Abstract

We consider an array of the meta-atom consisting of two cut-wires and a split-ring resonator coupled with an electromagnetic field with two polarization components. We show that the system can be taken as a classical analogue of the atomic medium of a double-Λ-type four-level configuration coupled with four laser fields and working under the condition of electromagnetically induced transparency, exhibits an effect of plasmon induced transparency (PIT), and displays a similar behavior of atomic four-wave mixing (FWM). We show also that with the PIT and FWM effects the system can support vector plasmonic dromions when a nonlinear varactor is mounted onto the each gap of the split-ring resonator. Our work not only gives a plasmonic analogue of the FWM in coherent atomic systems but also provides the possibility for obtaining new type of plasmonic excitations in metamaterials.

© 2017 Optical Society of America

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References

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2017 (2)

Z. Bai, Datang Xu, and G. Huang, “Storage and retrieval of electromagnetic waves with orbital angular momentum via plasmon-induced transparency,” Opt. Expr. 25, 785–798 (2017).
[Crossref]

S. Beck and I. E. Mazets, “Propagation of coupled dark-state polaritons and storage of light in a tripod medium,” Phys. Rev. A 95, 013818 (2017).
[Crossref]

2016 (2)

Z. Bai and G. Huang, “Plasmon dromions in a metamaterial via plasmon-induced transparency,” Phys. Rev. A 93, 013818 (2016).
[Crossref]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

2015 (5)

T. Nakanishi and M. Kitano, “Implementation of Electromagnetically Induced Transparency in a Metamaterial Controlled with Auxiliary Waves,” Phys. Rev. Appl. 4, 024013 (2015).
[Crossref]

Z. Bai, G. Huang, L. Liu, and S. Zhang, “Giant Kerr nonlinearity and low-power gigahertz solitons via plasmon-induced transparency,” Sci. Rep. 5, 13780 (2015).
[Crossref] [PubMed]

J. A. Souza, L. Cabral, R. R. Oliveira, and C. J. Villas-Boas, “Electromagnetically-induced-transparency-related phenomena and their mechanical analogs,” Phys. Rev. A 92, 023818 (2015).
[Crossref]

Y.-M. Liu, X.-D. Tian, D. Yan, Y. Zhang, C.-L. Cui, and J.-H. Wu, “Nonlinear modifications of photon correlations via controlled single and double Rydberg blockade,” Phys. Rev. A 91, 043802 (2015).
[Crossref]

C. Pelzman and S.-Y. Cho, “Polarization-selective optical transmission through a plasmonic metasurface,” Appl. Phys. Lett. 106, 251101 (2015).
[Crossref] [PubMed]

2014 (6)

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5, 5841 (2014).
[Crossref] [PubMed]

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriassov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Colloquium: Nonlinear metamaterials,” Rev. Mod. Phys. 86, 1093 (2014).
[Crossref]

J. Shao, J. Li, Y.-H. Wang, J.-Q. Li, Q. Chen, and Z.-G. Dong, “Polarization conversions of linearly and circularly polarized lights through a plasmon-induced transparent metasurface,” Appl. Phys. Lett. 115, 243503 (2014).

B. Peng, S. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

T. Matsui, M. Liu, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Electromagnetic tuning of resonant transmission in magnetoelastic metamaterials,” Appl. Phys. Lett. 104, 161117 (2014).
[Crossref]

2013 (4)

A. Kronwald and F. Marquardt, “Optomechanically Induced Transparency in the Nonlinear Quantum Regime,” Phys. Rev. Lett. 111, 133601 (2013).
[Crossref] [PubMed]

T. Nakanishi, T. Otani, Y. Tamayama, and M. Kitano, “Storage of electromagnetic waves in a metamaterial that mimics electromagnetically induced absorption in plasmonics,” Phys. Rev. B 87, 16110(R) (2013).
[Crossref]

Y. Sun, Y. Tong, C. Xue, Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic diode based on nonlinear electromagnetically induced transparency in metamaterials,” Appl. Phys. Lett. 103, 091904 (2013).
[Crossref]

J. Ruseckas, V. Kudriaskov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
[Crossref]

2012 (1)

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref] [PubMed]

2011 (6)

J. Chen, P. Wang, C. Chen, Y. Lu, H. Ming, and Q. Zhan, “Plasmonic EIT-like switching in bright-dark-bright plasmon resonators,” Opt. Expr. 19, 5970–5978 (2011).
[Crossref]

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisators, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407 (2011).
[Crossref] [PubMed]

Z. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Expr. 19, 3251–3257 (2011).
[Crossref]

J. Harden, A. Joshi, and J. D. Serna, “Demonstration of double EIT using coupled harmonic oscillators and RLC circuits,” Eur. J. Phys. 32, 541 (2011).
[Crossref]

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
[Crossref]

T. Jiang, K. Chang, L. Si, L. Ran, and H. Xin, “Active microwave negative-index metamaterial transmission line with gain,” Phys. Rev. Lett. 107, 205503 (2011).
[Crossref] [PubMed]

2010 (3)

U. Khadka, Y. Zhang, and Min Xiao, “Control of multitransparency windows via dark-state phase manipulation,” Phys. Rev. A 81, 023830 (2010).
[Crossref]

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Z. Dong, H. Liu, J. Cao, T. Li, S. Wang, S. Zhu, and X. Zhang, “Enhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterials,” Appl. Phys. Lett. 97, 114101 (2010).
[Crossref]

2009 (3)

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mat. 8, 758 (2009).
[Crossref]

C. Chen, I. Un, N. Tai, and T. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Expr. 17, 15372–15380 (2009).
[Crossref]

2008 (4)

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

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[Crossref] [PubMed]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

B. Wang, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Expr. 16, 16058–16063 (2008).
[Crossref]

2007 (1)

H.-j. Li and G. Huang, “Highly efficient four-wave mixing in a coherent six-level system in ultraslow propagation regime,” Phys. Rev. A 76, 043809 (2007).
[Crossref]

2006 (3)

I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Expr. 14, 9344–9349 (2006).
[Crossref]

S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
[Crossref]

H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Phase-controlled light switching at low light levels,” Phys. Rev. A 73, 011802R (2006).
[Crossref]

2005 (4)

A. Joshi and M. Xiao, “Generalized dark-state polaritons for photon memory in multilevel atomic media,” Phys. Rev. A 71, 041801 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

2004 (4)

D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
[Crossref]

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004).
[Crossref]

