Abstract

This study proposes a method of dynamically controlling the interference pattern of surface plasmon polaritons (SPPs) within a four-slit structure by changing the phase difference between multiple-incident Gaussian beams. The theoretical analysis of the controlling mechanism of the SPP interference field and the numerical simulation of the generation and movement of both one-dimensional and two-dimensional SPP interference fields are provided. In addition, through simulation, this study demonstrates using the controllable two-dimensional SPP interference bright spots field for manipulating particles in static liquids.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

C. F. Kuo and S. C. Chu, “Excitation of high density surface plasmon polariton vortex array,” J. Opt. 20(6), 065002 (2018).
[Crossref]

2017 (2)

2016 (1)

2015 (2)

2014 (1)

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

2013 (2)

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81(3), 224–230 (2013).
[Crossref]

2012 (1)

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

2011 (3)

Y. Guo, L. Yan, W. Pan, B. Luo, K. Wen, Z. Guo, H. Li, and X. Luo, “A plasmonic splitter based on slot cavity,” Opt. Express 19(15), 13831–13838 (2011).
[Crossref] [PubMed]

J. S. Q. Liu, R. A. Pala, F. Afshinmanesh, W. Cai, and M. L. Brongersma, “A submicron plasmonic dichroic splitter,” Nat. Commun. 2(1), 525 (2011).
[Crossref] [PubMed]

X. W. Li, Q. F. Tan, B. F. Bai, and G. F. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

2010 (1)

2009 (3)

P. Ginzburg, E. Hirshberg, and M. Orenstein, “Rigorous analysis of vectorial plasmonic diffraction: single- and double-slit experiments,” J. Opt. A, Pure Appl. Opt. 11(11), 114024 (2009).
[Crossref]

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

H. Kim and B. Lee, “Unidirectional Surface Plasmon Polariton Excitation on Single Slit with Oblique Backside Illumination,” Plasmonics 4(2), 153–159 (2009).
[Crossref]

2008 (2)

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

S. C. Chu, C. S. Yang, and K. Otsuka, “Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer,” Opt. Express 16(24), 19934–19949 (2008).
[Crossref] [PubMed]

2007 (2)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

2006 (2)

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

J. M. Steele, Z. Liu, Y. Wang, and X. Zhang, “Resonant and non-resonant generation and focusing of surface plasmons with circular gratings,” Opt. Express 14(12), 5664–5670 (2006).
[Crossref] [PubMed]

2005 (5)

S. H. Chang, S. Gray, and G. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13(8), 3150–3165 (2005).
[Crossref] [PubMed]

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005).
[Crossref]

R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005).
[Crossref] [PubMed]

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94(16), 160601 (2005).
[Crossref] [PubMed]

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” Appl. Phys. Lett. 86(3), 031106 (2005).
[Crossref]

2004 (1)

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

2003 (1)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

2002 (1)

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[Crossref] [PubMed]

1996 (2)

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[Crossref] [PubMed]

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

1986 (1)

1985 (1)

E. Dickinson, “Brownian Dynamics with Hydrodynamic Interactions - the Application to Protein Diffusional Problems,” Chem. Soc. Rev. 14(4), 421–455 (1985).
[Crossref]

1980 (1)

1978 (2)

D. L. Ermak and J. A. Mccammon, “Brownian Dynamics with Hydrodynamic Interactions,” J. Chem. Phys. 69(4), 1352–1360 (1978).
[Crossref]

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range −8 °C to 150 °C,” J. Phys. Chem. Ref. Data 7(3), 941–948 (1978).
[Crossref]

1970 (1)

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[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(1), 525 (2011).
[Crossref] [PubMed]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

Ashkin, A.

Babar, S.

Badenes, G.

Bai, B. F.

X. W. Li, Q. F. Tan, B. F. Bai, and G. F. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

Bielefeldt, H.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[Crossref] [PubMed]

Bjorkholm, J. E.

Bozhevolnyi, S. I.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Brínek, L.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

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(1), 525 (2011).
[Crossref] [PubMed]

Bu, J.

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(1), 525 (2011).
[Crossref] [PubMed]

Chang, S. H.

Chen, H.

