Abstract

Active, ultra-fast external control of the emission properties at the nanoscale is of great interest for chip-scale, tunable and efficient nanophotonics. Here we investigated the emission control of dipolar emitters coupled to a nanostructure made of an Au nanoantenna, and a thin vanadium dioxide (VO2) layer that changes from semiconductor to metallic state. If the emitters are sandwiched between the nanoantenna and the VO2 layer, the enhancement and/or suppression of the nanostructure’s magnetic dipole resonance enabled by the phase change behavior of the VO2 layer can provide a high contrast ratio of the emission efficiency. We show that a single nanoantenna can provide high magnetic field in the emission layer when VO2 is metallic, leading to high emission of the magnetic dipoles; this emission is then lowered when VO2 switches back to semiconductor. We finally optimized the contrast ratio by considering different orientation, distribution and nature of the dipoles, as well as the influence of a periodic Au nanoantenna pattern. As an example of a possible application, the design is optimized for the active control of an Er3+ doped SiO2 emission layer. The combination of the emission efficiency increase due to the plasmonic nanoantenna resonances and the ultra-fast contrast control due to the phase-changing medium can have important applications in tunable efficient light sources and their nanoscale integration.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

E. Petronijević, G. Leahu, V. Di Meo, A. Crescitelli, P. Dardano, E. Esposito, G. Coppola, I. Rendina, M. Miritello, M. G. Grimaldi, V. Torrisi, G. Compagnini, and C. Sibilia, “Near-infrared modulation by means of GeTe/SOI based metamaterial,” Opt. Lett. 44(6), 1508–1511 (2019).
[Crossref]

A. Vaskin, S. Mashhadi, M. Steinert, K. E. Chong, D. Keene, S. Nanz, A. Abass, E. Rusak, D. Y. Choi, I. Fernandez-Corbaton, T. Pertsch, C. Rockstuhl, M. A. Noginov, Y. S. Kivshar, D. N. Neshev, N. Noginova, and I. Staude, “Manipulation of magnetic dipole emission from Eu3+ with Mie-resonant dielectric metasurfaces,” Nano Lett. 19(2), 1015–1022 (2019).
[Crossref]

2018 (3)

B. Kalinic, T. Cesca, C. Scian, N. Michieli, I. G. Balasa, E. Trave, and G. Mattei, “Emission efficiency enhancement of Er3+ ions in silica by near-field coupling with plasmonic and pre-plasmonic nanostructures,” Phys. Status Solidi A 215(3), 1700437 (2018).
[Crossref]

N. Michieli, B. Kalinic, C. Scian, T. Cesca, and G. Mattei, “Emission rate modification and quantum efficiency enhancement of Er3+ emitters by near-field coupling with nanohole arrays,” ACS Photonics 5(6), 2189–2199 (2018).
[Crossref]

Y. Fan, C. Guo, Z. Zhu, W. Xu, F. Wu, X. Yuan, and S. Qin, “Monolayer-graphene-based broadband and wide-angle perfect absorption structures in the near infrared,” Sci. Rep. 8(1), 13709 (2018).
[Crossref]

2017 (9)

J. Pradhan, S. Anantha Ramakrishna, B. Rajeswaran, A. Umarji, V. Achanta, A. Agarwal, and A. Ghosh, “High contrast switchability of VO2 based metamaterial absorbers with ITO ground plane,” Opt. Express 25(8), 9116–9121 (2017).
[Crossref]

L. Yang, P. Zhou, T. Huang, G. Zhen, L. Zhang, L. Bi, X. Weng, J. Xie, and L. Deng, “Broadband thermal tunable infrared absorber based on the coupling between standing wave and magnetic resonance,” Opt. Mater. Express 7(8), 2767–2776 (2017).
[Crossref]

G. Leahu, E. Petronijevic, A. Belardini, M. Centini, R. Li Voti, T. Hakkarainen, E. Koivusalo, M. Guina, and C. Sibilia, “Photo-acoustic spectroscopy revealing resonant absorption of self-assembled GaAs-based nanowires,” Sci. Rep. 7(1), 2833 (2017).
[Crossref]

