Abstract

Mie theory is one of the main tools describing scattering of propagating electromagnetic waves by spherical particles. Evanescent optical fields are also scattered by particles and exert radiation forces which can be used for optical near-field manipulations. We show that the Mie theory can be naturally adopted for the scattering of evanescent waves via rotation of its standard solutions by a complex angle. This offers a simple and powerful tool for calculations of the scattered fields and radiation forces. Comparison with other, more cumbersome, approaches shows perfect agreement, thereby validating our theory. As examples of its application, we calculate angular distributions of the scattered far-field irradiance and radiation forces acting on dielectric and conducting particles immersed in an evanescent field.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]

2012 (2)

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012).
[CrossRef]

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[CrossRef]

2011 (1)

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[CrossRef]

2010 (2)

2008 (1)

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).
[CrossRef]

2007 (1)

2006 (2)

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[CrossRef] [PubMed]

M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006).
[CrossRef]

2005 (1)

H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005).
[CrossRef]

2004 (1)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004).
[CrossRef] [PubMed]

2003 (2)

2001 (1)

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001).
[CrossRef]

2000 (3)

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000).
[CrossRef]

K. Sasaki, J.-I. Hotta, K.-I. Wada, and H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25(18), 1385–1387 (2000).
[CrossRef] [PubMed]

L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000).
[CrossRef]

1999 (2)

J. Y. Walz, “Ray optics calculation of the radiation forces exerted on a dielectric sphere in an evanescent field,” Appl. Opt. 38(25), 5319–5330 (1999).
[CrossRef] [PubMed]

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999).
[CrossRef]

1998 (1)

M. Vilfan, I. Muševi?, and M. ?opi?, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43(1), 41–46 (1998).
[CrossRef]

1997 (1)

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

1996 (1)

1995 (2)

E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12(12), 2429–2438 (1995).
[CrossRef]

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995).
[CrossRef]

1993 (1)

1992 (1)

1979 (2)

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52(3), 133–201 (1979).
[CrossRef]

H. Chew, D.-S. Wang, and M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18(15), 2679–2687 (1979).
[CrossRef] [PubMed]

1972 (1)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields 5(4), 787–796 (1972).
[CrossRef]

1955 (1)

F. I. Fedorov, “To the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955) (reprinted in J. Opt. 15, 014002 (2013)).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Almaas, E.

Angelsky, O. V.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[CrossRef]

Arias-González, J. R.

Badenes, G.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[CrossRef] [PubMed]

Bekshaev, A.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[CrossRef]

Bekshaev, A. Y.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[CrossRef]

Bliokh, K.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012).
[CrossRef]

Brevik, I.

Chang, S.

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001).
[CrossRef]

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Chaumet, P. C.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004).
[CrossRef] [PubMed]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000).
[CrossRef]

Chew, H.

Cižmár, T.

M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006).
[CrossRef]

Copic, M.

M. Vilfan, I. Muševi?, and M. ?opi?, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43(1), 41–46 (1998).
[CrossRef]

Derouard, J.

Dholakia, K.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).
[CrossRef]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).
[CrossRef]

Fedeli, J. M.

Fedorov, F. I.

F. I. Fedorov, “To the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955) (reprinted in J. Opt. 15, 014002 (2013)).

Gaugiran, S.

Gétin, S.

Han, B. M.

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001).
[CrossRef]

Hanna, S.

Hanson, S. G.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[CrossRef]

Hellesø, O. G.

H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005).
[CrossRef]

Hotta, J.-I.

Imbert, C.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D Part. Fields 5(4), 787–796 (1972).
[CrossRef]

Jaising, H. Y.

H. Y. Jaising and O. G. Hellesø, “Radiation forces on a Mie particle in the evanescent field of an optical waveguide,” Opt. Commun. 246(4-6), 373–383 (2005).
[CrossRef]

Jo, J. H.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Kaiser, T.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995).
[CrossRef]

Kawata, S.

Kerker, M.

Kim, J. T.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Lange, S.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995).
[CrossRef]

Lee, S. S.

S. Chang, J. T. Kim, J. H. Jo, and S. S. Lee, “Optical force on a sphere caused by the evanescent field of a Gaussian beam; effects of multiple scattering,” Opt. Commun. 139(4-6), 252–261 (1997).
[CrossRef]

Liu, C.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995).
[CrossRef]

Luff, B. J.

L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000).
[CrossRef]

Masuhara, H.

Mazilu, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Muševic, I.

M. Vilfan, I. Muševi?, and M. ?opi?, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43(1), 41–46 (1998).
[CrossRef]

Ng, L. N.

L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas and J. J. Saenz, “Optical forces from an evanescent wave on a magnetodielectric small particle,” Opt. Lett. 35(23), 4078–4080 (2010).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004).
[CrossRef] [PubMed]

J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20(7), 1201–1209 (2003).
[CrossRef] [PubMed]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62(16), 11185–11191 (2000).
[CrossRef]

Nori, F.

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012).
[CrossRef]

Pack, A.

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999).
[CrossRef]

Petrov, D.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[CrossRef] [PubMed]

Prieve, D. C.