Chen, J. J.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Chon, J. W. M.

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

Chu, S.

Chu, S. C.

Cižmár, T.

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

Curry, J. J.

Dereux, A.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Devaux, E.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Dholakia, K.

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” Appl. Phys. Lett. 86(3), 031106 (2005).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Di Falco, A.

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

Dickinson, E.

E. Dickinson, “Brownian Dynamics with Hydrodynamic Interactions - the Application to Protein Diffusional Problems,” Chem. Soc. Rev. 14(4), 421–455 (1985).
[Crossref]

Dickinson, M. R.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Dorn, R.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005).
[Crossref]

Du, L.

Dub, P.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Dvorák, P.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Dziedzic, J. M.

Ebbesen, T. W.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Eftekhari, F.

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

Ermak, D. L.

D. L. Ermak and J. A. Mccammon, “Brownian Dynamics with Hydrodynamic Interactions,” J. Chem. Phys. 69(4), 1352–1360 (1978).
[Crossref]

Gan, X. S.

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

Garces-Chavez, V.

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

Garcés-Chávez, V.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” Appl. Phys. Lett. 86(3), 031106 (2005).
[Crossref]

Garcia-Vidal, F. J.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Ginzburg, P.

P. Ginzburg, E. Hirshberg, and M. Orenstein, “Rigorous analysis of vectorial plasmonic diffraction: single- and double-slit experiments,” J. Opt. A, Pure Appl. Opt. 11(11), 114024 (2009).
[Crossref]

Girard, C.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Gong, Q. H.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Gonzalez, M. U.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Gordon, R.

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

Gray, S.

Grier, D. G.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[Crossref] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Gu, M.

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

Guo, Y.

Guo, Z.

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

Haumonte, J. B.

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

Hecht, B.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[Crossref] [PubMed]

Hesselink, L.

M. A. Zaman, P. Padhy, and L. Hesselink, “Capturing range of a near-field optical trap,” Phys. Rev. A 96(4), 043825 (2017).
[Crossref]

Hirshberg, E.

P. Ginzburg, E. Hirshberg, and M. Orenstein, “Rigorous analysis of vectorial plasmonic diffraction: single- and double-slit experiments,” J. Opt. A, Pure Appl. Opt. 11(11), 114024 (2009).
[Crossref]

Inouye, Y.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[Crossref] [PubMed]

Jin, G. F.

X. W. Li, Q. F. Tan, B. F. Bai, and G. F. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

Juan, M. L.

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

Kalousek, R.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Kestin, J.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range −8 °C to 150 °C,” J. Phys. Chem. Ref. Data 7(3), 941–948 (1978).
[Crossref]

Kim, H.

H. Kim and B. Lee, “Unidirectional Surface Plasmon Polariton Excitation on Single Slit with Oblique Backside Illumination,” Plasmonics 4(2), 153–159 (2009).
[Crossref]

Korda, P. T.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[Crossref] [PubMed]

Krenn, J. R.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Kuo, C. F.

Lacasta, A. M.

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94(16), 160601 (2005).
[Crossref] [PubMed]

Lee, B.

H. Kim and B. Lee, “Unidirectional Surface Plasmon Polariton Excitation on Single Slit with Oblique Backside Illumination,” Plasmonics 4(2), 153–159 (2009).
[Crossref]

Lei, T.

Leuchs, G.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005).
[Crossref]

Levine, Z. H.

Li, H.

Li, X. W.

X. W. Li, Q. F. Tan, B. F. Bai, and G. F. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

Li, Z.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Liao, H. M.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Lindenberg, K.

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94(16), 160601 (2005).
[Crossref] [PubMed]

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(1), 525 (2011).
[Crossref] [PubMed]

Liu, Z.

López-Tejeira, F.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Luo, B.

Luo, X.

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Martin-Moreno, L.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Mazilu, M.

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

Mccammon, J. A.

D. L. Ermak and J. A. Mccammon, “Brownian Dynamics with Hydrodynamic Interactions,” J. Chem. Phys. 69(4), 1352–1360 (1978).
[Crossref]

Micheau, Y.

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

Neuman, T.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Novotny, L.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[Crossref] [PubMed]

Orenstein, M.