D. G. Baranov, R. S. Savelev, S. V. Li, A. E. Krasnok, and A. Alù, “Modifying magnetic dipole spontaneous emission with nanophotonic structures,” Laser Photonics Rev. 11(3), 1600268 (2017).
[Crossref]

J. Li, N. Verellen, and P. V. Dorpe, “Enhancing magnetic dipole emission by a nano-doughnut-shaped silicon disk,” ACS Photonics 4(8), 1893–1898 (2017).
[Crossref]

X. Han, F. Zhao, K. He, Z. He, and Z. Zhang, “Near-perfect absorber of infrared radiation based on Au nanoantenna arrays,” J. Nanophotonics 11(1), 016018 (2017).
[Crossref]

H. Sun, L. Yin, Z. Liu, Y. Zheng, F. Fan, S. Zhao, X. Feng, T. Li, and C. Z. Ning, “Giant optical gain in a single-crystal erbium chloride silicate nanowire,” Nat. Photonics 11(9), 589–593 (2017).
[Crossref]

J. Liang, X. Song, J. Li, K. Lan, and P. Li, “A visible-near infrared wavelength-tunable metamaterial absorber based on the structure of Au triangle arrays embedded in VO2 thin film,” J. Alloys Compd. 708, 999–1007 (2017).
[Crossref]

F. Kang, J. He, T. Sun, Z. Y. Bao, F. Wang, and D. Y. Lei, “Plasmonic dual-enhancement and precise color tuning of gold nanorod@SiO2 coupled core–shell–shell upconversion nanocrystals,” Adv. Funct. Mater. 27(36), 1701842 (2017).
[Crossref]

2016 (4)

B. Choi, M. Iwanaga, Y. Sugimoto, K. Sakoda, and H. T. Miyazaki, “Selective plasmonic enhancement of electric- and magnetic-dipole radiations of Er ions,” Nano Lett. 16(8), 5191–5196 (2016).
[Crossref]

M. Yang, Y. Yang, B. Hong, L. Wang, K. Hu, Y. Dong, H. Xu, H. Huang, J. Zhao, H. Chen, L. Song, H. Ju, J. Zhu, J. Bao, X. Li, Y. Gu, T. Yang, X. Gao, Z. Luo, and C. Gao, “Suppression of structural phase transition in VO2 by epitaxial strain in vicinity of metal-insulator transition,” Sci. Rep. 6(1), 23119 (2016).
[Crossref]

M. Rudé, V. Mkhitaryan, A. E. Cetin, T. A. Miller, A. Carrilero, S. Wall, F. J. García de Abajo, H. Altug, and V. Pruneri, “Ultrafast broadband tuning of resonant optical nanostructures using phase change materials,” Adv. Opt. Mater. 4(7), 1060–1066 (2016).
[Crossref]

E. Petronijevic and C. Sibilia, “All-optical tuning of EIT-like dielectric metasurfaces by means of chalcogenide phase change materials,” Opt. Express 24(26), 30411–30420 (2016).
[Crossref]

2015 (6)

T. Cesca, B. Kalinic, N. Michieli, C. Maurizio, A. Trapananti, C. Scian, G. Battaglin, P. Mazzoldi, and G. Mattei, “Au–Ag nanoalloy molecule-like clusters for enhanced quantum efficiency emission of Er3+ ions in silica,” Phys. Chem. Chem. Phys. 17(42), 28262–28269 (2015).
[Crossref]

T. Cesca, B. Kalinic, C. Maurizio, C. Scian, G. Battaglin, P. Mazzoldi, and G. Mattei, “Interatomic coupling of Au molecular clusters and Er3+ ions in silica,” ACS Photonics 2(1), 96–104 (2015).
[Crossref]

S. Cueff, D. Li, Y. Zhou, F. J. Wong, J. A. Kurvits, S. Ramanathan, and R. Zia, “Dynamic control of light emission faster than the lifetime limit using VO2 phase-change,” Nat. Commun. 6(1), 8636 (2015).
[Crossref]

H. Kocer, S. Butun, B. Banar, K. Wang, S. Tongay, J. Wu, and K. Aydin, “Thermal tuning of infrared resonant absorbers based on hybrid gold-VO2 nanostructures,” Appl. Phys. Lett. 106(16), 161104 (2015).
[Crossref]

