Abstract

Surface plasmon polariton (SPP) sources and launchers are highly demanded in various applications of nanophotonics. Here, we propose a general approach that can realize complete control of the complex extinction ratio (including amplitude and phase) of any two linearly independent SPP modes excited by any elementary SPP excitation architecture just by manipulating the incident polarization state. In an optical system, it suffices to simply tune the orientation angles of a linear polarizer and a quarter wave plate, which may greatly simplify the design and application of SPP launchers and diversify their functionalities. As an example to show the broad application prospect of this method, we design and realize a metaline consisting of Δ-shaped plasmonic nanoantennas, which can effectively realize dual functionalities, i.e., the tunable directional SPP excitation at an arbitrarily chosen wavelength and the complete unidirectional SPP excitation over a broad bandwidth. This general approach can also be extended to the control of the complex extinction ratio of any two linearly independent excited modes in many other linear optical systems, such as two modes in a waveguide or two diffraction orders in a grating, over a broad bandwidth.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
  3. X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
    [Crossref] [PubMed]
  4. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
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  5. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
    [Crossref] [PubMed]
  6. J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
    [Crossref] [PubMed]
  7. O. You, B. Bai, X. Wu, Z. Zhu, and Q. Wang., “A simple method for generating unidirectional surface plasmon polariton beams with arbitrary profiles,” Opt. Lett.,  40, 5486–5489 (2015).
    [Crossref] [PubMed]
  8. E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
    [Crossref]
  9. K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
    [Crossref]
  10. H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
    [Crossref] [PubMed]
  11. J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
    [Crossref] [PubMed]
  12. A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
    [Crossref] [PubMed]
  13. S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  16. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
    [Crossref] [PubMed]
  17. J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
    [Crossref] [PubMed]
  18. L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
    [Crossref]
  19. H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
    [Crossref]
  20. P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
    [Crossref]
  21. https://www.lumerical.com/ .
  22. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999)
    [Crossref]

2016 (2)

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

2015 (1)

2013 (6)

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
[Crossref] [PubMed]

2012 (5)

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

2011 (2)

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
[Crossref]

2007 (1)

2004 (1)

D. Egorov, B. S. Dennis, G. Blumberg, and M. I. Haftel, “Two-dimensional control of surface plasmons and directional beaming from arrays of subwavelength apertures,” Phys. Rev. B 70, 033404 (2004).
[Crossref]

Adam, P. M.

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

Alameh, K.

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

Antoniou, N.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

Bai, B.

O. You, B. Bai, X. Wu, Z. Zhu, and Q. Wang., “A simple method for generating unidirectional surface plasmon polariton beams with arbitrary profiles,” Opt. Lett.,  40, 5486–5489 (2015).
[Crossref] [PubMed]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Baron, A.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Blumberg, G.

D. Egorov, B. S. Dennis, G. Blumberg, and M. I. Haftel, “Two-dimensional control of surface plasmons and directional beaming from arrays of subwavelength apertures,” Phys. Rev. B 70, 033404 (2004).
[Crossref]

Bonod, N.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999)
[Crossref]

Bouillard, J.-S

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

Brongersma, M. L.

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
[Crossref] [PubMed]

Capasso, F.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

Chen, J.

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

Chen, X.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Chernov, B.

Christodoulides, D. N.

Cluzel, B.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

de Fornel, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

de la Cruz, S.

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

Dellinger, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

Dennis, B. S.

D. Egorov, B. S. Dennis, G. Blumberg, and M. I. Haftel, “Two-dimensional control of surface plasmons and directional beaming from arrays of subwavelength apertures,” Phys. Rev. B 70, 033404 (2004).
[Crossref]

Devaux, E.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Dickson, W.

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

Ebbesen, T. W.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Egorov, D.

D. Egorov, B. S. Dennis, G. Blumberg, and M. I. Haftel, “Two-dimensional control of surface plasmons and directional beaming from arrays of subwavelength apertures,” Phys. Rev. B 70, 033404 (2004).
[Crossref]

Genet, C.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Genevet, P.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

Georgi, P.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

Ginzburg, P.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

Gong, Q.

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

Haftel, M. I.

D. Egorov, B. S. Dennis, G. Blumberg, and M. I. Haftel, “Two-dimensional control of surface plasmons and directional beaming from arrays of subwavelength apertures,” Phys. Rev. B 70, 033404 (2004).
[Crossref]

Hong, J.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Huang, L.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Huang, X.

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
[Crossref] [PubMed]

Hugonin, J.-P

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Hugonin, J.P.

P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
[Crossref]

Jin, G.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Kim, H.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Kim, K.-Y

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Lalanne, P.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
[Crossref]

Lee, B.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Lee, G.-Y.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Lee, I.-M

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Lee, K.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Lee, S.-Y

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Lee, S.-Y.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Lee, W.

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Lee, Y.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Li, G.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

Li, K.

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

Li, L.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express 15(18), 11427–11432 (2007).
[Crossref] [PubMed]

Li, T.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

Li, Z.

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

Liao, H.

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

Lin, J.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

Liu, H.T.

