Abstract

The aim of this work is to develop a formal semi-analytical model using the modal expansion method in cylindrical coordinates to calculate the optical/electromagnetic (EM) radiation force-per-length experienced by an infinitely long electrically-conducting elliptical cylinder having a smooth or wavy/corrugated surface in EM plane progressive waves with different polarizations. In this analysis, one of the semi-axes of the elliptical cylinder coincides with the direction of the incident field. Initially, the modal matching method is used to determine the scattering coefficients by imposing appropriate boundary conditions and solving numerically a linear system of equations by matrix inversion. In this method, standard cylindrical (Bessel and Hankel) wave functions are used. Subsequently, simplified expressions leading to exact series expansions for the optical/EM radiation forces assuming either electric (TM) or magnetic (TE) plane wave incidences are provided without any approximations, in addition to integral equations demonstrating the direct relationship of the radiation force with the square of the scattered field magnitude. An important application of these integral equations concerns the accurate determination of the radiation force from the measurement of the scattered field by any 2D non-absorptive object of arbitrary shape in plane waves. Numerical computations for the non-dimensional radiation force function are performed for electrically conducting elliptic and circular cylinders having a smooth or ribbed/corrugated surface. Adequate convergence plots confirm the validity and correctness of the method to evaluate the radiation force with no limitation to a particular frequency range (i.e. Rayleigh, Mie, or geometrical optics regimes). Particular emphases are given on the aspect ratio, the non-dimensional size of the cylinder, the corrugation characteristic of its surface, and the polarization of the incident field. The results are particularly relevant in optical tweezers and other related applications in fluid dynamics, where the shape and stability of a cylindrical drop stressed by a uniform external electric/magnetic field are altered. Furthermore, a direct analogy with the acoustical counterpart is noted and discussed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  55. F. G. Mitri, “Acoustic attraction, repulsion and radiation force cancellation on a pair of rigid particles with arbitrary cross-sections in 2D: Circular cylinders example,” Ann. Phys. 386, 1–14 (2017).
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  59. F. G. Mitri, “Acoustic radiation force of attraction, cancellation and repulsion on a circular cylinder near a rigid corner space,” Applied Mathematical Modelling 64, 688–698 (2018).
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  60. F. G. Mitri, “Airy acoustical-sheet spinner tweezers,” J. Appl. Phys. 120(10), 104901 (2016).
    [Crossref]
  61. F. G. Mitri, “Extinction efficiency of “elastic–sheet” beams by a cylindrical (viscous) fluid inclusion embedded in an elastic medium and mode conversion—Examples of nonparaxial Gaussian and Airy beams,” J. Appl. Phys. 120(14), 144902 (2016).
    [Crossref]
  62. F. G. Mitri, “Acoustics of finite asymmetric exotic beams: Examples of Airy and fractional Bessel beams,” J. Appl. Phys. 122(22), 224903 (2017).
    [Crossref]
  63. F. G. Mitri, “Scattering of Airy elastic sheets by a cylindrical cavity in a solid,” Ultrasonics 81, 100–106 (2017).
    [Crossref]
  64. F. G. Mitri, “Pulling and spinning reversal of a subwavelength absorptive sphere in adjustable vector Airy light-sheets,” Appl. Phys. Lett. 110(18), 181112 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
  68. F. G. Mitri, “Cylindrical particle manipulation and negative spinning using a nonparaxial Hermite−Gaussian light-sheet beam,” J. Opt. 18(10), 105402 (2016).
    [Crossref]
  69. F. G. Mitri, “Acoustic radiation force and spin torque on a viscoelastic cylinder in a quasi-Gaussian cylindrically-focused beam with arbitrary incidence in a non-viscous fluid,” Wave Motion 66, 31–44 (2016).
    [Crossref]
  70. F. G. Mitri, “Acoustical spinner tweezers with nonparaxial Hermite-Gaussian acoustical-sheets and particle dynamics,” Ultrasonics 73, 236–244 (2017).
    [Crossref]

2018 (6)

F. G. Mitri, “Active electromagnetic invisibility cloaking and radiation force cancellation,” J. Quant. Spectrosc. Radiat. Transfer 207, 48–53 (2018).
[Crossref]

F. G. Mitri, “Pushing, pulling and electromagnetic radiation force cloaking by a pair of conducting cylindrical particles,” J. Quant. Spectrosc. Radiat. Transfer 206, 142–150 (2018).
[Crossref]

F. G. Mitri, “Acoustic radiation force on a cylindrical particle near a planar rigid boundary,” J. Phys. Commun. 2(4), 045019 (2018).
[Crossref]

F. G. Mitri, “Acoustic radiation force of attraction, cancellation and repulsion on a circular cylinder near a rigid corner space,” Applied Mathematical Modelling 64, 688–698 (2018).
[Crossref]

F. G. Mitri, “Adjustable vector Airy light-sheet single optical tweezers: negative radiation forces on a subwavelength spheroid and spin torque reversal,” Eur. Phys. J. D 72(1), 21 (2018).
[Crossref]

F. G. Mitri, “Electromagnetic binding and radiation force reversal on a pair of electrically conducting cylinders of arbitrary geometrical cross-section with smooth and corrugated surfaces,” OSA Continuum 1(2), 521–541 (2018).
[Crossref]

2017 (10)

F. G. Mitri, “Acoustics of finite asymmetric exotic beams: Examples of Airy and fractional Bessel beams,” J. Appl. Phys. 122(22), 224903 (2017).
[Crossref]