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[Crossref]

2003 (2)

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96–98 (2003).
[Crossref] [PubMed]

2002 (3)

M. G. Payne and L. Deng, “Consequences of induced transparency in a double-Λ scheme: Destructive interference in four-wave mixing,” Phys. Rev. A 65, 063806 (2002).
[Crossref]

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37 (2002).
[Crossref]

A. G. Litvak and M. D. Tokman, “Electromagnetically Induced Transparency in Ensembles of Classical Oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[Crossref] [PubMed]

2000 (2)

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient Nonlinear Frequency Conversion in an All-Resonant Double-Λ System,” Phys. Rev. Lett. 84, 5308 (2000).
[Crossref] [PubMed]

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

1999 (2)

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302 (1999).
[Crossref]

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996 (1999).
[Crossref]

1998 (1)

M. D. Lukin, P. R. Hemmer, M. Löffler, and M. O. Scully, “Resonant Enhancement of Parametric Process via Radiative Interference and Induced Coherence,” Phys. Rev. Lett. 81, 2675 (1998).
[Crossref]

1995 (1)

K. Nishinari and T. Yajima, “Numerical analyses of the collision of localized structures in the Davey-Stewartson equations,” Phys. Rev. E 51, 4986 (1995).
[Crossref]

Alivisators, A. P.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisators, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407 (2011).
[Crossref] [PubMed]

Alzar, C. L. G.

C. L. G. Alzar, M. A. G. Martinez, and P. Nussenzveig, “Classical analog of electromagnetically induced transparency,” Am. J. Phys. 70, 37 (2002).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Artoni, M.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Azad, A. K.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref] [PubMed]

Bai, Z.

Z. Bai, Datang Xu, and G. Huang, “Storage and retrieval of electromagnetic waves with orbital angular momentum via plasmon-induced transparency,” Opt. Expr. 25, 785–798 (2017).
[Crossref]

Z. Bai and G. Huang, “Plasmon dromions in a metamaterial via plasmon-induced transparency,” Phys. Rev. A 93, 013818 (2016).
[Crossref]

Z. Bai, G. Huang, L. Liu, and S. Zhang, “Giant Kerr nonlinearity and low-power gigahertz solitons via plasmon-induced transparency,” Sci. Rep. 5, 13780 (2015).
[Crossref] [PubMed]

Baluschev, S.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302 (1999).
[Crossref]

Beck, S.

S. Beck and I. E. Mazets, “Propagation of coupled dark-state polaritons and storage of light in a tripod medium,” Phys. Rev. A 95, 013818 (2017).
[Crossref]

Bozhevolnyi, S. I.

Z. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Expr. 19, 3251–3257 (2011).
[Crossref]

Cabral, L.

J. A. Souza, L. Cabral, R. R. Oliveira, and C. J. Villas-Boas, “Electromagnetically-induced-transparency-related phenomena and their mechanical analogs,” Phys. Rev. A 92, 023818 (2015).
[Crossref]

Cao, J.

Z. Dong, H. Liu, J. Cao, T. Li, S. Wang, S. Zhu, and X. Zhang, “Enhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterials,” Appl. Phys. Lett. 97, 114101 (2010).
[Crossref]

Capasso, F.

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
[Crossref]

Cataliotti, F.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
[Crossref] [PubMed]

Chang, K.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5, 5841 (2014).
[Crossref] [PubMed]

T. Jiang, K. Chang, L. Si, L. Ran, and H. Xin, “Active microwave negative-index metamaterial transmission line with gain,” Phys. Rev. Lett. 107, 205503 (2011).
[Crossref] [PubMed]

Chang, K.-F.

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriassov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

Chen, C.

J. Chen, P. Wang, C. Chen, Y. Lu, H. Ming, and Q. Zhan, “Plasmonic EIT-like switching in bright-dark-bright plasmon resonators,” Opt. Expr. 19, 5970–5978 (2011).
[Crossref]

C. Chen, I. Un, N. Tai, and T. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Expr. 17, 15372–15380 (2009).
[Crossref]

Chen, H.

Y. Sun, Y. Tong, C. Xue, Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic diode based on nonlinear electromagnetically induced transparency in metamaterials,” Appl. Phys. Lett. 103, 091904 (2013).
[Crossref]

Chen, H.-T.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref] [PubMed]

Chen, J.

J. Chen, P. Wang, C. Chen, Y. Lu, H. Ming, and Q. Zhan, “Plasmonic EIT-like switching in bright-dark-bright plasmon resonators,” Opt. Expr. 19, 5970–5978 (2011).
[Crossref]

Chen, K.-X.

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

Chen, Q.

J. Shao, J. Li, Y.-H. Wang, J.-Q. Li, Q. Chen, and Z.-G. Dong, “Polarization conversions of linearly and circularly polarized lights through a plasmon-induced transparent metasurface,” Appl. Phys. Lett. 115, 243503 (2014).

Chen, W.

B. Peng, S. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Chen, Y.-C.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Chen, Y.-F.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Chen, Y.-H.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Cho, H.-W.

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriassov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

Cho, S.-Y.

C. Pelzman and S.-Y. Cho, “Polarization-selective optical transmission through a plasmonic metasurface,” Appl. Phys. Lett. 106, 251101 (2015).
[Crossref] [PubMed]

Corbalán, R.

S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
[Crossref]

Cui, C.-L.

Y.-M. Liu, X.-D. Tian, D. Yan, Y. Zhang, C.-L. Cui, and J.-H. Wu, “Nonlinear modifications of photon correlations via controlled single and double Rydberg blockade,” Phys. Rev. A 91, 043802 (2015).
[Crossref]

Delèglise, S.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Deng, L.

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

M. G. Payne and L. Deng, “Consequences of induced transparency in a double-Λ scheme: Destructive interference in four-wave mixing,” Phys. Rev. A 65, 063806 (2002).
[Crossref]

Di Giuseppe, G.

S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
[Crossref]

Ding, Y.

Y. Sun, Y. Tong, C. Xue, Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic diode based on nonlinear electromagnetically induced transparency in metamaterials,” Appl. Phys. Lett. 103, 091904 (2013).
[Crossref]

Dong, Z.

Z. Dong, H. Liu, J. Cao, T. Li, S. Wang, S. Zhu, and X. Zhang, “Enhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterials,” Appl. Phys. Lett. 97, 114101 (2010).
[Crossref]

Dong, Z.-G.

