Abstract

The identification of a complete three-dimensional (3D) photonic band gap in real crystals typically employs theoretical or numerical models that invoke idealized crystal structures. Such an approach is prone to false positives (gap wrongly assigned) or false negatives (gap missed). Therefore, we propose a purely experimental probe of the 3D photonic band gap that pertains to any class of photonic crystals. We collect reflectivity spectra with a large aperture on exemplary 3D inverse woodpile structures that consist of two perpendicular nanopore arrays etched in silicon. We observe intense reflectivity peaks (R>90%) typical of high-quality crystals with broad stopbands. A resulting parametric plot of s-polarized versus p-polarized stopband width is linear ("y=x"), a characteristic of a 3D photonic band gap, as confirmed by simulations. By scanning the focus across the crystal, we track the polarization-resolved stopbands versus the volume fraction of high-index material and obtain many more parametric data to confirm that the high-NA stopband corresponds to the photonic band gap. This practical probe is model-free and provides fast feedback on the advanced nanofabrication needed for 3D photonic crystals and stimulates practical applications of band gaps in 3D silicon nanophotonics and photonic integrated circuits, photovoltaics, cavity QED, and quantum information processing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (3)

T. Tajiri, S. Takahashi, Y. Ota, K. Watanabe, S. Iwamoto, and Y. Arakawa, “Three-dimensional photonic crystal simultaneously integrating a nanocavity laser and waveguides,” Optica 6(3), 296–299 (2019).
[Crossref]

D. A. Grishina, C. A. M. Harteveld, A. Pacureanu, D. Devashish, A. Lagendijk, P. Cloetens, and W. L. Vos, “X-ray imaging non-destructively identifies functional 3D photonic nanostructures,” ACS Nano 13(12), 13932–13939 (2019).
[Crossref]

D. Devashish, O. S. Ojambati, S. B. Hasan, J. J. W. van der Vegt, and W. L. Vos, “Three-dimensional photonic band gap cavity with finite support: Enhanced energy density and optical absorption,” Phys. Rev. B 99(7), 075112 (2019).
[Crossref]

2018 (3)

K. P. Furlan, E. Larsson, A. Diaz, M. Holler, T. Krekeler, M. Ritter, A. Y. Petrov, M. Eich, R. Blick, G. A. Schneider, I. Greving, R. Zierold, and R. Janssen, “Photonic materials for high-temperature applications: Synthesis and characterization by x-ray ptychographic tomography,” Appl. Mater. Today 13, 359–369 (2018).
[Crossref]

P. Hong, O. S. Ojambati, A. Lagendijk, A. P. Mosk, and W. L. Vos, “Three-dimensional spatially resolved optical energy density enhanced by wavefront shaping,” Optica 5(7), 844–849 (2018).
[Crossref]

S. B. Hasan, A. P. Mosk, W. L. Vos, and A. Lagendijk, “Finite-size scaling of the density of states in photonic band gap crystals,” Phys. Rev. Lett. 120(23), 237402 (2018).
[Crossref]

2017 (2)

D. Devashish, S. B. Hasan, J. J. W. van der Vegt, and W. L. Vos, “Reflectivity calculated for a three-dimensional silicon photonic band gap crystal with finite support,” Phys. Rev. B 95(15), 155141 (2017).
[Crossref]

N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Photonic hyperuniform networks obtained by silicon double inversion of polymer templates,” Optica 4(3), 361–366 (2017).
[Crossref]

2016 (1)

C. Marichy, N. Muller, L. S. Froufe-Pérez, and F. Scheffold, “High-quality photonic crystals with a nearly complete band gap obtained by direct inversion of woodpile templates with titanium dioxide,” Sci. Rep. 6(1), 21818 (2016).
[Crossref]

2015 (2)

D. A. Grishina, C. A. M. Harteveld, L. A. Woldering, and W. L. Vos, “Method for making a single-step etch mask for 3D monolithic nanostructures,” Nanotechnology 26(50), 505302 (2015).
[Crossref]

A. F. Koenderink, A. Alú, and A. Polman, “Nanophotonics: Shrinking light-based technology,” Science 348(6234), 516–521 (2015).
[Crossref]

2013 (2)

K. Ishizaki, M. Koumura, K. Suzuki, K. Gondaira, and S. Noda, “Realization of three-dimensional guiding of photons in photonic crystals,” Nat. Photonics 7(2), 133–137 (2013).
[Crossref]

