Abstract

The dynamically tunable plasmon induced absorption (PIA) effect is demonstrated in a graphene-assisted metallodielectric grating structure. Two methods are employed to achieve the tunable PIA effect in the mid-infrared region: one is based on controlling the chemical potential of graphene by adjusting the gate voltage, the other is related to varying the refractive index of interlayer. Our calculated results reveal that high tunability in amplitude and bandwidth of the PIA effect can be achieved by using the above-mentioned methods. Compared with previous results, our scheme is much easier to fabricate and has significant applications in modulators, absorbers and sensors.

© 2017 Optical Society of America

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2017 (2)

M. Wen, L. Wang, X. Zhai, Q. Lin, and S. Xia, “Dynamically tunable plasmon-induced absorption in resonator-coupled graphene waveguide,” Europhys. Lett. 116(4), 44004 (2017).

I. T. Lin, C. Fan, and J. M. Liu, “Propagating and localized graphene surface plasmon polaritons on a grating structure,” IEEE J. Sel. Top. Quantum Electron. 23(1), 144–147 (2017).

2016 (4)

T. Zhang, X. Yin, L. Chen, and X. Li, “Ultra-compact polarization beam splitter utilizing a graphene-based asymmetrical directional coupler,” Opt. Lett. 41(2), 356–359 (2016).
[PubMed]

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

H.-J. Li, L.-L. Wang, and X. Zhai, “Plasmonically induced absorption and transparency based on MIM waveguides with concentric nanorings,” IEEE Photonics Technol. Lett. 28(13), 1454–1457 (2016).

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

2015 (8)

T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5, 11195 (2015).
[PubMed]

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Y. Li, B. An, S. Jiang, J. Gao, Y. Chen, and S. Pan, “Plasmonic induced triple-band absorber for sensor application,” Opt. Express 23(13), 17607–17612 (2015).
[PubMed]

J. He, P. Ding, J. Wang, C. Fan, and E. Liang, “Ultra-narrow band perfect absorbers based on plasmonic analog of electromagnetically induced absorption,” Opt. Express 23(5), 6083–6091 (2015).
[PubMed]

X. Han, T. Wang, X. Li, B. Liu, Y. He, and J. Tang, “Ultrafast and low-power dynamically tunable plasmon-induced transparencies in compact aperture-coupled rectangular resonators,” J. Lightwave Technol. 33(14), 3083–3090 (2015).

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Y. Liang, W. Peng, M. Lu, and S. Chu, “Narrow-band wavelength tunable filter based on asymmetric double layer metallic grating,” Opt. Express 23(11), 14434–14445 (2015).
[PubMed]

X. Han, T. Wang, X. Li, S. Xiao, and Y. Zhu, “Dynamically tunable plasmon induced transparency in a graphene-based nanoribbon waveguide coupled with graphene rectangular resonators structure on sapphire substrate,” Opt. Express 23(25), 31945–31955 (2015).
[PubMed]

2014 (3)

2013 (5)

2012 (7)

X. Piao, S. Yu, and N. Park, “Control of Fano asymmetry in plasmon induced transparency and its application to plasmonic waveguide modulator,” Opt. Express 20(17), 18994–18999 (2012).
[PubMed]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).

A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).

C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. 101(11), 111609 (2012).

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109(18), 187401 (2012).
[PubMed]

2011 (3)

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36(16), 3054–3056 (2011).
[PubMed]

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

2010 (1)

2009 (2)

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).

2008 (3)

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[PubMed]

G. Vincent, S. Collin, N. Bardou, J. L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).

C. Min, P. Wang, C. Chen, Y. Deng, Y. Lu, H. Ming, T. Ning, Y. Zhou, and G. Yang, “All-optical switching in subwavelength metallic grating structure containing nonlinear optical materials,” Opt. Lett. 33(8), 869–871 (2008).
[PubMed]

2007 (1)

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

2003 (1)

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003).
[PubMed]

1999 (1)

S. Link, C. Burda, M. Mohamed, B. Nikoobakht, and M. A. El-Sayed, “Laser photothermal melting and fragmentation of gold nanorods: energy and laser pulse-width dependence,” J. Phys. Chem. A 103(9), 1165–1170 (1999).