F. G. Mitri, “Scattering of Airy elastic sheets by a cylindrical cavity in a solid,” Ultrasonics 81, 100–106 (2017).
[Crossref]

F. G. Mitri, “Pulling and spinning reversal of a subwavelength absorptive sphere in adjustable vector Airy light-sheets,” Appl. Phys. Lett. 110(18), 181112 (2017).
[Crossref]

F. G. Mitri, “Nonparaxial scalar Airy light-sheets and their higher-order spatial derivatives,” Appl. Phys. Lett. 110(9), 091104 (2017).
[Crossref]

F. G. Mitri, “Acoustical spinner tweezers with nonparaxial Hermite-Gaussian acoustical-sheets and particle dynamics,” Ultrasonics 73, 236–244 (2017).
[Crossref]

S. Mhatre, “Dielectrophoretic motion and deformation of a liquid drop in an axisymmetric non-uniform AC electric field,” Sens. Actuators B 239, 1098–1108 (2017).
[Crossref]

F. G. Mitri, “Acoustic attraction, repulsion and radiation force cancellation on a pair of rigid particles with arbitrary cross-sections in 2D: Circular cylinders example,” Ann. Phys. 386, 1–14 (2017).
[Crossref]

F. G. Mitri, “Radiation force and torque of light-sheets,” J. Opt. 19(6), 065403 (2017).
[Crossref]

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

F. G. Mitri, “Axial acoustic radiation force on rigid oblate and prolate spheroids in Bessel vortex beams of progressive, standing and quasi-standing waves,” Ultrasonics 74, 62–71 (2017).
[Crossref]

2016 (8)

F. G. Mitri, “Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves,” Ultrasonics 66, 27–33 (2016).
[Crossref]

F. G. Mitri, “Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence,” Phys. Fluids 28(7), 077104 (2016).
[Crossref]

F. G. Mitri, “Extended optical theorem for scalar monochromatic acoustical beams of arbitrary wavefront in cylindrical coordinates,” Ultrasonics 67, 129–135 (2016).
[Crossref]

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

F. G. Mitri, “Airy acoustical-sheet spinner tweezers,” J. Appl. Phys. 120(10), 104901 (2016).
[Crossref]

F. G. Mitri, “Extinction efficiency of “elastic–sheet” beams by a cylindrical (viscous) fluid inclusion embedded in an elastic medium and mode conversion—Examples of nonparaxial Gaussian and Airy beams,” J. Appl. Phys. 120(14), 144902 (2016).
[Crossref]

F. G. Mitri, “Cylindrical particle manipulation and negative spinning using a nonparaxial Hermite−Gaussian light-sheet beam,” J. Opt. 18(10), 105402 (2016).
[Crossref]

F. G. Mitri, “Acoustic radiation force and spin torque on a viscoelastic cylinder in a quasi-Gaussian cylindrically-focused beam with arbitrary incidence in a non-viscous fluid,” Wave Motion 66, 31–44 (2016).
[Crossref]

2015 (9)

F. G. Mitri, “Interaction of an acoustical 2D-beam with an elastic cylinder with arbitrary location in a non-viscous fluid,” Ultrasonics 62, 244–252 (2015).
[Crossref]

F. G. Mitri, “Generalization of the optical theorem for monochromatic electromagnetic beams of arbitrary wavefront in cylindrical coordinates,” J. Quant. Spectrosc. Radiat. Transfer 166, 81–92 (2015).
[Crossref]

F. G. Mitri, “Optical theorem for two-dimensional (2D) scalar monochromatic acoustical beams in cylindrical coordinates,” Ultrasonics 62, 20–26 (2015).
[Crossref]

F. G. Mitri, “Acoustic radiation force on oblate and prolate spheroids in Bessel beams,” J. Wave Motion 57, 231–238 (2015).
[Crossref]

F. G. Mitri, “Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 62(10), 1809–1818 (2015).
[Crossref]

F. G. Mitri, “Acoustic scattering of a Bessel vortex beam by a rigid fixed spheroid,” Ann. Phys. 363, 262–274 (2015).
[Crossref]

F. G. Mitri, “Acoustical pulling force on rigid spheroids in single Bessel vortex tractor beams,” Europhys. Lett. 112(3), 34002 (2015).
[Crossref]

F. G. Mitri, “Acoustic scattering of a cylindrical quasi-Gaussian beam with arbitrary incidence focused on a rigid elliptical cylinder,” J. Appl. Phys. 118(18), 184902 (2015).
[Crossref]

F. G. Mitri, “Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves,” J. Appl. Phys. 118(21), 214903 (2015).
[Crossref]

2013 (1)

2012 (2)

K. Jiang, X. e. Han, and K. F. Ren, “Scattering from an elliptical cylinder by using the vectorial complex ray model,” Appl. Opt. 51(34), 8159–8168 (2012).
[Crossref]

S. D. Deshmukh and R. M. Thaokar, “Deformation, breakup and motion of a perfect dielectric drop in a quadrupole electric field,” Phys. Fluids 24(3), 032105 (2012).
[Crossref]

2010 (1)

F. G. Mitri, “Axial time-averaged acoustic radiation force on a cylinder in a nonviscous fluid revisited,” Ultrasonics 50(6), 620–627 (2010).
[Crossref]

2008 (2)

S. M. Hasheminejad and R. Sanaei, “Ultrasonic scattering by a fluid cylinder of elliptic cross section including viscous effects,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 55(2), 391–404 (2008).
[Crossref]