Quidant, R.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[CrossRef] [PubMed]

Quinten, M.

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999).
[CrossRef]

Rahmani, A.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Philos. Transact. A Math. Phys. Eng. Sci. 362(1817), 719–737 (2004).
[CrossRef] [PubMed]

Saenz, J. J.

Sasaki, K.

Schweiger, G.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117(5-6), 521–531 (1995).
[CrossRef]

Šerý, M.

M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006).
[CrossRef]

Šiler, M.

M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006).
[CrossRef]

Simpson, S. H.

Sivertsen, T. A.

Song, Y. G.

Y. G. Song, B. M. Han, and S. Chang, “Force of surface plasmon-coupled evanescent fields on Mie particles,” Opt. Commun. 198(1-3), 7–19 (2001).
[CrossRef]

Soskin, M.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[CrossRef]

Sugiura, T.

Tani, T.

Vilfan, M.

M. Vilfan, I. Muševi?, and M. ?opi?, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43(1), 41–46 (1998).
[CrossRef]

Volpe, G.

G. Volpe, R. Quidant, G. Badenes, and D. Petrov, “Surface plasmon radiation forces,” Phys. Rev. Lett. 96(23), 238101 (2006).
[CrossRef] [PubMed]

Wada, K.-I.

Walz, J. Y.

Wang, D.-S.

Wannemacher, R.

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999).
[CrossRef]

Wilkinson, J. S.

L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000).
[CrossRef]

Zemánek, P.

M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006).
[CrossRef]

Zenkova, C. Y.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[CrossRef]

Zervas, M. N.

L. N. Ng, M. N. Zervas, J. S. Wilkinson, and B. J. Luff, “Manipulation of colloidal gold nanoparticles in the evanescent field of a channel waveguide,” Appl. Phys. Lett. 76(15), 1993–1995 (2000).
[CrossRef]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Appl. Opt. (3)

Appl. Phys. B (2)

M. Šiler, T. ?ižmár, M. Šerý, and P. Zemánek, “Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,” Appl. Phys. B 84(1-2), 157–165 (2006).
[CrossRef]

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68(1), 87–92 (1999).
[CrossRef]

Appl. Phys. Lett. (1)

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Figures (5)

Fig. 1
Fig. 1

Schematic of the Mie scattering problem. Incident wave (blue), scattered field (green), and radiation force exerted on the particle (yellow) are shown. (a) standard Mie theory with the incident plane wave propagating along the z -axis. (b) Rotation of the field, Eqs. (2) and (5), by the complex angle γ=π/2 iα results in the modified Mie problem with evanescent incident wave Eq. (6). The parameter h indicates the distance to the surface where the evanescent wave is generated.

Fig. 2
Fig. 2

Angular diagrams for the scattering far-field irradiance I s ( θ,ϕ )=Re [ E s* ( θ,ϕ )× H s ( θ,ϕ ) ] r at rmax( a,λ ) for the s-polarized incident field, dielectric particle with m= n p /n=1.75 , and for different particle sizes. (a) Standard Mie scattering with incident plane wave (1) and (b) complex-angle Mie scattering for the incident evanescent wave Eq. (7) with sinhα=0.92 . Red, green, and blue curves in the polar plots represent cross-sections of the 3D scattering diagrams by azimuthal planes with ϕ=0 ( x,z ) , ϕ=π/4 , and ϕ=π/2 ( x,y ) , respectively. Diagrams in the (a) and (b) panels are related to each other via the complex-angle rotation (13).

Fig. 3
Fig. 3

Same as in Fig. 2, but for the p-polarized incident waves.

Fig. 4
Fig. 4

Dimensionless radiation force components F x,z / P 0 versus the particle-size parameter ka for a dielectric particle lying on a total-reflecting surface. The parameters of the particle and incident field are given by Eqs. (7), (10), (11), and (17). The cases of the s-polarization (solid lines) and p-polarization (dashed lines) are shown. The results of our calculations based on the complex-angle Mie theory Eqs. (12)(16) (red curves) are superimposed over the data taken from Figs. 8 and 9 of Ref [13]. (black curves).

Fig. 5
Fig. 5

Dimensionless radiation force components F x (black curves) and F z (red curves) versus the particle size parameter ka for the s-polarized (solid curves) and p-polarized (dashed curves) incident wave. The parameters are the same as in Eq. (17) but with (a) n=1.33 , n p =0.43+3.52i (gold particle in water at λ=650 nm [3]) and (b) n=1 , n p =i (“perfect metal” particle).

Tables (1)

Tables Icon

Table 1 Comparison of radiation forces for a particle with the parameters (21) and different permittivities ε p , s-polarized incident wave, calculated using: (i) the dipole approximation [23], (19) and (20), and (ii) the exact complex-angle Mie theory (10)–(16).