M. Mivelle, T. Grosjean, G. W. Burr, U. C. Fischer, and M. F. Garcia-Parajo, “Strong modification of magnetic dipole emission through diabolo nanoantennas,” ACS Photonics 2(8), 1071–1076 (2015).
[Crossref]

R. Hussain, S. S. Kruk, C. E. Bonner, A. M. Noginov, I. Staude, Y. S. Kivshar, N. Noginova, and D. N. Neshev, “Enhancing Eu3+ magnetic dipole emission by resonant plasmonic nanostructures,” Opt. Lett. 40(8), 1659–1662 (2015).
[Crossref]

2014 (1)

K. Appavoo, B. Wang, N. F. Brady, M. Seo, J. Nag, R. P. Prasankumar, D. J. Hilton, S. T. Pantelides, and R. F. Haglund, “Ultrafast phase transition via catastrophic phonon collapse driven by plasmonic hot-electron injection,” Nano Lett. 14(3), 1127–1133 (2014).
[Crossref]

2013 (3)

B. Gholipour, J. Zhang, K. F. MacDonald, D. W. Hewak, and N. I. Zheludev, “An all-optical, non-volatile, bidirectional, phase-change meta-switch,” Adv. Mater. 25(22), 3050–3054 (2013).
[Crossref]

J. Li, H. Guo, and Z. Li, “Microscopic and macroscopic manipulation of gold nanoantenna and its hybrid nanostructures,” Photonics Res. 1(1), 28–41 (2013).
[Crossref]

S. M. Hein and H. Giessen, “Tailoring magnetic dipole emission with plasmonic split-ring resonators,” Phys. Rev. Lett. 111(2), 026803 (2013).
[Crossref]

2012 (3)

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas - a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20(13), 13636–13650 (2012).
[Crossref]

C. M. Dodson and R. Zia, “Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: Calculated emission rates and oscillator strengths,” Phys. Rev. B 86(12), 125102 (2012).
[Crossref]

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B 85(24), 245432 (2012).
[Crossref]

2011 (3)

L. Zhao, T. Ming, H. Chen, Y. Liang, and J. Wang, “Plasmon-induced modulation of the emission spectra of the fluorescent molecules near gold nanoantennas,” Nanoscale 3(9), 3849–3859 (2011).
[Crossref]

M. D. Goldflam, T. Driscoll, B. Chapler, O. Khatib, N. Marie Jokerst, S. Palit, D. R. Smith, B.-J. Kim, G. Seo, H.-T. Kim, M. Di Ventra, and D. N. Basov, “Reconfigurable gradient index using VO2 memory metamaterials,” Appl. Phys. Lett. 99(4), 044103 (2011).
[Crossref]

T. Feng, Y. Zhou, D. Liu, and J. Li, “Controlling magnetic dipole transition with magnetic plasmonic structures,” Opt. Lett. 36(12), 2369–2371 (2011).
[Crossref]

2009 (3)

2008 (1)

G. W. Bryant, F. J. García de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8(2), 631–636 (2008).
[Crossref]

2007 (1)

S. Lysenko, A. Rúa, V. Vikhnin, F. Fernández, and H. Liu, “Insulator-to-metal phase transition and recovery processes in VO2 thin films after femtosecond laser excitation,” Phys. Rev. B 76(3), 035104 (2007).
[Crossref]

2006 (2)

H. Mertens and A. Polman, “Plasmon-enhanced erbium luminescence,” Appl. Phys. Lett. 89(21), 211107 (2006).
[Crossref]

Q. Thommen and P. Mandel, “Left-handed properties of erbium-doped crystals,” Opt. Lett. 31(12), 1803–1805 (2006).
[Crossref]

1996 (1)

E. Snoeks, P. G. Kik, and A. Polman, “Concentration quenching in erbium implanted alkali silicate glasses,” Opt. Mater. 5(3), 159–167 (1996).
[Crossref]

Abass, A.

