Abstract

Arthur Ashkin was awarded the 2018 Nobel Prize in Physics for the invention of optical tweezers. Since their first publication in 1986, optical tweezers have been used as a tool to measure forces and rheological properties of microscopic systems. For the calibration of these measurements, knowledge of the forces is fundamental. However, it is still common to deduce the optical forces from assumptions based on the particle size with respect to the trapping laser wavelength. This shows the necessity to develop a complete and accurate electromagnetic model that does not depend on early approximations of the force model. Furthermore, the model we have developed has several advantages, such as morphology-dependent resonances, size dependence for large spheres, and multipole effects from smaller particles, just to name a few. In this tutorial, we review and discuss the physical modeling of optical forces in optical tweezers, which are the resultant forces exerted by a trapping beam on a sphere of any size and composition.

© 2019 Optical Society of America

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [Crossref]
  2. 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, 288–290 (1986).
    [Crossref]
  3. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).
  4. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 335, 755–776 (1909).
    [Crossref]
  5. M. Kerker and E. M. Loebl, eds., The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  6. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
    [Crossref]
  7. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [Crossref]
  8. R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
    [Crossref]
  9. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [Crossref]
  10. A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
    [Crossref]
  11. A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006).
    [Crossref]
  12. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [Crossref]
  13. L. Lorenz, Lysbevægelsen i og uden for en af plane Lysbølger belyst Kugle, Kongelige Danske Videnskabernes Selskabs Skrifter (Naturvidenskabelig og Mathematisk Afdeling, 1890).
  14. N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
    [Crossref]
  15. G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [Crossref]
  16. 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. A 5, 1427–1443 (1988).
    [Crossref]
  17. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2017).
  18. 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, 303–312 (1983).
    [Crossref]
  19. P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984).
    [Crossref]
  20. 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, 4594–4602 (1989).
    [Crossref]
  21. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [Crossref]
  22. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [Crossref]
  23. P. A. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
    [Crossref]
  24. A. Rohrbach and E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2001).
    [Crossref]
  25. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
    [Crossref]
  26. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force,” Appl. Opt. 43, 2545–2554 (2004).
    [Crossref]
  27. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
    [Crossref]
  28. J. Chen, J. Ng, P. Wang, and Z. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding,” Opt. Lett. 35, 1674–1676 (2010).
    [Crossref]
  29. T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
    [Crossref]
  30. P. Jones, O. Marago, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge, 2015).
  31. J. M. Taylor and G. D. Love, “Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations,” J. Opt. Soc. Am. A 26, 278–282 (2009).
    [Crossref]
  32. A. Salandrino, S. Fardad, and D. N. Christodoulides, “Generalized Mie theory of optical forces,” J. Opt. Soc. Am. B 29, 855–866(2012).
    [Crossref]
  33. M. Šiler and P. Zemànek, “Optical trapping in secondary maxima of focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 162, 114–121 (2015).
    [Crossref]
  34. W. Lu, H. Chen, S. Liu, and Z. Lin, “Rigorous full-wave calculation of optical forces on dielectric and metallic microparticles immersed in a vector airy beam,” Opt. Express 25, 23238–23253 (2017).
    [Crossref]
  35. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  36. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [Crossref]
  37. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Science Series (Wiley, 2008).
  38. E. L. Hill, “The theory of vector spherical harmonics,” Am. J. Phys. 22, 211–214 (1954).
    [Crossref]
  39. W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1935).
    [Crossref]
  40. G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [Crossref]
  41. T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1005–1017 (2003).
    [Crossref]
  42. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
    [Crossref]
  43. W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
    [Crossref]
  44. L. Novotny, E. J. Sánchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998).
    [Crossref]
  45. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref]
  46. D. Ganic, X. Gan, and M. Gu, “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express 12, 2670–2675 (2004).
    [Crossref]
  47. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [Crossref]
  48. C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [Crossref]
  49. S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
    [Crossref]
  50. J. Kim, Y. Wang, and X. Zhang, “Calculation of vectorial diffraction in optical systems,” J. Opt. Soc. Am. A 35, 526–535 (2018).
    [Crossref]
  51. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [Crossref]
  52. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge, 2012).
  53. J. Peatross, M. Berrondo, D. Smith, and M. Ware, “Vector fields in a tight laser focus: comparison of models,” Opt. Express 25, 13990–14007 (2017).
    [Crossref]
  54. V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd1 (1919).
  55. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).
  56. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [Crossref]
  57. F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express 15, 11984–11998 (2007).
    [Crossref]
  58. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
    [Crossref]
  59. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [Crossref]
  60. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620–1631 (2005).
    [Crossref]
  61. A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
    [Crossref]
  62. S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Int. J. Math. Math. Sci. 2007, 1–5 (2007).
    [Crossref]
  63. P. J. Cregg and P. Svedlindh, “Comment on analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A 40, 14029–14031 (2007).
    [Crossref]
  64. S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
    [Crossref]
  65. Z. Gong, P. L. Marston, W. Li, and Y. Chai, “Multipole expansion of acoustical Bessel beams with arbitrary order and location,” J. Acoust. Soc. Am. 141, EL574–EL578 (2017).
    [Crossref]
  66. I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011).
    [Crossref]
  67. R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
    [Crossref]
  68. D. Braun and J. Martin, “Spontaneous emission from a two-level atom tunneling in a double-well potential,” Phys. Rev. A 77, 032102 (2008).
    [Crossref]
  69. A. P. Polychronakos and K. Sfetsos, “High spin limits and non-abelian T-duality,” Nucl. Phys. 843, 344–361 (2011).
    [Crossref]
  70. J. J. Blanco-Pillado and M. P. Salem, “Observable effects of anisotropic bubble nucleation,” J. Cosmol. Astropart. Phys. 2010, 007 (2010).
    [Crossref]
  71. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
    [Crossref]
  72. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
    [Crossref]
  73. J. A. Lock, “Angular spectrum and localized model of Davis-type beam,” J. Opt. Soc. Am. A 30, 489–500 (2013).
    [Crossref]
  74. A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
    [Crossref]
  75. A. A. R. Neves, “Força óptica em pinças ópticas: estudo teórico e experimental,” Ph.D. thesis (Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas (2006).
  76. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [Crossref]
  77. A. A. R. Neves and D. Pisignano, “Effect of finite terms on the truncation error of Mie series,” Opt. Lett. 37, 2418–2420 (2012).
    [Crossref]
  78. J. R. Allardice and E. C. L. Ru, “Convergence of Mie theory series: criteria for far-field and near-field properties,” Appl. Opt. 53, 7224–7229 (2014).
    [Crossref]
  79. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [Crossref]
  80. A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
    [Crossref]
  81. E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42, 3915–3926 (2003).
    [Crossref]
  82. K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45, 1812–1819 (2006).
    [Crossref]
  83. M. Jahnel, M. Behrndt, A. Jannasch, E. Schäffer, and S. W. Grill, “Measuring the complete force field of an optical trap,” Opt. Lett. 36, 1260–1262 (2011).
    [Crossref]
  84. I. A. Martínez and D. Petrov, “Force mapping of an optical trap using an acousto-optical deflector in a time-sharing regime,” Appl. Opt. 51, 5522–5526 (2012).
    [Crossref]
  85. I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
    [Crossref]
  86. M. Selmke, M. Braun, and F. Cichos, “Photothermal single-particle microscopy: detection of a nanolens,” ACS Nano 6, 2741–2749 (2012).
    [Crossref]
  87. W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).
  88. W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Opt. Express 24, 2370–2382 (2016).
    [Crossref]
  89. A. A. R. Neves, “Optical forces in optical tweezers,” https://github.com/aneves76/OpticalForcesInOpticalTweezers (2019).

