Abstract

We have investigated the local density of optical states (LDOS) in titania and silicon inverse opals—three-dimensional photonic crystals that have been realized experimentally. We used the H-field plane-wave expansion method to calculate the density of states and the projected LDOS, which are directly relevant for spontaneous emission dynamics and strong coupling. We present the first quantitative analysis of the frequency resolution and of the accuracy of the calculated LDOS. We have calculated the projected LDOS for many different emitter positions and orientations in inverse opals in order to supply a theoretical interpretation for recent emission experiments and as reference results for future experiments and theory by other workers. The results show that the LDOS in inverse opals strongly depends on the crystal lattice parameter as well as on the position and orientation of emitting dipoles.

© 2009 Optical Society of America

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2008 (2)

B. Julsgaard, J. Johansen, S. Stobbe, T. Stolberg-Rohr, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Decay dynamics of quantum dots influenced by the local density of optical states in two-dimensional photonic crystal membranes,” Appl. Phys. Lett. 93, 094102 (2008).
[CrossRef]

P. Kristensen, A. F. Koenderink, P. Lodahl, B. Tromborg, and J. Mørk, “Fractional decay of quantum dots in real photonic crystals,” Opt. Lett. 33, 1557-1559 (2008).
[CrossRef] [PubMed]

2007 (1)

I. S. Nikolaev, P. Lodahl, A. F. van Driel, A. F. Koenderink, and W. L. Vos, “Strongly nonexponential time-resolved fluorescence of quantum-dot ensembles in three-dimensional photonic crystals,” Phys. Rev. B 75, 115302 (2007).
[CrossRef]

2006 (1)

2005 (5)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158-1161 (2005).
[CrossRef] [PubMed]

A. Kress, F. Hofbauer, N. Reinelt, M. Kaniber, H. J. Krenner, R. Meyer, G. Böhm, and J. J. Finley, “Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals,” Phys. Rev. B 71, 241304 (2005).
[CrossRef]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296-1298 (2005).
[CrossRef] [PubMed]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

2004 (4)

S. Ogawa, M. Imada, S. Yoshimoto, M. Okato, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227-229 (2004).
[CrossRef] [PubMed]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,” Phys. Rev. E 70, 066608 (2004).
[CrossRef]

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444-1447 (2004).
[CrossRef] [PubMed]

2003 (2)

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

L. Rogobete, H. Schniepp, V. Sandoghdar, and C. Henkel, “Spontaneous emission in nanoscopic dielectric particles,” Opt. Lett. 28, 1736-1738 (2003).
[CrossRef] [PubMed]

2002 (3)

C. Hermann and O. Hess, “Modified spontaneous emission rate in an inverted-opal structure with complete photonic bandgap,” J. Opt. Soc. Am. B 19, 3013-3018 (2002).
[CrossRef]

N. Vats, S. John, and K. Busch, “Theory of fluorescence in photonic crystals,” Phys. Rev. A 65, 043808 (2002).
[CrossRef]

A. F. Koenderink, L. Bechger, H. P. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88, 143903 (2002).
[CrossRef] [PubMed]

2001 (4)

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, “Fabrication and characterization of large macroporous photonic crystals in titania,” Chem. Mater. 13, 4486-4499 (2001).
[CrossRef]

M. Grätzel, “Photoelectrochemical cells,” Nature 414, 338-344 (2001).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,” Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

2000 (4)

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17, 1438-1442 (2000).
[CrossRef]

W. L. Vos and H. M. van Driel, “Higher order Bragg diffraction by strongly photonic fcc crystals: onset of a photonic bandgap,” Phys. Lett. A 272, 101-106 (2000).
[CrossRef]

Z.-Y. Li, L.-L. Lin, and Z.-Q. Zhang, “Spontaneous emission from photonic crystals: full vectorial calculations,” Phys. Rev. Lett. 84, 4341-4344 (2000).
[CrossRef] [PubMed]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. López, F. Meseguer, H. Míguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