P. Ginzburg, E. Hirshberg, and M. Orenstein, “Rigorous analysis of vectorial plasmonic diffraction: single- and double-slit experiments,” J. Opt. A, Pure Appl. Opt. 11(11), 114024 (2009).
[Crossref]

Otsuka, K.

Padhy, P.

M. A. Zaman, P. Padhy, and L. Hesselink, “Capturing range of a near-field optical trap,” Phys. Rev. A 96(4), 043825 (2017).
[Crossref]

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(1), 525 (2011).
[Crossref] [PubMed]

Pan, W.

Pang, Y. J.

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

Petrov, D.

Ploschner, M.

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

Pohl, D. W.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77(9), 1889–1892 (1996).
[Crossref] [PubMed]

Quabis, S.

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005).
[Crossref]

Quidant, R.

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005).
[Crossref] [PubMed]

Radko, I. P.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Righini, M.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Roberts, N. W.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Rodrigo, S. G.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Romero, A. H.

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94(16), 160601 (2005).
[Crossref] [PubMed]

Šamoril, T.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Sancho, J. M.

A. M. Lacasta, J. M. Sancho, A. H. Romero, and K. Lindenberg, “Sorting on periodic surfaces,” Phys. Rev. Lett. 94(16), 160601 (2005).
[Crossref] [PubMed]

Schatz, G.

Sery, M.

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

Šikola, T.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Siler, M.

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

Sokolov, M.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range −8 °C to 150 °C,” J. Phys. Chem. Ref. Data 7(3), 941–948 (1978).
[Crossref]

Spalding, G. C.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” Appl. Phys. Lett. 86(3), 031106 (2005).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Steele, J. M.

Sun, C. W.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Tan, Q. F.

X. W. Li, Q. F. Tan, B. F. Bai, and G. F. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

Taylor, M. B.

P. T. Korda, M. B. Taylor, and D. G. Grier, “Kinetically locked-in colloidal transport in an array of optical tweezers,” Phys. Rev. Lett. 89(12), 128301 (2002).
[Crossref] [PubMed]

Varga, P.

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[Crossref] [PubMed]

Volpe, G.

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81(3), 224–230 (2013).
[Crossref]

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81(3), 224–230 (2013).
[Crossref]

Wakeham, W. A.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range −8 °C to 150 °C,” J. Phys. Chem. Ref. Data 7(3), 941–948 (1978).
[Crossref]

Wang, H. M.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Wang, Q.

Wang, Y.

Weaver, J. H.

Weeber, J. C.

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Wei, S.

Wen, K.

Yan, L.

Yang, C. S.

Yang, Y.

Yuan, X. C.

Zaman, M. A.

M. A. Zaman, P. Padhy, and L. Hesselink, “Capturing range of a near-field optical trap,” Phys. Rev. A 96(4), 043825 (2017).
[Crossref]

Zelenina, A. S.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

Zemanek, P.

T. Čižmár, M. Siler, M. Sery, P. Zemanek, V. Garces-Chavez, and K. Dholakia, “Optical sorting and detection of submicrometer objects in a motional standing wave,” Phys. Rev. B 74(3), 035105 (2006).
[Crossref]

Zhang, C.

Zhang, X.

Zhang, Y.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Zhang, Y. F.

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

Zhu, S. W.

Am. J. Phys. (1)

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81(3), 224–230 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

S. Quabis, R. Dorn, and G. Leuchs, “Generation of a radially polarized doughnut mode of high quality,” Appl. Phys. B 81(5), 597–600 (2005).
[Crossref]

Appl. Phys. Lett. (4)

M. Gu, J. B. Haumonte, Y. Micheau, J. W. M. Chon, and X. S. Gan, “Laser trapping and manipulation under focused evanescent wave illumination,” Appl. Phys. Lett. 84(21), 4236–4238 (2004).
[Crossref]

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticies on a surface,” Appl. Phys. Lett. 86(3), 031106 (2005).
[Crossref]

Y. F. Zhang, H. M. Wang, H. M. Liao, Z. Li, C. W. Sun, J. J. Chen, and Q. H. Gong, “Unidirectional launching of surface plasmons at the subwavelength scale,” Appl. Phys. Lett. 105(23), 231101 (2014).
[Crossref]