2018 (1)

2017 (3)

2016 (2)

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Opt. Express 24, 2370–2382 (2016).
[Crossref]

2015 (2)

R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
[Crossref]

M. Šiler and P. Zemànek, “Optical trapping in secondary maxima of focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 162, 114–121 (2015).
[Crossref]

2014 (2)

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

J. R. Allardice and E. C. L. Ru, “Convergence of Mie theory series: criteria for far-field and near-field properties,” Appl. Opt. 53, 7224–7229 (2014).
[Crossref]

2013 (2)

I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
[Crossref]

J. A. Lock, “Angular spectrum and localized model of Davis-type beam,” J. Opt. Soc. Am. A 30, 489–500 (2013).
[Crossref]

2012 (4)

2011 (3)

M. Jahnel, M. Behrndt, A. Jannasch, E. Schäffer, and S. W. Grill, “Measuring the complete force field of an optical trap,” Opt. Lett. 36, 1260–1262 (2011).
[Crossref]

A. P. Polychronakos and K. Sfetsos, “High spin limits and non-abelian T-duality,” Nucl. Phys. 843, 344–361 (2011).
[Crossref]

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011).
[Crossref]

2010 (2)

J. J. Blanco-Pillado and M. P. Salem, “Observable effects of anisotropic bubble nucleation,” J. Cosmol. Astropart. Phys. 2010, 007 (2010).
[Crossref]

J. Chen, J. Ng, P. Wang, and Z. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding,” Opt. Lett. 35, 1674–1676 (2010).
[Crossref]

2009 (1)

2008 (1)

D. Braun and J. Martin, “Spontaneous emission from a two-level atom tunneling in a double-well potential,” Phys. Rev. A 77, 032102 (2008).
[Crossref]

2007 (5)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Int. J. Math. Math. Sci. 2007, 1–5 (2007).
[Crossref]

P. J. Cregg and P. Svedlindh, “Comment on analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A 40, 14029–14031 (2007).
[Crossref]

F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express 15, 11984–11998 (2007).
[Crossref]

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

2006 (4)

2005 (2)

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use of the vector addition theorem,” J. Opt. Soc. Am. B 22, 1620–1631 (2005).
[Crossref]

2004 (3)

2003 (3)

E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42, 3915–3926 (2003).
[Crossref]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1005–1017 (2003).
[Crossref]

2002 (1)

2001 (1)

2000 (1)

P. A. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

1999 (1)

1998 (2)

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[Crossref]

L. Novotny, E. J. Sánchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998).
[Crossref]

1997 (1)

1996 (2)

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (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]

1995 (2)

1993 (1)

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

1992 (3)

1989 (2)

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, 4594–4602 (1989).
[Crossref]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

1988 (1)

1987 (1)

1986 (1)

1984 (1)

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984).
[Crossref]

1983 (1)

1982 (1)

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[Crossref]

1980 (1)

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

1977 (1)

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

1965 (1)

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

1964 (1)

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

1954 (1)

E. L. Hill, “The theory of vector spherical harmonics,” Am. J. Phys. 22, 211–214 (1954).
[Crossref]

1935 (1)

W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1935).
[Crossref]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 335, 755–776 (1909).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Ajito, K.

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[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, 4594–4602 (1989).
[Crossref]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

Allardice, J. R.

Antezza, M.

R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
[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.

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

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, 288–290 (1986).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).

Axner, O.

Barbosa, L. C.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006).
[Crossref]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

Barth, I.

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011).
[Crossref]

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, 4594–4602 (1989).
[Crossref]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

Behrndt, M.

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, 715–717 (1993).
[Crossref]

Berrondo, M.

Bjorkholm, J. E.

Blanco-Pillado, J. J.

J. J. Blanco-Pillado and M. P. Salem, “Observable effects of anisotropic bubble nucleation,” J. Cosmol. Astropart. Phys. 2010, 007 (2010).
[Crossref]

Bohren, C.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Science Series (Wiley, 2008).

Booker, G. R.

Borghese, F.

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

Braun, D.

