Abstract

Based on a single resonance, nanostructures often provide narrowband enhancement for magnetic dipole emissions. Here, tapered hollow hyperbolic metamaterial is designed in order to produce a multiband emission enhancement. Specifically, a series of coaxial magnetic hot spots is excited inside the structure in five discrete bands. Meanwhile, we demonstrate that the emission enhancement can be achieved at both multiple wavelengths and multiple spatial positions in one single device. An enhancement factor of radiative decay rate up to 694 is obtained. Results of this paper might open new possibilities for nanostructures to achieve multiband light emission enhancement in the magneto-optical domain.

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

Full Article  |  PDF Article
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References

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2019 (1)

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] [PubMed]

2018 (3)

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Y. L. Liao, Y. Zhao, S. Wu, and S. Feng, “Wide-angle broadband absorber based on uniform-sized hyperbolic metamaterial,” Opt. Mater. Express 8(9), 2484–2493 (2018).
[Crossref]

2017 (5)

P. Sethi, A. Haldar, and S. Kumar, “Ultra-compact low-loss broadband waveguide taper in silicon-on-insulator,” Opt. Express 25(9), 10196–10203 (2017).
[Crossref]

Y. Chen, Y. Chen, J. Chu, and X. Xu, “Bridged bowtie aperture antenna for producing an electromagnetic hot spot,” ACS Photonics 4(3), 567–575 (2017).
[Crossref]

D. G. Baranov, D. A. Zuev, S. I. Lepeshov, O. V. Kotov, A. E. Krasnok, A. B. Evlyukhin, and B. N. Chichkov, “All-dielectric nanophotonics: the quest for better materials and fabrication techniques,” Optica 4(7), 814–825 (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]

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

2016 (3)

2015 (2)

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]

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

2014 (4)

Y. Yang, H. T. Dai, and X. W. Sun, “Fractal diabolo antenna for enhancing and confining the optical magnetic field,” AIP Adv. 4(1), 017123 (2014).
[Crossref]

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

L. Ferrari, D. Lu, D. Lepage, and Z. Liu, “Enhanced spontaneous emission inside hyperbolic metamaterials,” Opt. Express 22(4), 4301–4306 (2014).
[Crossref] [PubMed]

J. Zhou, A. F. Kaplan, L. Chen, and L. J. Guo, “Experiment and theory of the broadband absorption by a tapered hyperbolic metamaterial array,” ACS Photonics 1(7), 618–624 (2014).
[Crossref]

2013 (5)

2012 (3)

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 Condens. Matter Mater. Phys. 86(12), 125102 (2012).
[Crossref]

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

2011 (2)

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

N. Zhou, E. C. Kinzel, and X. Xu, “Complementary bowtie aperture for localizing and enhancing optical magnetic field,” Opt. Lett. 36(15), 2764–2766 (2011).
[Crossref] [PubMed]

2010 (1)

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

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] [PubMed]

Alu, A.

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

Baida, F. I.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

Baranov, D. G.

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

D. G. Baranov, D. A. Zuev, S. I. Lepeshov, O. V. Kotov, A. E. Krasnok, A. B. Evlyukhin, and B. N. Chichkov, “All-dielectric nanophotonics: the quest for better materials and fabrication techniques,” Optica 4(7), 814–825 (2017).
[Crossref]

Brenny, B. J. M.

Burr, G. W.

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]

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

Chen, H. M.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Chen, L.

J. Zhou, A. F. Kaplan, L. Chen, and L. J. Guo, “Experiment and theory of the broadband absorption by a tapered hyperbolic metamaterial array,” ACS Photonics 1(7), 618–624 (2014).
[Crossref]

Chen, S.

Chen, Y.

Y. Chen, Y. Chen, J. Chu, and X. Xu, “Bridged bowtie aperture antenna for producing an electromagnetic hot spot,” ACS Photonics 4(3), 567–575 (2017).
[Crossref]

Y. Chen, Y. Chen, J. Chu, and X. Xu, “Bridged bowtie aperture antenna for producing an electromagnetic hot spot,” ACS Photonics 4(3), 567–575 (2017).
[Crossref]

Cheng, H.

Chichkov, B. N.

Childs, D. T. D.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Choi, D.-Y.

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] [PubMed]

Chong, K. E.

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] [PubMed]

Chu, J.

