Abstract

Under the framework of generalized Lorenz-Mie theory, we calculate the radiation force and torque exerted on a chiral sphere by a Gaussian beam. The theory and codes for axial radiation force are verified when the chiral sphere degenerates into an isotropic sphere. We discuss the influence of a chirality parameter on the radiation force and torque. Linearly and circularly polarized incident Gaussian beams are considered, and the corresponding radiation forces and torques are compared and analyzed. The polarization of the incident beam considerably influences radiation force of a chiral sphere. In trapping a chiral sphere, therefore, the polarization of incident beams should be chosen in accordance with the chirality. Unlike polarization, variation of chirality slightly affects radiation torque, except when the imaginary part of the chirality parameter is considered.

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    [CrossRef]
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    [CrossRef]
  38. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf.112(1), 1–27 (2011).
    [CrossRef]
  39. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A11(9), 2516–2525 (1994).
    [CrossRef]
  40. G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt.27(23), 4874–4883 (1988).
    [CrossRef] [PubMed]
  41. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral Localized Approximation in Generalized Lorenz-Mie Theory,” Appl. Opt.37(19), 4218–4225 (1998).
    [CrossRef] [PubMed]
  42. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A11(9), 2503–2515 (1994).
    [CrossRef]

2012

2011

L. A. Ambrosio and H. E. Hernández-Figueroa, “Radiation pressure cross sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt.50(22), 4489–4498 (2011).
[CrossRef] [PubMed]

Z. J. Li, Z. S. Wu, and Q. C. Shang, “Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam,” Opt. Express19(17), 16044–16057 (2011).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf.112(1), 1–27 (2011).
[CrossRef]

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron.41(6), 526–533 (2011).
[CrossRef]

2010

2007

T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online3(3), 338–342 (2007).
[CrossRef]

2005

2004

2001

A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A18(8), 1944–1953 (2001).
[CrossRef] [PubMed]

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun.142(1-3), 468–471 (2001).
[CrossRef]

1998

1997

1996

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]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt.35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

1995

1994

1993

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett.63(6), 715–717 (1993).
[CrossRef]

1992

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J.61(2), 569–582 (1992).
[CrossRef] [PubMed]

R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B9(10), 1922–1930 (1992).
[CrossRef]

1991

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry12(6), 479–485 (1991).
[CrossRef] [PubMed]

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci.26(6), 1393–1401 (1991).
[CrossRef]

1989

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

1988

1987

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

1986

1983

1982

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a gaussian laser beam,” Opt. Acta (Lond.)29(6), 801–806 (1982).
[CrossRef]

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A19(3), 1177–1179 (1979).
[CrossRef]

1977

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun.21(1), 189–194 (1977).
[CrossRef]

1974

F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974).
[CrossRef]

1971

A. Ashkin and J. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

1951

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

Ambrosio, L. A.

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.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J.61(2), 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science235(4795), 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11(5), 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin and J. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

Bai, L.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

Berns, M. W.

W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett.63(6), 715–717 (1993).
[CrossRef]

Bishop, A. I.

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun.142(1-3), 468–471 (2001).
[CrossRef]

Bjorkholm, J. E.

Bohren, F.

F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett.29(3), 458–462 (1974).
[CrossRef]

Branczyk, A.

T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online3(3), 338–342 (2007).
[CrossRef]

Brevik, I.

Cao, C. H.

Chu, S.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A19(3), 1177–1179 (1979).
[CrossRef]

de Grooth, B. G.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry12(6), 479–485 (1991).
[CrossRef] [PubMed]

Doicu, A.

Dziedzic, J.

A. Ashkin and J. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett.19(8), 283–285 (1971).
[CrossRef]

Dziedzic, J. M.

Frijlink, M.

Gauthier, R.

Gauthier, R. C.

Gouesbet, G.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf.112(1), 1–27 (2011).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral Localized Approximation in Generalized Lorenz-Mie Theory,” Appl. Opt.37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt.35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A11(9), 2503–2515 (1994).
[CrossRef]

K. F. Ren, G. Gréha, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun.108(4-6), 343–354 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A11(9), 2516–2525 (1994).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an arbitrary location,” Particle & Particle Systems Characterization5(1), 1–8 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A5(9), 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt.27(23), 4874–4883 (1988).
[CrossRef] [PubMed]

Gréha, G.

K. F. Ren, G. Gréha, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun.108(4-6), 343–354 (1994).
[CrossRef]

Grehan, G.

Gréhan, G.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf.112(1), 1–27 (2011).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral Localized Approximation in Generalized Lorenz-Mie Theory,” Appl. Opt.37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt.35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an arbitrary location,” Particle & Particle Systems Characterization5(1), 1–8 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A5(9), 1427–1443 (1988).
[CrossRef]

Greve, J.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry12(6), 479–485 (1991).
[CrossRef] [PubMed]

Gussgard, R.

