Abstract

Optical forces available on a chip that possess features of strong trapping at the subwavelength scale, in a coplanar geometry, and at specific and selective locations portend many useful applications. We demonstrate here a two-pronged approach to accomplish this. First, the plasmon fields emanating from a subwavelength aperture are manipulated so that they leak maximally to the sides on a surface through the use of tailored corrugations. Second, the location of secondary corrugation at some distance permits reflection of these leaky waves, which, with the coherence property of light used, generate optical standing wave fields capable of strong optical trapping. The linear optical forces generated with this scheme are presented here.

© 2011 Optical Society of America

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2010

2009

2008

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

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. Atom. Mol. Opt. Phys. 56, 261-337 (2008).

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480-484 (2008).
[CrossRef] [PubMed]

T. W. Ng, A. Neild, and P. Heeraman, “Continuous and fast sorting of Brownian particles,” Opt. Lett. 33, 584-586 (2008).
[CrossRef] [PubMed]

Y. Liu, H. Shi, C. Wang, C. Du, and X. Luo, “Multiple directional beaming effect of metallic subwavelength slit surrounded by periodically corrugated grooves,” Opt. Express 16, 4487-4493(2008).
[CrossRef] [PubMed]

D. R. Jackson, J. Chen, R. Qiang, F. Capolino, and A. A. Oliner, “The role of leaky plasmon waves in the directive beaming of light through a subwavelength aperture,” Opt. Express 16, 21271-21281 (2008).
[CrossRef] [PubMed]

Y. Gravel and Y. Sheng, “Rigorous solution for the transient surface plasmon polariton launched by subwavelength slit scattering,” Opt. Express 16, 21903-21913 (2008).
[CrossRef] [PubMed]

2006

T. Cizmar, 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, 035105 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

2004

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Trans. R. Soc. A 362, 719-737 (2004).
[CrossRef]

Z. Zhang and S. C. Glotzer, “Self-assembly of patchy particles,” Nano Lett. 4, 1407-1413 (2004).
[CrossRef]

2003

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

2002

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

P. Zemánek, A. Jonáš, and M. Liška, “Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave,” J. Opt. Soc. Am. A 19, 1025-1034 (2002).
[CrossRef]

2000

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

1999

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1900

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 306, 566-613 (1900).
[CrossRef]

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480-484 (2008).
[CrossRef] [PubMed]

Barnes, W. L.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

Capolino, F.

Chaumet, P. C.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Trans. R. Soc. A 362, 719-737 (2004).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Chen, J.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Cizmar, T.

T. Cizmar, 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, 035105 (2006).
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Devaux, E.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. Atom. Mol. Opt. Phys. 56, 261-337 (2008).

Dholakia , K.

T. Cizmar, 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, 035105 (2006).
[CrossRef]

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Dickinson, M. R.

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

Dintinger, J.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Drude, P.

P. Drude, “Zur elektronentheorie der metalle,” Ann. Phys. 306, 566-613 (1900).
[CrossRef]

Du, C.

Ebbesen, T. W.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Fang, Z.

Z. Fang, F. Lin, S. Huang, W. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94, 063306 (2009).
[CrossRef]

Gao, H.

Garces-Chavez, V.

T. Cizmar, 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, 035105 (2006).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Girard, C.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

Glotzer, S. C.

Z. Zhang and S. C. Glotzer, “Self-assembly of patchy particles,” Nano Lett. 4, 1407-1413 (2004).
[CrossRef]

Gravel, Y.

Gray, S. K.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
[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. Photon. 2, 365-370 (2008).
[CrossRef]

Heeraman, P.

Henzie, J.

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480-484 (2008).
[CrossRef] [PubMed]

Huang, S.

Z. Fang, F. Lin, S. Huang, W. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94, 063306 (2009).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

Jackson, D. R.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

Jonáš, A.

Kawata, S.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

Lee, M. H.

Lee, W. M.

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. Atom. Mol. Opt. Phys. 56, 261-337 (2008).

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Li, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480-484 (2008).
[CrossRef] [PubMed]

Lin, F.

Z. Fang, F. Lin, S. Huang, W. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94, 063306 (2009).
[CrossRef]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

Liška, M.

Liu, Y.

Luo, X.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science, 2007), pp. 42-52.

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-822 (2002).
[CrossRef] [PubMed]

McMahon, J. M.

Miret, J. J.

Murray, W. A.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Neild, A.

Ng, T. W.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Trans. R. Soc. A 362, 719-737 (2004).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Odom, T. W.

Okamoto, K.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83, 4534-4537 (1999).
[CrossRef]

Oliner, A. A.

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

Pernice, W. H. P.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480-484 (2008).
[CrossRef] [PubMed]

Petrov, D.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Qiang, R.

Quidant, R.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

Rahmani, A.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Trans. R. Soc. A 362, 719-737 (2004).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601 (2002).
[CrossRef] [PubMed]

Righini, M.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100, 186804 (2008).
[CrossRef] [PubMed]

Roberts, N. W.

