Versatile and tunable surface plasmon polariton excitation over a broad bandwidth with a simple metaline by external polarization modulation

Oubo You; Benfeng Bai; Lin Sun; Biyao Shen; Zhendong Zhu

doi:10.1364/OE.24.022061

Versatile and tunable surface plasmon polariton excitation over a broad bandwidth with a simple metaline by external polarization modulation

Oubo You, Benfeng Bai, Lin Sun, Biyao Shen, and Zhendong Zhu

Optics Express
Vol. 24,
Issue 19,
pp. 22061-22073
(2016)
•https://doi.org/10.1364/OE.24.022061

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Oubo You, Benfeng Bai, Lin Sun, Biyao Shen, and Zhendong Zhu, "Versatile and tunable surface plasmon polariton excitation over a broad bandwidth with a simple metaline by external polarization modulation," Opt. Express 24, 22061-22073 (2016)

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Related Topics
Table of Contents Category
- Plasmonics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optical switching devices (130.4815)
- Surface plasmons (240.6680)
- Interference (260.3160)
- Polarization (260.5430)

About this Article
History
- Original Manuscript: July 26, 2016
- Revised Manuscript: August 31, 2016
- Manuscript Accepted: August 31, 2016
- Published: September 13, 2016

Accessible

Open Access

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.

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References

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Publication

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]
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. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
[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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
https://www.lumerical.com/ .
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)

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]

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)

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
[Crossref] [PubMed]

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)

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]

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.

Alameh, K.

Antoniou, N.

Bai, B.

Baron, A.

Blumberg, G.

Bonod, N.

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]

Born, M.

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

Bouillard, J.-S

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.

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.

Chernov, B.

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]

Christodoulides, D. N.

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
[Crossref] [PubMed]

Cluzel, B.

de Fornel, F.

de la Cruz, S.

Dellinger, J.

Dennis, B. S.

Devaux, E.

Dickson, W.

Ebbesen, T. W.

Egorov, D.

Genet, C.

Genevet, P.

Georgi, P.

Ginzburg, P.

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.

Hong, J.

Huang, L.

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

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.

Kim, H.

Kim, K.-Y

Lalanne, 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]

Lee, B.

Lee, G.-Y.

Lee, I.-M

Lee, K.

Lee, S.-Y

Lee, S.-Y.

Lee, W.

Lee, Y.

Li, G.

Li, K.

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.

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.

Lu, F.

Macías, D.

Marino, G.

Martínez, A.

Méndez, E. R.

Mueller, J. P. B.

Mühlenbernd, H.

O’Connor, D.

Oh, S.

Palomba, S.

Park, J.

Park, Y.

Pholchai, N.

Popov, E.

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]

Rodier, J.-C.

Rodríguez-Fortuño, F. J.

Rousseau, E.

Salandrino, A.

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
[Crossref] [PubMed]

Salas-Montiel, R.

Song, E.-Y.

Tan, Q.

Vilain, S.

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.

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.

Xiao, F.

Xu, A.

Yin, X.

You, O.

Yuan, G.

Yuan, X.-C

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.

Zentgraf, T.

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.

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]

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)

Laser and Photonics Reviews (1)

Light-Sci. Appl. (1)

Nano Lett. (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]

New J. Phys. (1)

Opt. Express (1)

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]

Opt. Lett. (2)

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
[Crossref] [PubMed]

Phys. Rev. B (1)

Phys. Rev. Lett. (3)

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]

Phys. Status Solidi B (1)

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]

Science (2)

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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Fig. 1 Schematic illustrations of the excited SPP modes under the normal illumination of plane waves with (a) Ĵ_x polarization, (b) Ĵ_y polarization, and (c) an arbitrary combination of (a) and (b).

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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.

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Fig. 3 Schematic of a dielectric-metal interface. The dielectric and metal are both infinitely thick. ɛ_d and ɛ_m represent the permittivities of dielectric and metal, respectively. The dielectric-metal interface is located at z = z′.

