Abstract

The accuracy of the discrete dipole approximation (DDA) for computing forces and torques in optical trapping experiments is discussed in the context of dielectric spheres and a range of low symmetry particles, including particles with geometric anisotropy (spheroids), optical anisotropy (birefringent spheres) and structural inhomogeneity (core-shell spheres). DDA calculations are compared with the results of exact T-matrix theory. In each case excellent agreement is found between the two methods for predictions of optical forces, torques, trap stiffnesses and trapping positions. Since the DDA lends itself to calculations on particles of arbitrary shape, the study is augmented by considering more general systems which have received recent experimental interest. In particular, optical forces and torques on low symmetry letter-shaped colloidal particles, birefringent quartz cylinders and biphasic Janus particles are computed and the trapping behaviour of the particles is discussed. Very good agreement is found with the available experimental data. The efficiency of the DDA algorithm and methods of accelerating the calculations are also discussed.

© 2011 OSA

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2011 (2)

V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. 112, 1711–1725 (2011).
[CrossRef]

I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. 16, 84–95 (2011).
[CrossRef]

2010 (4)

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031,141 (2010).

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010).
[CrossRef]

S. H. Simpson and S. Hanna, “Holographic optical trapping of microrods and nanowires,” J. Opt. Soc. Am. A 27, 1255–1264 (2010).
[CrossRef]

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108(7), 073110 (2010).
[CrossRef]

2009 (2)

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009).
[CrossRef]

2008 (1)

J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. 81, 58005 (2008).
[CrossRef]

2007 (6)

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A: Pure Appl. Opt. 9(8), S228–S234 (2007).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[CrossRef]

S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A 24, 430–443 (2007).
[CrossRef]

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

2006 (1)

2005 (1)

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

2004 (1)

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

2003 (2)

2002 (1)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

2001 (2)

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

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. A 18, 1944–1953 (2001).
[CrossRef]

2000 (1)

D. A. White, “Numerical modeling of optical gradient traps using the vector finite element method,” J. Comp. Phys. 159, 13–37 (2000).
[CrossRef]

1999 (1)

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

1996 (2)

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains.1. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996).
[CrossRef]

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

1994 (1)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1986 (2)

1979 (1)

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

1941 (1)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Andreasson, J. O. L.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010).
[CrossRef]

Asakura, T.

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

Ashkin, A.

Axner, O.

Benito, D. C.

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

Billaudeau, C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

Bishop, A. I.

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

Bjorkholm, J. E.

Block, S. M.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010).
[CrossRef]

Bonessi, D.

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A: Pure Appl. Opt. 9(8), S228–S234 (2007).
[CrossRef]

Bonin, K.

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A: Pure Appl. Opt. 9(8), S228–S234 (2007).
[CrossRef]

Brevik, I.

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

Bryant, G. W.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: Application to a micro-propeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Chu, S.

Chui, S. T.

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

Dejgosha, S.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Deufel, C.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Doi, M.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1986).

Draine, B. T.

B. T. Draine and J. C. Weingartner, “Radiative torques on interstellar grains.1. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Dziedzic, J. M.

Edwards, S. F.

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1986).

Fällman, E.

Flatau, P. J.

Forth, S.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Frijlink, M.

Greenleaf, W. J.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010).
[CrossRef]

Gutierrez-Medina, B.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010).
[CrossRef]

Hanna, S.

Harada, Y.

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

Heckenberg, N. R.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

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

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[CrossRef]

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. A 18, 1944–1953 (2001).
[CrossRef]

Huang, L.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108(7), 073110 (2010).
[CrossRef]

Kahnert, F. M.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79, 775–824 (2003).
[CrossRef]

Kretzschmar, I.

I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. 16, 84–95 (2011).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

Laporta, A.

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. Laporta, and S. M. Block, “An optical apparatus for rotation and trapping,” Methods Enzymol. 474, 377–404 (2010).
[CrossRef]

Li, Z.-Y.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108(7), 073110 (2010).
[CrossRef]

Lin, Z. F.

Z. F. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[CrossRef]

Ling, L.

L. Ling, F. Zhou, L. Huang, and Z.-Y. Li, “Optical forces on arbitrary shaped particles in optical tweezers,” J. Appl. Phys. 108(7), 073110 (2010).
[CrossRef]

Loke, V. L. Y.

V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. 112, 1711–1725 (2011).
[CrossRef]

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

Mason, T. G.

J. N. Wilking and T. G. Mason, “Multiple trapped states and angular Kramers hopping of complex dielectric shapes in a simple optical trap,” Europhys. Lett. 81, 58005 (2008).
[CrossRef]

Mengüç, M. P.

V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. 112, 1711–1725 (2011).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

Nieminen, T. A.

V. L. Y. Loke, M. P. Mengüç, and T. A. Nieminen, “Discrete dipole approximation with surface interaction: Computational toolbox for MATLAB,” J. Quant. Spectrosc. Radiat. Transf. 112, 1711–1725 (2011).
[CrossRef]

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

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

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Rubinsztein-Dunlop, H.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

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

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Saghafi, S.

Sentenac, A.

P. C. Chaumet, A. Rahmani, A. Sentenac, and G. W. Bryant, “Efficient computation of optical forces with the coupled dipole method,” Phys. Rev. E 72, 046708 (2005).
[CrossRef]

Sheppard, C. J. R.

Simmons, C. R.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Simpson, S. H.

Sloot, P. M. A.