J. Shao, J. Li, Y.-H. Wang, J.-Q. Li, Q. Chen, and Z.-G. Dong, “Polarization conversions of linearly and circularly polarized lights through a plasmon-induced transparent metasurface,” Appl. Phys. Lett. 115, 243503 (2014).

Economou, E. N.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[Crossref] [PubMed]

Fedotov, V. A.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[Crossref] [PubMed]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mat. 8, 758 (2009).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

Gaburro, Z.

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
[Crossref]

Gao, J.-Y.

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

Gavartin, E.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Genevet, P.

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
[Crossref]

Genov, D. A.

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

Giessen, H.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisators, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407 (2011).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mat. 8, 758 (2009).
[Crossref]

Gu, J.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref] [PubMed]

Guo, X.-Z.

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

Hagley, E. W.

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Han, J.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref] [PubMed]

Han, Z.

Z. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Expr. 19, 3251–3257 (2011).
[Crossref]

Hang, C.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

Harden, J.

J. Harden, A. Joshi, and J. D. Serna, “Demonstration of double EIT using coupled harmonic oscillators and RLC circuits,” Eur. J. Phys. 32, 541 (2011).
[Crossref]

Harris, S. E.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient Nonlinear Frequency Conversion in an All-Resonant Double-Λ System,” Phys. Rev. Lett. 84, 5308 (2000).
[Crossref] [PubMed]

Hemmer, P. R.

M. D. Lukin, P. R. Hemmer, M. Löffler, and M. O. Scully, “Resonant Enhancement of Parametric Process via Radiative Interference and Induced Coherence,” Phys. Rev. Lett. 81, 2675 (1998).
[Crossref]

Hentschel, M.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisators, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407 (2011).
[Crossref] [PubMed]

Hernandez, G.

H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Phase-controlled light switching at low light levels,” Phys. Rev. A 73, 011802R (2006).
[Crossref]

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004).
[Crossref]

Hirota, R.

R. Hirota, The direct method in soliton theory (Cambridge University Press, Cambridge, 2004).
[Crossref]

Huang, G.

Z. Bai, Datang Xu, and G. Huang, “Storage and retrieval of electromagnetic waves with orbital angular momentum via plasmon-induced transparency,” Opt. Expr. 25, 785–798 (2017).
[Crossref]

Z. Bai and G. Huang, “Plasmon dromions in a metamaterial via plasmon-induced transparency,” Phys. Rev. A 93, 013818 (2016).
[Crossref]

Z. Bai, G. Huang, L. Liu, and S. Zhang, “Giant Kerr nonlinearity and low-power gigahertz solitons via plasmon-induced transparency,” Sci. Rep. 5, 13780 (2015).
[Crossref] [PubMed]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

H.-j. Li and G. Huang, “Highly efficient four-wave mixing in a coherent six-level system in ultraslow propagation regime,” Phys. Rev. A 76, 043809 (2007).
[Crossref]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
[Crossref]

Huss, A.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302 (1999).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

Jeffery, A.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

Jiang, H.

Y. Sun, Y. Tong, C. Xue, Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic diode based on nonlinear electromagnetically induced transparency in metamaterials,” Appl. Phys. Lett. 103, 091904 (2013).
[Crossref]

Jiang, T.

T. Jiang, K. Chang, L. Si, L. Ran, and H. Xin, “Active microwave negative-index metamaterial transmission line with gain,” Phys. Rev. Lett. 107, 205503 (2011).
[Crossref] [PubMed]

Jiang, Y.

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

Joshi, A.

J. Harden, A. Joshi, and J. D. Serna, “Demonstration of double EIT using coupled harmonic oscillators and RLC circuits,” Eur. J. Phys. 32, 541 (2011).
[Crossref]

A. Joshi and M. Xiao, “Generalized dark-state polaritons for photon memory in multilevel atomic media,” Phys. Rev. A 71, 041801 (2005).
[Crossref]

Juzeliunas, G.

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriassov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

J. Ruseckas, V. Kudriaskov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
[Crossref]

Kang, H.

H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Phase-controlled light switching at low light levels,” Phys. Rev. A 73, 011802R (2006).
[Crossref]

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004).
[Crossref]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mat. 8, 758 (2009).
[Crossref]

Kats, M. A.

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
[Crossref]

Kawahawa, T.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

Khadka, U.

U. Khadka, Y. Zhang, and Min Xiao, “Control of multitransparency windows via dark-state phase manipulation,” Phys. Rev. A 81, 023830 (2010).
[Crossref]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Kitano, M.

T. Nakanishi and M. Kitano, “Implementation of Electromagnetically Induced Transparency in a Metamaterial Controlled with Auxiliary Waves,” Phys. Rev. Appl. 4, 024013 (2015).
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T. Nakanishi, T. Otani, Y. Tamayama, and M. Kitano, “Storage of electromagnetic waves in a metamaterial that mimics electromagnetically induced absorption in plasmonics,” Phys. Rev. B 87, 16110(R) (2013).
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Kivshar, Y. S.

T. Matsui, M. Liu, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Electromagnetic tuning of resonant transmission in magnetoelastic metamaterials,” Appl. Phys. Lett. 104, 161117 (2014).
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M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Colloquium: Nonlinear metamaterials,” Rev. Mod. Phys. 86, 1093 (2014).
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I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Expr. 14, 9344–9349 (2006).
[Crossref]

Korsunsky, E. A.

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996 (1999).
[Crossref]

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302 (1999).
[Crossref]

Kosachiov, D. V.

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996 (1999).
[Crossref]

Koschny, T.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[Crossref] [PubMed]

B. Wang, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Expr. 16, 16058–16063 (2008).
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Kronwald, A.

A. Kronwald and F. Marquardt, “Optomechanically Induced Transparency in the Nonlinear Quantum Regime,” Phys. Rev. Lett. 111, 133601 (2013).
[Crossref] [PubMed]

Kudriaskov, V.

J. Ruseckas, V. Kudriaskov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
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S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Weiss, T.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisators, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407 (2011).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mat. 8, 758 (2009).
[Crossref]

Welch, G. R.

Windholz, L.

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302 (1999).
[Crossref]

Wu, J.-H.

Y.-M. Liu, X.-D. Tian, D. Yan, Y. Zhang, C.-L. Cui, and J.-H. Wu, “Nonlinear modifications of photon correlations via controlled single and double Rydberg blockade,” Phys. Rev. A 91, 043802 (2015).
[Crossref]

Wu, Y.