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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}^{-}

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Fig. 5 (a) Amplitudes of α_x, (b) amplitudes of α_y, (c) phases of α_x, and (d) phases of α_y at different wavelengths. Real parts (e) and imaginary parts (f) of gold index at different wavelengths.

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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 ψ_t formed by the fast axis of the QWP and the long axis of the vibration ellipse

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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 (θ_LP) and QWP (θ_QWP) 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.

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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.

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Equations (16)

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

{Ext}_{21} = \frac{C_{1} α_{12} + C_{2} α_{22}}{C_{1} α_{11} + C_{2} α_{21}} = \frac{α_{12} + C_{21} α_{22}}{α_{11} + C_{21} α_{21}},

α_{11} α_{22} - α_{12} α_{21} \neq 0,

{Ext}_{+ -} = \frac{α_{x} + C_{y x} α_{y}}{- α_{x} + C_{y x} α_{y}},

C_{y x} = \frac{α_{x} ({Ext}_{+ -} + 1)}{α_{y} ({Ext}_{+ -} - 1)} .

{\begin{matrix} 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)] \end{matrix} .

Ψ_{SPP}^{-} (x, z) = exp (i β x) \cdot {\begin{matrix} exp (i κ_{d} (z - z^{'})) [- 1, \frac{- κ_{d}}{ω ε_{0} ε_{d}}, \frac{β}{ω ε_{0} ε_{d}}] (z > z^{'}) \\ exp (- i κ_{m} (z - z^{'})) [- 1, \frac{κ_{m}}{ω ε_{0} ε_{m}}, \frac{β}{ω ε_{0} ε_{m}}] (z \leq z^{'}) \end{matrix},

Ψ_{SPP}^{-} (x, z) = exp (- i β x) \cdot {\begin{matrix} exp (i κ_{d} (z - z^{'})) [1, \frac{κ_{d}}{ω ε_{0} ε_{d}}, \frac{β}{ω ε_{0} ε_{d}}] (z > z^{'}) \\ exp (- i κ_{m} (z - z^{'})) [1, \frac{- κ_{m}}{ω ε_{0} ε_{m}}, \frac{β}{ω ε_{0} ε_{m}}] (z \leq z^{'}) \end{matrix} .

\begin{matrix} α^{+} = N^{- 1} \int_{- \infty}^{\infty} [E_{z} (x, z) H_{y}^{-} (x, z) - E_{z}^{-} (x, z) H_{y} (x, z)] d z \\ α^{-} = N^{- 1} \int_{- \infty}^{\infty} [H_{y} (x, z) E_{z}^{+} (x, z) - H_{x}^{+} (x, z) E_{z} (x, z)] d z \end{matrix},

{\hat{J}}_{x y} = [\begin{matrix} a_{x} exp (j δ_{x}) \\ a_{y} exp (j δ_{y}) \end{matrix}] = [\begin{matrix} J_{x} \\ J_{y} \end{matrix}],

{\hat{E}}_{x y} = {[exp (j τ) {\hat{J}}_{x y}]}^{*} = exp (- j τ) [\begin{matrix} a_{x} exp (- j δ_{x}) \\ a_{y} exp (- j δ_{y}) \end{matrix}] = exp (- j τ) [\begin{matrix} J_{x}^{*} \\ J_{y}^{*} \end{matrix}],

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} = \pm J_{b} sin (τ + δ_{0}) .

tan χ = \mp J_{b} / J_{a} .

[\begin{matrix} J_{σ} \\ J_{τ} \end{matrix}] = [\begin{matrix} cos (φ) exp (j Γ / 2) \\ sin (φ) exp (- j Γ / 2) \end{matrix}]

\begin{matrix} tan 2 ψ_{t} = (tan 2 φ) cos (- Γ) \\ sin 2 χ = (sin 2 φ) sin (- Γ) \end{matrix} .