1999 (2)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, “Inhibited light propagation and broad band reflection in photonic air-sphere crystals,” Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

1998 (4)

H. Miyazaki and K. Ohtaka, “Near-field images of a monolayer of periodically arrayed dielectric spheres,” Phys. Rev. B 58, 6920-6937 (1998).
[CrossRef]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802-804 (1998).
[CrossRef]

A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khayrullin, S. O. Dantas, J. Marti, and V. G. Ralchenko, “Carbon structures with three-dimensional periodicity at optical wavelengths,” Science 282, 897-901 (1998).
[CrossRef] [PubMed]

1996 (3)

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265-270 (1996).
[CrossRef]

A. A. Krokhin and P. Halevi, “Influence of weak dissipation on the photonic band structure of periodic composites,” Phys. Rev. B 53, 1205-1214 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

1995 (1)

1994 (2)

P. E. Blöchl, O. Jepsen, and O. K. Andersen, “Improved tetrahedron method for Brillouin-zone integrations,” Phys. Rev. B 49, 16223-16233 (1994).
[CrossRef]

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835-16844 (1994).
[CrossRef]

1992 (1)

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

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]

1976 (1)

H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integration,” Phys. Rev. B 13, 5188-5192 (1976).
[CrossRef]

1975 (1)

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861-871 (1975).
[CrossRef]

1972 (2)

G. Gilat, “Analysis of methods for calculating spectral properties in solids,” J. Comput. Phys. 10, 432-465 (1972).
[CrossRef]

V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. JETP 35, 269-273 (1972).

Andersen, O. K.

P. E. Blöchl, O. Jepsen, and O. K. Andersen, “Improved tetrahedron method for Brillouin-zone integrations,” Phys. Rev. B 49, 16223-16233 (1994).
[CrossRef]

Arakawa, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

Asano, T.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296-1298 (2005).
[CrossRef] [PubMed]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

Atatüre, M.

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158-1161 (2005).
[CrossRef] [PubMed]

Badolato, A.

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158-1161 (2005).
[CrossRef] [PubMed]

Baek, J.-H.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444-1447 (2004).
[CrossRef] [PubMed]

Baughman, R. H.

A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khayrullin, S. O. Dantas, J. Marti, and V. G. Ralchenko, “Carbon structures with three-dimensional periodicity at optical wavelengths,” Science 282, 897-901 (1998).
[CrossRef] [PubMed]

Bechger, L.

A. F. Koenderink, L. Bechger, H. P. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88, 143903 (2002).
[CrossRef] [PubMed]

J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, “Fabrication and characterization of large macroporous photonic crystals in titania,” Chem. Mater. 13, 4486-4499 (2001).
[CrossRef]

Blanco, A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. López, F. Meseguer, H. Míguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Blöchl, P. E.

P. E. Blöchl, O. Jepsen, and O. K. Andersen, “Improved tetrahedron method for Brillouin-zone integrations,” Phys. Rev. B 49, 16223-16233 (1994).
[CrossRef]

Bo, X. Z.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Böhm, G.

A. Kress, F. Hofbauer, N. Reinelt, M. Kaniber, H. J. Krenner, R. Meyer, G. Böhm, and J. J. Finley, “Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals,” Phys. Rev. B 71, 241304 (2005).
[CrossRef]

Busch, K.

N. Vats, S. John, and K. Busch, “Theory of fluorescence in photonic crystals,” Phys. Rev. A 65, 043808 (2002).
[CrossRef]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Bykov, V. P.

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861-871 (1975).
[CrossRef]

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O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Ogawa, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okato, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227-229 (2004).
[CrossRef] [PubMed]

Ohtaka, K.

H. Miyazaki and K. Ohtaka, “Near-field images of a monolayer of periodically arrayed dielectric spheres,” Phys. Rev. B 58, 6920-6937 (1998).
[CrossRef]

Okato, M.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okato, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227-229 (2004).
[CrossRef] [PubMed]

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Ozin, G. A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. López, F. Meseguer, H. Míguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Pack, J. D.