P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
[Crossref]

Liu, Y.

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

Lu, F.

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

Macías, D.

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

Marino, G.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

Martínez, A.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

Méndez, E. R.

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

Mueller, J. P. B.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

Mühlenbernd, H.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

O’Connor, D.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

Oh, S.

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Palomba, S.

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

Park, J.

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

Park, Y.

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

Pholchai, N.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

Popov, E.

Rodier, J.-C.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Rodríguez-Fortuño, F. J.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

Rousseau, E.

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

Salandrino, A.

Salas-Montiel, R.

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

Song, E.-Y.

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Tan, Q.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Vilain, S.

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

Wang, B.

P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
[Crossref]

Wang, Q.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

Wang, S. M.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

Wang., Q.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999)
[Crossref]

Wu, X.

Wurtz, G. A.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

Xiao, F.

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

Xu, A.

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

Yin, X.

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

You, O.

Yuan, G.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

Yuan, X.-C

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

Yue, S.

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

Zayats, A. V.

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

Zentgraf, T.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

Zhang, C.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

Zhang, S.

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Zhang, X.

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

Zhu, S. N.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

Zhu, Z.

ACS Photonics (1)

H. Mühlenbernd, P. Georgi, N. Pholchai, L. Huang, G. Li, S. Zhang, and T. Zentgraf, “Amplitude- and phase-controlled surface plasmon polariton excitation with metasurfaces,” ACS Photonics 3(1), 124–129 (2016).
[Crossref]

Laser and Photonics Reviews (1)

E.-Y. Song, S.-Y. Lee, J. Hong, K. Lee, Y. Lee, G.-Y. Lee, H. Kim, and B. Lee, “A double-lined metasurface for plasmonic complex-field generation,” Laser and Photonics Reviews,  10, 299–308 (2016).
[Crossref]

Light-Sci. Appl. (1)

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Nano Lett. (3)

Y. Liu, S. Palomba, Y. Park, T. Zentgraf, X. Yin, and X. Zhang, “Compact magnetic antennas for directional excitation of surface plasmons,” Nano Lett. 12(9), 4853–4858 (2012).
[Crossref] [PubMed]

X. Huang and M. L. Brongersma, “Compact aperiodic metallic groove arrays for unidirectional launching of surface plasmons,” Nano Lett. 13(11), 5420–5424 (2013).
[Crossref] [PubMed]

A. Baron, E. Devaux, J.-C. Rodier, J.-P Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11(10), 4207–4212 (2011).
[Crossref] [PubMed]

New J. Phys. (1)

K. Li, F. Xiao, F. Lu, K. Alameh, and A. Xu, “Unidirectional coupling of surface plasmons with ultra-broadband and wide-angle efficiency: potential applications in sensing,” New J. Phys. 15, 113040 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (1)

D. Egorov, B. S. Dennis, G. Blumberg, and M. I. Haftel, “Two-dimensional control of surface plasmons and directional beaming from arrays of subwavelength apertures,” Phys. Rev. B 70, 033404 (2004).
[Crossref]

Phys. Rev. Lett. (3)

S.-Y Lee, I.-M Lee, J. Park, S. Oh, W. Lee, K.-Y Kim, and B. Lee, “Role of magnetic induction currents in nanoslit excitation of surface plasmon polaritons,” Phys. Rev. Lett. 108, 213907 (2012).
[Crossref] [PubMed]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref] [PubMed]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref] [PubMed]

Phys. Status Solidi B (1)

S. de la Cruz, E. R. Méndez, D. Macías, R. Salas-Montiel, and P. M. Adam, “Compact surface structures for the efficient excitation of surface plasmon-polaritons,” Phys. Status Solidi B 249(6), 1178–1187 (2012).
[Crossref]

Sci. Rep. (2)

H. Liao, Z. Li, J. Chen, X. Zhang, S. Yue, and Q. Gong, “A submicron broadband surface-plasmon-polariton unidirectional coupler,” Sci. Rep. 3, 1918 (2013).
[Crossref] [PubMed]

J.-S Bouillard, S. Vilain, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Broadband and broadangle SPP antennas based on plasmonic crystals with linear chirp,” Sci. Rep. 2, 829 (2012).
[Crossref] [PubMed]

Science (2)

F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013).
[Crossref] [PubMed]

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013).
[Crossref] [PubMed]

Surf. Sci. Rep (1)

P. Lalanne, J.P. Hugonin, H.T. Liu, and B. Wang, “A microscopic view of the electromagnetic properties of sub-λ metallic surfaces,” Surf. Sci. Rep 64(10), 453–469 (2009).
[Crossref]

Other (2)

https://www.lumerical.com/ .