Y. Chen, Y. Chen, J. Chu, and X. Xu, “Bridged bowtie aperture antenna for producing an electromagnetic hot spot,” ACS Photonics 4(3), 567–575 (2017).
[Crossref]

Cortes, C. L.

Cui, Y.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Dai, H. T.

Y. Yang, H. T. Dai, and X. W. Sun, “Fractal diabolo antenna for enhancing and confining the optical magnetic field,” AIP Adv. 4(1), 017123 (2014).
[Crossref]

Y. Yang, H. T. Dai, and X. W. Sun, “Split ring aperture for optical magnetic field enhancement by radially polarized beam,” Opt. Express 21(6), 6845–6850 (2013).
[Crossref] [PubMed]

Dodson, C. M.

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 Condens. Matter Mater. Phys. 86(12), 125102 (2012).
[Crossref]

Dorpe, P. V.

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]

Evlyukhin, A. B.

Fang, N. X.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Feng, S.

Feng, T.

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41(21), 5011–5014 (2016).
[Crossref] [PubMed]

Fernandez-Corbaton, I.

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] [PubMed]

Ferrari, L.

Fischer, U. C.

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]

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

Fu, Y.

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

Fullerton, E. E.

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

Fung, K. H.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Garcia-Parajo, M. F.

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]

Grosjean, T.

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]

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

Guo, L. J.

J. Zhou, A. F. Kaplan, L. Chen, and L. J. Guo, “Experiment and theory of the broadband absorption by a tapered hyperbolic metamaterial array,” ACS Photonics 1(7), 618–624 (2014).
[Crossref]

Haldar, A.

He, S.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Hogg, R. A.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Hou, C. C.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Hu, X.

Huang, Y. Q.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Jacob, Z.

Jiang, X.

Jin, Y.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Kan, J. J.

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

Kaplan, A. F.

J. Zhou, A. F. Kaplan, L. Chen, and L. J. Guo, “Experiment and theory of the broadband absorption by a tapered hyperbolic metamaterial array,” ACS Photonics 1(7), 618–624 (2014).
[Crossref]

Keene, D.

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] [PubMed]

Kinzel, E. C.

Kivshar, Y. S.

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] [PubMed]

Kotov, O. V.

Krasnok, A. E.

D. G. Baranov, D. A. Zuev, S. I. Lepeshov, O. V. Kotov, A. E. Krasnok, A. B. Evlyukhin, and B. N. Chichkov, “All-dielectric nanophotonics: the quest for better materials and fabrication techniques,” Optica 4(7), 814–825 (2017).
[Crossref]

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

Krishna, K. H.

Kumar, S.

Lepage, D.

Lepeshov, S. I.

Li, J.

Li, S. V.

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

Li, X.

Li, Z.

Liang, Q.

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

Liang, Z.

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41(21), 5011–5014 (2016).
[Crossref] [PubMed]

Liao, Y. L.

Liu, F. Q.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Liu, Z.

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

L. Ferrari, D. Lu, D. Lepage, and Z. Liu, “Enhanced spontaneous emission inside hyperbolic metamaterials,” Opt. Express 22(4), 4301–4306 (2014).
[Crossref] [PubMed]

J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, and J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. 35(1), 16–18 (2010).
[Crossref] [PubMed]

Lu, D.

L. Ferrari, D. Lu, D. Lepage, and Z. Liu, “Enhanced spontaneous emission inside hyperbolic metamaterials,” Opt. Express 22(4), 4301–4306 (2014).
[Crossref] [PubMed]

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

Lu, Z.

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

Ma, H.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Mashhadi, S.

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] [PubMed]

Miroshnichenko, A. E.

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

Mivelle, M.

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]

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

Nanz, S.

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] [PubMed]

Neshev, D. N.

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] [PubMed]

Newman, W. D.

Ning, J. Q.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Noginov, M. 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] [PubMed]

Noginova, N.

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] [PubMed]

Pertsch, T.

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] [PubMed]

Polman, A.

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Rho, J.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

Rockstuhl, C.

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] [PubMed]

Rusak, E.

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] [PubMed]

Savelev, R. S.

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

Sethi, P.

Song, Q.

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

Sreekanth, K. V.

Staude, I.

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] [PubMed]

Steinert, M.

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] [PubMed]

Strangi, G.