D. Braun and J. Martin, “Spontaneous emission from a two-level atom tunneling in a double-well potential,” Phys. Rev. A 77, 032102 (2008).
[Crossref]

Braun, M.

M. Selmke, M. Braun, and F. Cichos, “Photothermal single-particle microscopy: detection of a nanolens,” ACS Nano 6, 2741–2749 (2012).
[Crossref]

Brevik, I.

Bui, A. A.

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

Camposeo, A.

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

Cesar, C.

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

Cesar, C. L.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Opt. Express 24, 2370–2382 (2016).
[Crossref]

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

Chai, Y.

Z. Gong, P. L. Marston, W. Li, and Y. Chai, “Multipole expansion of acoustical Bessel beams with arbitrary order and location,” J. Acoust. Soc. Am. 141, EL574–EL578 (2017).
[Crossref]

Chen, H.

Chen, J.

Chillce, E.

Christodoulides, D. N.

Chu, S.

Cichos, F.

M. Selmke, M. Braun, and F. Cichos, “Photothermal single-particle microscopy: detection of a nanolens,” ACS Nano 6, 2741–2749 (2012).
[Crossref]

Cingolani, R.

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

Cregg, P. J.

P. J. Cregg and P. Svedlindh, “Comment on analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A 40, 14029–14031 (2007).
[Crossref]

Crichton, J. H.

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984).
[Crossref]

Cruz, C. H. B.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
[Crossref]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

de Brito Cruz, C. H.

de Paula, A. M.

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

de Thomaz, A. A.

A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006).
[Crossref]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 335, 755–776 (1909).
[Crossref]

Denti, P.

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Doicu, A.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Dziedzic, J. M.

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, 288–290 (1986).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

Euser, T.

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

Euser, T. G.

Fällman, E.

Fardad, S.

Fontes, A.

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006).
[Crossref]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

Furuya, K.

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

Gan, X.

Ganic, D.

Garbos, M.

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

Garbos, M. K.

Gong, Z.

Z. Gong, P. L. Marston, W. Li, and Y. Chai, “Multipole expansion of acoustical Bessel beams with arbitrary order and location,” J. Acoust. Soc. Am. 141, EL574–EL578 (2017).
[Crossref]

Gouesbet, G.

Gréhan, G.

Grill, S. W.

Gu, M.

Guizal, B.

R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
[Crossref]

Gussgard, R.

Hansen, W. W.

W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1935).
[Crossref]

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]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge, 2012).

Heckenberg, N.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1005–1017 (2003).
[Crossref]

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

Hell, S.

Hill, E. L.

E. L. Hill, “The theory of vector spherical harmonics,” Am. J. Phys. 22, 211–214 (1954).
[Crossref]

Huffman, D.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Science Series (Wiley, 2008).

Iatì, M. A.

Ignatowsky, V.

V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd1 (1919).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Jahnel, M.

Jannasch, A.

Jones, P.

P. Jones, O. Marago, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge, 2015).

Kim, J.

Kim, J. S.

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

Koumandos, S.

S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Int. J. Math. Math. Sci. 2007, 1–5 (2007).
[Crossref]

Koyama, S.

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

Laczik, Z.

Lee, S. S.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Li, W.

Z. Gong, P. L. Marston, W. Li, and Y. Chai, “Multipole expansion of acoustical Bessel beams with arbitrary order and location,” J. Acoust. Soc. Am. 141, EL574–EL578 (2017).
[Crossref]

Lin, Z.

Lindmo, T.

Liu, S.

Lock, J. A.

Logan, N. A.

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

Loke, V. L.

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

Lorenz, L.

L. Lorenz, Lysbevægelsen i og uden for en af plane Lysbølger belyst Kugle, Kongelige Danske Videnskabernes Selskabs Skrifter (Naturvidenskabelig og Mathematisk Afdeling, 1890).

Love, G. D.

Lu, W.

Mackowski, D. W.

Maheu, B.

Maia Neto, P. A.

R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Marago, O.

P. Jones, O. Marago, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge, 2015).

Marston, P. L.

Z. Gong, P. L. Marston, W. Li, and Y. Chai, “Multipole expansion of acoustical Bessel beams with arbitrary order and location,” J. Acoust. Soc. Am. 141, EL574–EL578 (2017).
[Crossref]

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984).
[Crossref]

Martin, J.

D. Braun and J. Martin, “Spontaneous emission from a two-level atom tunneling in a double-well potential,” Phys. Rev. A 77, 032102 (2008).
[Crossref]

Martínez, I. A.

Martnez, I. A.

I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
[Crossref]

Matthews, H. J.

McCutchen, C. W.

Messina, R.

R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
[Crossref]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Mishchenko, M. I.

Moine, O.

Moreira, W.

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

Moreira, W. L.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Opt. Express 24, 2370–2382 (2016).
[Crossref]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

Neto, P. A. M.

P. A. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

Neves, A.

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

Neves, A. A. R.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Opt. Express 24, 2370–2382 (2016).
[Crossref]

A. A. R. Neves and D. Pisignano, “Effect of finite terms on the truncation error of Mie series,” Opt. Lett. 37, 2418–2420 (2012).
[Crossref]

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. de Y. Pozzo, A. A. de Thomaz, E. Chillce, E. Rodriguez, L. C. Barbosa, and C. L. Cesar, “Electromagnetic forces for an arbitrary optical trapping of a spherical dielectric,” Opt. Express 14, 13101–13106 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref]

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

A. A. R. Neves, “Força óptica em pinças ópticas: estudo teórico e experimental,” Ph.D. thesis (Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas (2006).

Ng, J.

Nieminen, T.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1005–1017 (2003).
[Crossref]

Nieminen, T. A.

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

Novotny, L.

L. Novotny, E. J. Sánchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge, 2012).

Nussenzveig, H. M.

P. A. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

Padilha, L. A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[Crossref]

Parrondo, J. M. R.

I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
[Crossref]

Peatross, J.