A. Frölich, J. Fischer, T. Zebrowski, K. Busch, and M. Wegener, “Titania woodpiles with complete three-dimensional photonic bandgaps in the visible,” Adv. Mater. 25(26), 3588–3592 (2013).
[Crossref]

2012 (5)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6(5), 283–292 (2012).
[Crossref]

J. M. van den Broek, L. A. Woldering, R. W. Tjerkstra, F. B. Segerink, I. D. Setija, and W. L. Vos, “Inverse-woodpile photonic band gap crystals with a cubic diamond-like structure made from single-crystalline silicon,” Adv. Funct. Mater. 22(1), 25–31 (2012).
[Crossref]

A. David, H. Benisty, and C. Weisbuch, “Photonic crystal light-emitting sources,” Rep. Prog. Phys. 75(12), 126501 (2012).
[Crossref]

J. Wang and M. Qi, “Design of a compact mode and polarization converter in three-dimensional photonic crystals,” Opt. Express 20(18), 20356–20367 (2012).
[Crossref]

R. B. Wehrspohn and J. Üpping, “3d photonic crystals for photon management in solar cells,” J. Opt. 14(2), 024003 (2012).
[Crossref]

2011 (6)

A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics 5(2), 91–94 (2011).
[Crossref]

J. F. Galisteo Lòpez, M. Ibisate, R. Sapienza, L. S. Froufe-Pérez, À. Blanco, and C. Lòpez, “Self-assembled photonic structures,” Adv. Mater. 23(1), 30–69 (2011).
[Crossref]

M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited Spontaneous Emission of Quantum Dots Observed in a 3D Photonic Band Gap,” Phys. Rev. Lett. 107(19), 193903 (2011).
[Crossref]

R. W. Tjerkstra, L. A. Woldering, J. M. van den Broek, F. Roozeboom, I. D. Setija, and W. L. Vos, “Method to pattern etch masks in two inclined planes for three-dimensional nano- and microfabrication,” J. Vac. Sci. Technol., B 29(6), 061604 (2011).
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S. R. Huisman, R. V. Nair, L. A. Woldering, M. D. Leistikow, A. P. Mosk, and W. L. Vos, “Signature of a three-dimensional photonic band gap observed on silicon inverse woodpile photonic crystals,” Phys. Rev. B 83(20), 205313 (2011).
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G. Subramania, Q. Li, Y. J. Lee, J. J. Figiel, G. T. Wang, and A. J. Fische, “Gallium nitride based logpile photonic crystals,” Nano Lett. 11(11), 4591–4596 (2011).
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2010 (3)

I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett. 35(7), 1094 (2010).
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G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J. M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82(19), 195330 (2010).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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2009 (2)

L. A. Woldering, A. P. Mosk, R. W. Tjerkstra, and W. L. Vos, “The influence of fabrication deviations on the photonic band gap of three-dimensional inverse woodpile nanostructures,” J. Appl. Phys. 105(9), 093108 (2009).
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S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, “Direct creation of three-dimensional photonic crystals by a top-down approach,” Nat. Mater. 8(9), 721–725 (2009).
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2008 (3)

T. G. Euser, A. J. Molenaar, J. G. Fleming, B. Gralak, A. Polman, and W. L. Vos, “All-optical octave-broad ultrafast switching of Si woodpile photonic band gap crystals,” Phys. Rev. B 77(11), 115214 (2008).
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S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2(1), 52–56 (2008).
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K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2(11), 688–692 (2008).
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2007 (4)

2005 (1)

J. Schilling, J. White, A. Scherer, G. Stupian, R. Hillebrand, and U. Gösele, “Three-dimensional macroporous silicon photonic crystal with large photonic band gap,” Appl. Phys. Lett. 86(1), 011101 (2005).
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2004 (4)

M. Maldovan and E. L. Thomas, “Diamond-structured photonic crystals,” Nat. Mater. 3(9), 593–600 (2004).
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S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305(5681), 227–229 (2004).
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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 430(7000), 654–657 (2004).
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M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429(6991), 538–542 (2004).
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2003 (5)