Sun, Q.

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

Sun, S.

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

Sun, X. W.

Y. Yang, H. T. Dai, and X. W. Sun, “Fractal diabolo antenna for enhancing and confining the optical magnetic field,” AIP Adv. 4(1), 017123 (2014).
[Crossref]

Y. Yang, H. T. Dai, and X. W. Sun, “Split ring aperture for optical magnetic field enhancement by radially polarized beam,” Opt. Express 21(6), 6845–6850 (2013).
[Crossref] [PubMed]

Tian, J.

van de Groep, J.

van de Haar, M. A.

Vaskin, 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] [PubMed]

Verellen, N.

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]

Wang, T.

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

Wang, Z. G.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Wu, S.

Xiao, S.

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

Xu, J.

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

Xu, X.

Y. Chen, Y. Chen, J. Chu, and X. Xu, “Bridged bowtie aperture antenna for producing an electromagnetic hot spot,” ACS Photonics 4(3), 567–575 (2017).
[Crossref]

N. Zhou, E. C. Kinzel, and X. Xu, “Complementary bowtie aperture for localizing and enhancing optical magnetic field,” Opt. Lett. 36(15), 2764–2766 (2011).
[Crossref] [PubMed]

Xu, Y.

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41(21), 5011–5014 (2016).
[Crossref] [PubMed]

Yang, J.

Yang, X.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

Yang, Y.

Y. Yang, H. T. Dai, and X. W. Sun, “Fractal diabolo antenna for enhancing and confining the optical magnetic field,” AIP Adv. 4(1), 017123 (2014).
[Crossref]

Y. Yang, H. T. Dai, and X. W. Sun, “Split ring aperture for optical magnetic field enhancement by radially polarized beam,” Opt. Express 21(6), 6845–6850 (2013).
[Crossref] [PubMed]

Yao, J.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

Yi, N.

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

Yin, X.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

Yu, P.

Yu, W.

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

Zhang, J. C.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Zhang, W.

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41(21), 5011–5014 (2016).
[Crossref] [PubMed]

Zhang, X.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

Zhang, Z. Y.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Zhao, Y.

Zhou, J.

J. Zhou, A. F. Kaplan, L. Chen, and L. J. Guo, “Experiment and theory of the broadband absorption by a tapered hyperbolic metamaterial array,” ACS Photonics 1(7), 618–624 (2014).
[Crossref]

Zhou, N.

Zhuo, N.

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Zi, J.

Zia, R.

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 Condens. Matter Mater. Phys. 86(12), 125102 (2012).
[Crossref]

Zuev, D. A.

ACS Photonics (5)

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]

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]

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic Magnetic Purcell Effect,” ACS Photonics 5(3), 678–683 (2018).
[Crossref]

Y. Chen, Y. Chen, J. Chu, and X. Xu, “Bridged bowtie aperture antenna for producing an electromagnetic hot spot,” ACS Photonics 4(3), 567–575 (2017).
[Crossref]

J. Zhou, A. F. Kaplan, L. Chen, and L. J. Guo, “Experiment and theory of the broadband absorption by a tapered hyperbolic metamaterial array,” ACS Photonics 1(7), 618–624 (2014).
[Crossref]

Adv. Opt. Mater. (1)

Q. Liang, T. Wang, Z. Lu, Q. Sun, Y. Fu, and W. Yu, “Metamaterial-based two dimensional plasmonic subwavelength structures offer the broadest waveband light harvesting,” Adv. Opt. Mater. 1(1), 43–49 (2013).
[Crossref]

AIP Adv. (1)

Y. Yang, H. T. Dai, and X. W. Sun, “Fractal diabolo antenna for enhancing and confining the optical magnetic field,” AIP Adv. 4(1), 017123 (2014).
[Crossref]

Carbon (1)

N. Yi, Z. Liu, S. Sun, Q. Song, and S. Xiao, “Mid-infrared tunable magnetic response in graphene-based diabolo nanoantennas,” Carbon 94, 501–506 (2015).
[Crossref]

J. Opt. Soc. Am. B (2)

Laser Photonics Rev. (1)

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

Light Sci. Appl. (1)