Petrov, D.

I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
[Crossref]

I. A. Martínez and D. Petrov, “Force mapping of an optical trap using an acousto-optical deflector in a time-sharing regime,” Appl. Opt. 51, 5522–5526 (2012).
[Crossref]

Pisignano, D.

A. A. R. Neves and D. Pisignano, “Effect of finite terms on the truncation error of Mie series,” Opt. Lett. 37, 2418–2420 (2012).
[Crossref]

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

Polychronakos, A. P.

A. P. Polychronakos and K. Sfetsos, “High spin limits and non-abelian T-duality,” Nucl. Phys. 843, 344–361 (2011).
[Crossref]

Pozzo, L. de Y.

Preez-Wilkinson, N. D.

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

Ren, K. F.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Rodriguez, E.

Rohrbach, A.

Roldán, E.

I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
[Crossref]

Ru, E. C. L.

Rubinsztein-Dunlop, H.

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1005–1017 (2003).
[Crossref]

Russell, P. S. J.

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

Saija, R.

Salandrino, A.

Salem, M. P.

J. J. Blanco-Pillado and M. P. Salem, “Observable effects of anisotropic bubble nucleation,” J. Cosmol. Astropart. Phys. 2010, 007 (2010).
[Crossref]

Sánchez, E. J.

L. Novotny, E. J. Sánchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998).
[Crossref]

Saruwatari, H.

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

Schäffer, E.

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, 4594–4602 (1989).
[Crossref]

Schmidt, C. F.

Selmke, M.

M. Selmke, M. Braun, and F. Cichos, “Photothermal single-particle microscopy: detection of a nanolens,” ACS Nano 6, 2741–2749 (2012).
[Crossref]

Sfetsos, K.

A. P. Polychronakos and K. Sfetsos, “High spin limits and non-abelian T-duality,” Nucl. Phys. 843, 344–361 (2011).
[Crossref]

Sheppard, C. J. R.

Shimauchi, S.

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

Šiler, M.

M. Šiler and P. Zemànek, “Optical trapping in secondary maxima of focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 162, 114–121 (2015).
[Crossref]

Smirnova, O.

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011).
[Crossref]

Smith, D.

Sonek, G. J.

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

Stelzer, E. H. K.

Stienen, G. J. M.

Stilgoe, A. B.

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

Stout, B.

Svedlindh, P.

P. J. Cregg and P. Svedlindh, “Comment on analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A 40, 14029–14031 (2007).
[Crossref]

Taylor, J. M.

Török, P.

Varga, P.

Vermeulen, K. C.

Volpe, G.

P. Jones, O. Marago, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge, 2015).

Wakayama, K.

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

Wang, P.

Wang, Y.

Ware, M.

Wiscombe, W. J.

Wolf, E.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Wriedt, T.

Wright, W. H.

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

Wuite, G. J. L.

Xie, X. S.

L. Novotny, E. J. Sánchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998).
[Crossref]

Zemànek, P.

M. Šiler and P. Zemànek, “Optical trapping in secondary maxima of focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 162, 114–121 (2015).
[Crossref]

Zhang, X.

ACS Nano (1)

M. Selmke, M. Braun, and F. Cichos, “Photothermal single-particle microscopy: detection of a nanolens,” ACS Nano 6, 2741–2749 (2012).
[Crossref]

Am. J. Phys. (1)

E. L. Hill, “The theory of vector spherical harmonics,” Am. J. Phys. 22, 211–214 (1954).
[Crossref]

Ann. Phys. (2)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 335, 755–776 (1909).
[Crossref]

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Appl. Opt. (11)

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
[Crossref]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. II. On-axis trapping force,” Appl. Opt. 43, 2545–2554 (2004).
[Crossref]

G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[Crossref]

A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
[Crossref]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz-Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[Crossref]

I. A. Martínez and D. Petrov, “Force mapping of an optical trap using an acousto-optical deflector in a time-sharing regime,” Appl. Opt. 51, 5522–5526 (2012).
[Crossref]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[Crossref]

J. R. Allardice and E. C. L. Ru, “Convergence of Mie theory series: criteria for far-field and near-field properties,” Appl. Opt. 53, 7224–7229 (2014).
[Crossref]

A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
[Crossref]

E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42, 3915–3926 (2003).
[Crossref]

K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45, 1812–1819 (2006).
[Crossref]

Appl. Phys. Lett. (1)

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

Biophys. J. (1)

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

Europhys. Lett. (1)

P. A. M. Neto and H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

Int. J. Math. Math. Sci. (1)

S. Koumandos, “On a class of integrals involving a Bessel function times Gegenbauer polynomials,” Int. J. Math. Math. Sci. 2007, 1–5 (2007).
[Crossref]

J. Acoust. Soc. Am. (2)

S. Koyama, K. Furuya, K. Wakayama, S. Shimauchi, and H. Saruwatari, “Analytical approach to transforming filter design for sound field recording and reproduction using circular arrays with a spherical baffle,” J. Acoust. Soc. Am. 139, 1024–1036 (2016).
[Crossref]

Z. Gong, P. L. Marston, W. Li, and Y. Chai, “Multipole expansion of acoustical Bessel beams with arbitrary order and location,” J. Acoust. Soc. Am. 141, EL574–EL578 (2017).
[Crossref]

J. Appl. Phys. (2)

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, 4594–4602 (1989).
[Crossref]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

J. Cosmol. Astropart. Phys. (1)

J. J. Blanco-Pillado and M. P. Salem, “Observable effects of anisotropic bubble nucleation,” J. Cosmol. Astropart. Phys. 2010, 007 (2010).
[Crossref]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

J. Opt. (Paris) (1)

G. Gouesbet and G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[Crossref]

J. Opt. A (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[Crossref]

S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
[Crossref]

J. Kim, Y. Wang, and X. Zhang, “Calculation of vectorial diffraction in optical systems,” J. Opt. Soc. Am. A 35, 526–535 (2018).
[Crossref]