A. Vaskin, S. Mashhadi, M. Steinert, K. E. Chong, D. Keene, S. Nanz, A. Abass, E. Rusak, D. Y. Choi, I. Fernandez-Corbaton, T. Pertsch, C. Rockstuhl, M. A. Noginov, Y. S. Kivshar, D. N. Neshev, N. Noginova, and I. Staude, “Manipulation of magnetic dipole emission from Eu3+ with Mie-resonant dielectric metasurfaces,” Nano Lett. 19(2), 1015–1022 (2019).
[Crossref]

Achanta, V.

Agarwal, A.

Aizpurua, J.

Altug, H.

M. Rudé, V. Mkhitaryan, A. E. Cetin, T. A. Miller, A. Carrilero, S. Wall, F. J. García de Abajo, H. Altug, and V. Pruneri, “Ultrafast broadband tuning of resonant optical nanostructures using phase change materials,” Adv. Opt. Mater. 4(7), 1060–1066 (2016).
[Crossref]

Alù, A.

D. G. Baranov, R. S. Savelev, S. V. Li, A. E. Krasnok, and A. Alù, “Modifying magnetic dipole spontaneous emission with nanophotonic structures,” Laser Photonics Rev. 11(3), 1600268 (2017).
[Crossref]

Anantha Ramakrishna, S.

Appavoo, K.

K. Appavoo, B. Wang, N. F. Brady, M. Seo, J. Nag, R. P. Prasankumar, D. J. Hilton, S. T. Pantelides, and R. F. Haglund, “Ultrafast phase transition via catastrophic phonon collapse driven by plasmonic hot-electron injection,” Nano Lett. 14(3), 1127–1133 (2014).
[Crossref]

Atwater, H.

Aydin, K.

H. Kocer, S. Butun, B. Banar, K. Wang, S. Tongay, J. Wu, and K. Aydin, “Thermal tuning of infrared resonant absorbers based on hybrid gold-VO2 nanostructures,” Appl. Phys. Lett. 106(16), 161104 (2015).
[Crossref]

M. Dicken, K. Aydin, I. Pryce, L. Sweatlock, E. Boyd, S. Walavalkar, J. Ma, and H. Atwater, “Frequency tunable near-infrared metamaterials based on VO2 phase transition,” Opt. Express 17(20), 18330–18339 (2009).
[Crossref]

Balasa, I. G.

B. Kalinic, T. Cesca, C. Scian, N. Michieli, I. G. Balasa, E. Trave, and G. Mattei, “Emission efficiency enhancement of Er3+ ions in silica by near-field coupling with plasmonic and pre-plasmonic nanostructures,” Phys. Status Solidi A 215(3), 1700437 (2018).
[Crossref]

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A. Vaskin, S. Mashhadi, M. Steinert, K. E. Chong, D. Keene, S. Nanz, A. Abass, E. Rusak, D. Y. Choi, I. Fernandez-Corbaton, T. Pertsch, C. Rockstuhl, M. A. Noginov, Y. S. Kivshar, D. N. Neshev, N. Noginova, and I. Staude, “Manipulation of magnetic dipole emission from Eu3+ with Mie-resonant dielectric metasurfaces,” Nano Lett. 19(2), 1015–1022 (2019).
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A. Vaskin, S. Mashhadi, M. Steinert, K. E. Chong, D. Keene, S. Nanz, A. Abass, E. Rusak, D. Y. Choi, I. Fernandez-Corbaton, T. Pertsch, C. Rockstuhl, M. A. Noginov, Y. S. Kivshar, D. N. Neshev, N. Noginova, and I. Staude, “Manipulation of magnetic dipole emission from Eu3+ with Mie-resonant dielectric metasurfaces,” Nano Lett. 19(2), 1015–1022 (2019).
[Crossref]

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B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B 85(24), 245432 (2012).
[Crossref]

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Sugimoto, Y.

B. Choi, M. Iwanaga, Y. Sugimoto, K. Sakoda, and H. T. Miyazaki, “Selective plasmonic enhancement of electric- and magnetic-dipole radiations of Er ions,” Nano Lett. 16(8), 5191–5196 (2016).
[Crossref]

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H. Sun, L. Yin, Z. Liu, Y. Zheng, F. Fan, S. Zhao, X. Feng, T. Li, and C. Z. Ning, “Giant optical gain in a single-crystal erbium chloride silicate nanowire,” Nat. Photonics 11(9), 589–593 (2017).
[Crossref]

Sun, T.