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999)
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic illustrations of the excited SPP modes under the normal illumination of plane waves with (a) Ĵx polarization, (b) Ĵy polarization, and (c) an arbitrary combination of (a) and (b).
Fig. 2
Fig. 2 (a) Schematic of the experimental optical setup for characterizing the SPP excitation. (b) SEM picture of a sample fabricated by focused ion beam milling.
Fig. 3
Fig. 3 Schematic of a dielectric-metal interface. The dielectric and metal are both infinitely thick. ɛd and ɛm represent the permittivities of dielectric and metal, respectively. The dielectric-metal interface is located at z = z′.
Fig. 4
Fig. 4 (a) |E|2 distribution of the electromagnetic field in the xz plane excited by an x-polarized illumination. (b) |E|2 distribution of the electromagnetic field in the xz plane excited by a y-polarized illumination. (c) The amplitudes of the modal excitation coefficients α x + and α x of the excited Ψ SPP + and Ψ SPP modes under x-polarized illumination. (d) The amplitudes of the modal excitation coefficients α y + and α y of the excited Ψ SPP + and Ψ SPP modes under y-polarized illumination (b). (e) The same as (c), but for the phases of the modal excitation coefficients. (f) The same as (d), but for the phases of the modal excitation coefficients. The red lines stand for Ψ SPP + and the blue lines stand for Ψ SPP .
Fig. 5
Fig. 5 (a) Amplitudes of αx, (b) amplitudes of αy, (c) phases of αx, and (d) phases of αy at different wavelengths. Real parts (e) and imaginary parts (f) of gold index at different wavelengths.
Fig. 6
Fig. 6 (a) Schematic illustration of ψ, where x and y axes represent the original coordinate system, and a and b represent the rotated coordinate system. The long axis of the vibration ellipse of the electric field is along a, and the short axis is along b. (b) Schematic illustration of the angle φ formed by the polarizing direction of the LP and the fast axis of the QWP, and the angle ψt formed by the fast axis of the QWP and the long axis of the vibration ellipse
Fig. 7
Fig. 7 Comparison of the desired (blue triangles) and the experimentally measured (green circles) arctan|Ext+−|2 of the tunable directional SPP excitation. The required polarization states of incident light and the corresponding rotation angles of the LP (θLP) and QWP (θQWP) for seven typical experimental points are also given below the curves. The inset CCD images show the unidirectional and symmetrical SPP excitations at three special points.
Fig. 8
Fig. 8 Comparison of the desired (blue triangles) and experimentally measured (green circles) |Ext+−|2 in a broadband range for wavelength from 710 nm to 1000 nm with a step of 10 nm for complete unidirectional SPP excitation. The required incident polarization states at different wavelengths are illustrated by red ellipses. The two CCD images in the bottom show the unidirectional SPP excitation effect at wavelengths 710 nm and 1000 nm. The y-axis stands for |Ext+−|2.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Ext 21 = C 1 α 12 + C 2 α 22 C 1 α 11 + C 2 α 21 = α 12 + C 21 α 22 α 11 + C 21 α 21 ,
α 11 α 22 α 12 α 21 0 ,
Ext + = α x + C y x α y α x + C y x α y ,
C y x = α x ( Ext + + 1 ) α y ( Ext + 1 ) .
{ Ψ SPP + ( x , z ) = [ H y + ( x , z ) , E x + ( x , z ) , E z + ( x , z ) ] Ψ SPP ( x , z ) = [ H y ( x , z ) , E x ( x , z ) , E z ( x , z ) ] .
Ψ SPP ( x , z ) = exp ( i β x ) { exp ( i κ d ( z z ) ) [ 1 , κ d ω ε 0 ε d , β ω ε 0 ε d ] ( z > z ) exp ( i κ m ( z z ) ) [ 1 , κ m ω ε 0 ε m , β ω ε 0 ε m ] ( z z ) ,
Ψ SPP ( x , z ) = exp ( i β x ) { exp ( i κ d ( z z ) ) [ 1 , κ d ω ε 0 ε d , β ω ε 0 ε d ] ( z > z ) exp ( i κ m ( z z ) ) [ 1 , κ m ω ε 0 ε m , β ω ε 0 ε m ] ( z z ) .
α + = N 1 [ E z ( x , z ) H y ( x , z ) E z ( x , z ) H y ( x , z ) ] d z α = N 1 [ H y ( x , z ) E z + ( x , z ) H x + ( x , z ) E z ( x , z ) ] d z ,
J ^ x y = [ a x exp ( j δ x ) a y exp ( j δ y ) ] = [ J x J y ] ,
E ^ x y = [ exp ( j τ ) J ^ x y ] * = exp ( j τ ) [ a x exp ( j δ x ) a y exp ( j δ y ) ] = exp ( j τ ) [ J x * J y * ] ,
C y x * = J y / J x = a y exp ( j δ ) / a x = tan α exp ( j δ ) ,
tan 2 ψ = ( tan 2 α ) cos δ sin 2 χ = ( sin 2 α ) sin δ .
E a = J a cos ( τ + δ 0 ) E b = ± J b sin ( τ + δ 0 ) .
tan χ = J b / J a .
[ J σ J τ ] = [ cos ( φ ) exp ( j Γ / 2 ) sin ( φ ) exp ( j Γ / 2 ) ]
tan 2 ψ t = ( tan 2 φ ) cos ( Γ ) sin 2 χ = ( sin 2 φ ) sin ( Γ ) .

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