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
[Crossref]

J. A. Lock, “Angular spectrum and localized model of Davis-type beam,” J. Opt. Soc. Am. A 30, 489–500 (2013).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[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. A 5, 1427–1443 (1988).
[Crossref]

A. Rohrbach and E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2001).
[Crossref]

J. M. Taylor and G. D. Love, “Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations,” J. Opt. Soc. Am. A 26, 278–282 (2009).
[Crossref]

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[Crossref]

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

J. Phys. A (2)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Analytical results for a Bessel function times Legendre polynomials class integrals,” J. Phys. A 39, L293–L296 (2006).
[Crossref]

P. J. Cregg and P. Svedlindh, “Comment on analytical results for a Bessel function times Legendre polynomials class integrals’,” J. Phys. A 40, 14029–14031 (2007).
[Crossref]

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

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1005–1017 (2003).
[Crossref]

M. Šiler and P. Zemànek, “Optical trapping in secondary maxima of focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 162, 114–121 (2015).
[Crossref]

T. A. Nieminen, N. D. Preez-Wilkinson, A. B. Stilgoe, V. L. Loke, A. A. Bui, and H. Rubinsztein-Dunlop, “Optical tweezers: theory and modelling,” J. Quant. Spectrosc. Radiat. Transfer 146, 59–80 (2014).
[Crossref]

Nucl. Phys. (1)

A. P. Polychronakos and K. Sfetsos, “High spin limits and non-abelian T-duality,” Nucl. Phys. 843, 344–361 (2011).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (6)

Opt. Lett. (5)

Phys. Rev. (1)

W. W. Hansen, “A new type of expansion in radiation problems,” Phys. Rev. 47, 139–143 (1935).
[Crossref]

Phys. Rev. A (5)

I. Barth and O. Smirnova, “Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations,” Phys. Rev. A 84, 063415 (2011).
[Crossref]

R. Messina, P. A. Maia Neto, B. Guizal, and M. Antezza, “Casimir interaction between a sphere and a grating,” Phys. Rev. A 92, 062504 (2015).
[Crossref]

D. Braun and J. Martin, “Spontaneous emission from a two-level atom tunneling in a double-well potential,” Phys. Rev. A 77, 032102 (2008).
[Crossref]

P. L. Marston and J. H. Crichton, “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A 30, 2508–2516 (1984).
[Crossref]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Phys. Rev. E (3)

A. Fontes, K. Ajito, A. A. R. Neves, W. L. Moreira, A. A. de Thomaz, L. C. Barbosa, A. M. de Paula, and C. L. Cesar, “Raman, hyper-Raman, hyper-Rayleigh, two-photon luminescence and morphology-dependent resonance modes in a single optical tweezers system,” Phys. Rev. E 72, 012903 (2005).
[Crossref]

A. A. R. Neves, A. Fontes, C. L. Cesar, A. Camposeo, R. Cingolani, and D. Pisignano, “Axial optical trapping efficiency through a dielectric interface,” Phys. Rev. E 76, 061917 (2007).
[Crossref]

I. A. Martnez, E. Roldán, J. M. R. Parrondo, and D. Petrov, “Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,” Phys. Rev. E 87, 032159 (2013).
[Crossref]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[Crossref]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[Crossref]

Proc. IEEE (1)

N. A. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[Crossref]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Q. Appl. Math. (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
[Crossref]

Ultramicroscopy (1)

L. Novotny, E. J. Sánchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998).
[Crossref]

Other (13)

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley Science Series (Wiley, 2008).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge, 2012).

V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd1 (1919).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

A. A. R. Neves, “Força óptica em pinças ópticas: estudo teórico e experimental,” Ph.D. thesis (Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas (2006).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

P. Jones, O. Marago, and G. Volpe, Optical Tweezers: Principles and Applications (Cambridge, 2015).

A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific, 2006).

M. Kerker and E. M. Loebl, eds., The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

L. Lorenz, Lysbevægelsen i og uden for en af plane Lysbølger belyst Kugle, Kongelige Danske Videnskabernes Selskabs Skrifter (Naturvidenskabelig og Mathematisk Afdeling, 1890).

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2017).

W. Moreira, A. Neves, M. Garbos, T. Euser, P. S. J. Russell, and C. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” arXiv:1003.2392 (2010).

A. A. R. Neves, “Optical forces in optical tweezers,” https://github.com/aneves76/OpticalForcesInOpticalTweezers (2019).

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Figures (1)

Fig. 1.
Fig. 1. Illustration of the variables from the incoming beam (red rays), mapped onto a reference sphere and later focused (converging red rays). The focus is shifted from the origin of coordinates located at the center of the spherical scatterer.

Equations (97)

Equations on this page are rendered with MathJax. Learn more.