J. J. Xiao and C. T. Chan, “Calculation of the optical force on an infinite cylinder with arbitrary cross section by the boundary element method,” J. Opt. Soc. Am. B 25(9), 1553–1561 (2008).
[Crossref]

2007 (2)

S. M. Hasheminejad and R. Sanaei, “Acoustic radiation force and torque on a solid elliptic cylinder,” J. Comp. Acous. 15(03), 377–399 (2007).
[Crossref]

A. Zinke and A. Weber, “Light Scattering from Filaments,” IEEE Trans. Visual. Comput. Graphics 13(2), 342–356 (2007).
[Crossref]

2006 (1)

S. M. Hasheminejad and R. Sanaei, “Ultrasonic Scattering by a Viscoelastic Fiber of Elliptic Cross-Section Suspended in a Viscous Fluid Medium,” J. Dispersion Sci. Technol. 27(8), 1165–1179 (2006).
[Crossref]

2005 (1)

2004 (1)

C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. 6(10), 921–931 (2004).
[Crossref]

2001 (1)

2000 (1)

1999 (1)

1998 (2)

1997 (1)

1993 (1)

S. Sankaran and D. A. Saville, “Experiments on the stability of a liquid bridge in an axial electric field,” Phys. Fluids A 5(4), 1081–1083 (1993).
[Crossref]

1989 (1)

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 (1)

1987 (1)

N. Dodgson and C. Sozou, “The deformation of a liquid drop by an electric field,” Z. Angew. Math. Phys. 38(3), 424–432 (1987).
[Crossref]

1983 (1)

1976 (1)

1963 (1)

R. Barakat, “Diffraction of Plane Waves by an Elliptic Cylinder,” J. Acoust. Soc. Am. 35(12), 1990–1996 (1963).
[Crossref]

1961 (1)

K. Harumi, “Scattering of Plane Waves by a Rigid Ribbon in a Solid,” J. Appl. Phys. 32(8), 1488–1497 (1961).
[Crossref]

1938 (1)

P. M. Morse and P. J. Rubenstein, “The Diffraction of Waves by Ribbons and by Slits,” Phys. Rev. 54(11), 895–898 (1938).
[Crossref]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335(11), 57–136 (1909).
[Crossref]

Adler, C. L.

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]

Barakat, R.

R. Barakat, “Diffraction of Plane Waves by an Elliptic Cylinder,” J. Acoust. Soc. Am. 35(12), 1990–1996 (1963).
[Crossref]

Barbosa, S.

F. Onofri and S. Barbosa, “Optical Particle Characterization,” in Laser Metrology in Fluid Mechanics (John Wiley & Sons, Inc., 2013), pp. 67–158.

Barton, J. P.

J. P. Barton, “Electromagnetic-field calculations for irregularly shaped, layered cylindrical particles with focused illumination,” Appl. Opt. 36(6), 1312–1319 (1997).
[Crossref]

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]

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and acoustic scattering by simples shapes (John Wiley & Sons, 1969).

Chai, Y.

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

Chan, C. T.

DeBenedetti, S.

S. DeBenedetti, Nuclear Interactions (John Wiley & Sons, Incorporated, 1964).

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335(11), 57–136 (1909).
[Crossref]

Deshmukh, S. D.

S. D. Deshmukh and R. M. Thaokar, “Deformation, breakup and motion of a perfect dielectric drop in a quadrupole electric field,” Phys. Fluids 24(3), 032105 (2012).
[Crossref]

Dodgson, N.

N. Dodgson and C. Sozou, “The deformation of a liquid drop by an electric field,” Z. Angew. Math. Phys. 38(3), 424–432 (1987).
[Crossref]

Eyges, L.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill Book Co., 1953), Vol. 2.

Gong, Z.

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

Gouesbet, G.

Grehan, G.

G. Gouesbet and G. Grehan, Generalized Lorenz-Mie Theories, 1st ed. (Springer, 2011).

Gréhan, G.

Han, X. e.

Harumi, K.

K. Harumi, “Scattering of Plane Waves by a Rigid Ribbon in a Solid,” J. Appl. Phys. 32(8), 1488–1497 (1961).
[Crossref]

Hasheminejad, S. M.

S. M. Hasheminejad and R. Sanaei, “Ultrasonic scattering by a fluid cylinder of elliptic cross section including viscous effects,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 55(2), 391–404 (2008).
[Crossref]

S. M. Hasheminejad and R. Sanaei, “Acoustic radiation force and torque on a solid elliptic cylinder,” J. Comp. Acous. 15(03), 377–399 (2007).
[Crossref]

S. M. Hasheminejad and R. Sanaei, “Ultrasonic Scattering by a Viscoelastic Fiber of Elliptic Cross-Section Suspended in a Viscous Fluid Medium,” J. Dispersion Sci. Technol. 27(8), 1165–1179 (2006).
[Crossref]

Herzig, H. P.

C. Rockstuhl and H. P. Herzig, “Calculation of the torque on dielectric elliptical cylinders,” J. Opt. Soc. Am. A 22(1), 109–116 (2005).
[Crossref]

C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. 6(10), 921–931 (2004).
[Crossref]

Jiang, K.

Kim, J. S.

Lee, S. S.

Li, W.

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

Lock, J. A.

Maheu, B.

Mees, L.

Mhatre, S.

S. Mhatre, “Dielectrophoretic motion and deformation of a liquid drop in an axisymmetric non-uniform AC electric field,” Sens. Actuators B 239, 1098–1108 (2017).
[Crossref]

Mitri, F. G.

