Abstract

Unavoidable structural disorder in photonic crystals causes multiple scattering of light, resulting in extinction of coherent beams and generation of diffuse light. We demonstrate experimentally that the diffusely transmitted intensity is distributed over exit angles in a strikingly non-Lambertian manner, depending strongly on frequency. The angular redistribution of diffuse light reveals both photonic gaps and the diffuse extrapolation length, as confirmed by a quantitative diffusion theory that includes photonic band structures. Total transmission corrected for internal reflection shows that extinction increases slower with frequency than Rayleigh’s law predicts. Hence disorder affects the high-frequency photonic bandgap of fcc crystals less severely than expected previously.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  47. T. Yoshiyama and I. Sogami, "Kossel images as direct manifestations of the gap structure of the dispersion surface for colloidal crystals," Phys. Rev. Lett. 56, 1609-1612 (1986).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  52. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
    [CrossRef]
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  55. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001). The authors report a resolution of <5% in diameter and <1% in distance between airholes in Si photonic crystal slabs.
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  56. S. Ogawa, K. Tomoda, and S. Noda, "Effects of structural fluctuations on three-dimensional photonic crystals operating at near infrared wavelengths," J. Appl. Phys. 91, 513-515 (2002). The authors report alignment errors less than 7% relative to the wood pile periodicity for optimal strongly photonic GaAs layer-by-layer crystals operating at a 1.2-µm wavelength.
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    [CrossRef] [PubMed]
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    [CrossRef]

2004 (1)

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 (London) 430, 654-657 (2004).
[CrossRef]

2003 (3)

C. Lopez, "Recent review on fabrication of photonic crystals," Adv. Mater. (Weinheim, Ger.) 15, 1679-1704 (2003).

S. J. McNab, N. Moll, and Y. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express 11, 2927-2939 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

A. F. Koenderink and W. L. Vos, "Light exiting real photonic band gap crystals is diffuse and strongly directional," Phys. Rev. Lett. 91, 213902 (2003).
[CrossRef]

2002 (6)

J. F. Galisteo Lòpez and W. L. Vos, "Angle resolved reflectivity of single-domain photonic crystals: effects of disorder," Phys. Rev. E 66, 036616 (2002).
[CrossRef]

S. Ogawa, K. Tomoda, and S. Noda, "Effects of structural fluctuations on three-dimensional photonic crystals operating at near infrared wavelengths," J. Appl. Phys. 91, 513-515 (2002). The authors report alignment errors less than 7% relative to the wood pile periodicity for optimal strongly photonic GaAs layer-by-layer crystals operating at a 1.2-µm wavelength.
[CrossRef]

R. F. Service, "Building better photonic crystals," Science 295, 2399 (2002).
[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]

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystal," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

A. F. Koenderink, P. M. Johnson, J. F. Galisteo Lopez, and W. L. Vos, "Three-dimensional photonic crystals as a cage for light," C. R. Phys. 3, 65-77 (2002).
[CrossRef]

2001 (9)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001). The authors report a resolution of <5% in diameter and <1% in distance between airholes in Si photonic crystal slabs.
[CrossRef]

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

J. Gómez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
[CrossRef]

S. G. Romanov, T. Maka, C. M. S. Torres, M. Muller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, "Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals," Phys. Rev. E 63, 056603 (2001).
[CrossRef]

K. Busch and C. M. Soukoulis, "Energy-density CPA: a new effective medium theory for classical waves," Physica B 296, 56-61 (2001).
[CrossRef]

J. Huang, N. Eradat, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, "Anomalous coherent backscattering of light from opal photonic crystals," Phys. Rev. Lett. 86, 4815-4818 (2001).
[CrossRef] [PubMed]

H. P. Schriemer, H. M. van Driel, A. F. Koenderink, and W. L. Vos, "Modified spontaneous emission spectra of laser dye in inverse opal photonic crystals," Phys. Rev. A 63, 011801 (2001).
[CrossRef]

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. Megens and W. L. Vos, "Particle excursions in colloidal crystals," Phys. Rev. Lett. 86, 4855-4858 (2001).
[CrossRef] [PubMed]

2000 (10)

H. M. van Driel and W. L. Vos, "Multiple Bragg wave coupling in photonic band-gap crystals," Phys. Rev. B 62, 9872-9875 (2000).
[CrossRef]

A. F. Koenderink, M. Megens, G. van Soest, W. L. Vos, and A. Lagendijk, "Enhanced backscattering from photonic crystals," Phys. Lett. A 268, 104-111 (2000).
[CrossRef]

Z. Y. Li and Z. Q. Zhang, "Fragility of photonic band gaps in inverse-opal photonic crystals," Phys. Rev. B 62, 1516-1519 (2000).
[CrossRef]