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[Crossref]

Xiao, M.

A. Joshi and M. Xiao, “Generalized dark-state polaritons for photon memory in multilevel atomic media,” Phys. Rev. A 71, 041801 (2005).
[Crossref]

Xiao, Min

U. Khadka, Y. Zhang, and Min Xiao, “Control of multitransparency windows via dark-state phase manipulation,” Phys. Rev. A 81, 023830 (2010).
[Crossref]

Xin, H.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5, 5841 (2014).
[Crossref] [PubMed]

T. Jiang, K. Chang, L. Si, L. Ran, and H. Xin, “Active microwave negative-index metamaterial transmission line with gain,” Phys. Rev. Lett. 107, 205503 (2011).
[Crossref] [PubMed]

Xu, Datang

Z. Bai, Datang Xu, and G. Huang, “Storage and retrieval of electromagnetic waves with orbital angular momentum via plasmon-induced transparency,” Opt. Expr. 25, 785–798 (2017).
[Crossref]

Xue, C.

Y. Sun, Y. Tong, C. Xue, Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic diode based on nonlinear electromagnetically induced transparency in metamaterials,” Appl. Phys. Lett. 103, 091904 (2013).
[Crossref]

Yajima, T.

K. Nishinari and T. Yajima, “Numerical analyses of the collision of localized structures in the Davey-Stewartson equations,” Phys. Rev. E 51, 4986 (1995).
[Crossref]

Yan, D.

Y.-M. Liu, X.-D. Tian, D. Yan, Y. Zhang, C.-L. Cui, and J.-H. Wu, “Nonlinear modifications of photon correlations via controlled single and double Rydberg blockade,” Phys. Rev. A 91, 043802 (2015).
[Crossref]

Yang, L.

B. Peng, S. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

Yang, S.-H.

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

Yang, X.

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[Crossref]

Ye, D.

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5, 5841 (2014).
[Crossref] [PubMed]

Yen, T.

C. Chen, I. Un, N. Tai, and T. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Expr. 17, 15372–15380 (2009).
[Crossref]

Yin, G. Y.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient Nonlinear Frequency Conversion in an All-Resonant Double-Λ System,” Phys. Rev. Lett. 84, 5308 (2000).
[Crossref] [PubMed]

Yu, I.

Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

Yu, I. A.

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriassov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

J. Ruseckas, V. Kudriaskov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
[Crossref]

Yu, N.

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
[Crossref]

Zhan, Q.

J. Chen, P. Wang, C. Chen, Y. Lu, H. Ming, and Q. Zhan, “Plasmonic EIT-like switching in bright-dark-bright plasmon resonators,” Opt. Expr. 19, 5970–5978 (2011).
[Crossref]

Zhang, J.

H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Phase-controlled light switching at low light levels,” Phys. Rev. A 73, 011802R (2006).
[Crossref]

Zhang, L.

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
[Crossref] [PubMed]

Zhang, S.

Z. Bai, G. Huang, L. Liu, and S. Zhang, “Giant Kerr nonlinearity and low-power gigahertz solitons via plasmon-induced transparency,” Sci. Rep. 5, 13780 (2015).
[Crossref] [PubMed]

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
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S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
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Zhang, W.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
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Zhang, X.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
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Z. Dong, H. Liu, J. Cao, T. Li, S. Wang, S. Zhu, and X. Zhang, “Enhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterials,” Appl. Phys. Lett. 97, 114101 (2010).
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S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
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Zhang, Y.

Y.-M. Liu, X.-D. Tian, D. Yan, Y. Zhang, C.-L. Cui, and J.-H. Wu, “Nonlinear modifications of photon correlations via controlled single and double Rydberg blockade,” Phys. Rev. A 91, 043802 (2015).
[Crossref]

U. Khadka, Y. Zhang, and Min Xiao, “Control of multitransparency windows via dark-state phase manipulation,” Phys. Rev. A 81, 023830 (2010).
[Crossref]

Zhao, B.

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
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Zheludev, N. I.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
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Zhou, J. F.

B. Wang, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Expr. 16, 16058–16063 (2008).
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Zhu, S.

Z. Dong, H. Liu, J. Cao, T. Li, S. Wang, S. Zhu, and X. Zhang, “Enhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterials,” Appl. Phys. Lett. 97, 114101 (2010).
[Crossref]

Zhu, Y.

H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Phase-controlled light switching at low light levels,” Phys. Rev. A 73, 011802R (2006).
[Crossref]

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004).
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Zubairy, M. S.

Am. J. Phys. (1)

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Appl. Phys. Lett. (5)

Z. Dong, H. Liu, J. Cao, T. Li, S. Wang, S. Zhu, and X. Zhang, “Enhanced sensing performance by the plasmonic analog of electromagnetically induced transparency in active metamaterials,” Appl. Phys. Lett. 97, 114101 (2010).
[Crossref]

Y. Sun, Y. Tong, C. Xue, Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic diode based on nonlinear electromagnetically induced transparency in metamaterials,” Appl. Phys. Lett. 103, 091904 (2013).
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T. Matsui, M. Liu, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Electromagnetic tuning of resonant transmission in magnetoelastic metamaterials,” Appl. Phys. Lett. 104, 161117 (2014).
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J. Shao, J. Li, Y.-H. Wang, J.-Q. Li, Q. Chen, and Z.-G. Dong, “Polarization conversions of linearly and circularly polarized lights through a plasmon-induced transparent metasurface,” Appl. Phys. Lett. 115, 243503 (2014).