F. Kang, J. He, T. Sun, Z. Y. Bao, F. Wang, and D. Y. Lei, “Plasmonic dual-enhancement and precise color tuning of gold nanorod@SiO2 coupled core–shell–shell upconversion nanocrystals,” Adv. Funct. Mater. 27(36), 1701842 (2017).
[Crossref]

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Thommen, Q.

Tongay, S.

H. Kocer, S. Butun, B. Banar, K. Wang, S. Tongay, J. Wu, and K. Aydin, “Thermal tuning of infrared resonant absorbers based on hybrid gold-VO2 nanostructures,” Appl. Phys. Lett. 106(16), 161104 (2015).
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Trapananti, A.

T. Cesca, B. Kalinic, N. Michieli, C. Maurizio, A. Trapananti, C. Scian, G. Battaglin, P. Mazzoldi, and G. Mattei, “Au–Ag nanoalloy molecule-like clusters for enhanced quantum efficiency emission of Er3+ ions in silica,” Phys. Chem. Chem. Phys. 17(42), 28262–28269 (2015).
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M. Yang, Y. Yang, B. Hong, L. Wang, K. Hu, Y. Dong, H. Xu, H. Huang, J. Zhao, H. Chen, L. Song, H. Ju, J. Zhu, J. Bao, X. Li, Y. Gu, T. Yang, X. Gao, Z. Luo, and C. Gao, “Suppression of structural phase transition in VO2 by epitaxial strain in vicinity of metal-insulator transition,” Sci. Rep. 6(1), 23119 (2016).
[Crossref]

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L. Zhao, T. Ming, H. Chen, Y. Liang, and J. Wang, “Plasmon-induced modulation of the emission spectra of the fluorescent molecules near gold nanoantennas,” Nanoscale 3(9), 3849–3859 (2011).
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H. Sun, L. Yin, Z. Liu, Y. Zheng, F. Fan, S. Zhao, X. Feng, T. Li, and C. Z. Ning, “Giant optical gain in a single-crystal erbium chloride silicate nanowire,” Nat. Photonics 11(9), 589–593 (2017).
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B. Gholipour, J. Zhang, K. F. MacDonald, D. W. Hewak, and N. I. Zheludev, “An all-optical, non-volatile, bidirectional, phase-change meta-switch,” Adv. Mater. 25(22), 3050–3054 (2013).
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H. Sun, L. Yin, Z. Liu, Y. Zheng, F. Fan, S. Zhao, X. Feng, T. Li, and C. Z. Ning, “Giant optical gain in a single-crystal erbium chloride silicate nanowire,” Nat. Photonics 11(9), 589–593 (2017).
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Zhou, Y.

S. Cueff, D. Li, Y. Zhou, F. J. Wong, J. A. Kurvits, S. Ramanathan, and R. Zia, “Dynamic control of light emission faster than the lifetime limit using VO2 phase-change,” Nat. Commun. 6(1), 8636 (2015).
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[Crossref]

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Y. Fan, C. Guo, Z. Zhu, W. Xu, F. Wu, X. Yuan, and S. Qin, “Monolayer-graphene-based broadband and wide-angle perfect absorption structures in the near infrared,” Sci. Rep. 8(1), 13709 (2018).
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Other (1)

Lumerical Solutions, Inc. http://www.lumerical.com/tcad-products/fdtd/

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Figures (10)