C. Lòpez, “Materials aspects of photonic crystals,” Adv. Mater. 15(20), 1679–1704 (2003).
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A. F. Koenderink, L. Bechger, A. Lagendijk, and W. L. Vos, “An experimental study of strongly modified emission in inverse opal photonic crystals,” Phys. Stat. Sol. (a) 197(3), 648–661 (2003).
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Z. Y. Li and K. M. Ho, “Waveguides in three-dimensional layer-by-layer photonic crystals,” J. Opt. Soc. Am. B 20(5), 801–809 (2003).
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R. Hillebrand, S. Senz, W. Hergert, and U. Gösele, “Macroporous-silicon-based three-dimensional photonic crystal with a large complete band gap,” J. Appl. Phys. 94(4), 2758–2760 (2003).
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Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, “Optical properties of inverted opal photonic band gap crystals with stacking disorder,” Phys. Rev. E 67(1), 016612 (2003).
[Crossref]

2002 (3)

A. V. Petukhov, D. G. A. L. Aarts, I. P. Dolbnya, E. H. A. de Hoog, K. Kassapidou, G. J. Vroege, W. Bras, and H. N. W. Lekkerkerker, “High-resolution small-angle x-ray diffraction study of long-range order in hard-sphere colloidal crystals,” Phys. Rev. Lett. 88(20), 208301 (2002).
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A. F. Koenderink, L. Bechger, H. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88(14), 143903 (2002).
[Crossref]

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer, C. Lòpez, and J. Sánchez-Dehesa, “Optical study of the full photonic band gap in silicon inverse opals,” Appl. Phys. Lett. 81(26), 4925–4927 (2002).
[Crossref]

2001 (3)

Y. A. Vlasov, X.-Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly of silicon photonic bandgap crystals,” Nature 414(6861), 289–293 (2001).
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S. G. Romanov, T. Maka, C. M. Sotomayor Torres, M. Müller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, “Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals,” Phys. Rev. E 63(5), 056603 (2001).
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J. E. G. J. Wijnhoven, L. Bechger, and W. L. Vos, “Fabrication and characterization of large macroporous photonic crystals in titania,” Chem. Mater. 13(12), 4486–4499 (2001).
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2000 (5)

Z. Y. Li and Z. Q. Zhang, “Fragility of photonic band gaps in inverse-opal photonic crystals,” Phys. Rev. B 62(3), 1516–1519 (2000).
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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(15), 9872–9875 (2000).
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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(1-2), 101–106 (2000).
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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(5479), 604–606 (2000).
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A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lòpez, F. Meseguer, H. Miguez, J. P. Mondla, 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(6785), 437–440 (2000).
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1999 (2)

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(14), 2730–2733 (1999).
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G. S. Smith, M. P. Kesler, and J. G. Maloney, “Dipole antennas used with all-dielectric, woodpile photonic-bandgap reflectors: Gain, field patterns, and input impedance,” Microw. Opt. Technol. Lett. 21(3), 191–196 (1999).
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1998 (2)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
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J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in titania,” Science 281(5378), 802–804 (1998).
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1997 (1)

W. L. Vos, M. Megens, C. M. van Kats, and P. Bösecke, “X-ray diffraction of photonic colloidal single crystals,” Langmuir 13(23), 6004–6008 (1997).
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1996 (1)

W. L. Vos, R. Sprik, A. van Blaaderen, A. Imhof, A. Lagendijk, and G. H. Wegdam, “Strong effects of photonic band structures on the diffraction of colloidal crystals,” Phys. Rev. B 53(24), 16231–16235 (1996).
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1994 (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
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1993 (1)

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Effective dielectric constant of periodic composite structures,” Phys. Rev. B 48(20), 14936–14943 (1993).
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1992 (1)

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68(13), 2023–2026 (1992).
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1990 (1)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms,” Phys. Rev. Lett. 64(20), 2418–2421 (1990).
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1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
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1972 (1)

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

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K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2(11), 688–692 (2008).
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Arakawa, Y.

T. Tajiri, S. Takahashi, Y. Ota, K. Watanabe, S. Iwamoto, and Y. Arakawa, “Three-dimensional photonic crystal simultaneously integrating a nanocavity laser and waveguides,” Optica 6(3), 296–299 (2019).
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A. Tandaechanurat, S. Ishida, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Lasing oscillation in a three-dimensional photonic crystal nanocavity with a complete bandgap,” Nat. Photonics 5(2), 91–94 (2011).
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K. Aoki, D. Guimard, M. Nishioka, M. Nomura, S. Iwamoto, and Y. Arakawa, “Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity,” Nat. Photonics 2(11), 688–692 (2008).
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Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68(13), 2023–2026 (1992).
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A. F. Koenderink, L. Bechger, A. Lagendijk, and W. L. Vos, “An experimental study of strongly modified emission in inverse opal photonic crystals,” Phys. Stat. Sol. (a) 197(3), 648–661 (2003).
[Crossref]

A. F. Koenderink, L. Bechger, H. Schriemer, A. Lagendijk, and W. L. Vos, “Broadband fivefold reduction of vacuum fluctuations probed by dyes in photonic crystals,” Phys. Rev. Lett. 88(14), 143903 (2002).
[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(12), 4486–4499 (2001).
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A. David, H. Benisty, and C. Weisbuch, “Photonic crystal light-emitting sources,” Rep. Prog. Phys. 75(12), 126501 (2012).
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J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals (Springer, Verlag, 2008).