C. Pelzman and S.-Y. Cho, “Polarization-selective optical transmission through a plasmonic metasurface,” Appl. Phys. Lett. 106, 251101 (2015).
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Nat. Commun. (4)

B. Peng, S. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5, 5082 (2014).
[Crossref] [PubMed]

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat. Commun. 3, 1151 (2012).
[Crossref] [PubMed]

D. Ye, K. Chang, L. Ran, and H. Xin, “Microwave gain medium with negative refractive index,” Nat. Commun. 5, 5841 (2014).
[Crossref] [PubMed]

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriassov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
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Nat. Mat. (1)

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mat. 8, 758 (2009).
[Crossref]

Opt. Expr. (7)

C. Chen, I. Un, N. Tai, and T. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Expr. 17, 15372–15380 (2009).
[Crossref]

J. Chen, P. Wang, C. Chen, Y. Lu, H. Ming, and Q. Zhan, “Plasmonic EIT-like switching in bright-dark-bright plasmon resonators,” Opt. Expr. 19, 5970–5978 (2011).
[Crossref]

Z. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Expr. 19, 3251–3257 (2011).
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Z. Bai, Datang Xu, and G. Huang, “Storage and retrieval of electromagnetic waves with orbital angular momentum via plasmon-induced transparency,” Opt. Expr. 25, 785–798 (2017).
[Crossref]

M. A. Kats, N. Yu, P. Genevet, Z. Gaburro, and F. Capasso, “Effect of radiation damping on the spectral response of plasmonic components,” Opt. Expr. 19, 21748–21753 (2011).
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I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, “Tunable split-ring resonators for nonlinear negative-index metamaterials,” Opt. Expr. 14, 9344–9349 (2006).
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B. Wang, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Nonlinear properties of split-ring resonators,” Opt. Expr. 16, 16058–16063 (2008).
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Opt. Lett. (1)

Phys. Rev. A (20)

S. Rebić, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase-gate operation based on nonlinear optics,” Phys. Rev. A 74, 032301 (2006).
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C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
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J. Ruseckas, V. Kudriaskov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
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D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and cross-phase modulation via coherently driven four-level atoms in a tripod configuration,” Phys. Rev. A 70, 023822 (2004).
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S. Rebić, D. Vitali, C. Ottaviani, P. Tombesi, M. Artoni, F. Cataliotti, and R. Corbalán, “Polarization phase gate with a tripod atomic system,” Phys. Rev. A 70, 032317 (2004).
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S. Beck and I. E. Mazets, “Propagation of coupled dark-state polaritons and storage of light in a tripod medium,” Phys. Rev. A 95, 013818 (2017).
[Crossref]

A. Joshi and M. Xiao, “Generalized dark-state polaritons for photon memory in multilevel atomic media,” Phys. Rev. A 71, 041801 (2005).
[Crossref]

J.-Y. Gao, S.-H. Yang, D. Wang, X.-Z. Guo, K.-X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[Crossref]

U. Khadka, Y. Zhang, and Min Xiao, “Control of multitransparency windows via dark-state phase manipulation,” Phys. Rev. A 81, 023830 (2010).
[Crossref]

Y.-M. Liu, X.-D. Tian, D. Yan, Y. Zhang, C.-L. Cui, and J.-H. Wu, “Nonlinear modifications of photon correlations via controlled single and double Rydberg blockade,” Phys. Rev. A 91, 043802 (2015).
[Crossref]

H.-j. Li and G. Huang, “Highly efficient four-wave mixing in a coherent six-level system in ultraslow propagation regime,” Phys. Rev. A 76, 043809 (2007).
[Crossref]

Z. Bai and G. Huang, “Plasmon dromions in a metamaterial via plasmon-induced transparency,” Phys. Rev. A 93, 013818 (2016).
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M. G. Payne and L. Deng, “Consequences of induced transparency in a double-Λ scheme: Destructive interference in four-wave mixing,” Phys. Rev. A 65, 063806 (2002).
[Crossref]

H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Phys. Rev. A 70, 061804(R) (2004).
[Crossref]

Y. Wu and X. Yang, “Highly efficient four-wave mixing in double-Λ system in ultraslow propagation regime,” Phys. Rev. A 70, 053818 (2004).
[Crossref]

E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, “Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 59, 2302 (1999).
[Crossref]

E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996 (1999).
[Crossref]

Y. Wu, “Two-color ultraslow optical solitons via four-wave mixing in cold-atom media,” Phys. Rev. A 71, 053820 (2005).
[Crossref]

H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Phase-controlled light switching at low light levels,” Phys. Rev. A 73, 011802R (2006).
[Crossref]

J. A. Souza, L. Cabral, R. R. Oliveira, and C. J. Villas-Boas, “Electromagnetically-induced-transparency-related phenomena and their mechanical analogs,” Phys. Rev. A 92, 023818 (2015).
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Phys. Rev. Appl. (1)

T. Nakanishi and M. Kitano, “Implementation of Electromagnetically Induced Transparency in a Metamaterial Controlled with Auxiliary Waves,” Phys. Rev. Appl. 4, 024013 (2015).
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Phys. Rev. B (1)

T. Nakanishi, T. Otani, Y. Tamayama, and M. Kitano, “Storage of electromagnetic waves in a metamaterial that mimics electromagnetically induced absorption in plasmonics,” Phys. Rev. B 87, 16110(R) (2013).
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Phys. Rev. E (2)

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, “Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-Λ system,” Phys. Rev. E 72, 055601(R) (2005).
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K. Nishinari and T. Yajima, “Numerical analyses of the collision of localized structures in the Davey-Stewartson equations,” Phys. Rev. E 51, 4986 (1995).
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Phys. Rev. Lett. (10)

T. Jiang, K. Chang, L. Si, L. Ran, and H. Xin, “Active microwave negative-index metamaterial transmission line with gain,” Phys. Rev. Lett. 107, 205503 (2011).
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C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, “Polarization Qubit Phase Gate in Driven Atomic Media,” Phys. Rev. Lett. 90, 197902 (2003).
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Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
[Crossref] [PubMed]

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient Nonlinear Frequency Conversion in an All-Resonant Double-Λ System,” Phys. Rev. Lett. 84, 5308 (2000).
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S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[Crossref] [PubMed]

P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009).
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Rev. Mod. Phys. (2)

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Sci. Rep. (1)

Z. Bai, G. Huang, L. Liu, and S. Zhang, “Giant Kerr nonlinearity and low-power gigahertz solitons via plasmon-induced transparency,” Sci. Rep. 5, 13780 (2015).
[Crossref] [PubMed]

Science (2)

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisators, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407 (2011).
[Crossref] [PubMed]

S. Weis, R. Rivière, S. Delèglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520 (2010).
[Crossref] [PubMed]

Other (8)

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

R. Hirota, The direct method in soliton theory (Cambridge University Press, Cambridge, 2004).
[Crossref]

Bright state (dark state) is an eigenstate of the Hamiltonian that involves (does not involve) the upper states |3〉 and |4〉.

In quantum mechanics, a two-level atom is equivalent to an oscillator. The double-Λ-type atom has four levels, and hence is equivalent to three oscillators.

The frequency and wave number of l th probe field are given by ωpl + ω and kpl + Ka(ω) (l = 1, 2), respectively. Thus ω = 0 corresponds to the central frequency of the probe field.