1998 (1)

Agarwal, G. S.

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Albrecht, W.

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

An, B.

Avouris, P.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Baets, R.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Bai, W.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).

Bals, S.

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

Bardou, N.

E. Sakat, G. Vincent, P. Ghenuche, N. Bardou, S. Collin, F. Pardo, J.-L. Pelouard, and R. Haïdar, “Guided mode resonance in subwavelength metallodielectric free-standing grating for bandpass filtering,” Opt. Lett. 36(16), 3054–3056 (2011).
[PubMed]

G. Vincent, S. Collin, N. Bardou, J. L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).

Biaggio, I.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Bouchon, P.

Brosi, J. M.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Burda, C.

S. Link, C. Burda, M. Mohamed, B. Nikoobakht, and M. A. El-Sayed, “Laser photothermal melting and fragmentation of gold nanorods: energy and laser pulse-width dependence,” J. Phys. Chem. A 103(9), 1165–1170 (1999).

Cai, L.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).

Casiraghi, C.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Chai, Z.

X. Yang, X. Hu, Z. Chai, C. Lu, H. Yang, and Q. Gong, “Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies,” Appl. Phys. Lett. 104(22), 221114 (2014).

Z. Chai, X. Hu, Y. Zhu, F. Zhang, H. Yang, and Q. Gong, “Low-power and ultrafast all-optical tunable plasmon-induced transparency in plasmonic nanostructures,” Appl. Phys. Lett. 102(20), 201119 (2013).

Chandra, B.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Chen, C.

Chen, L.

T. Zhang, X. Yin, L. Chen, and X. Li, “Ultra-compact polarization beam splitter utilizing a graphene-based asymmetrical directional coupler,” Opt. Lett. 41(2), 356–359 (2016).
[PubMed]

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5, 11195 (2015).
[PubMed]

T. Zhang, L. Chen, and X. Li, “Graphene-based tunable broadband hyperlens for far-field subdiffraction imaging at mid-infrared frequencies,” Opt. Express 21(18), 20888–20899 (2013).
[PubMed]

Chen, Y.

Chen, Z.

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Christ, A.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003).
[PubMed]

Chu, S.

Collin, S.

Cui, T. J.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).

Deng, T.-S.

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

Deng, Y.

Diederich, F.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Ding, P.

Dong, Z. G.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).

Dumon, P.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

El-Sayed, M. A.

S. Link, C. Burda, M. Mohamed, B. Nikoobakht, and M. A. El-Sayed, “Laser photothermal melting and fragmentation of gold nanorods: energy and laser pulse-width dependence,” J. Phys. Chem. A 103(9), 1165–1170 (1999).

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Fan, C.

I. T. Lin, C. Fan, and J. M. Liu, “Propagating and localized graphene surface plasmon polaritons on a grating structure,” IEEE J. Sel. Top. Quantum Electron. 23(1), 144–147 (2017).

J. He, P. Ding, J. Wang, C. Fan, and E. Liang, “Ultra-narrow band perfect absorbers based on plasmonic analog of electromagnetically induced absorption,” Opt. Express 23(5), 6083–6091 (2015).
[PubMed]

Ferrari, A. C.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Frank, B.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Freitag, M.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Freude, W.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Fu, J. S.

Fu, Y.

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication range,” Sci. Rep. 3, 2338 (2013).
[PubMed]

Gan, C. H.

C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. 101(11), 111609 (2012).

Gan, Q.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).

Gao, J.

Gao, W.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[PubMed]

Ghenuche, P.

Giessen, H.

R. Taubert, M. Hentschel, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption: simulations, experiments, and coupled oscillator analysis,” J. Opt. Soc. Am. B 30(12), 3123–3134 (2013).

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003).
[PubMed]

Giessen, H. W.

R. Taubert, M. Hentschel, J. Kästel, and H. W. Giessen, “Classical analog of electromagnetically induced absorption in plasmonics,” in Quantum Electronics and Laser Science Conference, (Optical Society of America, 2012), QW1B. 1.

Gippius, N. A.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003).
[PubMed]

Gokus, T.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Gong, Q.