X. W. Li, Q. F. Tan, B. F. Bai, and G. F. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).
[Crossref]

Chem. Soc. Rev. (1)

E. Dickinson, “Brownian Dynamics with Hydrodynamic Interactions - the Application to Protein Diffusional Problems,” Chem. Soc. Rev. 14(4), 421–455 (1985).
[Crossref]

J. Chem. Phys. (1)

D. L. Ermak and J. A. Mccammon, “Brownian Dynamics with Hydrodynamic Interactions,” J. Chem. Phys. 69(4), 1352–1360 (1978).
[Crossref]

J. Opt. (1)

C. F. Kuo and S. C. Chu, “Excitation of high density surface plasmon polariton vortex array,” J. Opt. 20(6), 065002 (2018).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

P. Ginzburg, E. Hirshberg, and M. Orenstein, “Rigorous analysis of vectorial plasmonic diffraction: single- and double-slit experiments,” J. Opt. A, Pure Appl. Opt. 11(11), 114024 (2009).
[Crossref]

J. Phys. Chem. Ref. Data (1)

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range −8 °C to 150 °C,” J. Phys. Chem. Ref. Data 7(3), 941–948 (1978).
[Crossref]

Nano Lett. (2)

M. Ploschner, T. Čižmár, M. Mazilu, A. Di Falco, and K. Dholakia, “Bidirectional Optical Sorting of Gold Nanoparticles,” Nano Lett. 12(4), 1923–1927 (2012).
[Crossref] [PubMed]

P. Dvořák, T. Neuman, L. Břínek, T. Šamořil, R. Kalousek, P. Dub, P. Varga, and T. Šikola, “Control and Near-Field Detection of Surface Plasmon Interference Patterns,” Nano Lett. 13(6), 2558–2563 (2013).
[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(1), 525 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[Crossref]

Nat. Phys. (3)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nat. Phys. 3(7), 477–480 (2007).
[Crossref]

M. L. Juan, R. Gordon, Y. J. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[Crossref]

F. López-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007).
[Crossref]

Nature (1)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[Crossref]

Opt. Express (8)

J. J. Curry and Z. H. Levine, “Continuous-feed optical sorting of aerosol particles,” Opt. Express 24(13), 14100–14123 (2016).
[Crossref] [PubMed]

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Four-slit structure and axis diagrams: (A) Top view of the four-slit structure and definition of the coordinate axes, (B) Side view of the four-slit structure and the relationship between the incident light and structure.
Fig. 2
Fig. 2 The SPP interference field between the four metal slits when four ideal SPP line sources are in different phase relationships: (A) SPP interference bright spot array; (B) - (D) several transitional SPP interference fields between the SPP bright spot array and the SPP dark spot array; (E) SPP interference dark spot array. The (ϕ1, ϕ2, ϕ3, ϕ4) values of Figs. (A)-(E) (0, 0, 0, 0), (π/8, 0, π/8, 0), (2π/8, 0, 2π/8, 0), (3π/8, 0, 3π/8, 0) and (4π/8, 0, 4π/8, 0), respectively.
Fig. 3
Fig. 3 (A) the schematic diagram of the relationship between the two incident Gaussian beams; (B)-(F) SPP interference fields between the slits when the phases of two incident Gaussian beams are as the values listed in the columns (B)-(F) of Table 1, respectively. The translucent white box in Fig. (A) shows the four metal slits, and the white double arrows indicate the polarization direction of the incident light. The range of the displayed SPP field in Figs. (B)-(F) is the area within the green box in Fig. (A). The Δϕy values in the upper right corner of the five pictures shows the difference of the phase constant of the two SPP line sources, ϕ2 - ϕ4, in Eq. (7). The white dashed line shows the line y = 0 in the defined coordinate of Fig. 1(A), which is plotted for convenient observing of the movement of the SPP interference field.
Fig. 4
Fig. 4 (A) The schematic diagram of the relationship between the four incident Gaussian beams; (B)-(F) SPP interference fields between the slits when the phases of four incident Gaussian beams are as the values listed in the columns (B)-(F) in Table 2, respectively. The translucent white box in Fig. (A) shows the four metal slits. The range of the displayed SPP field in Figs. (B)-(F) is the area within the green box in Fig. (A). The Δϕx and Δϕy values in the upper right corner of the five pictures show the difference of the phase constant between two of the four SPP line sources in Eq. (5), i.e., ϕ1 - ϕ3, and ϕ2 - ϕ4, respectively. The two white dashed lines show the lines y = 0 and x = 0 in the defined coordinate of Fig. 1(A), which are plotted for convenient observing of the movement of the SPP interference bright spot array.
Fig. 5
Fig. 5 Simulated trajectories of a single particle in the SPP interference bright spot array as the incident light field phase changes over time. The white circle indicates the particle and its position, and the blue line in the picture indicates the experienced trajectory of the particle at the time. Figures (A)-(H) plot the particle experienced trajectories at different times and the SPP light field distribution at different time. The time values are marked in the lower right corner of each picture. They are 0.0, 13.0, 26.0, 39, 52.0, 65.0, 78.0, 92.0 ms, respectively.
Fig. 6
Fig. 6 Simulated trajectories of two particles in the SPP interference bright spot array as the incident light field phase changes over time. The white circle indicates two particles and their positions. The blue and orange line in the picture indicates the experienced trajectories of two particles at the time. Figures (A)-(H) plot the particle experienced trajectories at different time and the SPP light field distribution at different times. The time values are marked in the lower right corner of each picture. They are 0.0, 13.0, 26.0, 39.0, 52.0, 65.0, 78.0, 92.0 ms, respectively.
Fig. 7
Fig. 7 The intensity distribution of a fundamental Gaussian beam and a bright spot in the SPP interference field.