Equations (27)

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E=( E E 0 )exp( ikz ), H=( H H 0 )exp( ikz )= ε μ ( E E 0 )exp( ikz ).
E( r ) R ^ y ( γ )E[ R ^ y ( γ )r ], H( r ) R ^ y ( γ )H[ R ^ y ( γ )r ],
R ^ y ( γ )=( cosγ 0 sinγ 0 1 0 sinγ 0 cosγ )
E=( E cosγ E E sinγ )exp[ ik( zcosγ+xsinγ ) ], H= ε μ ( E cosγ E E sinγ )exp[ ik( zcosγ+xsinγ ) ].
γ= π 2 iα, α>0.
R ^ y ( π 2 +iα )=( isinhα 0 coshα 0 1 0 coshα 0 isinhα ),
E=( i E sinhα E E coshα )exp( ikxcoshαkzsinhα ), H= ε μ ( i E sinhα E E coshα )exp( ikxcoshαkzsinhα ),
w= g 2 ( ε | E | 2 +μ | H | 2 ), p= g c Re( E * ×H ),
w=εg cosh 2 α( | E | 2 + | E | 2 )exp( 2kzsinhα ), p x = gn cμ coshα( | E | 2 + | E | 2 )exp( 2kzsinhα ), p z =0, p y =2 gn cμ sinhαcoshαIm( E * E )exp( 2kzsinhα ).
coshα= n 1 n sin θ 1 , sinhα= ( n 1 n ) 2 sin 2 θ 1 1 ,
E = 2 n n 1 cos θ 1 ε ε 1 cos θ 1 +i n n 1 sinhα e khsinhα E 0 , E = 2 μ μ 1 cos θ 1 μ μ 1 cos θ 1 +i n n 1 sinhα e khsinhα E 0 .
E s ( r )= S ^ E ( r )E, H s ( r )= S ^ H ( r )E,
E s ( r ) R ^ y ( π 2 +iα ) E s [ R ^ y ( π 2 iα )r ], H s ( r ) R ^ y ( π 2 +iα ) H s [ R ^ y ( π 2 iα )r ].
E tot =E+ E s , H tot =H+ H s .
T ij =gRe[ ε E i tot* E j tot +μ H i tot* H j tot 1 2 δ ij ( ε | E tot | 2 +μ | H tot | 2 ) ].
F= A T ^ ndA = R 2 S T ^ ndΩ ,
μ 1 =μ= μ p =1, n 1 =1.75, n=1, n p =1.5, θ 1 =51°.
P 0 = a 2 4π ( | E 0 | 2 + | E 0 | 2 ).
F x = 1 2 [ | E | 2 + | E | 2 ( 2 cosh 2 α1 ) ]k a 3 coshα ×[ Im( ε p ε ε p +2ε )+ 2 3 ( ka ) 3 | ε p ε ε p +2ε | 2 ] e 2khsinhα ,
F z = 1 2 [ | E | 2 + | E | 2 ( 2 cosh 2 α1 ) ]k a 3 sinhαRe( ε p ε ε p +2ε ) e 2khsinhα .
n 1 =1.5, n=1, θ 1 =42°, λ=632.8 nm, a=10 nm (ka0.1),
( E ˜ E ˜ )=( cosϕ sinϕ sinϕ cosϕ )( E E ), ( H ˜ H ˜ )= ε μ ( cosϕ sinϕ sinϕ cosϕ )( E E )
E θ s = E ˜ 1 r =1 A ( i a ξ τ b ξ π ) , H θ s = E ˜ 1 r ε μ =1 A ( i b ξ τ a ξ π ) , E ϕ s = E ˜ 1 r =1 A ( b ξ τ i a ξ π ) , H ϕ s = E ˜ 1 r ε μ =1 A ( i b ξ π a ξ τ ) , E r s = E ˜ sinθ 1 r 2 =1 A ( +1 )i a ξ π , H r s = E ˜ sinθ 1 r 2 ε μ =1 A ( +1 )i b ξ π .
ξ ( kr )=kr h ( 1 ) ( kr ), ξ ( kr )= d[ kr h ( 1 ) ( kr ) ] d( kr ) , π ( cosθ )= P 1 ( cosθ ) sinθ , τ ( cosθ )= d P 1 ( cosθ ) dθ .
a = m ε ψ ( mχ ) ψ ( χ ) m μ ψ ( χ ) ψ ( mχ ) m ε ψ ( mχ ) ξ ( χ ) m μ ξ ( χ ) ψ ( mχ ) , b = m μ ψ ( mχ ) ψ ( χ ) m ε ψ ( χ ) ψ ( mχ ) m μ ψ ( mχ ) ξ ( χ ) m ε ξ ( χ ) ψ ( mχ ) .
m ε = ε p ε , m μ = μ p μ , m= n p n , ψ ( u )=u j ( u ), ψ ( u )= d[ u j ( u ) ] du ,
E s =( E x s E y s E z s )= R ^ ( θ,ϕ )( E θ s E ϕ s E r s ), H s =( H x s H y s H z s )= R ^ ( θ,ϕ )( H θ s H ϕ s H r s ), R ^ ( θ,ϕ )= R ^ z ( ϕ ) R ^ y ( θ )=( cosθcosϕ sinϕ sinθcosϕ cosθsinϕ cosϕ sinθsinϕ sinθ 0 cosθ ).

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