H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integration,” Phys. Rev. B 13, 5188-5192 (1976).
[CrossRef]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Park, H.-G.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444-1447 (2004).
[CrossRef] [PubMed]

Petroff, P. M.

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158-1161 (2005).
[CrossRef] [PubMed]

Pincemin, F.

V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835-16844 (1994).
[CrossRef]

Ralchenko, V. G.

A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khayrullin, S. O. Dantas, J. Marti, and V. G. Ralchenko, “Carbon structures with three-dimensional periodicity at optical wavelengths,” Science 282, 897-901 (1998).
[CrossRef] [PubMed]

Reinelt, N.

A. Kress, F. Hofbauer, N. Reinelt, M. Kaniber, H. J. Krenner, R. Meyer, G. Böhm, and J. J. Finley, “Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals,” Phys. Rev. B 71, 241304 (2005).
[CrossRef]

Rogobete, L.

Sandoghdar, V.

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Schniepp, H.

Schriemer, H. P.

A. F. Koenderink, L. Bechger, H. P. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88, 143903 (2002).
[CrossRef] [PubMed]

Solomon, G.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

Sorensen, D. C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM Publications, 1998).

Soukoulis, C. M.

A. F. Koenderink, M. Kafesaki, C. M. Soukoulis, and V. Sandoghdar, “Spontaneous emission control in two-dimensional photonic crystal membranes,” J. Opt. Soc. Am. B 23, 1196-1206 (2006).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Sözüer, H. S.

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Sprik, R.

M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, “Inhibited light propagation and broad band reflection in photonic air-sphere crystals,” Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265-270 (1996).
[CrossRef]

Stobbe, S.

B. Julsgaard, J. Johansen, S. Stobbe, T. Stolberg-Rohr, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Decay dynamics of quantum dots influenced by the local density of optical states in two-dimensional photonic crystal membranes,” Appl. Phys. Lett. 93, 094102 (2008).
[CrossRef]

Stolberg-Rohr, T.

B. Julsgaard, J. Johansen, S. Stobbe, T. Stolberg-Rohr, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Decay dynamics of quantum dots influenced by the local density of optical states in two-dimensional photonic crystal membranes,” Appl. Phys. Lett. 93, 094102 (2008).
[CrossRef]

Sturm, J. C.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Sünner, T.

B. Julsgaard, J. Johansen, S. Stobbe, T. Stolberg-Rohr, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Decay dynamics of quantum dots influenced by the local density of optical states in two-dimensional photonic crystal membranes,” Appl. Phys. Lett. 93, 094102 (2008).
[CrossRef]

Suzuki, T.

Takahashi, S.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296-1298 (2005).
[CrossRef] [PubMed]

Tanaka, Y.

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296-1298 (2005).
[CrossRef] [PubMed]

Thijssen, M. S.

M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, “Inhibited light propagation and broad band reflection in photonic air-sphere crystals,” Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

Toader, O.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. López, F. Meseguer, H. Míguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Tromborg, B.

van Driel, A. F.

I. S. Nikolaev, P. Lodahl, A. F. van Driel, A. F. Koenderink, and W. L. Vos, “Strongly nonexponential time-resolved fluorescence of quantum-dot ensembles in three-dimensional photonic crystals,” Phys. Rev. B 75, 115302 (2007).
[CrossRef]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

van Driel, H. M.

W. L. Vos and H. M. van Driel, “Higher order Bragg diffraction by strongly photonic fcc crystals: onset of a photonic bandgap,” Phys. Lett. A 272, 101-106 (2000).
[CrossRef]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. López, F. Meseguer, H. Míguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

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R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265-270 (1996).
[CrossRef]

Vanmaekelbergh, D.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

Vats, N.