E inc ( r ) = E 0 l = 1 m = l l ( G l m TE M l m ( 1 ) ( r ) + G l m TM N l m ( 1 ) ( r ) ) ,
Z H inc ( r ) = E 0 l = 1 m = l l ( G l m TM M l m ( 1 ) ( r ) G l m TE N l m ( 1 ) ( r ) ) ,
E sca ( r ) = E 0 l = 1 m = l l ( b l m M l m ( 3 ) ( r ) + a l m N l m ( 3 ) ( r ) ) ,
Z H sca ( r ) = E 0 l = 1 m = l l ( a l m M l m ( 3 ) ( r ) b l m N l m ( 3 ) ( r ) ) ,
E int ( r ) = E 0 l = 1 m = l l ( d l m M l m ( 1 ) ( r ) + c l m N l m ( 1 ) ( r ) ) ,
Z H int ( r ) = E 0 l = 1 m = l l ( c l m M l m ( 1 ) ( r ) d l m N l m ( 1 ) ( r ) ) ,
[ E inc ( r ) + E sca ( r ) E int ( r ) ] × r ^ | r = a = 0 ,
[ H inc ( r ) + H sca ( r ) H int ( r ) ] × r ^ | r = a = 0 .
x = k 2 a = 2 π n 2 a λ , M x = n 1 n 2 k 2 a = k 1 a ,
l = 1 m = l l ( G l m TE j l ( x ) + b l m h l ( 1 ) ( x ) d l m j l ( M x ) ) X l m ( r ^ ) × r ^ = 0 .
G l m TE j l ( x ) + b l m h l ( 1 ) ( x ) d l m j l ( M x ) = 0 .
G l m TM j l ( x ) + a l m h l ( 1 ) ( x ) Z 1 Z 2 c l m j l ( M x ) = 0 .
N l m ( r ) × r ^ = i k ( × ( z l ( k r ) X l m ( r ^ ) ) ) × r ^ = i k r ( k r z l ( k r ) ) k r .
G l m TM ( k 1 r j l ( k 1 r ) ) k 1 r + a l m ( k 1 r h l ( 1 ) ) k 1 r c l m 1 M ( k 2 r j l ( k 2 r ) ) k 2 r = 0 .
G l m TE ( k 1 r j l ( k 1 r ) ) k 1 r + b l m ( k 1 r h l ( 1 ) ) k 1 r Z 1 Z 2 d l m 1 M ( k 2 r j l ( k 2 r ) ) k 2 r = 0 .
ψ l ( y ) = y j l ( y ) , ξ l ( y ) = y h l ( 1 ) ( y ) .
G l m TM ψ l ( x ) + a l m ξ l ( x ) c l m 1 M ψ l ( M x ) = 0
G l m TE ψ l ( x ) + b l m ξ l ( x ) Z 1 Z 2 1 M d l m ψ l ( M x ) = 0 .
Z 1 Z 2 1 M = μ 1 ϵ 2 ϵ 1 μ 2 n 1 n 2 = μ 1 ϵ 2 ϵ 1 μ 2 μ 1 ϵ 1 μ 2 ϵ 2 1 .
G l m TE ψ l ( x ) + b l m ξ l ( x ) 1 M d l m ψ l ( M x ) = 0 ,
G l m TM ψ l ( x ) + a l m ξ l ( x ) c l m ψ l ( M x ) = 0 ,
G l m TM ψ l ( x ) + a l m ξ l ( x ) 1 M c l m ψ l ( M x ) = 0 ,
G l m TE ψ l ( x ) + b l m ξ l ( x ) d l m ψ l ( M x ) = 0 .
a l m = G l m TM [ M ψ l ( x ) ψ l ( M x ) ψ l ( x ) ψ l ( M x ) M ξ l ( x ) ψ l ( M x ) ξ l ( x ) ψ l ( M x ) ] = G l m TM a l .
b l m = G l m TE [ M ψ l ( x ) ψ l ( M x ) ψ l ( x ) ψ l ( M x ) M ξ l ( x ) ψ l ( M x ) ξ l ( x ) ψ l ( M x ) ] = G l m TE b l ,
a l = M ψ l ( x ) ψ l ( M x ) ψ l ( x ) ψ l ( M x ) M ξ l ( x ) ψ l ( M x ) ξ l ( x ) ψ l ( M x ) ,
b l = M ψ l ( x ) ψ l ( M x ) ψ l ( x ) ψ l ( M x ) M ξ l ( x ) ψ l ( M x ) ξ l ( x ) ψ l ( M x ) .
Y l m * Y p q d Ω = δ l p δ m q ,
Y l m = 2 l + 1 4 π ( l m ) ! ( l + m ) ! P l m ( cos ( θ ) ) e i m ϕ .
r · E inc ( r ) = E 0 l = 1 m = l l G l m TM r · N l m ( 1 ) ( r ) .
r · N l m ( 1 ) ( r ) = j l ( k r ) k l ( l + 1 ) Y l m .
r · E inc ( r ) = E 0 k l = 1 m = l l G l m TM j l ( k r ) l ( l + 1 ) Y l m .
Y p q * r · E inc ( r ) d Ω = E 0 k l = 1 m = l l G l m TM j l ( k r ) l ( l + 1 ) Y p q * Y l m d Ω .
Y p q * r · E inc ( r ) d Ω = E 0 k G p q TM j p ( k r ) p ( p + 1 )
j p ( k r ) G p q TM = k r 1 E 0 p ( p + 1 ) Y p q * r ^ · E inc ( r ) d Ω .
j p ( k r ) G p q TE = k r Z E 0 p ( p + 1 ) Y p q * r ^ · H inc ( r ) d Ω .
E inc = E 0 e i k z x ^ .
r ^ · E inc ( r ) = E 0 e i k r cos θ sin θ cos ϕ .
G l m TM = k r j l ( k r ) 1 l ( l + 1 ) Y l m * e i k r cos θ sin θ cos ϕ d Ω = k r j l ( k r ) 1 l ( l + 1 ) 2 l + 1 4 π ( l m ) ! ( l + m ) ! θ = 0 π P l m ( cos θ ) e i k r cos θ sin 2 θ d θ ϕ = 0 2 π e i m ϕ cos ϕ d ϕ .
ϕ = 0 2 π e i m ϕ cos ϕ d ϕ = π ( δ m , + 1 + δ m , 1 ) .
G l , + 1 TM = k r j l ( k r ) π l ( l + 1 ) 2 l + 1 4 π θ = 0 π P l 1 ( cos θ ) e i k r cos θ sin 2 θ d θ .
θ = 0 π sin 2 θ e i k r cos θ P l 1 ( cos θ ) d θ = 2 i l + 1 l ( l + 1 ) j l ( k r ) k r .
G l , + 1 TM = π ( 2 l + 1 ) i l + 1 .
G l , 1 TM = π ( 2 l + 1 ) i l + 1 .