Z. Y. Li, X. D. Zhang, and Z. Q. Zhang, "Disordered photonic crystals understood by a perturbation formalism," Phys. Rev. B 61, 15738-15748 (2000).
[CrossRef]

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (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]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
[CrossRef]

O. D. Velev and E. W. Kaler, "Structured porous materials via colloidal crystal templating: from inorganic oxides to metals," Adv. Mater. (Weinheim, Ger.) 12, 531-534 (2000).
[CrossRef]

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604-606 (2000).
[CrossRef] [PubMed]

Y. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystal," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

1999 (7)

O. J. 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. Megens, J. E. G. J. Wijnhoven, A. Lagendijk, and W. L. Vos, "Light sources inside photonic crystals," J. Opt. Soc. Am. B 16, 1403-1408 (1999).
[CrossRef]

J. F. Bertone, P. Jiang, K. S. Hwang, D. M. Mittleman, and V. L. Colvin, "Thickness dependence of the optical properties of ordered silica-air and air-polymer photonic crystals," Phys. Rev. Lett. 83, 300-303 (1999).
[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 broadband reflection in photonic air-sphere crystals," Phys. Rev. Lett. 83, 2730-2733 (1999).
[CrossRef]

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, "Strongly photonic macroporous gallium phosphide networks," Science 284, 141-143 (1999).
[CrossRef] [PubMed]

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, R. Biswas, and K. M. Ho, "Effect of disorder on photonic band gaps," Phys. Rev. B 59, 12767-12770 (1999).
[CrossRef]

Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, "Different regimes of light localization in a disordered photonic crystal," Phys. Rev. B 60, 1555-1562 (1999).
[CrossRef]

1998 (3)

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]

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

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

1996 (3)

Ì. Ì. Tarhan and G. H. Watson, "Photonic band structure of fcc colloidal crystals," Phys. Rev. Lett. 76, 315-319 (1996).
[CrossRef] [PubMed]

W. L. Vos, M. Megens, C. M. van Kats, and P. Bösecke, "Transmission and diffraction by photonic colloidal crystals," J. Phys.: Condens. Matter 8, 9503-9507 (1996).

M. U. Vera and D. J. Durian, "Angular distribution of diffusely transmitted light," Phys. Rev. E 53, 3215-3224 (1996).
[CrossRef]

1994 (1)

D. J. Durian, "Influence of boundary reflection and refraction on diffusive photon transport," Phys. Rev. E 50, 857-866 (1994).
[CrossRef]

1991 (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, "Internal reflection of diffusive light in random media," Phys. Rev. A 44, 3948-3959 (1991).
[CrossRef] [PubMed]

1989 (1)

A. Lagendijk, R. Vreeker, and P. de Vries, "Influence of internal reflection on diffusive transport in strongly scattering media," Phys. Lett. A 136, 81-88 (1989).
[CrossRef]

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]

1986 (1)

T. Yoshiyama and I. Sogami, "Kossel images as direct manifestations of the gap structure of the dispersion surface for colloidal crystals," Phys. Rev. Lett. 56, 1609-1612 (1986).
[CrossRef] [PubMed]

1985 (1)

P. A. Lee and T. V. Ramakrishnan, "Disordered electronic systems," Rev. Mod. Phys. 57, 287-337 (1985).
[CrossRef]

1984 (1)

T. Yoshiyama and I. Sogami, "Kossel line analysis on colloidal crystals in semidilute aqueous solutions," Phys. Rev. Lett. 53, 2153-2156 (1984).
[CrossRef]

Adawi, A. M.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystal," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehard & Winston, New York, 1976), pp. 616-620.

Astratov, V. N.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystal," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Y. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystal," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Baba, T.

T. Baba and N. Fukaya, "Light propagation characteristics of defect waveguides in a photonic crystal slab," in Ref. , pp. 105-116 (2000). In pioneering work to quantify propagation losses in state-of-the-art silicon-on-insulator 2D photonic crystals, the authors quote an airhole nonuniformity of <4%.

Baryshev, A. V.

Y. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystal," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Baughman, R. H.

J. Huang, N. Eradat, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, "Anomalous coherent backscattering of light from opal photonic crystals," Phys. Rev. Lett. 86, 4815-4818 (2001).
[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]

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A. F. Koenderink, A. Lagendijk, and W. L. Vos, "Optical loss due to intrinsic structural variations in photonic crystals," preprint, http://arxiv.org/abs/physics/0406052 (2004).