N. Vats, S. John, and K. Busch, “Theory of fluorescence in photonic crystals,” Phys. Rev. A 65, 043808 (2002).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Vos, W. L.

I. S. Nikolaev, P. Lodahl, A. F. van Driel, A. F. Koenderink, and W. L. Vos, “Strongly nonexponential time-resolved fluorescence of quantum-dot ensembles in three-dimensional photonic crystals,” Phys. Rev. B 75, 115302 (2007).
[CrossRef]

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
[CrossRef]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
[CrossRef] [PubMed]

A. F. Koenderink, L. Bechger, H. P. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88, 143903 (2002).
[CrossRef] [PubMed]

J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, “Fabrication and characterization of large macroporous photonic crystals in titania,” Chem. Mater. 13, 4486-4499 (2001).
[CrossRef]

W. L. Vos and H. M. van Driel, “Higher order Bragg diffraction by strongly photonic fcc crystals: onset of a photonic bandgap,” Phys. Lett. A 272, 101-106 (2000).
[CrossRef]

M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, “Inhibited light propagation and broad band reflection in photonic air-sphere crystals,” Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802-804 (1998).
[CrossRef]

Vuckovic, J.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

Waks, E.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

Wang, R.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

Wang, X.-H.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

Wijnhoven, J. E. G. J.

J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, “Fabrication and characterization of large macroporous photonic crystals in titania,” Chem. Mater. 13, 4486-4499 (2001).
[CrossRef]

M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, “Inhibited light propagation and broad band reflection in photonic air-sphere crystals,” Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802-804 (1998).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Xu, Y.

Yablonovitch, E.

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

Yamamoto, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

Yang, C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM Publications, 1998).

Yang, G.-Z.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

Yang, J.-K.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444-1447 (2004).
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Yariv, A.

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17, 1438-1442 (2000).
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O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Yoshimoto, S.

S. Ogawa, M. Imada, S. Yoshimoto, M. Okato, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227-229 (2004).
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Yu, P. K. L.

Zakhidov, A. A.

A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khayrullin, S. O. Dantas, J. Marti, and V. G. Ralchenko, “Carbon structures with three-dimensional periodicity at optical wavelengths,” Science 282, 897-901 (1998).
[CrossRef] [PubMed]

Zhang, B.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
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Zhang, Z.-Q.

Z.-Y. Li, L.-L. Lin, and Z.-Q. Zhang, “Spontaneous emission from photonic crystals: full vectorial calculations,” Phys. Rev. Lett. 84, 4341-4344 (2000).
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Appl. Phys. Lett. (1)

B. Julsgaard, J. Johansen, S. Stobbe, T. Stolberg-Rohr, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Decay dynamics of quantum dots influenced by the local density of optical states in two-dimensional photonic crystal membranes,” Appl. Phys. Lett. 93, 094102 (2008).
[CrossRef]

Chem. Mater. (1)

J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, “Fabrication and characterization of large macroporous photonic crystals in titania,” Chem. Mater. 13, 4486-4499 (2001).
[CrossRef]

Europhys. Lett. (1)

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265-270 (1996).
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J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (4)

Nature (4)

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. López, F. Meseguer, H. Míguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature 405, 437-440 (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, 289-293 (2001).
[CrossRef] [PubMed]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654-657 (2004).
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M. Grätzel, “Photoelectrochemical cells,” Nature 414, 338-344 (2001).
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Opt. Express (1)

Opt. Lett. (2)

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W. L. Vos and H. M. van Driel, “Higher order Bragg diffraction by strongly photonic fcc crystals: onset of a photonic bandgap,” Phys. Lett. A 272, 101-106 (2000).
[CrossRef]

Phys. Rev. A (1)

N. Vats, S. John, and K. Busch, “Theory of fluorescence in photonic crystals,” Phys. Rev. A 65, 043808 (2002).
[CrossRef]

Phys. Rev. B (10)