A similar model was also considered in Ref. [20], but in which a different frequency region was chosen and no atomic FWM analogue and no study of nonlinear excitations were given.

D. Steck, 87Rb D Line Data, http://steck.us/alkalidata .

For the chosen geometry and parameters of the PIT-based metamaterial shown in Fig. 2, the resonance frequencies of the CWs and the SRR are approximately equal, and the damping rate of the dark oscillator (i.e. γ3) is much smaller than those of the bright oscillators (i.e. γ1 and γ2); see Appendix B.

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Figures (5)

Fig. 1
Fig. 1 (a) Double-Λ-type four-level atomic system with the atomic states |j〉 (j = 1, 2, 3, 4), coupled with two probe fields (with Rabi frequency Ωpn) and two strong control fields (with Rabi frequency Ωcn) (n = 1, 2). Δ3, Δ2, and Δ4 are respectively the one, two, and three-photon detunings. (b) Im ( K a + ) [imaginary part of K a +] as a function of ω for Ωc1 = Ωc2 = 20 MHz (red dashed line) and Ωc1 = Ωc2 = 60 MHz (green dashed-dot line). EIT transparency window is opened near the central frequency of the probe fields (i.e. at ω = 0). The blue solid cure is Im ( K a ), which has always a large absorption peak at ω = 0 for arbitrary Ωc1 and Ωc2.
Fig. 2
Fig. 2 (a) Schematic of the plasmonic metamaterial, which is an array of meta-atoms. (b) The meta-atom consists of two CWs (indicated by “A” and “B”) and a SRR, where the parameters dx, dy, Lx, Ly, wb, wg, and ws are given in the text. For generating nonlinear excitations, four hyperabrupt tuning varactors are mounted onto the slits of the SRR. (see Sec. 3). (c) The numerical result (blue dashed lines) of the normalized absorption spectrum of the EM wave as a function of frequency by taking y0 = −x0, dx = dy = 4.0 mm (first panel), and dx = dy = 3.4 mm (second panel). (d) The numerical result (blue dashed line) of normalized absorption spectrum for y0 = x0, dx = dy = 4.0 mm. Red solid lines in (c) and (d) are analytical results obtained from the formula Im(q10) given by Eq. (26) in Appendix B.
Fig. 3
Fig. 3 (a) Linear dispersion relation of the K m +-mode (PIT-mode). Im ( K m + ) (blue dashed line) and Re ( K m + ) (red solid line) are plotted as functions of ω for κ2 = −κ1 = 50 GHz2 (first panel) and κ2 = −κ1 = 250 GHz2 (second panel). (b) Linear dispersion relation of the K m -mode (non-PIT-mode) for arbitrary κ1 (κ2 = −κ1).
Fig. 4
Fig. 4 FWM conversion efficiency η as a function of the dimensionless optical depth (κ0gf1/γ1)L for Δ1 = Δ2 = 0 (blue dashed line), and Δ1 = Δ2 = 5γ1 (red solid line). Inset: FWM conversion efficiency η for optical depth up to 300 for Δ1 = Δ2 = 5γ1.
Fig. 5
Fig. 5 Plasmonic dromions and their interaction. (a) [(b)] is the intensity profile of the shortwave |u|2 (longwave |v1|2) as functions of ξ1 and τ1 at s = 0. (c1), (c2), (c3), (c4) [(d1), (d2), (d3), (d4)] are intensity profiles of the shortwave |u|2 (longwave |v1|2) during the interaction between two dromions, respectively at sz/(2Ldiff) = 0, 1, 2, 3. System parameters are given in the text.

Equations (80)

