Abstract

We report a plasmon steering method that enables us to dynamically control the direction of surface plasmons generated by a two-mode slit in a thin metal film. By varying the phase between different coherent beams that are incident on the slit, individual waveguide modes are excited. Different linear combinations of the two modes lead to different diffracted fields at the exit of the slit. As a result, the direction in which surface plasmons are launched can be controlled. Experiments confirm that it is possible to distribute an approximately constant surface plasmon intensity in any desired proportion over the two launching directions. We also find that the anti-symmetric mode generates surface plasmons more efficiently than the fundamental symmetric mode.

© 2012 OSA

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2012 (1)

2011 (3)

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98, 251109 (2011).
[CrossRef]

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

2010 (1)

2009 (7)

Y. Wang, X. Zhang, H. Tang, K. Yang, Y. Wang, Y. Song, T. Wei, and C. H. Wang, “A tunable unidirectional surface plasmon polaritons source,” Opt. Express 17, 20457–20464 (2009).
[CrossRef] [PubMed]

Q. Gan, Y. Gao, and F. J. Bartoli, “Vertical plasmonic Mach-Zehnder interferometer for sensitive optical sensing,” Opt. Express 17, 20747–20755 (2009).
[CrossRef] [PubMed]

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[CrossRef]

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

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

2008 (1)

H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

2007 (4)

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

H. A. Atwater, “The promise of plasmonics,” Scientific American 296, 56–62 (2007.
[CrossRef] [PubMed]

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

2006 (1)

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2, 551–556, (2006).
[CrossRef]

2004 (1)

2003 (2)

F.J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

2001 (1)

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Afshinmanesh, F.

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

Atwater, H. A.

H. A. Atwater, “The promise of plasmonics,” Scientific American 296, 56–62 (2007.
[CrossRef] [PubMed]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Aussenegg, F. R.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

Aydin, K.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Bai, B.

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98, 251109 (2011).
[CrossRef]

Baida, F. I.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Bakker, R.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Baron, A.

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

Bartoli, F. J.

Belgrave, A.M.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Besbes, M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Bienstman, P.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Bilotti, F.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brongersma, M. L.

Brongersma, M.L.

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

Cai, W.

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

Cakmak, A. O.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Catrysse, P. B.

Degiron, A.

Dereux, A.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Devaux, E.

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

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Ditlbacher, H.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

Ebbesen, T.

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

Ebbesen, T. W.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12, 3694–3700 (2004).
[CrossRef] [PubMed]

F.J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Fan, S.

Gan, Q.

Gao, Y.

García-Vidal, F. J.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

García-Vidal, F.J.

F.J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

Genet, C.

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

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

González, M. U.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Granet, G.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Guizal, B.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Helfert, S.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Herz, E.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Hohenau, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hugonin, J. P.

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

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

H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2, 551–556, (2006).
[CrossRef]

Janssen, O. T. A.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Jin, G.

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98, 251109 (2011).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Krenn, J. R.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

Lalanne, P.

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

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

H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2, 551–556, (2006).
[CrossRef]

Leitner, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

Lezec, H. J.

F.J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Li, X.

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98, 251109 (2011).
[CrossRef]

Li, Z.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Linke, R. A.

Liu, H.

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

H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

Liu, J.S.Q.

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

López-Tejeira, F.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

MacDonald, K. F.

K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Martín-Moreno, L.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

F.J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. 90, 213901 (2003).
[CrossRef] [PubMed]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Miyata, M.

Moreau, A.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Narimanov, E.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Noginov, M.A.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Nugrowati, A. M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Ozbay, E.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Pala, R.A.

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

Pellerin, K. M.

Pereira, S. F.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Radko, I. P.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

Requicha, A. A. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nature Materials 2, 229–232 (2003).
[CrossRef] [PubMed]

Rodier, J. C.

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

Rodrigo, S. G.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Rousseau, E.

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

Sahin, L.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Sámson, Z. L.

K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Schuller, J. A.

Seideman, T.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Shalaev, V.M.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Song, Y.

Steinberger, B.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

Stockman, M. I.

K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Stout, S.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Sukharev, M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Suteewong, T.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Takahara, J.

Tan, Q.

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98, 251109 (2011).
[CrossRef]

Tang, H.

Thio, T.

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Urbach, H.P.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

van de Nes, A. S.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

van Haver, S.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Van Labeke, D.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

Vegni, L.

K. Aydin, A. O. Cakmak, L. Sahin, Z. Li, F. Bilotti, L. Vegni, and E. Ozbay, “Split-ring-resonator-coupled enhanced transmission through a single subwavelength aperture,” Phys. Rev. Lett. 102, 013904 (2009).
[CrossRef] [PubMed]

Verslegers, L.