A. Kress, F. Hofbauer, N. Reinelt, M. Kaniber, H. J. Krenner, R. Meyer, G. Böhm, and J. J. Finley, “Manipulation of the spontaneous emission dynamics of quantum dots in two-dimensional photonic crystals,” Phys. Rev. B 71, 241304 (2005).
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I. S. Nikolaev, P. Lodahl, A. F. van Driel, A. F. Koenderink, and W. L. Vos, “Strongly nonexponential time-resolved fluorescence of quantum-dot ensembles in three-dimensional photonic crystals,” Phys. Rev. B 75, 115302 (2007).
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A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B 72, 153102 (2005).
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A. A. Krokhin and P. Halevi, “Influence of weak dissipation on the photonic band structure of periodic composites,” Phys. Rev. B 53, 1205-1214 (1996).
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H. Miyazaki and K. Ohtaka, “Near-field images of a monolayer of periodically arrayed dielectric spheres,” Phys. Rev. B 58, 6920-6937 (1998).
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P. E. Blöchl, O. Jepsen, and O. K. Andersen, “Improved tetrahedron method for Brillouin-zone integrations,” Phys. Rev. B 49, 16223-16233 (1994).
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H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integration,” Phys. Rev. B 13, 5188-5192 (1976).
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V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B 50, 16835-16844 (1994).
[CrossRef]

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

H. S. Sözüer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Phys. Rev. E (2)

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,” Phys. Rev. E 70, 066608 (2004).
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K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Phys. Rev. Lett. (7)

Z.-Y. Li, L.-L. Lin, and Z.-Q. Zhang, “Spontaneous emission from photonic crystals: full vectorial calculations,” Phys. Rev. Lett. 84, 4341-4344 (2000).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

A. F. Koenderink, L. Bechger, H. P. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88, 143903 (2002).
[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]

M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, “Inhibited light propagation and broad band reflection in photonic air-sphere crystals,” Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
[CrossRef] [PubMed]

Science (7)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444-1447 (2004).
[CrossRef] [PubMed]

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281, 802-804 (1998).
[CrossRef]

A. A. Zakhidov, R. H. Baughman, Z. Iqbal, C. Cui, I. Khayrullin, S. O. Dantas, J. Marti, and V. G. Ralchenko, “Carbon structures with three-dimensional periodicity at optical wavelengths,” Science 282, 897-901 (1998).
[CrossRef] [PubMed]

S. Ogawa, M. Imada, S. Yoshimoto, M. Okato, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227-229 (2004).
[CrossRef] [PubMed]

M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296-1298 (2005).
[CrossRef] [PubMed]

A. Badolato, K. Hennessy, M. Atatüre, J. Dreiser, E. Hu, P. M. Petroff, and A. Imamoğlu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science 308, 1158-1161 (2005).
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Sov. Phys. JETP (1)

V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. JETP 35, 269-273 (1972).

Other (8)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Photonic Crystals and Light Localization in the 21st Century, C.M.Soukoulis, ed. (Kluwer, 2001).

Photonic Crystals. Advances in Design, Fabrication, and Characterization, K.Busch, S.Lölkes, R.B.Wehrspohn, and H.Föll, ed. (Wiley-VCH Verlag GmbH, 2004).
[CrossRef]

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM Publications, 1998).

A. F. Koenderink, “Emission and transport of light in photonic crystals,” Ph.D. thesis (University of Amsterdam, 2003). ISBN 90-9016903-2, available from http://www.koenderink.info.

P. J. Harding, “Photonic crystals modified by optically resonant systems,” Ph.D. thesis (University of Twente, 2008), ISBN 978-90-365-2683-8, available from http://www.photonicbandgaps.com.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).

Supplementary Material (13)

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Figures (11)

Fig. 1
Fig. 1

DOS per volume in units 4 a 2 c for vacuum modeled as an “empty” f c c crystal. The calculated DOS shown by histogram bars is compared to the analytically derived ω 2 behavior (curve). In vacuum the DOS per volume equals the dipole-averaged LDOS.