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Biswas, R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
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K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
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Blanco, A.

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer, C. Lòpez, and J. Sánchez-Dehesa, “Optical study of the full photonic band gap in silicon inverse opals,” Appl. Phys. Lett. 81(26), 4925–4927 (2002).
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A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lòpez, F. Meseguer, H. Miguez, J. P. Mondla, 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(6785), 437–440 (2000).
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J. F. Galisteo Lòpez, M. Ibisate, R. Sapienza, L. S. Froufe-Pérez, À. Blanco, and C. Lòpez, “Self-assembled photonic structures,” Adv. Mater. 23(1), 30–69 (2011).
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K. P. Furlan, E. Larsson, A. Diaz, M. Holler, T. Krekeler, M. Ritter, A. Y. Petrov, M. Eich, R. Blick, G. A. Schneider, I. Greving, R. Zierold, and R. Janssen, “Photonic materials for high-temperature applications: Synthesis and characterization by x-ray ptychographic tomography,” Appl. Mater. Today 13, 359–369 (2018).
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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(6861), 289–293 (2001).
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Bösecke, P.

W. L. Vos, M. Megens, C. M. van Kats, and P. Bösecke, “X-ray diffraction of photonic colloidal single crystals,” Langmuir 13(23), 6004–6008 (1997).
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Bras, W.

A. V. Petukhov, D. G. A. L. Aarts, I. P. Dolbnya, E. H. A. de Hoog, K. Kassapidou, G. J. Vroege, W. Bras, and H. N. W. Lekkerkerker, “High-resolution small-angle x-ray diffraction study of long-range order in hard-sphere colloidal crystals,” Phys. Rev. Lett. 88(20), 208301 (2002).
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Braun, P. V.

S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2(1), 52–56 (2008).
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F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic band gap,” Adv. Mater. 19(12), 1567–1570 (2007).
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Brener, I.

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68(13), 2023–2026 (1992).
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Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
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A. Frölich, J. Fischer, T. Zebrowski, K. Busch, and M. Wegener, “Titania woodpiles with complete three-dimensional photonic bandgaps in the visible,” Adv. Mater. 25(26), 3588–3592 (2013).
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I. Staude, M. Thiel, S. Essig, C. Wolff, K. Busch, G. von Freymann, and M. Wegener, “Fabrication and characterization of silicon woodpile photonic crystals with a complete bandgap at telecom wavelengths,” Opt. Lett. 35(7), 1094 (2010).
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Bykov, V. P.

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

Cassagne, D.

S. G. Romanov, T. Maka, C. M. Sotomayor Torres, M. Müller, R. Zentel, D. Cassagne, J. Manzanares-Martinez, and C. Jouanin, “Diffraction of light from thin-film polymethylmethacrylate opaline photonic crystals,” Phys. Rev. E 63(5), 056603 (2001).
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Chan, C. T.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, “Optical properties of inverted opal photonic band gap crystals with stacking disorder,” Phys. Rev. E 67(1), 016612 (2003).
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K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: New layer-by-layer periodic structures,” Solid State Commun. 89(5), 413–416 (1994).
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S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Effective dielectric constant of periodic composite structures,” Phys. Rev. B 48(20), 14936–14943 (1993).
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Chen, Z.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, “Optical properties of inverted opal photonic band gap crystals with stacking disorder,” Phys. Rev. E 67(1), 016612 (2003).
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Chomski, E.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lòpez, F. Meseguer, H. Miguez, J. P. Mondla, 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(6785), 437–440 (2000).
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Chutinan, A.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full Three-Dimensional Photonic Bandgap Crystals at Near- Infrared Wavelengths,” Science 289(5479), 604–606 (2000).
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Claudon, J.

G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J. M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82(19), 195330 (2010).
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Clem, P.

Clerk, A. A.