F. G. Mitri, “Adjustable vector Airy light-sheet single optical tweezers: negative radiation forces on a subwavelength spheroid and spin torque reversal,” Eur. Phys. J. D 72(1), 21 (2018).
[Crossref]

F. G. Mitri, “Electromagnetic binding and radiation force reversal on a pair of electrically conducting cylinders of arbitrary geometrical cross-section with smooth and corrugated surfaces,” OSA Continuum 1(2), 521–541 (2018).
[Crossref]

F. G. Mitri, “Pushing, pulling and electromagnetic radiation force cloaking by a pair of conducting cylindrical particles,” J. Quant. Spectrosc. Radiat. Transfer 206, 142–150 (2018).
[Crossref]

F. G. Mitri, “Acoustic radiation force on a cylindrical particle near a planar rigid boundary,” J. Phys. Commun. 2(4), 045019 (2018).
[Crossref]

F. G. Mitri, “Acoustic radiation force of attraction, cancellation and repulsion on a circular cylinder near a rigid corner space,” Applied Mathematical Modelling 64, 688–698 (2018).
[Crossref]

F. G. Mitri, “Active electromagnetic invisibility cloaking and radiation force cancellation,” J. Quant. Spectrosc. Radiat. Transfer 207, 48–53 (2018).
[Crossref]

F. G. Mitri, “Radiation force and torque of light-sheets,” J. Opt. 19(6), 065403 (2017).
[Crossref]

F. G. Mitri, “Acoustic attraction, repulsion and radiation force cancellation on a pair of rigid particles with arbitrary cross-sections in 2D: Circular cylinders example,” Ann. Phys. 386, 1–14 (2017).
[Crossref]

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

F. G. Mitri, “Axial acoustic radiation force on rigid oblate and prolate spheroids in Bessel vortex beams of progressive, standing and quasi-standing waves,” Ultrasonics 74, 62–71 (2017).
[Crossref]

F. G. Mitri, “Acoustical spinner tweezers with nonparaxial Hermite-Gaussian acoustical-sheets and particle dynamics,” Ultrasonics 73, 236–244 (2017).
[Crossref]

F. G. Mitri, “Acoustics of finite asymmetric exotic beams: Examples of Airy and fractional Bessel beams,” J. Appl. Phys. 122(22), 224903 (2017).
[Crossref]

F. G. Mitri, “Scattering of Airy elastic sheets by a cylindrical cavity in a solid,” Ultrasonics 81, 100–106 (2017).
[Crossref]

F. G. Mitri, “Pulling and spinning reversal of a subwavelength absorptive sphere in adjustable vector Airy light-sheets,” Appl. Phys. Lett. 110(18), 181112 (2017).
[Crossref]

F. G. Mitri, “Nonparaxial scalar Airy light-sheets and their higher-order spatial derivatives,” Appl. Phys. Lett. 110(9), 091104 (2017).
[Crossref]

F. G. Mitri, “Cylindrical particle manipulation and negative spinning using a nonparaxial Hermite−Gaussian light-sheet beam,” J. Opt. 18(10), 105402 (2016).
[Crossref]

F. G. Mitri, “Acoustic radiation force and spin torque on a viscoelastic cylinder in a quasi-Gaussian cylindrically-focused beam with arbitrary incidence in a non-viscous fluid,” Wave Motion 66, 31–44 (2016).
[Crossref]

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

F. G. Mitri, “Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves,” Ultrasonics 66, 27–33 (2016).
[Crossref]

F. G. Mitri, “Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence,” Phys. Fluids 28(7), 077104 (2016).
[Crossref]

F. G. Mitri, “Airy acoustical-sheet spinner tweezers,” J. Appl. Phys. 120(10), 104901 (2016).
[Crossref]

F. G. Mitri, “Extinction efficiency of “elastic–sheet” beams by a cylindrical (viscous) fluid inclusion embedded in an elastic medium and mode conversion—Examples of nonparaxial Gaussian and Airy beams,” J. Appl. Phys. 120(14), 144902 (2016).
[Crossref]

F. G. Mitri, “Extended optical theorem for scalar monochromatic acoustical beams of arbitrary wavefront in cylindrical coordinates,” Ultrasonics 67, 129–135 (2016).
[Crossref]

F. G. Mitri, “Interaction of an acoustical 2D-beam with an elastic cylinder with arbitrary location in a non-viscous fluid,” Ultrasonics 62, 244–252 (2015).
[Crossref]

F. G. Mitri, “Generalization of the optical theorem for monochromatic electromagnetic beams of arbitrary wavefront in cylindrical coordinates,” J. Quant. Spectrosc. Radiat. Transfer 166, 81–92 (2015).
[Crossref]

F. G. Mitri, “Optical theorem for two-dimensional (2D) scalar monochromatic acoustical beams in cylindrical coordinates,” Ultrasonics 62, 20–26 (2015).
[Crossref]

F. G. Mitri, “Acoustic scattering of a cylindrical quasi-Gaussian beam with arbitrary incidence focused on a rigid elliptical cylinder,” J. Appl. Phys. 118(18), 184902 (2015).
[Crossref]

F. G. Mitri, “Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves,” J. Appl. Phys. 118(21), 214903 (2015).
[Crossref]

F. G. Mitri, “Acoustic radiation force on oblate and prolate spheroids in Bessel beams,” J. Wave Motion 57, 231–238 (2015).
[Crossref]