X. Yang, X. Hu, Z. Chai, C. Lu, H. Yang, and Q. Gong, “Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies,” Appl. Phys. Lett. 104(22), 221114 (2014).

Z. Chai, X. Hu, Y. Zhu, F. Zhang, H. Yang, and Q. Gong, “Low-power and ultrafast all-optical tunable plasmon-induced transparency in plasmonic nanostructures,” Appl. Phys. Lett. 102(20), 201119 (2013).

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication range,” Sci. Rep. 3, 2338 (2013).
[PubMed]

Goris, B.

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

Grigorenko, A.

A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).

Guo, H.

Haïdar, R.

Han, J.

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Han, X.

Hartschuh, A.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Harutyunyan, H.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

He, J.

He, M.

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

He, Y.

He, Z.

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Hentschel, M.

R. Taubert, M. Hentschel, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption: simulations, experiments, and coupled oscillator analysis,” J. Opt. Soc. Am. B 30(12), 3123–3134 (2013).

R. Taubert, M. Hentschel, J. Kästel, and H. W. Giessen, “Classical analog of electromagnetically induced absorption in plasmonics,” in Quantum Electronics and Laser Science Conference, (Optical Society of America, 2012), QW1B. 1.

Héron, S.

Hu, R.

Hu, X.

X. Yang, X. Hu, Z. Chai, C. Lu, H. Yang, and Q. Gong, “Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies,” Appl. Phys. Lett. 104(22), 221114 (2014).

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication range,” Sci. Rep. 3, 2338 (2013).
[PubMed]

Z. Chai, X. Hu, Y. Zhu, F. Zhang, H. Yang, and Q. Gong, “Low-power and ultrafast all-optical tunable plasmon-induced transparency in plasmonic nanostructures,” Appl. Phys. Lett. 102(20), 201119 (2013).

Hu, Y.

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

Huan, C. H. A.

Jain, A.

P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109(18), 187401 (2012).
[PubMed]

Jiang, S.

Kästel, J.

R. Taubert, M. Hentschel, J. Kästel, and H. W. Giessen, “Classical analog of electromagnetically induced absorption in plasmonics,” in Quantum Electronics and Laser Science Conference, (Optical Society of America, 2012), QW1B. 1.

Koos, C.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Koschny, T.

P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109(18), 187401 (2012).
[PubMed]

Kuhl, J.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003).
[PubMed]

Lei, L.

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

Leuthold, J.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Li, B.

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Li, H.

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Li, H.-J.

H.-J. Li, L.-L. Wang, and X. Zhai, “Plasmonically induced absorption and transparency based on MIM waveguides with concentric nanorings,” IEEE Photonics Technol. Lett. 28(13), 1454–1457 (2016).

Li, X.

Li, Y.

Liang, E.

Liang, Y.

Liao, H. B.

Lidorikis, E.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Lin, I. T.

I. T. Lin, C. Fan, and J. M. Liu, “Propagating and localized graphene surface plasmon polaritons on a grating structure,” IEEE J. Sel. Top. Quantum Electron. 23(1), 144–147 (2017).

Lin, Q.

M. Wen, L. Wang, X. Zhai, Q. Lin, and S. Xia, “Dynamically tunable plasmon-induced absorption in resonator-coupled graphene waveguide,” Europhys. Lett. 116(4), 44004 (2017).

Link, S.

S. Link, C. Burda, M. Mohamed, B. Nikoobakht, and M. A. El-Sayed, “Laser photothermal melting and fragmentation of gold nanorods: energy and laser pulse-width dependence,” J. Phys. Chem. A 103(9), 1165–1170 (1999).

Liu, B.

Liu, J. M.

I. T. Lin, C. Fan, and J. M. Liu, “Propagating and localized graphene surface plasmon polaritons on a grating structure,” IEEE J. Sel. Top. Quantum Electron. 23(1), 144–147 (2017).

Liu, M.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[PubMed]

Lu, C.

X. Yang, X. Hu, Z. Chai, C. Lu, H. Yang, and Q. Gong, “Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies,” Appl. Phys. Lett. 104(22), 221114 (2014).

Lu, M.

Lu, W. B.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).

Lu, Y.

Meng, Z.

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

Min, C.