Fig. 2
Fig. 2

Average absolute deviation of the calculated DOS from the exact total DOS N ( ω ) for an “empty crystal.” The average runs over the frequency range 0 < ω < 2 π c a , and the deviation is in units of the DOS N ( ω ) per volume at ω a 2 π c = 1 , i.e., in units 4 a 2 c . In accordance with Eq. (11), the error is inversely proportional to the ratio Δ ω Δ k of the histogram bin width Δ ω to the integration grid spacing Δ k . Symbols correspond to integration using N k = 280 , 770, 1300, 2480, 2992, and 3570 (◻, ◼, ○, ●, ◇, ◆) k-points in the irreducible wedge of the Brillouin zone, with various Δ ω .

Fig. 3
Fig. 3

DOS per volume in an f c c crystal consisting of spheres with ϵ = 7.35 in a medium with ϵ = 1.77 and with a filling fraction of the spheres of 25 vol%. The solid dotted curve represents calculations from [31]. Our result is plotted as a histogram.

Fig. 4
Fig. 4

Dipole-averaged LDOS in the same photonic crystal as in Fig. 3 at a position r = ( 1 4 , 1 4 , 0 ) . Histogram: our calculations (Media 1). Solid curve: results from [33]. This relative LDOS is the ratio of the LDOS to that in vacuum at a λ = 0.495 .

Fig. 5
Fig. 5

Rendering of the dielectric function in one f c c unit cell that models the Ti O 2 inverse-opal structure in Section 4: an f c c lattice of air spheres of radius r = 0.25 2 a with a being the cubic lattice parameter. The spheres are covered by shells with ϵ = 6.5 and outer radius 1.09 r . Neighboring air spheres are connected by windows of radius 0.4 r . The letters a d indicate four different positions at the Ti O 2 –air interface: a = ( 1 , 0 , 0 ) ( 2 2 ) , b = ( 1 , 1 , 2 ) ( 4 3 ) , c = ( 1 , 1 , 1 ) ( 2 6 ) , d = ( 0.33 , 0.13 , 0 ) (points shown are symmetry-equivalents). The dashed–dotted line shows the body diagonal of the cubic unit cell.

Fig. 6
Fig. 6

Left side: Photonic band structure (Media 2) for the Ti O 2 inverse opal shown in Fig. 5. The gray rectangles indicate stopgaps in the Γ L direction and one stopgap in the Γ X direction for the inverse opal. Right side: The stopgaps result in the decreased DOS as shown by open circles (Media 3) at corresponding frequencies compared to the DOS in a homogeneous medium with n av = 1.27 as shown by the solid curve.

Fig. 7
Fig. 7

Relative LDOS in the inverse opal shown in Fig. 5 at three different positions: (a) (Media 4) r = ( 0 , 0 , 0 ) [the center of an air sphere, solid curve], r = 1 4 ( 1 , 1 , 1 ) [among three air spheres, dashed–dotted curve], r = ( 1 2 , 0 , 0 ) [midway between two spheres along [1,0,0] direction, dotted curve]. (b) (Media 5) Relative LDOS at r = 1 4 ( 1 , 1 , 0 ) [in the window between two spheres] projected on e d = [ 1 , 1 , 0 ] , [ 1 , 1 , 0 ] , [0,0,1] directions shown by solid, dashed–dotted, dotted curves, respectively.

Fig. 8
Fig. 8

Relative LDOS at four key frequencies a λ = 0.535 , 0.725, 0.865, 1.295 in the inverse opal as a function of position r on the line from r = ( 0 , 0 , 0 ) to r = 1 2 ( 1 , 1 , 1 ) . The hatched boxes indicate the position of the dielectric Ti O 2 shell. The LDOS is projected on two dipole orientations: (a) e d = [ 1 , 1 , 1 ] perpendicular to the dielectric–air interface, (b) e d = [ 1 , 1 , 0 ] parallel to the interface. The LDOS projected on the e d = [ 1 , 1 , 2 ] and e d = [ 1 , 1 , 0 ] directions are equal. For r a between 3 2 and 3 the LDOS is mirror-symmetric to that in the region from 0 to 3 2 .