G l m TE = k r j l ( k r ) 1 l ( l + 1 ) Y p q * e i k r cos θ sin θ sin ϕ d Ω = k r j l ( k r ) 1 l ( l + 1 ) 2 l + 1 4 π ( l m ) ! ( l + m ) ! θ = 0 π P l m ( cos θ ) e i k r cos θ sin 2 θ d θ ϕ = 0 2 π e i m ϕ sin ϕ d ϕ .
ϕ = 0 2 π e i m ϕ sin ϕ d ϕ = i π ( δ m , + 1 δ m , 1 ) .
G l , ± 1 TE = i π ( 2 l + 1 ) i l + 1 .
r ^ · E inc ( r ) = E 0 l = 1 π ( 2 l + 1 ) i l + 1 r ^ · ( N l , + 1 ( r ) + N l , 1 ( r ) ) = E 0 l = 1 j l ( k r ) k r l ( l + 1 ) π ( 2 l + 1 ) i l + 1 ( Y l , + 1 + Y l , 1 ) = i E 0 k r cos ϕ sin θ d d ( cos θ ) l = 1 j l ( k r ) i l ( 2 l + 1 ) P l ( cos θ ) = E 0 e i k r cos θ sin θ cos ϕ .
E inc ( ρ , ϕ , z ) = i k f e i k f 2 π 0 θ NA 0 2 π E ( θ , ϕ ) e i k z cos θ e i k ρ sin θ cos ( ϕ ϕ ) sin θ d θ d ϕ ,
E = n b n a cos θ ( [ E par · ϕ ^ ] ϕ ^ + [ E par · ρ ^ ] θ ^ ) ,
ρ cos ϕ = r sin θ cos ϕ ρ 0 cos ϕ 0 ,
ρ sin ϕ = r sin θ sin ϕ ρ 0 sin ϕ 0 ,
z = r cos θ z 0 .
E inc ( ρ , ϕ , z ) = i k f e i k f 2 π 0 α NA d α sin α e i k r cos α cos θ e i k z 0 cos α 0 2 π d β E ( α , β ) e i k r sin α sin θ cos ( β ϕ ) e i k ρ 0 sin α cos ( β ϕ 0 ) .
E par = E 0 e f 2 sin 2 α / ω 2 ( p x x ^ + p y y ^ ) ,
Z H par = z ^ × E par = E 0 e f 2 sin 2 α / ω 2 ( p x y ^ p y x ^ ) .
r ^ · E = E x , sin θ cos ϕ + E y , sin θ sin ϕ + E z , cos θ .
r ^ · E ( α , β ) = n b n a cos α E 0 e f 2 sin 2 α / ω 2 ( [ sin β [ p x sin β + p y cos β ] + cos β cos α [ p x cos β + p y sin β ] ] sin θ cos ϕ + [ cos β [ p x sin β + p y cos β ] + sin β cos α [ p x cos β + p y sin β ] ] sin θ sin ϕ sin α [ p x cos β + p y sin β ] cos θ ) ,
r ^ · E inc ( ρ , ϕ , z ) = i k f e i k f E 0 2 π n b n a 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α e i k r cos α cos θ 0 2 π d β e i k r sin α sin θ cos ( β ϕ ) e i k ρ 0 sin α cos ( β ϕ 0 ) { [ sin β ( p x sin β + p y cos β ) + cos β cos α ( p x cos β + p y sin β ) ] sin θ cos ϕ + [ cos β ( p x sin β + p y cos β ) + sin β cos α ( p x cos β + p y sin β ) ] sin θ sin ϕ sin α ( p x cos β + p y sin β ) cos θ } .
G l m TM = k r j l ( k r ) 1 E 0 l ( l + 1 ) 2 l + 1 4 π ( l m ) ! ( l + m ) ! 0 π d θ P l m ( cos θ ) sin θ 0 2 π d ϕ e i m ϕ r ^ · E inc ( r ) = k r j l ( k r ) i k f e i k f 2 π n b n a 2 l + 1 4 π l ( l + 1 ) ( l m ) ! ( l + m ) ! 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α 0 2 π d β e i k ρ 0 sin α cos ( β ϕ 0 ) 0 π d θ P l m ( cos θ ) sin θ e i k r cos α cos θ 0 2 π d ϕ e i m ϕ e i k r sin α sin θ cos ( β ϕ ) { [ sin β ( p x sin β + p y cos β ) + cos β cos α ( p x cos β + p y sin β ) ] sin θ cos ϕ + [ cos β ( p x sin β + p y cos β ) + sin β cos α ( p x cos β + p y sin β ) ] sin θ sin ϕ sin α [ p x cos β + p y sin β ] cos θ } .
0 2 π d ϕ e i m ϕ e i x cos ( β ϕ ) ( cos ϕ sin ϕ 1 ) = 2 π i m e i m β ( ( sin β m J m ( x ) x + i cos β d J m ( x ) d x ) ( cos β m J m ( x ) x + i sin β d J m ( x ) d x ) J m ( x ) ) ,
A = i m + 1 k f e i k f k r j l ( k r ) n b n a 2 l + 1 4 π l ( l + 1 ) ( l m ) ! ( l + m ) ! , B = [ p x sin β + p y cos β ] , C = [ p x cos β + p y sin β ] .
G l m TM = A 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α 0 2 π d β e i k ρ 0 sin α cos ( β ϕ 0 ) e i m β 0 π d θ P l m ( cos θ ) sin θ e i k r cos α cos θ ( B sin θ m J m ( y ) y + i C cos α sin θ d J m ( y ) d y + C sin α cos θ J m ( y ) ) .
G l m TM = A 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α 1 k r 0 2 π d β e i k ρ 0 sin α cos ( β ϕ 0 ) e i q β [ m B sin α + i C d d α ] 0 π d θ P l m ( cos θ ) sin θ e i k r cos α cos θ J m ( k r sin α sin θ ) .
0 π d θ P l m ( cos θ ) sin θ e i k r cos α cos θ J m ( k r sin α sin θ ) = 2 i l m P l m ( cos α ) j l ( k r ) .