Ming, H.

Mohamed, M.

S. Link, C. Burda, M. Mohamed, B. Nikoobakht, and M. A. El-Sayed, “Laser photothermal melting and fragmentation of gold nanorods: energy and laser pulse-width dependence,” J. Phys. Chem. A 103(9), 1165–1170 (1999).

Nikoobakht, B.

S. Link, C. Burda, M. Mohamed, B. Nikoobakht, and M. A. El-Sayed, “Laser photothermal melting and fragmentation of gold nanorods: energy and laser pulse-width dependence,” J. Phys. Chem. A 103(9), 1165–1170 (1999).

Ning, T.

Novoselov, K.

A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).

Novoselov, K. S.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Oulton, R. F.

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).

Pan, S.

Pardo, F.

Park, N.

Pelouard, J. L.

G. Vincent, S. Collin, N. Bardou, J. L. Pelouard, and R. Haïdar, “Large-area dielectric and metallic freestanding gratings for midinfrared optical filtering applications,” J. Vac. Sci. Technol. B 26(6), 1852–1855 (2008).

Pelouard, J.-L.

Peng, W.

Piao, X.

Polini, M.

A. Grigorenko, M. Polini, and K. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).

Qian, H.

C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, and A. C. Ferrari, “Rayleigh imaging of graphene and graphene layers,” Nano Lett. 7(9), 2711–2717 (2007).
[PubMed]

Qiu, C.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Qu, K.

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Sakat, E.

Scimeca, M. L.

J. Leuthold, W. Freude, J. M. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon Organic Hybrid Technology—A Platform for Practical Nonlinear Optics,” Proc. IEEE 97(7), 1304–1316 (2009).

Shu, J.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Singh, R.

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Song, G.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).

Soukoulis, C. M.

P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109(18), 187401 (2012).
[PubMed]

Sum, T. C.

Tang, J.

Tassin, P.

P. Tassin, L. Zhang, R. Zhao, A. Jain, T. Koschny, and C. M. Soukoulis, “Electromagnetically induced transparency and absorption in metamaterials: the radiating two-oscillator model and its experimental confirmation,” Phys. Rev. Lett. 109(18), 187401 (2012).
[PubMed]

Taubert, R.

R. Taubert, M. Hentschel, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption: simulations, experiments, and coupled oscillator analysis,” J. Opt. Soc. Am. B 30(12), 3123–3134 (2013).

R. Taubert, M. Hentschel, J. Kästel, and H. W. Giessen, “Classical analog of electromagnetically induced absorption in plasmonics,” in Quantum Electronics and Laser Science Conference, (Optical Society of America, 2012), QW1B. 1.

Tian, Z.

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Tikhodeev, S. G.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003).
[PubMed]

Tulevski, G.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

van Blaaderen, A.

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

van Huis, M. A.

W. Albrecht, T.-S. Deng, B. Goris, M. A. van Huis, S. Bals, and A. van Blaaderen, “Single particle deformation and analysis of silica-coated gold nanorods before and after femtosecond laser pulse excitation,” Nano Lett. 16(3), 1818–1825 (2016).
[PubMed]

Vincent, G.

Wang, B.

T. Zhang, L. Chen, B. Wang, and X. Li, “Tunable broadband plasmonic field enhancement on a graphene surface using a normal-incidence plane wave at mid-infrared frequencies,” Sci. Rep. 5, 11195 (2015).
[PubMed]

Wang, H.

Wang, J.

Wang, L.

M. Wen, L. Wang, X. Zhai, Q. Lin, and S. Xia, “Dynamically tunable plasmon-induced absorption in resonator-coupled graphene waveguide,” Europhys. Lett. 116(4), 44004 (2017).

Wang, L.-L.

H.-J. Li, L.-L. Wang, and X. Zhai, “Plasmonically induced absorption and transparency based on MIM waveguides with concentric nanorings,” IEEE Photonics Technol. Lett. 28(13), 1454–1457 (2016).

Wang, P.

Wang, T.

Wang, Y.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[PubMed]

Wen, K.

K. Wen, Y. Hu, L. Chen, J. Zhou, M. He, L. Lei, and Z. Meng, “Plasmonic-induced absorption and transparency based on a compact ring-groove joint MIM waveguide structure,” IEEE Photonics J. 8(5), 1–8 (2016).

