8 at Ω = 0.78 and (b) band 9 at Ω = 0.84. The curvature K̅ is mapped onto the IFSs (see color bars).

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