Song, J. H. K.

I. Kretzschmar and J. H. K. Song, “Surface-anisotropic spherical colloids in geometric and field confinement,” Curr. Opin. Colloid Interface Sci. 16, 84–95 (2011).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

Walker, T.

D. Bonessi, K. Bonin, and T. Walker, “Optical forces on particles of arbitrary shape and size,” J. Opt. A: Pure Appl. Opt. 9(8), S228–S234 (2007).
[CrossRef]

Wang, M. D.

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007).
[CrossRef] [PubMed]

Waters, L. B. F. M.

Weingartner, J. C.

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Figures (9)

Fig. 1
Fig. 1

Schematic showing the coordinate axes, and (a) a trapped sphere, (b) a prolate spheroid and (c) a Janus sphere.

Fig. 2
Fig. 2

(a) The equilibrium trapping height, z eqm, and (b) stiffness, kx , for dielectric spheres of radii r = 100,200,300 and 400nm, as a function of the DDA lattice parameter, δ. z eqm is measured from the focal plane.

Fig. 3
Fig. 3

(a) The equilibrium trapping height, z eqm, and (b) the translational stiffness kx , for dielectric spheres as a function of radius, for four values of the DDA lattice parameter, δ. The curves for δ ≥ 30nm are offset vertically, as they would otherwise be superimposed. (c) z eqm and (d) kx for dielectric spheres, comparing the DDA with δ = 15nm (black curves) and the T-matrix method (red curves), for three values of n m.

Fig. 4
Fig. 4

Stiffness coefficients for prolate silica spheroids with an aspect ratio of two, as a function of equivalent radius. Calculations are performed using both the DDA and the T-matrix method.

Fig. 5
Fig. 5

The forces and torques acting on silica letters, computed using the DDA. The dimensions are: ‘N’, 4 × 5μm; ‘A’, 3 × 4.6μm; ‘O’, 3 × 5μm. All letters have a thickness of 0.5μm in the y-direction, and lie in the xz-plane. (a) The y component of torque, τy , when rotating a letter ‘N’ about the y-axis through the two centres shown. (b) τy for rotation of a letter ‘A’ about the y-axis through the centre of the letter. (c) The x component of force, Fx , for a letter ‘O’ being translated parallel to the x-axis, for the two orientations shown. (d) A schematic illustration of the preferred trapping orientations of the three letters.

Fig. 6
Fig. 6

(a) The azimuthal stiffness, K z z r r , of homogeneous birefringent spheres in Gaussian beams as a function of sphere radius for several dielectric anisotropies, Δɛ. Calculations are performed using both T-matrix and DDA methods. (b) The azimuthal stiffness of quartz cylinders of various radii, as a function of length. The cylinders are oriented with long axes parallel to the beam axis. n o = 1.544 parallel to the cylinder axis and n e = 1.553 parallel to a radius. Calculations were performed using the DDA method only.

Fig. 7
Fig. 7

(a) Schematic of the PTFE/silica core-shell spheres. (b) Equilibrium trapping heights and (c) trap stiffnesses for the core-shell spheres as a function of ra . (d) Expanded view of the rectangular region in part (c), plotted as a function of the shell thickness, rb ra . In all cases, the shell radius, rb , is held constant at 0.75μm; the results from T-matrix and DDA calculations are compared.

Fig. 8
Fig. 8

(a) Longitudinal (xz-plane, y = 0) and (b) transverse (xy-plane, z = 0) vector force fields for a PTFE/silica core-shell sphere with an outer radius rb = 1μm and core radius ra = 0.33μm, in a Gaussian optical trap. The force amplitudes are highlighted through the background colour map; a black background indicates zero force. In (a) the trapping position is indicated by a cross.

Fig. 9
Fig. 9

The torques acting on biphasic Janus spheres, positioned at the focal point of a Gaussian beam, as a function of rotation angle about (a) the y-axis and (b) the x-axis. The accompanying lateral forces are shown for rotations about (c) the y-axis and (d) the x-axis. Calculations were performed using the DDA method, for spheres with radii in the range 0.3 to 0.7 μm. The refractive indices used were n 1 = 1.05n m and n 2 = 1.1n m.

Equations (14)

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E i n c = n = 1 m = n n ( a n , m Rg M n , m + b n , m Rg N n , m )
a n , 1 = a n , 1 = b n , 1 = b n , 1 = I n , m
I n , m = i n + 1 γ n , m 0 π d θ e κ z 0 cos θ ( 1 + cos θ ) [ P n m ( cos θ ) + sin θ d d θ P n m ( cos θ ) ]
γ n , m = [ ( 2 n + 1 ) ( n m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 2
α = α CM ( 1 i 2 3 k 3 α CM ) where α CM = a 3 ( ɛ s ɛ m ) ( ɛ s + 2 ɛ m )
F i cell = 1 2 Re ( p j i E j )
T cell = r × F cell + 1 2 Re ( p × E )
F = K ( q q eqm ) ,
K = [ F ( q q eqm ) ] T .
K = [ K tt K tr K rt K rr ] .
K x x t t = F x x ; K y y r r = τ y θ y ; K x y t r = F x θ y ; K y x r t = τ y x
f = 1 2 Re [ ( J × B * ) 1 2 E j E k * ɛ j k ] ,
t = r × f + 1 2 Re ( D × E * ) .
f = 1 4 | E | 2 ɛ δ ( n ) n ^

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