Wang, B.

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

Wang, C. H.

Wang, Y.

Weeber, J. C.

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

Wei, T.

Wiesner, U.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

Xu, M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Yang, K.

Yang, X.

H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

Yu, Z.

Zhang, X.

Zheludev, N. I.

K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Zhu, G.

M.A. Noginov, G. Zhu, A.M. Belgrave, R. Bakker, V.M. Shalaev, E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature (London) 460, 1110–1113 (2009).
[CrossRef]

Appl. Phys. Lett. (2)

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91, 081111 (2007).
[CrossRef]

X. Li, Q. Tan, B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98, 251109 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H. Liu, P. Lalanne, X. Yang, and J. P. Hugonin “Surface plasmon generation by subwavelength isolated objects,” IEEE J. Sel. Top. Quantum Electron. 14, 1522–1529 (2008).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H.P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, “Numerical analysis of a slit-groove diffraction problem,” J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

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

Nat. Commun. (1)

J.S.Q. Liu, R.A. Pala, F. Afshinmanesh, W. Cai, and M.L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2:525 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

K. F. MacDonald, Z. L. Sámson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3, 55–58 (2009).
[CrossRef]

Nat. Phys. (2)

F. López-Tejeira, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, M. U. González, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
[CrossRef]

P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nat. Phys. 2, 551–556, (2006).
[CrossRef]

Nature (London) (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Principle of the proposed surface plasmon steering method. (a) A subwavelength slit of width w in a gold film supports only two TM modes for λ/2 ≲ wλ: a symmetric mode (s, green curve) and an anti-symmetric mode (a, blue curve). Three coherent beams, A, −A (with opposite angle of incidence compared to A and π-phase shifted), and B are incident on the slit from the glass substrate. At the dark side of the film, SPs are launched to the left and to the right with amplitude β(l) and β(r), respectively. A series of grooves at 8 μm from either side of the slit converts the SPs back to freely propagating fields that are detected in the far field. (b) Illustrating how a coherent superposition of the a and s modes can lead to unidirectional SP launching at a gold-air interface. The first two panels show the intensity of the magnetic field when the slit is illuminated with either the s or the a mode. Superposed dotted blue curves show the total magnetic field scattered on the interface. The length of the white bar in the first panel indicates the illumination wavelength in vacuum (λ= 600 nm), and the slit width w is λ/2.

Fig. 2
Fig. 2

The calculated SP cross sections σa and σs as defined in Eqs. (1) and (2) for a unit Poynting vector of each of the incident beams. (a) Variation of σa and σs with slit width w. The angles of incidence of the plane waves A, −A are taken to be θ = ±30°. The two insets indicate the setup for calculating σa and σs, respectively. (b) Variation of σa with angle of incidence θ for slit widths w = 300, 320, 350, and 450 nm. The refractive index of gold nAu = 0.18 + i2.99 for λ = 632.8 nm, is taken from Ref. [17], and the thickness of the gold film is 200 nm.

Fig. 3
Fig. 3

Sketch of the experimental setup. The sample is illuminated from the glass-substrate side.

Fig. 4
Fig. 4

Typical line trace of the CCD camera screen, perpendicular to the slits. The first low peak on the left (near pixel 70) is the signal from the indicator slit. The second and fourth peak are the intensities I(l)(δ) and I(r)(δ) from surface plasmons scattered by the left-hand grooves and right-hand grooves, respectively. The highest peak is the intensity transmitted by the central slit. The insets show sample details of the the central slit and the plasmon grooves made by a scanning electron microscope.

Fig. 5
Fig. 5

Experimental results of the proposed plasmon switching method. The SP intensities I(l)(δ) (red curve) and I(r)(δ) (blue curve) are shown as a function of the phase δ of arm B or, equivalently, as a function of the voltage across the piezo element. The total intensity I(l)(δ) + I(r)(δ) is shown as a dotted grey curve. The error bars indicate the standard deviation of ten independent measurements.

Equations (6)

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σ s = 2 | B s | 2 ,
σ a = 2 | A a | 2 ,
β ( l ) ( δ ) = B s e i δ + A a ,
β ( r ) ( δ ) = B s e i δ A a ,
I ( l ) ( δ ) = | β ( l ) ( δ ) | 2 = 2 | B s | 2 ( 1 + cos δ ) ,
I ( r ) ( δ ) = | β ( r ) ( δ ) | 2 = 2 | B s | 2 ( 1 cos δ ) .

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