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H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
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M. Megens, J. E. G. J. Wijnhoven, A. Lagendijk, and W. L. Vos, "Light sources inside photonic crystals," J. Opt. Soc. Am. B 16, 1403-1408 (1999).
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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 (London) 430, 654-657 (2004).
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A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
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S. G. Romanov, T. Maka, C. M. S. Torres, M. Muller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, "Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals," Phys. Rev. E 63, 056603 (2001).
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S. G. Romanov, T. Maka, C. M. S. Torres, M. Muller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, "Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals," Phys. Rev. E 63, 056603 (2001).
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A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
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A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
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J. F. Bertone, P. Jiang, K. S. Hwang, D. M. Mittleman, and V. L. Colvin, "Thickness dependence of the optical properties of ordered silica-air and air-polymer photonic crystals," Phys. Rev. Lett. 83, 300-303 (1999).
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Mondia, J. P.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
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S. G. Romanov, T. Maka, C. M. S. Torres, M. Muller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, "Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals," Phys. Rev. E 63, 056603 (2001).
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M. S. Thijssen, R. Sprik, J. E. G. J. Wijnhoven, M. Megens, T. Narayanan, A. Lagendijk, and W. L. Vos, "Inhibited light propagation and broadband reflection in photonic air-sphere crystals," Phys. Rev. Lett. 83, 2730-2733 (1999).
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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 (London) 430, 654-657 (2004).
[CrossRef]

I. S. Nikolaev, P. Lodahl, and W. L. Vos, "Quantitative analysis of directional spontaneous emission spectra from light sources in photonic crystals," preprint, http://arxiv.org/abs/physics/0410056 (2004).

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Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, "Different regimes of light localization in a disordered photonic crystal," Phys. Rev. B 60, 1555-1562 (1999).
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S. Ogawa, K. Tomoda, and S. Noda, "Effects of structural fluctuations on three-dimensional photonic crystals operating at near infrared wavelengths," J. Appl. Phys. 91, 513-515 (2002). The authors report alignment errors less than 7% relative to the wood pile periodicity for optimal strongly photonic GaAs layer-by-layer crystals operating at a 1.2-µm wavelength.
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S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, "Full three-dimensional photonic bandgap crystals at near-infrared wavelengths," Science 289, 604-606 (2000).
[CrossRef] [PubMed]

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J. Gómez Rivas, R. Sprik, A. Lagendijk, L. D. Noordam, and C. W. Rella, "Static and dynamic transport of light close to the Anderson localization transition," Phys. Rev. E 63, 046613 (2001).
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Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
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O. J. 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).
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S. Ogawa, K. Tomoda, and S. Noda, "Effects of structural fluctuations on three-dimensional photonic crystals operating at near infrared wavelengths," J. Appl. Phys. 91, 513-515 (2002). The authors report alignment errors less than 7% relative to the wood pile periodicity for optimal strongly photonic GaAs layer-by-layer crystals operating at a 1.2-µm wavelength.
[CrossRef]

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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 (London) 430, 654-657 (2004).
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A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
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O. J. 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).
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S. G. Romanov, T. Maka, C. M. S. Torres, M. Muller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, "Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals," Phys. Rev. E 63, 056603 (2001).
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Adv. Mater. (Weinheim, Ger.) (2)

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C. R. Phys. (1)

A. F. Koenderink, P. M. Johnson, J. F. Galisteo Lopez, and W. L. Vos, "Three-dimensional photonic crystals as a cage for light," C. R. Phys. 3, 65-77 (2002).
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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).
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J. Appl. Phys. (1)

S. Ogawa, K. Tomoda, and S. Noda, "Effects of structural fluctuations on three-dimensional photonic crystals operating at near infrared wavelengths," J. Appl. Phys. 91, 513-515 (2002). The authors report alignment errors less than 7% relative to the wood pile periodicity for optimal strongly photonic GaAs layer-by-layer crystals operating at a 1.2-µm wavelength.
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J. Opt. Soc. Am. B (1)

J. Phys.: Condens. Matter (1)

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Nature (London) (3)

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, 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 micrometers," Nature (London) 405, 437-440 (2000).
[CrossRef]

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature (London) 414, 289-293 (2001).
[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 (London) 430, 654-657 (2004).
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Opt. Express (1)

Phys. Lett. A (3)

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).
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A. F. Koenderink, M. Megens, G. van Soest, W. L. Vos, and A. Lagendijk, "Enhanced backscattering from photonic crystals," Phys. Lett. A 268, 104-111 (2000).
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[CrossRef]

Phys. Rev. A (2)

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H. P. Schriemer, H. M. van Driel, A. F. Koenderink, and W. L. Vos, "Modified spontaneous emission spectra of laser dye in inverse opal photonic crystals," Phys. Rev. A 63, 011801 (2001).
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Phys. Rev. B (8)

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H. M. van Driel and W. L. Vos, "Multiple Bragg wave coupling in photonic band-gap crystals," Phys. Rev. B 62, 9872-9875 (2000).
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Figures (8)

Fig. 1
Fig. 1

As an incident light beam I 0 ( ω ) impinges on a photonic crystal, a fraction R front ( ω ; γ ) is Bragg reflected, which depends on the frequency ω and the angle of incidence γ. In the sample the light diffuses with typical step length l ( ω ) . The diffuse glow on the transmission side is measured as a function of cos α = μ e . The depth of stop gaps in the escape function is determined by l and the Bragg attenuation length L B (see Section 5).