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

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687-702 (2010).
[CrossRef]

Schatz, G. C.

Sery, M.

T. Cizmar, 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, 035105 (2006).
[CrossRef]

Sheng, Y.

Shi, H.

Siler, M.

T. Cizmar, 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, 035105 (2006).
[CrossRef]

Song, W.

Z. Fang, F. Lin, S. Huang, W. Song, and X. Zhu, “Focusing surface plasmon polariton trapping of colloidal particles,” Appl. Phys. Lett. 94, 063306 (2009).
[CrossRef]

Sun, K.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407(2006).
[CrossRef]

Tang, H. X.

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Supplementary Material (8)

» Media 1: MPEG (2408 KB)     
» Media 2: MPEG (2299 KB)     
» Media 3: MPEG (1605 KB)     
» Media 4: MPEG (1471 KB)     
» Media 5: MPEG (2801 KB)     
» Media 6: MPEG (2807 KB)     
» Media 7: MPEG (2830 KB)     
» Media 8: MPEG (2964 KB)     

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Figures (7)

Fig. 1
Fig. 1

(a) Optical scheme of illuminating metallic structures in a whole-field manner to create localized plasmon field for trapping [9, 10, 11]. (b) Envisaged approach of having illumination delivered from individually addressable waveguides within a substrate to subwavelength apertures in the metallic layer to create switchable surface plasmon sites for trapping; which in the process will facilitate many lab-on-a-chip procedures.

Fig. 2
Fig. 2

(a) The real and imaginary components of the dielectric function for silver at the wavelength range of λ = [ 300 nm , 2000 nm ] . (b) The blue curve shows the dispersion curves of modes supported by the silver–air interface. The modes that decay into air are located to the right of the light curve (green curve). The plasmon resonance phenomena is observed around ω 5.0 .

Fig. 3
Fig. 3

(a) Simulation geometry with a lattice constant, groove height, and width of p, h, and w, respectively. The Gaussian source, located to the left, has a center wavelength of λ = 1.5 μm and a pulse width of λ = 1 μm . (b) Simulation geometry with a subwavelength slit and tailored surface corrugations on a silver substrate. Throughout this study the subwavelength aperture width, s, and slab thickness, t, are fixed at 140 and 1000 nm , respectively.

Fig. 4
Fig. 4

Transmission spectra for a grating period of (a) 400, (b) 500, and (c)  600 nm , respectively, at different h and w configurations. (d) Combined transmission spectra outlines the input Gaussian spectra. The highest resonance is observed when h = 150 nm , w = 250 nm , and p = 500 nm .

Fig. 5
Fig. 5

Snapshots of the electric field intensity, E 2 , shown as a contour plot along the metallic structure shown in Fig. 3b for (a)  p = 500 nm , λ = 1380 nm , h = 150 nm , and w = 250 nm (Media 1); (b)  p = 500 nm , λ = 1464 nm , h = 150 nm , and w = 250 nm (Media 2); (c)  p = 600 nm , λ = 1587 nm , h = 150 nm , and w = 200 nm (Media 3); (d)  p = 600 nm , λ = 1715 nm , h = 150 nm , and w = 200 nm (Media 4). Media 1 to Media 4 depict the corresponding animation sequences of (a) to (d).

Fig. 6
Fig. 6

Snapshots of the electric field intensity, E 2 , shown as a contour plot along the metallic structure shown in Fig. 3b where p = 500 nm , λ = 1380 nm , h = 150 nm , w = 250 nm , p = 500 nm , h = 125 nm , w = 300 nm , and (a)  q = 0 (Media 5); (b)  q = 105 nm (Media 6); (c)  q = 155 nm (Media 7); (d)  q = 205 nm (Media 8). Media 5 to Media 8 depict the corresponding animation sequences of (a) to (d).

Fig. 7
Fig. 7

Optical trapping force developed on a spherical polystyrene particle with a radius of 80 nm and refractive index of 2.5 in water inserted into the design producing the electric intensity field of Fig. 6b.

Equations (6)

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k spp 2 = ε 1 ( ω ) ε 2 ( ω ) ε 1 ( ω ) + ε 2 ( ω ) ( ω c ) 2 ,
λ spp Re { ε 1 ( ω ) } + Re { ε 2 ( ω ) } Re { ε 1 ( ω ) } · Re { ε 2 ( ω ) } .
ε Drude ( ω ) = ε ω p 2 ω 2 + Γ 2 + i Γ ω p 2 ω ( ω 2 + Γ 2 ) ,
k spp = k sin θ ± 2 π n p ,
d P mec d t + d P field d t = s T · n d s .
F = d P mec d t = T = = [ ε 0 ε E E μ 0 μ H H 1 2 ( ε 0 ε E 2 + μ 0 μ H 2 ) I ]

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