Wen, M.

M. Wen, L. Wang, X. Zhai, Q. Lin, and S. Xia, “Dynamically tunable plasmon-induced absorption in resonator-coupled graphene waveguide,” Europhys. Lett. 116(4), 44004 (2017).

Wong, G. K.

Wong, K. S.

Wu, Y.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Xia, F.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Xia, S.

M. Wen, L. Wang, X. Zhai, Q. Lin, and S. Xia, “Dynamically tunable plasmon-induced absorption in resonator-coupled graphene waveguide,” Europhys. Lett. 116(4), 44004 (2017).

Xiao, R. F.

Xiao, S.

Xie, L.

Xing, G.

Xu, H.

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Xu, H. J.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).

Xu, N.

X. Zhang, N. Xu, K. Qu, Z. Tian, R. Singh, J. Han, G. S. Agarwal, and W. Zhang, “Electromagnetically induced absorption in a three-resonator metasurface system,” Sci. Rep. 5, 10737 (2015).
[PubMed]

Xu, Q.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Xu, Y.

J. Zhang, W. Bai, L. Cai, Y. Xu, G. Song, and Q. Gan, “Observation of ultra-narrow band plasmon induced transparency based on large-area hybrid plasmon-waveguide systems,” Appl. Phys. Lett. 99(18), 181120 (2011).

Yan, H.

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[PubMed]

Yang, G.

Yang, H.

X. Yang, X. Hu, Z. Chai, C. Lu, H. Yang, and Q. Gong, “Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies,” Appl. Phys. Lett. 104(22), 221114 (2014).

Z. Chai, X. Hu, Y. Zhu, F. Zhang, H. Yang, and Q. Gong, “Low-power and ultrafast all-optical tunable plasmon-induced transparency in plasmonic nanostructures,” Appl. Phys. Lett. 102(20), 201119 (2013).

Y. Zhu, X. Hu, Y. Fu, H. Yang, and Q. Gong, “Ultralow-power and ultrafast all-optical tunable plasmon-induced transparency in metamaterials at optical communication range,” Sci. Rep. 3, 2338 (2013).
[PubMed]

Yang, X.

X. Yang, X. Hu, Z. Chai, C. Lu, H. Yang, and Q. Gong, “Tunable ultracompact chip-integrated multichannel filter based on plasmon-induced transparencies,” Appl. Phys. Lett. 104(22), 221114 (2014).

Yin, X.

Yu, S.

Zentgraf, T.

T. Zentgraf, S. Zhang, R. F. Oulton, and X. Zhang, “Ultranarrow coupling-induced transparency bands in hybrid plasmonic systems,” Phys. Rev. B 80(19), 195415 (2009).

Zhai, X.

M. Wen, L. Wang, X. Zhai, Q. Lin, and S. Xia, “Dynamically tunable plasmon-induced absorption in resonator-coupled graphene waveguide,” Europhys. Lett. 116(4), 44004 (2017).

H.-J. Li, L.-L. Wang, and X. Zhai, “Plasmonically induced absorption and transparency based on MIM waveguides with concentric nanorings,” IEEE Photonics Technol. Lett. 28(13), 1454–1457 (2016).

Zhan, S.

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “Tunable multi-switching in plasmonic waveguide with Kerr nonlinear resonator,” Sci. Rep. 5, 15837 (2015).
[PubMed]

Zhang, F.

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Zhang, J.

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X. Han, T. Wang, X. Li, S. Xiao, and Y. Zhu, “Dynamically tunable plasmon induced transparency in a graphene-based nanoribbon waveguide coupled with graphene rectangular resonators structure on sapphire substrate,” Opt. Express 23(25), 31945–31955 (2015).
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H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).