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H ^ int = j = 1 4 Δ j | j j | [ Ω p 1 | 3 1 | + Ω p 2 | 4 1 | + Ω c 1 | 3 2 | + Ω c 2 | 4 2 | + H . c . ] ,
Ω p 1 Ω c 2 Ω p 2 Ω c 1 = 0 .
( i t + d 31 ) σ 31 + Ω c 1 σ 21 + Ω p 1 = 0 ,
( i t + d 41 ) σ 41 + Ω c 2 σ 21 + Ω p 2 = 0 ,
( i t + d 21 ) σ 21 + Ω c 1 * σ 31 + Ω c 2 * σ 41 = 0 ,
i ( z + 1 c t ) Ω p 1 + κ 13 σ 31 = 0 ,
i ( z + 1 c t ) Ω p 2 + κ 14 σ 41 = 0 ,
K a ± ( ω ) = ω c + ( κ 14 D 3 + κ 13 D 4 ) ± ( κ 14 D 3 κ 13 D 4 ) 2 + 4 κ 13 κ 14 | Ω c 1 Ω c 2 | 2 2 [ | Ω c 1 | 2 ( ω + d 41 ) + | Ω c 2 | 2 ( ω + d 31 ) ( ω + d 21 ) ( ω + d 31 ) ( ω + d 41 ) ] ,
2 q 1 t 2 + γ 1 q 1 t + ω 1 2 q 1 κ 1 q 3 = g 1 E x ,
2 q 2 t 2 + γ 2 q 2 t + ω 2 2 q 2 κ 2 q 3 = g 2 E y ,
2 q 3 t 2 + γ 3 q 3 t + ω 3 2 q 3 κ 1 q 1 κ 2 q 2 = 0 ,
κ 2 y 0 κ 1 x 0 = 0 .
2 E x ( y ) 1 c 2 2 E x ( y ) t 2 = 1 ε 0 c 2 2 P x ( y ) t 2 ,
( i t + d 1 ) q ˜ 1 + κ 1 2 ω p q ˜ 3 + g 1 2 ω p x = 0 ,
( i t + d 2 ) q ˜ 2 + κ 2 2 ω p q ˜ 3 + g 2 2 ω p y = 0 ,
( i t + d 3 ) q ˜ 3 + κ 1 2 ω p q ˜ 1 + κ 2 2 ω p q ˜ 2 = 0 ,
i ( z + n D c t ) x + κ 0 q ˜ 1 = 0 ,
i ( z + n D c t ) y + κ 0 q ˜ 2 = 0 ,
K m ± ( ω ) = n D c ω + κ 0 ( R 1 g f 2 + R 2 g f 1 ) ± ( R 1 g f 2 R 2 g f 1 ) 2 + 4 κ f 1 2 κ f 2 2 g f 1 g f 2 2 [ κ f 1 2 ( ω + d 2 ) + κ f 2 2 ( ω + d 1 ) ( ω + d 3 ) ( ω + d 1 ) ( ω + d 2 ) ] ,
x ( z , t ) = 1 2 π + d ω [ F 0 + e i ( K m + z ω t ) + F 0 e i ( K m z ω t ) ] ,
y ( z , t ) = 1 2 π + d ω [ G + F 0 + e i ( K m + z ω t ) + G F 0 e i ( K m z ω t ) ] ,
x ( z , t ) = 1 2 π + d ω G + e i ( K m z ω t ) G e i ( K m + z ω t ) G + G ˜ x ( 0 , ω ) ,
y ( z , t ) = 1 2 π + d ω G + G G + G [ e i ( K m z ω t ) e i ( K m + z ω t ) ] ˜ x ( 0 , ω ) ,
x ( z , t ) = G 0 + x ( 0 , τ ) e i K 0 z G 0 x ( 0 , τ + ) e i K 0 + z G 0 + G 0 ,
y ( z , t ) = G 0 + G 0 G 0 + G 0 [ x ( 0 , τ ) e i K 0 z x ( 0 , τ + ) e i K 0 + z ] ,
x ( z , t ) = G 0 G 0 G 0 + x ( 0 , τ + ) e i K 0 + z ,
y ( z , t ) = G 0 G 0 + G 0 G 0 + x ( 0 , τ + ) e i K 0 + z .
η ( L ) = | G 0 + G 0 | 2 | G 0 + G 0 | 2 | exp ( i K 0 + L ) | 2 .
2 q 3 t 2 + γ 3 q 3 t + ω 3 2 q 3 κ 2 q 1 κ 2 q 2 + α q 3 2 + β q 3 3 = 0 ,
i F + z 2 K 2 + 2 2 F + t 1 2 + c 2 ω p n D ( 2 x 1 2 + 2 y 1 2 ) F + + ω p R 0 2 c n D χ + ( 2 ) Q + F + + ω p 2 c n D χ + ( 3 ) | F + | 2 F + = 0 ,
χ + ( 2 ) = N m e ε 0 2 α ( ω 2 2 κ 1 g 1 + ω 1 2 κ 2 g 2 G + ) g 2 G + ( G + G ) ( ω 1 2 ω 2 2 ω 3 2 ω 2 2 κ 1 2 ω 1 2 κ 2 2 ) | D 2 κ 1 g 1 + D 1 κ 2 g 2 G + D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 | 2 ,
χ + ( 3 ) = N m e ε 0 ( D 2 κ 1 g 1 + D 1 κ 2 g 2 G + ) 2 | D 2 κ 1 g 1 + D 1 κ 2 g 2 G + | 2 g 2 G + ( G + G ) ( D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 ) 2 | D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 | 2 × [ ( 4 α 2 ω 1 2 ω 2 2 ω 1 2 ω 2 2 ω 3 2 ω 2 2 κ 1 2 ω 1 2 κ 2 2 + 2 α 2 H 1 H 2 H 1 H 2 H 3 H 2 κ 1 2 H 1 κ 2 2 ) 3 β ] .
( 2 x 1 2 + 2 y 1 2 ) Q + + [ ( 1 V g + ) 2 ( 1 V p + ) 2 ] 2 Q + t 1 2 χ + ( 2 ) c 2 2 | F + | 2 t 1 2 = 0 ,
1 V p + = n D c + N m e 2 ε 0 c n D ( g 1 X 2 + g 2 X 1 ) + ( g 1 X 2 g 2 X 1 ) 2 + 4 κ 1 2 κ 2 2 g 1 g 2 ω 1 2 ω 2 2 ω 3 2 ω 1 2 κ 2 2 ω 2 2 κ 1 2 ,
i u s + ( 2 ξ 2 + g d 0 2 η 2 + g d 1 2 τ 2 ) u + 2 g 1 | u | 2 u + g 2 v u = 0 ,
g d 2 2 v τ 2 ( 2 ξ 2 + g d 0 2 η 2 ) v + g 3 2 | u | 2 τ 2 = 0 .
E ( r , t ) e x κ 1 + e y κ 2 κ 1 2 + κ 2 2 [ ( U 0 u e i k p z i ω p t + c . c . ) + V 0 v ] .
i σ 11 t i Γ 13 σ 33 i Γ 14 σ 44 Ω p 1 σ 31 * Ω p 2 σ 41 * + Ω p 1 * σ 31 + Ω p 2 * σ 41 = 0 ,
i σ 22 t i Γ 23 σ 33 i Γ 24 σ 44 Ω c 1 σ 32 * Ω c 2 σ 42 * + Ω c 1 * σ 32 + Ω c 2 * σ 42 = 0 ,
i ( t + Γ 13 + Γ 23 ) σ 33 + Ω p 1 σ 31 * + Ω c 1 σ 31 * Ω p 1 * σ 31 Ω c 1 * σ 32 = 0 ,
i ( t + Γ 14 + Γ 24 ) σ 44 + Ω p 2 σ 41 * + Ω c 2 σ 42 * Ω p 2 * σ 41 Ω c 2 * σ 42 = 0