F. G. Mitri, “Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 62(10), 1809–1818 (2015).
[Crossref]

F. G. Mitri, “Acoustic scattering of a Bessel vortex beam by a rigid fixed spheroid,” Ann. Phys. 363, 262–274 (2015).
[Crossref]

F. G. Mitri, “Acoustical pulling force on rigid spheroids in single Bessel vortex tractor beams,” Europhys. Lett. 112(3), 34002 (2015).
[Crossref]

F. G. Mitri, “Cylindrical quasi-Gaussian beams,” Opt. Lett. 38(22), 4727–4730 (2013).
[Crossref]

F. G. Mitri, “Axial time-averaged acoustic radiation force on a cylinder in a nonviscous fluid revisited,” Ultrasonics 50(6), 620–627 (2010).
[Crossref]

F. G. Mitri, “Acoustic radiation force and torque on a viscous fluid cylindrical particle nearby a planar rigid wall,” https://arxiv.org/abs/1801.04863 (2017).

F. G. Mitri, See the response to some misleading and obtuse comments in: “Reply to: Comments on [Phys. Fluids 28, 077104 (2016)],” https://arxiv.org/abs/1703.00487 (2017).

Morse, P. M.

P. M. Morse and P. J. Rubenstein, “The Diffraction of Waves by Ribbons and by Slits,” Phys. Rev. 54(11), 895–898 (1938).
[Crossref]

P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill Book Co., 1953), Vol. 2.

Nash, J. K.

Nelson, A.

Onofri, F.

F. Onofri and S. Barbosa, “Optical Particle Characterization,” in Laser Metrology in Fluid Mechanics (John Wiley & Sons, Inc., 2013), pp. 67–158.

Phipps, D.

Ren, K. F.

Rockstuhl, C.

C. Rockstuhl and H. P. Herzig, “Calculation of the torque on dielectric elliptical cylinders,” J. Opt. Soc. Am. A 22(1), 109–116 (2005).
[Crossref]

C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. 6(10), 921–931 (2004).
[Crossref]

Roman, P.

P. Roman, Introduction to Quantum Field Theory (Wiley, 1969).

Rubenstein, P. J.

P. M. Morse and P. J. Rubenstein, “The Diffraction of Waves by Ribbons and by Slits,” Phys. Rev. 54(11), 895–898 (1938).
[Crossref]

Sanaei, R.

S. M. Hasheminejad and R. Sanaei, “Ultrasonic scattering by a fluid cylinder of elliptic cross section including viscous effects,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 55(2), 391–404 (2008).
[Crossref]

S. M. Hasheminejad and R. Sanaei, “Acoustic radiation force and torque on a solid elliptic cylinder,” J. Comp. Acous. 15(03), 377–399 (2007).
[Crossref]

S. M. Hasheminejad and R. Sanaei, “Ultrasonic Scattering by a Viscoelastic Fiber of Elliptic Cross-Section Suspended in a Viscous Fluid Medium,” J. Dispersion Sci. Technol. 27(8), 1165–1179 (2006).
[Crossref]

Sankaran, S.

S. Sankaran and D. A. Saville, “Experiments on the stability of a liquid bridge in an axial electric field,” Phys. Fluids A 5(4), 1081–1083 (1993).
[Crossref]

Saunders, K. W.

Saville, D. A.

S. Sankaran and D. A. Saville, “Experiments on the stability of a liquid bridge in an axial electric field,” Phys. Fluids A 5(4), 1081–1083 (1993).
[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]

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and acoustic scattering by simples shapes (John Wiley & Sons, 1969).

Sozou, C.

N. Dodgson and C. Sozou, “The deformation of a liquid drop by an electric field,” Z. Angew. Math. Phys. 38(3), 424–432 (1987).
[Crossref]

Stone, B. R.

Thaokar, R. M.

S. D. Deshmukh and R. M. Thaokar, “Deformation, breakup and motion of a perfect dielectric drop in a quadrupole electric field,” Phys. Fluids 24(3), 032105 (2012).
[Crossref]

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and acoustic scattering by simples shapes (John Wiley & Sons, 1969).

van de Hulst, H. C.

H. C. van de Hulst, Light scattering by small particles, Light scattering by small particles (John Wiley and Sons, Inc., 1957), p. xiii + 470p.

Weber, A.

A. Zinke and A. Weber, “Light Scattering from Filaments,” IEEE Trans. Visual. Comput. Graphics 13(2), 342–356 (2007).
[Crossref]

Xiao, J. J.

Zajak, P. D.

Zhao, Y.

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

Zinke, A.

A. Zinke and A. Weber, “Light Scattering from Filaments,” IEEE Trans. Visual. Comput. Graphics 13(2), 342–356 (2007).
[Crossref]

Ann. Phys. (3)

F. G. Mitri, “Acoustic scattering of a Bessel vortex beam by a rigid fixed spheroid,” Ann. Phys. 363, 262–274 (2015).
[Crossref]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335(11), 57–136 (1909).
[Crossref]

F. G. Mitri, “Acoustic attraction, repulsion and radiation force cancellation on a pair of rigid particles with arbitrary cross-sections in 2D: Circular cylinders example,” Ann. Phys. 386, 1–14 (2017).
[Crossref]

Appl. Opt. (6)

Appl. Phys. Lett. (2)

F. G. Mitri, “Pulling and spinning reversal of a subwavelength absorptive sphere in adjustable vector Airy light-sheets,” Appl. Phys. Lett. 110(18), 181112 (2017).
[Crossref]