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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D. A. Grishina, C. A. M. Harteveld, A. Pacureanu, D. Devashish, A. Lagendijk, P. Cloetens, and W. L. Vos, “X-ray imaging non-destructively identifies functional 3D photonic nanostructures,” ACS Nano 13(12), 13932–13939 (2019).
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G. Ctistis, A. Hartsuiker, E. van der Pol, J. Claudon, W. L. Vos, and J. M. Gérard, “Optical characterization and selective addressing of the resonant modes of a micropillar cavity with a white light beam,” Phys. Rev. B 82(19), 195330 (2010).
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S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Effective dielectric constant of periodic composite structures,” Phys. Rev. B 48(20), 14936–14943 (1993).
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A. David, H. Benisty, and C. Weisbuch, “Photonic crystal light-emitting sources,” Rep. Prog. Phys. 75(12), 126501 (2012).
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A. V. Petukhov, D. G. A. L. Aarts, I. P. Dolbnya, E. H. A. de Hoog, K. Kassapidou, G. J. Vroege, W. Bras, and H. N. W. Lekkerkerker, “High-resolution small-angle x-ray diffraction study of long-range order in hard-sphere colloidal crystals,” Phys. Rev. Lett. 88(20), 208301 (2002).
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D. Devashish, O. S. Ojambati, S. B. Hasan, J. J. W. van der Vegt, and W. L. Vos, “Three-dimensional photonic band gap cavity with finite support: Enhanced energy density and optical absorption,” Phys. Rev. B 99(7), 075112 (2019).
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D. A. Grishina, C. A. M. Harteveld, A. Pacureanu, D. Devashish, A. Lagendijk, P. Cloetens, and W. L. Vos, “X-ray imaging non-destructively identifies functional 3D photonic nanostructures,” ACS Nano 13(12), 13932–13939 (2019).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82(2), 1155–1208 (2010).
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A. V. Petukhov, D. G. A. L. Aarts, I. P. Dolbnya, E. H. A. de Hoog, K. Kassapidou, G. J. Vroege, W. Bras, and H. N. W. Lekkerkerker, “High-resolution small-angle x-ray diffraction study of long-range order in hard-sphere colloidal crystals,” Phys. Rev. Lett. 88(20), 208301 (2002).
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Euser, T. G.

T. G. Euser, A. J. Molenaar, J. G. Fleming, B. Gralak, A. Polman, and W. L. Vos, “All-optical octave-broad ultrafast switching of Si woodpile photonic band gap crystals,” Phys. Rev. B 77(11), 115214 (2008).
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Fan, S.

F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, “A germanium inverse woodpile structure with a large photonic band gap,” Adv. Mater. 19(12), 1567–1570 (2007).
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G. Subramania, Q. Li, Y. J. Lee, J. J. Figiel, G. T. Wang, and A. J. Fische, “Gallium nitride based logpile photonic crystals,” Nano Lett. 11(11), 4591–4596 (2011).
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Figures (8)