Fig. 9
Fig. 9

Relative LDOS in the inverse opal at four different positions on the Ti O 2 –air interface shown in Fig. 5. At each position the LDOS is projected on three mutually orthogonal dipole orientations e d . (a) (Media 6) Point a for e d = [ 1 , 0 , 0 ] and [0,1,0]; the LDOS for e d = [ 0 , 0 , 1 ] is identical to that at e d = [ 0 , 1 , 0 ] . (b) (Media 7) Relative LDOS at point b for e d = [ 1 , 1 , 2 ] , [ 1 , 1 , 0 ] , [ 1 , 1 , 1 ] . (c) (Media 8) Point c for e d = [ 1 , 1 , 1 ] and [ 1 , 1 , 0 ] ; LDOS at e d = [ 1 , 1 , 2 ] is equal to that at e d = [ 1 , 1 , 0 ] . (d) (Media 9) Relative LDOS at point d for e d = [ 0.33 , 0.13 , 0 ] , [ 0.13 , 0 , 0.33 ] , [0,0,1].

Fig. 10
Fig. 10

Left side: Photonic band structure (Media 10) for an inverse opal from silicon ( ϵ = 11.9 ) . Right side: The total DOS in the Si inverse opal (Media 11). The DOS is strongly depleted for frequencies near Γ L and Γ X stopgaps (gray rectangles). A photonic bandgap (gray bar) occurs between bands 8 and 9, as also reflected in the vanishing DOS.

Fig. 11
Fig. 11

Relative LDOS in a Si inverse opal at (a) (Media 12) r = ( 0 , 0 , 0 ) [the center of an air sphere, solid curve], r = 1 4 ( 1 , 1 , 1 ) [among three air spheres, dashed–dotted curve], r = ( 1 2 , 0 , 0 ) [midway between two spheres along [1,0,0] direction, dotted curve]; (b) (Media 13) Relative LDOS at r = 1 4 ( 1 , 1 , 0 ) [in the window between two air spheres] projected on e d = [ 1 , 1 , 0 ] , [ 1 , 1 , 0 ] , and e d = [ 0 , 0 , 1 ] directions shown by solid, dashed–dotted, dotted curves, respectively.

Equations (11)

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N ( r , ω , e d ) = 1 ( 2 π ) 3 n BZ d k δ ( ω ω n , k ) e d E n , k ( r ) 2 ,
× [ ϵ ( r ) 1 × H ( r ) ] = ω 2 c 2 H ( r ) ,
H k ( r ) = e i k r u k ( r ) .
ϵ ( r ) 1 = η ( r ) = G η G e i G r ,
H k ( r ) = G u G k e i ( k + G ) r ,
G η G G ( k + G ) × [ ( k + G ) × u G n , k ] = ω n 2 ( k ) c 2 u G n , k .
G G η G G k + G k + G { [ e k + G 2 e k + G 2 e k + G 2 e k + G 1 e k + G 1 e k + G 2 e k + G 1 e k + G 1 ] ( u G , 1 n , k u G , 2 n , k ) } = ω n 2 ( k ) c 2 ( u G , 1 n , k u G , 2 n , k ) , G G .
E n , k ( r ) = 1 ω n ( k ) ϵ 0 G , G G η G G k + G [ ( u G , 1 n , k e k + G 2 u G , 2 n , k e k + G 1 ) e i ( k + G ) r ] .
BZ H n , k ( r ) H n , k * ( r ) d r = δ ( k k ) δ n , n ,
BZ ϵ ( r ) E n , k ( r ) E n , k * ( r ) d r = δ ( k k ) δ n , n .
Δ ω Δ k k ω ,

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