F. G. Mitri, “Nonparaxial scalar Airy light-sheets and their higher-order spatial derivatives,” Appl. Phys. Lett. 110(9), 091104 (2017).
[Crossref]

Applied Mathematical Modelling (1)

F. G. Mitri, “Acoustic radiation force of attraction, cancellation and repulsion on a circular cylinder near a rigid corner space,” Applied Mathematical Modelling 64, 688–698 (2018).
[Crossref]

Eur. Phys. J. D (1)

F. G. Mitri, “Adjustable vector Airy light-sheet single optical tweezers: negative radiation forces on a subwavelength spheroid and spin torque reversal,” Eur. Phys. J. D 72(1), 21 (2018).
[Crossref]

Europhys. Lett. (1)

F. G. Mitri, “Acoustical pulling force on rigid spheroids in single Bessel vortex tractor beams,” Europhys. Lett. 112(3), 34002 (2015).
[Crossref]

IEEE Trans. Ultrason., Ferroelect., Freq. Contr. (2)

S. M. Hasheminejad and R. Sanaei, “Ultrasonic scattering by a fluid cylinder of elliptic cross section including viscous effects,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 55(2), 391–404 (2008).
[Crossref]

F. G. Mitri, “Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 62(10), 1809–1818 (2015).
[Crossref]

IEEE Trans. Visual. Comput. Graphics (1)

A. Zinke and A. Weber, “Light Scattering from Filaments,” IEEE Trans. Visual. Comput. Graphics 13(2), 342–356 (2007).
[Crossref]

J. Acoust. Soc. Am. (1)

R. Barakat, “Diffraction of Plane Waves by an Elliptic Cylinder,” J. Acoust. Soc. Am. 35(12), 1990–1996 (1963).
[Crossref]

J. Appl. Phys. (7)

K. Harumi, “Scattering of Plane Waves by a Rigid Ribbon in a Solid,” J. Appl. Phys. 32(8), 1488–1497 (1961).
[Crossref]

F. G. Mitri, “Acoustic scattering of a cylindrical quasi-Gaussian beam with arbitrary incidence focused on a rigid elliptical cylinder,” J. Appl. Phys. 118(18), 184902 (2015).
[Crossref]

F. G. Mitri, “Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves,” J. Appl. Phys. 118(21), 214903 (2015).
[Crossref]

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]

F. G. Mitri, “Airy acoustical-sheet spinner tweezers,” J. Appl. Phys. 120(10), 104901 (2016).
[Crossref]

F. G. Mitri, “Extinction efficiency of “elastic–sheet” beams by a cylindrical (viscous) fluid inclusion embedded in an elastic medium and mode conversion—Examples of nonparaxial Gaussian and Airy beams,” J. Appl. Phys. 120(14), 144902 (2016).
[Crossref]

F. G. Mitri, “Acoustics of finite asymmetric exotic beams: Examples of Airy and fractional Bessel beams,” J. Appl. Phys. 122(22), 224903 (2017).
[Crossref]

J. Comp. Acous. (1)

S. M. Hasheminejad and R. Sanaei, “Acoustic radiation force and torque on a solid elliptic cylinder,” J. Comp. Acous. 15(03), 377–399 (2007).
[Crossref]

J. Dispersion Sci. Technol. (1)

S. M. Hasheminejad and R. Sanaei, “Ultrasonic Scattering by a Viscoelastic Fiber of Elliptic Cross-Section Suspended in a Viscous Fluid Medium,” J. Dispersion Sci. Technol. 27(8), 1165–1179 (2006).
[Crossref]

J. Opt. (2)

F. G. Mitri, “Radiation force and torque of light-sheets,” J. Opt. 19(6), 065403 (2017).
[Crossref]

F. G. Mitri, “Cylindrical particle manipulation and negative spinning using a nonparaxial Hermite−Gaussian light-sheet beam,” J. Opt. 18(10), 105402 (2016).
[Crossref]

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

C. Rockstuhl and H. P. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A: Pure Appl. Opt. 6(10), 921–931 (2004).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

J. Phys. Commun. (1)

F. G. Mitri, “Acoustic radiation force on a cylindrical particle near a planar rigid boundary,” J. Phys. Commun. 2(4), 045019 (2018).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (3)

F. G. Mitri, “Pushing, pulling and electromagnetic radiation force cloaking by a pair of conducting cylindrical particles,” J. Quant. Spectrosc. Radiat. Transfer 206, 142–150 (2018).
[Crossref]

F. G. Mitri, “Generalization of the optical theorem for monochromatic electromagnetic beams of arbitrary wavefront in cylindrical coordinates,” J. Quant. Spectrosc. Radiat. Transfer 166, 81–92 (2015).
[Crossref]

F. G. Mitri, “Active electromagnetic invisibility cloaking and radiation force cancellation,” J. Quant. Spectrosc. Radiat. Transfer 207, 48–53 (2018).
[Crossref]

J. Sound. Vib. (1)

Z. Gong, W. Li, F. G. Mitri, Y. Chai, and Y. Zhao, “Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio,” J. Sound. Vib. 383, 233–247 (2016).
[Crossref]

J. Wave Motion (1)

F. G. Mitri, “Acoustic radiation force on oblate and prolate spheroids in Bessel beams,” J. Wave Motion 57, 231–238 (2015).
[Crossref]

Ocean Eng. (1)