Fig. 1.
Fig. 1. (a) Band structures calculated for an inverse woodpile photonic crystal for $r/a$ = 0.19 and relative permittivity $\varepsilon _{Si}=11.68$. The abscissa is the wave vector in the 1st Brillouin zone (see inset). The experimentally relevant $\Gamma Z$ high-symmetry direction is enlarged for clarity. The $\Gamma Z$ stop gaps for $s$ and $p$-polarized light are indicated by the yellow and hatched bars, respectively. The $p$-polarized bands are shown in blue and $s$ bands in red [26]. The pink bar is the 3D photonic band gap. (b) The $\Gamma Z$ stop gaps and 3D photonic band gap as a function of the reduced pore radius $r/a$, with the corresponding air volume fraction as the top abscissa. The solid curves are the edges of the 3D band gap. The $\Gamma Z$ stop gap edges are shown as the blue and red dotted curves ($p$-polarization) and the green and magenta dashed curves ($s$-polarization). The left ordinate is for a lattice parameter $a = 680$ nm.
Fig. 2.
Fig. 2. Parametric plot of relative stopband width for p-polarization versus relative stopband width for s-polarization measured with NA $= 0.85$ at the same position on crystals with a range of volume fractions (blue circles). Black dashed-dotted line is the linear "y=x" dependence characteristic of the 3D photonic band gap. The red dashed curve pertains to the $\Gamma Z$ stop gap, as obtained from band structures. The cyan cross and green asterisk are numerical results for normal incidence (NA $= 0$) and angle-averaged (NA $= 0.65$) stopbands for $r/a=0.19$, respectively, and the magenta star is the band gap width simulated for a finite-thickness crystal with $r/a=0.19$ that are connected by the gray dotted line as a guide to the eye [26].
Fig. 3.
Fig. 3. (a) Scanning electron microscopy (SEM) image of the edge of the silicon beam with a cubic 3D inverse woodpile crystal in perspective view. The crystal consists of two sets of perpendicular pores along the $X$ and $Z$ directions with design radius $r_{d} = 160$ nm. The coordinate system used in the paper is shown with the origin at the lower right corner of the crystal. The crystal has lattice parameters $a = 680$ nm in the Y-direction, and $c$ in the X and Z-directions with $c=a/\sqrt {2}$. Top: The incident light cone is centered around the wave vector $\vec {k_{in}}$ in the $\Gamma Z$ direction. The polarization is shown: light is $p$-polarized when the incident E-field is parallel to the $X$-directed pores, and $s$-polarized when the incident E-field is perpendicular to the $X$-directed pores. (b) Image of the $XY$-surface of one of the 3D inverse woodpile crystals taken with the IR camera in the optical setup with near infrared LED illumination, with two partly visible neighboring crystals below and above. The bright spot on the crystal is the focus of the incident light from the supercontinuum source filtered by the monochromator. The dotted red line shows the position scan of the focus across the crystal as shown in Fig. 6.
Fig. 4.
Fig. 4. Setup to measure position-resolved microscopic broadband reflectivity. The Fianium SC is the broadband supercontinuum source, the long-pass glass filter F blocks the visible light at $\lambda\;<\;850$ nm, the monochromator filters the light to a narrow band, HWP are half-wave plates, P are polarizers, and BS are beam splitters. Incident light is focused on the sample with a 100$\times$ objective that also collects the reflected light; the coordinate system is shown at top right. The NIR camera views the sample in reflection with an effective magnification of 250$\times$. The photodiodes PD1 and PD2 monitor the incident light power and measure signal from the crystal, respectively.
Fig. 5.
Fig. 5. Reflectivity spectra of three different 3D photonic crystals with three designed pore radii $r_d = 130, 140$ and $160$ nm $(r_d/a = 0.191, 0.206, 0.235)$ (red circles, yellow diamonds and blue triangles, respectively). The stopbands appear at different frequency ranges. The gray squares represent reflectivity from bulk Si on the beam away from the crystals.
Fig. 6.
Fig. 6. Reflectivity measured as a function of Y-position on a crystal with design pore radius $r_d = 130$ nm (or $r_d/a = 0.191$), measured with $p$-polarized light. (a) Maximum peak reflectivity $(R_m)$ and minimum reflectivity below the stopband $(R_l)$. (b) Upper edges (magenta diamonds) and lower edges (blue triangles) of the stopband obtained from the half heights of the reflectivity peaks. The right ordinate is absolute frequency for a lattice parameter $a = 680$ nm. (c) Relative radii $r/a$ derived by comparing the lower edge of the stopband with data shown in Fig. 1(b). The grey areas at $Y\;<\;0\;\mu$m and $Y\;>\;10\;\mu$m indicate bulk silicon outside the crystal with a constant reflectivity near $31\%$.
Fig. 7.
Fig. 7. Evolution of the stopband edges versus pore radius. The red and blue triangles represent upper edge of the stopband for $s$ and $p$-polarized light respectively. The red and blue circles represent the lower edge of the stopband for $s$ and $p$ polarized light. The stopband edges are inferred from the reflectivity peak measured on 11 crystals. The solid lines indicate the edges of the photonic band gap. The upper edge of the $\Gamma Z$ stop gap for $s$ and $p$ polarized light are plotted as the red and blue dotted curves, respectively. The right ordinate is absolute frequency for a lattice parameter $a = 680$ nm.
Fig. 8.
Fig. 8. Measured relative stopband width (gap width to midgap, $\Delta \omega / \omega _{c}$) versus $r/a$ for (a) $s$-polarized (red circles), (b) p-polarized (blue circles) input light. Yellow diamonds in (a) are data from an older Si beam. The cyan crosses, green asterisks, and magenta stars are numerical results for normal incidence, angle-averaged stopband, and complete band gap at $r/a=0.19$, respectively for both polarizations [26]. The dashed red and dash dotted blue curves represent the width of the $\Gamma Z$ stop gap obtained from band structures for $s$ and $p$ polarized light, respectively. The magenta solid curve is the 3D photonic band gap from band structures.

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