Fig. 1.
Fig. 1. Complex refractive indices [10] of VO2 in (a) semiconductor (cold), and (b) metallic (hot) state. Insets show the principle of magnetic dipole emission control (low and high emission in cold and hot state, respectively). (c) Schematic of the investigated hybrid nanostructure: the Au nanoantenna is defined by its length L, width W and thickness t, and it lies above a multilayer structure, made of tem thick Er:SiO2, and tpc thick VO2 layers upon SiO2 substrate. (d,e) x-polarized plane-wave simulations for the two VO2 states; the parameters of the nanostructure are L = 340 nm, W = 50 nm, t = 40 nm, tem=50 nm, and tpc=60 nm: (d) absorption efficiency, and (e) magnetic field intensity at 1540 nm, monitored in xz cross-section, at y = 0 nm.
Fig. 2.
Fig. 2. (a) Spectral dependence of the magnetic field intensity normalized to the intensity of the incident magnetic field H0, at the point Δx=Δy = 0 nm, Δz=-10 nm, for the two VO2 states. (b) Normalized Hy intensity for the two VO2 states at 1540 nm, monitored in yz cross-section, at x = 0 nm.
Fig. 3.
Fig. 3. (a) Sketch of the dipole excitation simulations. θ and φ are the emission cone half-angle, and azimuthal angle, respectively. (b-c) Spectra of the total power emitted to z+ far-field for the two VO2 states, for different dipole orientations of a dipole located at the center (Δx=Δy = 0 nm), at Δz=-10 nm, for (b) magnetic dipole; (c) electric dipole.
Fig. 4.
Fig. 4. Normalized total power emitted at 1540 nm for a y-oriented dipole, positioned (a) at Δz=-10 nm, as a function of the distance from the center |Δx| or |Δy|, and (b) at Δx=Δy = 0 nm, as a function of |Δz|. The power is normalized to the maximum power emitted for the resonant dipole at 1540 nm (y-oriented dipole and the hot VO2 state).
Fig. 5.
Fig. 5. (a) Distribution of power emitted to the z+ far-field, for a magnetic dipole at 1540 nm, in the two VO2 states and three different dipole orientations. The dipole is located at the center, at Δz=-10 nm. All the maps are normalized to the maximum value (power emitted to far-field θ=φ=0 for the y-oriented dipole in the hot state). (b) Far-field emission distribution above the structure, averaged over the three orientations of the magnetic dipole which is positioned centrally at Δz=-10 nm, and emits at 1540 nm.
Fig. 6.
Fig. 6. (a) Simulation sketch for a periodic structure, with dipoles positioned in a unit cell (green dots). (b) Far-field averaged over the three dipole orientations with homogeneous distribution in the unit cell at Δz=-10 nm, for p = 900 nm, and p = 800 nm. (c) Far-field response of the dipoles positioned away from the center for both VO2 states. The randomly oriented magnetic dipole is positioned at (left) the border and (right) the corner of the unit cell. (d) Far-field averaged over the three dipole orientations and positions in the unit cell (xy plane) at Δz=-10 nm, with dipoles only under the nanoantennas (patterned distribution). All maps are normalized to the hot state maximum.
Fig. 7.
Fig. 7. Far-field directional efficiency ratio γff,hotff,cold for the two states as a function of p and collection angle θ for a distribution of randomly oriented magnetic dipoles positioned under the nanoantennas, at Δz=-10 nm.
Fig. 8.
Fig. 8. Far-field power averaged over the three orientations and dipole positions for which |Δx|≤L/2, |Δy|≤W/2, and –40 nm≤Δz≤-10 nm, for the nanoantenna metamaterial with p = 900 nm at 1540 nm. Both maps are normalized to the hot state maximum (θ=φ=0).
Fig. 9.
Fig. 9. (a) Sketch of the periodic structure. (b) Absorption dependence on the period when VO2 is hot and (c) when VO2 is cold. (d) Normalized absorption modulation depth dependence on the period.
Fig. 10.
Fig. 10. Scatter plot of MDA as a function of manufacturing tolerances of the parameter L. The device parameters are taken as Gaussian distributions defined as: L = 340 ± 5 nm, tpc=60 ± 5 nm, t = 40 ± 2 nm, and W = 50 ± 2 nm.

Tables (2)

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Table 1. Modulation depth (MD) and contrast ratio (CR) for different p and collection angles θ.

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Table 2. Modulation depth (MD) and contrast ratio (CR) for the set of dipoles under the nanoantenna for p = 900 nm, at different Δz and collection angles θ.

Equations (3)

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γ f f = P θ P r a d = | Δ x | , | Δ y | f f ( θ ) P d s ( θ ) | Δ x | , | Δ y | ( z + P d s + z P d s ) ,
M D = 100 P θ , h o t P θ , c o l d P θ , h o t + P θ , c o l d , C R = P θ , h o t P θ , c o l d .
M D A = 100 A h o t A c o l d A h o t + A c o l d .