Fig. 2
Fig. 2

Overview of the diffuse transmission setup. The output of a tungsten–halogen source is passed through a Fourier-transform spectrometer (FTS). The beam is focused onto a pinhole AP 1 by lens L 1 . The pinhole acts as a point source that is imaged on the front sample surface by use of lens L 2 and camera objective L 3 . The angle of incidence γ is controlled by rotating the sample. The detector angle α is changed independently by rotating the diode together with aperture AP 2 and lens L 4 , which determine the angular acceptance.

Fig. 3
Fig. 3

Photon escape function as a function of frequency for an inverse opal with lattice parameter a = 930 nm for exit angles α = 15 ° , 25 ° , 35 ° , 45 ° , 55 ° , 65 ° , 75 ° . The incidence angle is γ = 0 ° . The top axis shows normalized frequency units a λ where λ is the wavelength in vacuum. No relative scaling or offset was applied to the curves.

Fig. 4
Fig. 4

Photon escape function as a function of the cosine μ e of the escape angle α for an inverse opal with lattice parameter a = 930 nm for frequencies ω = 6270 , 9400 , 10,750 , 12,300 cm 1 (filled circles, diamonds, squares, triangles, respectively) as extracted from a white-lamp data set. These frequencies correspond to a λ = 0.58 , 0.87, 1.0, and 1.14. Open diamonds show a measurement obtained from the same sample with a Nd:YVO laser beam ( ω = 9400 cm 1 ) . An angular scale is shown on the top axis. The shaded region corresponds to half of the range of the exit angle α relative to the surface normal. The dashed curve partially obscured by filled circles corresponds to calibration measurements on a dilute colloidal suspension.

Fig. 5
Fig. 5

(a) Contour plot of the measured photon escape function as a function of exit angle α and the optical frequency corresponding to an inverse opal with a = 930 nm . Right-hand panel (b): contour plot of the fitted escape function, according to the diffusion model [Eq. (3)] combined with an internal reflection coefficient derived from the band structure [Eq. (5)]. The lowest six bands along the LU direction are plotted in white; we used the effective index to transform the internal propagation angles into external propagation angles. The band structure alone can not fully explain the frequency and angle dependence of the escape function. A common gray scale is displayed on the right.

Fig. 6
Fig. 6

Solid curve is the fitted peak internal reflection coefficient of the lowest-order stop band versus exit angle α pertaining to the fit [Fig. 5b] to the experimental data shown in Figs. 3, 4, 5a. The peak internal reflection coefficient typically decreases with cos ( α ) (dashed curve).

Fig. 7
Fig. 7

Total diffuse transmission as a function of optical frequency for a sample with lattice parameter a = 930 nm for incidence angles γ = 0 ° , 15 ° , 30 ° (solid black, gray, and light gray curves, respectively). For a sample with a = 800 nm , the stop gap at γ = 0 ° (L gap) is shifted to a higher frequency (dashed curve).

Fig. 8
Fig. 8

τ e = z e l pertaining to the fit to the data in Fig. 5. Upper panel: T * = T ( 1 + τ e [ 1 2 T ] ) for incidence angles γ = 0 ° , 15 ° , 30 ° (black, dark gray, and light gray curves, respectively). Dashed curves represent the power laws ω 3 (short dashes) and ω 2 (long dashes).

Equations (6)

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z e ( ω ) = 2 3 [ 1 + R ¯ D ( ω ) 1 R ¯ D ( ω ) ] l ( ω )
I T ( ω ; γ ) = I 0 ( ω ) T ( ω ; γ ) = I 0 ( ω ) [ 1 R front ( ω ; γ ) ] l ( ω ) + z e ( ω ) L + 2 z e ( ω ) .
P ( ω ; μ e ) d μ e = 3 2 μ e [ τ e ( ω ) + μ i ] [ 1 R D ( ω ; μ i ) ] d μ e .
R ¯ D ( ω ) = 3 C 2 ( ω ) + 2 C 1 ( ω ) 3 C 2 ( ω ) 2 C 1 ( ω ) + 2 with C n ( ω ) = d μ μ n R D ( ω ; μ ) ,
I ( ω ; μ e , γ ) d μ e = I 0 ( ω ) T ( ω ; γ ) P ( ω ; μ e ) d μ e .
R D ( ω ; μ i ) = R 1 ( μ i ) exp { [ ω ω 1 ( μ i ) ] 2 2 Δ ω 1 ( μ i ) 2 } + R 2 ( μ i ) exp { [ ω ω 2 ( μ i ) ] 2 2 Δ ω 2 ( μ i ) 2 } ,

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