Z. Gong, W. Li, Y. Chai, Y. Zhao, and F. G. Mitri, “T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps,” Ocean Eng. 129, 507–519 (2017).
[Crossref]

Opt. Lett. (1)

OSA Continuum (1)

Phys. Fluids (2)

F. G. Mitri, “Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence,” Phys. Fluids 28(7), 077104 (2016).
[Crossref]

S. D. Deshmukh and R. M. Thaokar, “Deformation, breakup and motion of a perfect dielectric drop in a quadrupole electric field,” Phys. Fluids 24(3), 032105 (2012).
[Crossref]

Phys. Fluids A (1)

S. Sankaran and D. A. Saville, “Experiments on the stability of a liquid bridge in an axial electric field,” Phys. Fluids A 5(4), 1081–1083 (1993).
[Crossref]

Phys. Rev. (1)

P. M. Morse and P. J. Rubenstein, “The Diffraction of Waves by Ribbons and by Slits,” Phys. Rev. 54(11), 895–898 (1938).
[Crossref]

Sens. Actuators B (1)

S. Mhatre, “Dielectrophoretic motion and deformation of a liquid drop in an axisymmetric non-uniform AC electric field,” Sens. Actuators B 239, 1098–1108 (2017).
[Crossref]

Ultrasonics (8)

F. G. Mitri, “Optical theorem for two-dimensional (2D) scalar monochromatic acoustical beams in cylindrical coordinates,” Ultrasonics 62, 20–26 (2015).
[Crossref]

F. G. Mitri, “Extended optical theorem for scalar monochromatic acoustical beams of arbitrary wavefront in cylindrical coordinates,” Ultrasonics 67, 129–135 (2016).
[Crossref]

F. G. Mitri, “Axial time-averaged acoustic radiation force on a cylinder in a nonviscous fluid revisited,” Ultrasonics 50(6), 620–627 (2010).
[Crossref]

F. G. Mitri, “Interaction of an acoustical 2D-beam with an elastic cylinder with arbitrary location in a non-viscous fluid,” Ultrasonics 62, 244–252 (2015).
[Crossref]

F. G. Mitri, “Axial acoustic radiation force on rigid oblate and prolate spheroids in Bessel vortex beams of progressive, standing and quasi-standing waves,” Ultrasonics 74, 62–71 (2017).
[Crossref]

F. G. Mitri, “Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves,” Ultrasonics 66, 27–33 (2016).
[Crossref]

F. G. Mitri, “Acoustical spinner tweezers with nonparaxial Hermite-Gaussian acoustical-sheets and particle dynamics,” Ultrasonics 73, 236–244 (2017).
[Crossref]

F. G. Mitri, “Scattering of Airy elastic sheets by a cylindrical cavity in a solid,” Ultrasonics 81, 100–106 (2017).
[Crossref]

Wave Motion (1)

F. G. Mitri, “Acoustic radiation force and spin torque on a viscoelastic cylinder in a quasi-Gaussian cylindrically-focused beam with arbitrary incidence in a non-viscous fluid,” Wave Motion 66, 31–44 (2016).
[Crossref]

Z. Angew. Math. Phys. (1)

N. Dodgson and C. Sozou, “The deformation of a liquid drop by an electric field,” Z. Angew. Math. Phys. 38(3), 424–432 (1987).
[Crossref]

Other (10)

H. C. van de Hulst, Light scattering by small particles, Light scattering by small particles (John Wiley and Sons, Inc., 1957), p. xiii + 470p.

F. Onofri and S. Barbosa, “Optical Particle Characterization,” in Laser Metrology in Fluid Mechanics (John Wiley & Sons, Inc., 2013), pp. 67–158.

P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill Book Co., 1953), Vol. 2.

S. DeBenedetti, Nuclear Interactions (John Wiley & Sons, Incorporated, 1964).

P. Roman, Introduction to Quantum Field Theory (Wiley, 1969).

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and acoustic scattering by simples shapes (John Wiley & Sons, 1969).

Acoustic, electromagnetic and elastic wave scattering - Focus on the T-matrix approach, V. K. Varadan and V. V. Varadan, eds. (Pergamon, 1980).

F. G. Mitri, See the response to some misleading and obtuse comments in: “Reply to: Comments on [Phys. Fluids 28, 077104 (2016)],” https://arxiv.org/abs/1703.00487 (2017).

F. G. Mitri, “Acoustic radiation force and torque on a viscous fluid cylindrical particle nearby a planar rigid wall,” https://arxiv.org/abs/1801.04863 (2017).

G. Gouesbet and G. Grehan, Generalized Lorenz-Mie Theories, 1st ed. (Springer, 2011).

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Figures (8)