( i t + d 21 ) σ 21 Ω p 1 σ 32 * Ω p 2 σ 42 * + Ω c 1 * σ 31 + Ω c 2 * σ 41 = 0 ,
( i t + d 31 ) σ 31 Ω p 1 ( σ 33 σ 11 ) Ω p 2 σ 43 * + Ω c 1 σ 21 = 0 ,
( i t + d 32 ) σ 32 Ω p 2 ( σ 44 σ 11 ) Ω c 2 σ 43 * + Ω p 1 σ 21 * = 0 ,
( i t + d 41 ) σ 41 Ω c 1 ( σ 33 σ 22 ) Ω p 1 σ 43 + Ω c 2 σ 21 = 0 ,
( i t + d 42 ) σ 42 Ω c 2 ( σ 44 σ 22 ) Ω c 1 σ 43 + Ω p 2 σ 21 * = 0 ,
( i t + d 43 ) σ 43 Ω p 1 * σ 41 Ω c 1 * σ 42 + Ω p 2 σ 31 * + Ω c 2 σ 32 * = 0 ,
q 10 = ( D 3 D 2 κ 2 2 ) g 1 x 0 + κ 1 κ 2 g 2 y 0 D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 ,
q 20 = κ 2 κ 1 g 1 x 0 + ( D 3 D 1 κ 1 2 ) g 2 y 0 D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 ,
q 30 = D 2 κ 1 g 1 x 0 + D 1 κ 2 g 2 y 0 D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 ,
( i t + d f 1 ) q f 1 + κ 1 2 ω p q f 3 + g 1 2 ω p f x = 0 ,
( i t + d f 2 ) q f 2 + κ 2 2 ω p q f 3 + g 2 2 ω p f y = 0 ,
( i t + d f 3 ) q f 3 + κ 1 2 ω p q f 1 + κ 2 2 ω p q f 2 1 2 ω p [ 2 α ( q d 2 q f 3 + q s 3 q f 3 * ) + 3 β | q f 3 | 2 q f 3 ] = 0 ,
( 2 t 2 + γ 1 t ω 1 2 ) q d 1 κ 1 q d 3 g 1 d x = 0 ,
( 2 t 2 + γ 2 t ω 2 2 ) q d 2 κ 2 q d 3 g 2 d y = 0 ,
( 2 t 2 + γ 3 t ω 3 2 ) q d 3 κ 2 q d 1 κ 2 q d 2 + 2 α | q f 3 | 2 = 0 ,
( i t + d s 1 + 3 4 ω 1 ) q s 1 + κ 1 4 ω p q s 3 + g 1 4 ω p s x e i Δ κ z = 0 ,
( i t + d s 2 + 3 4 ω 2 ) q s 2 + κ 2 4 ω p q s 3 + g 2 4 ω p s y e i Δ κ z = 0 ,
( i t + d s 3 + 3 4 ω 3 ) q s 3 + κ 1 4 ω p q s 1 + κ 2 4 ω p q s 2 + α 4 ω p q f 3 2 = 0 ,
i ( z + n D c + t ) f x + c 2 ω p n D + ( 2 x 2 + 2 y 2 ) f x + κ 0 q f 1 = 0 ,
i ( z + n D c t ) f y + c 2 ω p n D + ( 2 x 2 + 2 y 2 ) f y + κ 0 q f 2 = 0 ,
( 2 z 2 n D 2 c 2 2 t 2 ) d x + ( 2 x 2 + 2 y 2 ) d x N m e ε 0 c 2 2 t 2 q d 1 = 0 ,
( 2 z 2 n D 2 c 2 2 t 2 ) d y + ( 2 x 2 + 2 y 2 ) d y N m e ε 0 c 2 2 t 2 q d 2 = 0 ,
i ( z + n D c t ) s x + c 4 ω p n D + ( 2 x 2 + 2 y 2 ) s x + 2 κ 0 q s 1 e i Δ k z = 0 ,
i ( z + n D c t ) s y + c 4 ω p n D + ( 2 x 2 + 2 y 2 ) s y + 2 κ 0 q s 2 e i Δ k z = 0 .
q f 1 ( 1 ) = 1 κ 0 ( K m + n D c ω ) F + e i θ + ,
q f 2 ( 1 ) = G + κ 0 ( K m + n D c ω ) F + e i θ + ,
q f 3 ( 1 ) = D 2 κ 1 g 1 + D 2 κ 2 g 2 G + D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 F + e i θ + ,
q f 2 ( 2 ) = 1 κ 0 ( K m + ω n D c ) ( i t 1 ) F + e i θ + ,
q f 2 ( 2 ) = G + κ 0 ( K m + ω n D c ) ( i t 1 ) F + e i θ + .
q d 1 ( 2 ) = ( N m e 2 ε 0 n D ) 1 [ n m + ( 0 ) n D ] Q + + 2 α ω 2 2 κ 1 ω 1 2 ω 2 2 ω 3 2 ω 2 2 κ 1 2 ω 1 2 κ 2 2 | q f 3 ( 1 ) | 2 ,
q d 2 ( 2 ) = ( N m e 2 ε 0 n D ) 1 G + [ n m + ( 0 ) n D ] Q + + 2 α ω 1 2 κ 2 ω 1 2 ω 2 2 ω 3 2 ω 2 2 κ 1 2 ω 1 2 κ 2 2 | q f 3 ( 1 ) | 2 ,
q d 3 ( 2 ) = ω 2 2 κ 1 g 1 + ω 1 2 κ 2 g 2 G + ω 1 2 ω 2 2 ω 3 2 ω 2 2 κ 1 2 ω 1 2 κ 2 2 Q + + 2 α ω 1 2 ω 2 2 | q f 3 ( 1 ) | 2 ω 1 2 ω 2 2 ω 3 2 ω 2 2 κ 1 2 ω 1 2 κ 2 2 ,
q s 3 ( 2 ) = α H 1 H 2 H 1 H 2 H 3 H 2 κ 1 2 H 1 κ 2 2 [ q f 3 ( 1 ) ] 2 ,
q f 1 ( 3 ) = 1 2 κ 0 2 K m + ω 2 2 F + t 1 2 e i θ + + D 2 κ 1 [ 2 α [ q d 3 ( 2 ) q f 3 ( 1 ) + q s 3 ( 2 ) q f 3 ( 1 ) * ] + 3 β | q f 3 ( 1 ) | 2 q f 3 ( 1 ) ] D 1 D 2 D 3 D 2 κ 1 2 D 2 κ 2 2 ,
q f 1 ( 3 ) = G + 2 κ 0 2 K m + ω 2 2 F + t 1 2 e i θ + + D 2 κ 2 [ 2 α [ q d 3 ( 2 ) q f 3 ( 1 ) + q s 3 ( 2 ) q f 3 ( 1 ) * ] + 3 β | q f 3 ( 1 ) | 2 q f 3 ( 1 ) ] D 1 D 2 D 3 D 2 κ 1 2 D 2 κ 2 2 .
R 0 = ( D 2 κ 1 g 1 + D 1 κ 2 g 2 G + ) 2 | D 2 κ 1 g 1 + D 1 κ 2 g 2 G + | 2 | D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 | 2 ( D 1 D 2 D 3 D 2 κ 1 2 D 1 κ 2 2 ) 2 .
2 u 1 t 2 + γ 1 u 1 t + ω 1 2 u 1 κ 1 u 3 = g 1 Q 0 E x ,
2 u 2 t 2 + γ 2 u 2 t + ω 2 2 u 2 κ 2 u 3 = g 1 Q 0 E y ,
2 u 3 t 2 + γ 3 u 3 t + ω 3 2 u 3 κ 1 u 1 κ 2 u 2 + α Q 0 u 3 2 + β Q 0 2 u 3 3 = 0 ,

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