C. C. Hou, H. M. Chen, J. C. Zhang, N. Zhuo, Y. Q. Huang, R. A. Hogg, D. T. D. Childs, J. Q. Ning, Z. G. Wang, F. Q. Liu, and Z. Y. Zhang, “Near-infrared and mid-infrared semiconductor broadband light emitters,” Light Sci. Appl. 7(3), 17170 (2018).
[Crossref] [PubMed]

Nano Lett. (3)

Y. Cui, K. H. Fung, J. Xu, H. Ma, Y. Jin, S. He, and N. X. Fang, “Ultrabroadband light absorption by a sawtooth anisotropic metamaterial slab,” Nano Lett. 12(3), 1443–1447 (2012).
[Crossref] [PubMed]

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11(3), 1009–1013 (2011).
[Crossref] [PubMed]

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] [PubMed]

Nat. Nanotechnol. (1)

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6(7), 450–454 (2012).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Opt. Mater. Express (1)

Optica (1)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. B Condens. Matter Mater. Phys. (1)

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 Condens. Matter Mater. Phys. 86(12), 125102 (2012).
[Crossref]

Other (2)

T. C. Choy, Effective Medium Theory: Principles and Applications (Oxford University Press on Demand, 1999).

Lumerical FDTD Solution, http://www.lumerical.com/ .

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Figures (6)

Fig. 1
Fig. 1 (a) 3D perspective view of tapered hollow hyperbolic metamaterial (THHM). (b) The normalized absorption cross section of original tapered hyperbolic metamaterial (THM, red curve), THHM (blue curve) and its effective medium counterpart (EMC, green curve). The magnetic resonant modes at partial marked peaks are illustrated in Fig. 2.
Fig. 2
Fig. 2 Magnetic field intensity distribution under resonances for THHM (Wh = 90 nm) at (a) 2.01 μm, (b) 2.60 μm, (c) 3.04 μm. Magnetic field intensity distribution for THM (Wh = 0) at (d) 2.31 μm and EMC of THHM (Wh = 90 nm) at (f) 2.76 μm. All the resonances are marked in Fig. 1(b) by different signs. All subfigures are Hx components of the resonant modes. (e) Isofrequency contours (IFC) for the multilayer structures. The blue and green curves are plotted under the wavelengths of 2.6 μm and 2.76 μm, which correspond to the resonant wavelengths in panels (b) and (f). The dashed and solid lines represent IFCs predicted by the effective medium theory (EMT) and transfer matrix method (TMM) in despite of colors.
Fig. 3
Fig. 3 Normalized magnetic field intensity at three positions on the axis of the structure: point A (y = −400 nm), point B (y = 0 nm, the original point) and point C (y = 400 nm). The THHM structure starts from −500 nm to 500 nm along y axis. Five resonant peaks indicated by blue marks are in identical wavelengths with those by the same marks in Fig. 1(b).
Fig. 4
Fig. 4 (a) Radiative decay rate enhancement of the MD emitters located at points A, B and C shown in Fig. 3. (b) Radiative decay rate enhancement as function of y position of MD emitter.
Fig. 5
Fig. 5 Radiative decay rate enhancement for THHMs with different heights while the MD emitter is located at –400 nm (a) and 0 nm (b). Wh in (a) and (b) are kept at 90 nm. Dependence of the radiative decay rate enhancement on the hole size Wh (c) and hole shape (d). The dotted and solid curves in (d) correspond to the structures with a circular hole of 30 nm diameter and a square hole with 30 nm side length, respectively. Inset of (c) shows the linear relationship between wavelength of the main resonances and the hole area. The height of the whole structures in (c) and (d) are kept at 1 μm. All the abscissas and ordinates in Fig. 5 are wavelengths and radiative decay rate enhancement.
Fig. 6
Fig. 6 (a) Quantum efficiencies for THHMs with different hole dimension Wh. (b) Radiative decay rate enhancement when Wh = 90 nm. Shadow region in (b) shows the wavelength range with a quantum efficiency higher than 40%.

Equations (4)

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Yc Yt k y (y,λ)dy =nπ
V mode = ( V H 2 dV ) 2 V H 4 dV
cos[ k y ( t m + t d )]=cos( q 1y t m )cos( q 2y t d ) 1 2 ( q 1y ε d q 2y ε m + q 2y ε m q 1y ε d )sin( q 1y t m )sin( q 2y t d )
γ m = π μ 0 ω 0 | m | 2 ρ m ( r 0 , ω 0 )

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