Fig. 1.
Fig. 1. A sketch describing the interaction of an incident electric field polarized along the axial z-direction perpendicular to the plane of the figure (known also as TM polarization) with a perfect electrically conducting elliptical smooth cross-section in a homogeneous medium. The semi-axes of the ellipse are denoted by a and b, respectively. The cylindrical coordinate system (r, θ, z) is referenced to the center of the elliptical cylinder.
Fig. 2.
Fig. 2. Panel (a) displays the convergence plots for the dimensionless radiation force function versus the maximum truncation order |nmax| assuming TM polarized plane waves at kb = 5 for various aspect ratios of the elliptical cylinder with a smooth surface. The circular cylinder case corresponds to the plot having a/b = 1. Panel (b) displays similar convergence plots assuming TE polarized plane progressive waves.
Fig. 3.
Fig. 3. Panel (a) displays the plots for the dimensionless radiation force function assuming TM polarization of the incident plane waves for Rayleigh electrically conducting elliptical cylinders of various aspect ratios a/b versus kb. Panel (b) corresponds to the same plots but in a larger kb-range.
Fig. 4.
Fig. 4. The same as in Fig. 3, but the incident plane wave field is TE polarized.
Fig. 5.
Fig. 5. Panel (a) displays the plot for the surface shape function of a corrugated elliptical cylinder having an aspect ratio a/b = 0.5 and $\ell$  = 10. Panel (b) corresponds to the case where a/b = 1, while panel (c) is for an aspect ratio a/b = 1.5. In all the panels, d/a = 0.1.
Fig. 6.
Fig. 6. Convergence plots for the dimensionless radiation force function versus the maximum truncation order nmax assuming TE polarized plane waves at kb = 5 for various aspect ratios of the elliptical cylinder with a corrugated surface having d/a = 0.1 and $\ell$  = 10.
Fig. 7.
Fig. 7. The plots for the dimensionless radiation force function assuming TE polarization of the incident plane waves for electrically conducting elliptical cylinders with corrugated surfaces [as shown in Fig. 5] of various aspect ratios a/b versus kb.
Fig. 8.
Fig. 8. Panel (a) shows a comparison of the plots for the dimensionless radiation force function assuming TE polarization of the incident plane waves for an electrically conducting elliptical cylinder having an aspect ratio a/b = 0.5 without (d/a = 0) and with corrugations (d/a = 0.1). Panels (b) and (c) display similar plots but for aspect ratios a/b = 1 and 1.5, respectively.

Equations (28)

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E z i n c = E 0 e i ω t n i n J n ( k r ) e i n θ ,
E z s c a = E 0 e i ω t n i n C n TM H n ( 1 ) ( k r ) e i n θ ,
A θ = [ ( cos θ / a ) 2 + ( sin θ / b ) 2 ] 1 / 2 + d cos ( θ ) ,
( n × E ) e θ | r = A θ = 0 ,
n  =  e r ( 1 A θ ) d A θ d θ e θ ,
n i n [ Γ n TM ( θ ) + C n TM Υ n TM ( θ ) ] = 0 ,
{ Γ n TM ( θ ) Υ n TM ( θ ) } = e i n θ { J n ( k A θ ) H n ( 1 ) ( k A θ ) } .
n i n [ Γ n TM ( θ ) + C n TM Υ n TM ( θ ) ] = n [ ψ n TM + C n TM Ω n TM ] e i n θ = 0.
0 2 π e i ( n m ) θ d θ = 2 π δ n , m ,
m [ ψ n , m TM + C n TM Ω n , m TM ] = 0 ,
{ ψ n , m TM Ω n , m TM } = 1 2 π n i n 0 2 π { Γ n TM ( θ ) Υ n TM ( θ ) } e i m θ d θ .
( n × E ) e z | r = A θ = 0 ,
m [ ψ n , m TE + C n TE Ω n , m TE ] = 0 ,
{ ψ n , m TE Ω n , m TE } = 1 2 π n i n 0 2 π { Γ n TE ( θ ) Υ n TE ( θ ) } e i m θ d θ ,
{ Γ n TE ( θ ) Υ n TE ( θ ) } = e i n θ [ k { J n ( k A θ ) H n ( 1 ) ( k A θ ) } i ( n A θ 2 ) d A θ d θ { J n ( k A θ ) H n ( 1 ) ( k A θ ) } ] .
{ f TM ( k , θ ) f TE ( k , θ ) } = ( a e f f / a ) ( 2 r / a ) { E z s c a / E 0 H z s c a / H 0 } e i ( k r ω t ) = 2 i π k a ( a e f f / a ) n { C n TM C n TE } e i n θ ,
F TM = k r 1 2 c { S [ E z , i n c H θ , s c a + E z , s c a H θ , i n c + E z , s c a H θ , s c a ] d S } ,
F TM = k r ε 2 c S { E i s } d S ,
F TE = k r 1 2 c { S [ E θ , i n c H z , s c a + E θ , s c a H z , i n c + E θ , s c a H z , s c a ] d S } .
F TE = k r 1 2 c ε S { H i s } d S ,
F x TM L = ε 2 c r 0 2 π { E z i n c E z s c a ( 1 + cos θ ) + | E z s c a | 2 cos θ } d θ .
0 2 π { E z i n c E z s c a ( 1 + cos θ ) } d θ = 0 2 π | E z s c a | 2 d θ .
F x TM = ε 2 c 0 2 π | E z s c a | 2 ( 1 cos θ ) d S .
Y p TM = ( a / a e f f ) 1 4 0 2 π | f TM ( k , θ ) | 2 ( 1 cos θ ) d θ ,
F x TE = 1 2 c ε 0 2 π | H z s c a | 2 ( 1 cos θ ) d S ,
Y p TE = ( a / a e f f ) 1 4 0 2 π | f TE ( k , θ ) | 2 ( 1 cos θ ) d θ .
Y p { TM,TE } = 1 k a { n C n { TM,TE } ( C n + 1 { TM,TE } + C n 1 { TM,TE } + 2 ) } .
Y p { TM,TE } = 1 k a { n sin δ n { TM,TE } e i δ n { TM,TE } × ( sin δ n + 1 { TM,TE } e i δ n + 1 { TM,TE } + sin δ n 1 { TM,TE } e i δ n 1 { TM,TE } + 2 ) } .

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