Tables (2)

Tables Icon

Table 1 Phase values of two incident Gaussian beams used in a simulation calculation when two Gaussian beams are used to generate SPP interference fringes between slits. The values listed in columns (B)-(F) of the table correspond to the phase values of two Gaussian beams used in the simulation results in Figs. 3(B)-3(F), respectively.

Tables Icon

Table 2 Phase values of four incident Gaussian beams used in a simulation calculation when four Gaussian beams are used to generate an SPP interference bright spot array between slits. The values listed in columns (B)-(F) of the table correspond to the phase values of two Gaussian beams used in the simulation results in Figs. 4(B)-4(F), respectively.

Equations (9)

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E z spp ( r;t ) E s exp[ i( k spp rωt )+iϕ ]exp( r 2 L spp ),
k spp = ω c ( ε 1 ε 2 ε 1 + ε 2 ) 1 2 ,
E quad (x,y,z;t)=[ E(xD/2 ,y,z;t) e i ϕ 1 ' +E(x+D/2 ,y,z;t) e i ϕ 3 ' ] x ^ +[ E(x,yD/2 ,z;t) e i ϕ 2 ' +E(x,y+D/2 ,z;t) e i ϕ 4 ' ] y ^ ,
E z ( x,y;t ) E s exp(iωt){ exp[ i k spp ( d 2 /2 +x )+i ϕ 1 ]+exp[ i k spp ( d 2 /2 x )+i ϕ 3 ] +exp[ i k spp ( d 1 /2 +y )+i ϕ 2 ]+exp[ i k spp ( d 1 /2 y )+i ϕ 4 ] }.
E z ( x,y;t ) E 0 [ cos( k spp y+ ϕ 2 ϕ 4 2 ) +exp[ i ( ϕ 1 + ϕ 3 )( ϕ 2 + ϕ 4 ) 2 ]cos( k spp x+ ϕ 1 ϕ 3 2 ) ],
E dual (x,y,z;t)=[ E(x,yD/2 ,z;t) e i ϕ 2 ' +E(x,y+D/2 ,z;t) e i ϕ 4 ' ] y ^ .
E z ( x,y;t ) E 0 [ cos( k spp y+ ϕ 2 ϕ 4 2 ) ],
r( t )=r( tΔt )+ DΔt k B T F( t )+ 2DΔt w( t ).
F grad ( r )= 2π n 2 R 3 c ( m 2 1 m 2 +2 )I( r ).

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