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Figures (7)

Fig. 1
Fig. 1 Structure schematic of the proposed GMG structure.
Fig. 2
Fig. 2 (a) The schematic of the MG structure. (b) The simulated transmission spectrums when a TM-polarized light normally incident onto the MG structure with (blue solid line) and without (black dashed line) the interlayer upon the gold grating. Spatial magnetic field distributions for the MG structure with interlayer at λ = 4.53 μm (c) and λ = 4.7 μm (d).
Fig. 3
Fig. 3 (a) Calculated dispersion curves by employing Eqs. (6) and (7) for AGIS (two red lines) and AGIG (two blue lines) models with μc = 0.79 eV and μc = 0.83 eV, respectively. (b) Utilizing the calculated result in (a) and Eq. (9), the calculated band structure for the designed GMG structure with μc = 0.79 eV (blue dashed line) and μc = 0.83 eV (red solid line), respectively.
Fig. 4
Fig. 4 (a) When normally incident electromagnetic wave onto the GMG structure, calculated transmission, reflection and absorption spectrums. (b-d) Evolutions of the transmission (b), reflection (c) and absorption (d) spectrum versus chemical potential of graphene and wavelength. (e, f) Distributions of the magnetic field corresponding to two transmission peaks and one transmission dip represented by A, C, and B points in (a).
Fig. 5
Fig. 5 Calculated transmission, reflection and absorption spectrums of the GMG structure for different thickness of the interlayer (a-c) and the SiNx dielectric layer (d-f). The inset in (a) denotes the transmission spectrums for different relaxation times of graphene.
Fig. 6
Fig. 6 (a) Calculated transmission, reflection and absorption spectrum of the GMG structure for different pump intensities under the condition of normal incidence. Evolution of the transmission (b), reflection (c) and absorption (d) spectrums versus the refractive index of the interlayer and wavelength.
Fig. 7
Fig. 7 (a) Transmittance of the dip at central wavelength of the PIA effect (blue line), and the green line denotes the intensity of a square pump pulse. (b-d) Calculated transmission, reflection and absorption spectrums of the GMG structure for different incident angles.

Equations (14)

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× E = j ω μ 0 H
× H = j ω ε 0 ε ¯ ¯ m E
ε ¯ ¯ m = [ ε m x 0 0 0 ε m y 0 0 0 ε m z ]
ε 0 x = ε 0 y = ε 0 z = ε 0
ε g x = ε g y = ε g , ε g z = 2.5
ε i x = ε i y = ε i z = ε i
ε s x = ε s y = ε s z = ε s
H y = { A e j β x e k s z , B e j β x e k i z + C e j β x e k i z , D e j β x e k g z + E e j β x e k g z , F e j β x e k 0 z , z < d g d i d A u d g d i d A u < z < d g d g < z < 0 z > 0
E z = { A β ω ε 0 ε s z e j β x e k s z , B β ω ε 0 ε i z e j β x e k i z C β ω ε 0 ε i z e j β x e k i z , D β ω ε 0 ε g z e j β x e k g z E β ω ε 0 ε g z e j β x e k g z , F β ω ε 0 ε 0 z e j β x e k 0 z , z < d g d i d A u d g d i d A u < z < d g d g < z < 0 z > 0
e 2 k i ( d i + d A u ) = 1 + ε i k s ε s k i 1 ε i k s ε s k i ( 1 + ε i k g ε g k i ) ( 1 + ε g k 0 ε 0 k g ) + ( 1 ε i k g ε g k i ) ( 1 ε g k 0 ε 0 k g ) e 2 k g d g ( 1 ε i k g ε g k i ) ( 1 + ε g k 0 ε 0 k g ) + ( 1 + ε i k g ε g k i ) ( 1 ε g k 0 ε 0 k g ) e 2 k g d g
1 + ( ε 0 ε i β ω ε i n e 2 μ c π 2 ω ) tan h ( β d i ) = 0
σ g , s = i e 2 k B T 4 π 2 ( ω + i τ 1 ) [ μ c k B T + 2 ln ( exp ( μ c k B T ) + 1 ) ] + i e 2 4 π ln [ 2 | μ c | ( ω + i τ 1 ) 2 | μ c | + ( ω + i τ 1 ) ]
c o s ( β Λ ) = cos ( k 1 w 1 ) cos ( k 2 w 2 ) 1 2 ( n 1 n 2 + n 2 n 1 ) sin ( k 1 w 1 ) sin ( k 2 w 2 )
Re ( β ) = k 0 sin θ + q 2 π Λ

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