Guzatov, D.

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron.41(6), 526–533 (2011).
[CrossRef]

Halas, N. J.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
[CrossRef]

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]

Heckenberg, N.

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun.142(1-3), 468–471 (2001).
[CrossRef]

Hernández-Figueroa, H. E.

Hesselink, G.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry12(6), 479–485 (1991).
[CrossRef] [PubMed]

Hoekstra, A. G.

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys.22(10), 1242–1246 (1951).
[CrossRef]

Kim, J. S.

J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am.73(3), 303–312 (1983).
[CrossRef]

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a gaussian laser beam,” Opt. Acta (Lond.)29(6), 801–806 (1982).
[CrossRef]

Klimov, V. V.

D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron.41(6), 526–533 (2011).
[CrossRef]

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online3(3), 338–342 (2007).
[CrossRef]

Lee, S. S.

J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am.73(3), 303–312 (1983).
[CrossRef]

J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a gaussian laser beam,” Opt. Acta (Lond.)29(6), 801–806 (1982).
[CrossRef]

Li, Z. J.

Li, Z.-J.

Lindmo, T.

Lock, J. A.

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online3(3), 338–342 (2007).
[CrossRef]

Maheu, B.

Nieminen, T.

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun.142(1-3), 468–471 (2001).
[CrossRef]

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online3(3), 338–342 (2007).
[CrossRef]

Ren, K. F.

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral Localized Approximation in Generalized Lorenz-Mie Theory,” Appl. Opt.37(19), 4218–4225 (1998).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt.35(15), 2702–2710 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gréha, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory, and associated resonance effects,” Opt. Commun.108(4-6), 343–354 (1994).
[CrossRef]

Roosen, G.

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun.21(1), 189–194 (1977).
[CrossRef]

Rubinsztein-Dunlop, H.

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun.142(1-3), 468–471 (2001).
[CrossRef]

Sarkar, D.

D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics56(1), 1102–1112 (1997).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4602 (1989).
[CrossRef]

Schut, T. C. B.

T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry12(6), 479–485 (1991).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Comparison with results of axial radiation force from literature.

Fig. 2
Fig. 2

Axial radiation force exerted on a chiral sphere versus z0.

Fig. 3
Fig. 3

Transverse radiation force exerted on a chiral sphere versus x0. (a) Fx; (b) Fy.

Fig. 4
Fig. 4

Axial radiation force versus chirality parameter. (a) z0 = 0 μm; (b) z0 = 10 μm.

Fig. 5
Fig. 5

Transverse radiation force versus chirality parameter. (a) x0 = 1.0μm; (b) x0 = 0.5μm.

Fig. 6
Fig. 6

Axial radiation torque versus z0.

Fig. 7
Fig. 7

Radiation torque versus transverse position x0. (a) Nz; (b) Nx; (c) Ny.

Equations (25)

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D= ε c E+iκ ε 0 μ 0 H, B=iκ ε 0 μ 0 E+ μ c H,
E int = n=1 m=n n [ A mn M mn (1) (r, k 1 )+ A mn N mn (1) (r, k 1 ) + B mn M mn (1) (r, k 2 ) B mn N mn (1) (r, k 2 ) ],
H int =Q n=1 m=n n [ A mn N mn (1) (r, k 1 )+ A mn M mn (1) (r, k 1 )+ B mn N mn (1) (r, k 2 ) B mn M mn (1) (r, k 2 ) ] ,
E ip = E 0 n=1 m=n n [ a mn ip M mn (1) (r,k)+ b mn ip N mn (1) (r,k) ] ,
H ip = k E 0 iωμ n=1 m=n n [ a mn ip N mn (1) (r,k)+ b mn ip M mn (1) (r,k) ] ,
E s = E 0 n=1 m=n n [ A mn s M mn (3) (r,k)+ B mn s N mn (3) (r,k) ] ,
H s = k E 0 iωμ n=1 m=n n [ A mn s N mn (3) (r,k)+ B mn s M mn (3) (r,k) ] ,
A mn s = A n sa a mn ip + A n sb b mn ip , B mn s = B n sa a mn ip + B n sb b mn ip ,
A n sa = ψ n ( x 0 ) ξ n ( x 0 ) D n (1) ( x 1 ) η r D n (1) ( x 0 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) + D n (1) ( x 2 ) η r D n (1) ( x 0 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) η r D n (3) ( x 0 ) D n (1) ( x 1 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) + η r D n (3) ( x 0 ) D n (1) ( x 2 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) ,
A n sb = ψ n ( x 0 ) ξ n ( x 0 ) η r D n (1) ( x 1 ) D n (1) ( x 0 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) η r D n (1) ( x 2 ) D n (1) ( x 0 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) η r D n (3) ( x 0 ) D n (1) ( x 1 ) η r D n (1) ( x 1 ) D n (3) ( x 0 ) + η r D n (3) ( x 0 ) D n (1) ( x 2 ) η r D n (1) ( x 2 ) D n (3) ( x 0 ) ,
B n sa = A n sb ,
B n sb = ψ n ( x 0 ) ξ n ( x 0 ) η r D n (1) ( x 1 ) D n (1) ( x 0 ) D n (1) ( x 1 ) η r D n (3) ( x 0 ) + η r D n (1) ( x 2 ) D n (1) ( x 0 ) D n (1) ( x 2 ) η r D n (3) ( x 0 ) D n (3) ( x 0 ) η r D n (1) ( x 1 ) D n (1) ( x 1 ) η r D n (3) ( x 0 ) + D n (3) ( x 0 ) η r D n (1) ( x 2 ) D n (1) ( x 2 ) η r D n (3) ( x 0 ) .
a mn ix = C nm (i g n,TE m ), b mn ix = C nm g n,TM m ,
C nm ={ i n1 2n+1 n(n+1) , m0 (1) | m | (n+| m |)! (n| m |)! i n1 2n+1 n(n+1) ,m<0 .
[ g n,TM m i g n,TE m ]= (1) m1 K nm ψ 0 e i k 0 z 0 1 2 [ e i(m1) φ 0 J m1 (2 Q ¯ ρ 0 ρ n w 0 2 )± e i(m+1) φ 0 J m+1 (2 Q ¯ ρ 0 ρ n w 0 2 ) ],
ψ 0 =i Q ¯ exp( i Q ¯ ρ 0 2 w 0 2 )exp( i Q ¯ (n+0.5) 2 k 0 2 w 0 2 ),
K nm ={ (i) | m | i (n+0.5) | m |1 ,m0 n(n+1) n+0.5 ,m=0 ,
ρ n = (n+0.5) / k 0 , Q ¯ = (i2 z 0 /l) 1 , ρ 0 = x 0 2 + y 0 2 , φ 0 =arctan( x 0 / y 0 ), l=k w 0 2 .
a mn iR = 2 ( a mn ix + b mn ix )/2, b mn iR = 2 ( b mn ix + a mn ix )/2.
F= S n ^ T dS ,
F= 1 2 Re 0 2π 0 π [ ε E r E+μ H r H 1 2 (ε E 2 +μ H 2 ) r ^ ] r 2 sinθdθdϕ ,
F x +i F y = n 0 P πc k 2 w 0 2 n=1 m=n n N mn 1 × [ (nm)(n+m+1) N m+1n 1 × ( a mn ix B m+1n S* + b mn ix A m+1n S* + a m+1n ix* B mn S + b m+1n ix* A mn S +2 A mn S B m+1n S* +2 B mn S A m+1n S* ) i (nm1)(nm) (2n1)(2n+1) (n1)(n+1) N m+1n1 1 ×, ( a mn ix A m+1n1 S* + b mn ix B m+1n1 S* + a m+1n1 ix* A mn S + b m+1n1 ix* B mn S +2 A mn S A m+1n1 S* +2 B mn S B m+1n1 S* ) i (n+m+1)(n+m+2) (2n+1)(2n+3) n(n+2) N m+1n+1 1 × ( a mn ix A m+1n+1 S* + b mn ix B m+1n+1 S* + a m+1n+1 ix* A mn S + b m+1n+1 ix* B mn S +2 A mn S A m+1n+1 S* +2 B mn S B m+1n+1 S* ) ]
F z = 2n 0 P πc k 2 w 0 2 n=1 m=n n Re [ in(n+2) (nm+1)(n+m+1) (2n+1)(2n+3) N mn 1 N mn+1 1 × ( A mn S* a mn+1 ix + a mn ix* A mn+1 S + B mn S* b mn+1 ix + b mn ix* B mn+1 S +2 A mn+1 S A mn S* . +2 B mn+1 S B mn S* )m N mn 2 ( a mn ix B mn S* + b mn ix A mn S* +2 A mn S B mn S* ) ]
N x +i N y = n 0 P πc k 3 w 0 2 n=1 m=n n n(n+1) (nm)(n+m+1) N mn 1 N m+1n 1 × , [ ( b mn i b m+1n s* + a mn i a m+1n s* )+( b mn s b m+1n i* + a mn s a m+1n i* )+2( b mn s b m+1n s* + a mn s a m+1n s* ) ]
N z = 2 n 0 P πc k 3 w 0 2 n=1 m=n n mn(n+1) N mn 1 N mn 1 [ Re( a mn i a mn s* + b mn i b mn s* )+(| a mn s | 2 +| b mn s | 2 ) ] ,

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