G l m TM = 2 i l m j l ( k r ) k r A 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α 0 2 π d β e i k ρ 0 sin α cos ( β ϕ 0 ) e i m β [ m B P l m ( cos α ) sin α + i C d P l m ( cos α ) d α ] .
π l m ( α ) = m P l m ( cos α ) sin α , τ l m ( α ) = d P l m ( cos α ) d α .
G l m TM = i l + 1 k f e i k f n b n a 2 l + 1 π l ( l + 1 ) ( l m ) ! ( l + m ) ! 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α 0 2 π d β e i k ρ 0 sin α cos ( β ϕ 0 ) e i m β [ ( p x π l m ( α ) + i p y τ l m ( α ) ) sin β + ( p y π l m ( α ) + i p x τ l m ( α ) ) cos β ] .
0 2 π d β e i x cos ( β ϕ 0 ) e i m β ( cos β sin β ) = 2 π i m e i m ϕ 0 ( sin ϕ 0 m J m ( x ) x + i cos ϕ 0 d J m ( x ) d x cos ϕ 0 m J m ( x ) x + i sin ϕ 0 d J m ( x ) d x ) ,
G l m TM = i l m + 1 k f e i k f e i m ϕ 0 n b n a 4 π ( 2 l + 1 ) l ( l + 1 ) ( l m ) ! ( l + m ) ! 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α { [ p x π l m ( α ) + i p y τ l m ( α ) ] [ cos ϕ 0 m J m ( x ) x + i sin ϕ 0 d J m ( x ) d x ] + [ p y π l m ( α ) + i p x τ l m ( α ) ] [ sin ϕ 0 m J m ( x ) x + i cos ϕ 0 d J m ( x ) d x ] } .
G l m TE = i l m + 1 k f e i k f e i m ϕ 0 n b n a 4 π ( 2 l + 1 ) l ( l + 1 ) ( l m ) ! ( l + m ) ! 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α { [ p y π l m ( α ) + i p x τ l m ( α ) ] [ cos ϕ 0 m J m ( x ) x + i sin ϕ 0 d J m ( x ) d x ] + [ p x π l m ( α ) i p y τ l m ( α ) ] [ sin ϕ 0 m J m ( x ) x + i cos ϕ 0 d J m ( x ) d x ] } .
G l m TM , TE = i l m + 1 k f e i k f e i m ϕ 0 n b n a 4 π ( 2 l + 1 ) l ( l + 1 ) ( l m ) ! ( l + m ) ! 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α ( [ m J m ( k ρ 0 sin α ) k ρ 0 sin α π l m ( α ) + J m ( k ρ 0 sin α ) τ l m ( α ) ] , i [ m J m ( k ρ 0 sin α ) k ρ 0 sin α τ l m ( α ) + J m ( k ρ 0 sin α ) π l m ( α ) ] ) · ( cos ϕ 0 sin ϕ 0 sin ϕ 0 cos ϕ 0 ) · ( p x , p y p y , p x ) .
lim x 0 d J m ( x ) d x = 1 2 ( δ m , + 1 δ m , 1 ) , lim x 0 m J m ( x ) x = 1 2 ( δ m , + 1 + δ m , 1 ) .
G l , ± 1 TM = i l ( k f e i k f ) l ( l + 1 ) n b n a π ( 2 l + 1 ) ( p x + i p y ) 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α ( π l 1 ( α ) + τ l 1 ( α ) ) ,
G l , ± 1 TE = i l ( k f e i k f ) l ( l + 1 ) n b n a π ( 2 p + 1 ) ( p y ± i p x ) 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α ( π l 1 ( α ) + τ l 1 ( α ) ) ,
G l , ± 1 TM = ( p x + i p y ) G l , G l , ± 1 TE = ( p y ± i p x ) G l ,
G l = i l ( k f e i k f ) l ( l + 1 ) n b n a π ( 2 l + 1 ) 0 α NA d α sin α cos α e f 2 sin 2 α / ω 2 e i k z 0 cos α ( π l 1 ( α ) + τ l 1 ( α ) ) .
F i = T i j n j d A = 1 2 R [ ϵ E i E j * + μ H i H j * 1 2 ( ϵ E · E * + μ H · H * ) δ i j ] n j d A ,
F = r 2 4 R ( ϵ E · E * + μ H · H * ) r ^ d Ω .
× ( z l ( k r ) X l m ( r ^ ) ) = k z l ( k r ) ( r ^ × X l m ( r ^ ) ) + z l ( k r ) × X l m ( r ^ ) .
× ( z l ( k r ) X l m ( r ^ ) ) = k z l ( k r ) ( r ^ × X l m ( r ^ ) ) .
j l ( k r ) = ( i ) l + 1 e i k r 2 k r + i l + 1 e i k r 2 k r ,
j l ( k r ) = ( i ) l e i k r 2 k r + i l e i k r 2 k r ,
h l ( 1 ) ( k r ) = ( i ) l + 1 e i k r k r ,
h l ( 1 ) ( k r ) = ( i ) l e i k r k r .
( r ^ × X l m ) · ( r ^ × X p q * ) = r ^ · [ X p q * × ( r ^ × X l m ) ] = r ^ · [ r ^ X l m · X p q * X l m ( r ^ · X p q * ) ] = X l m · X p q *
X p q * · ( r ^ × X l m ) = X p q * · ( X l m × r ^ ) = X l m · ( r ^ × X p q * ) .
F = ϵ | E 0 | 2 4 k 2 R l = 1 m = l l p = 1 q = p p i p l { [ G l m TM G p q TM * ( a l + a p * 2 a l a p * ) + G l m TE G p q TE * ( b l + b p * 2 b l b p * ) ] X l m · X p q * r ^ d Ω + [ G l m TM G p q TE * ( a l + b p * 2 a l b p * ) G l m TE G p q TM * ( b l + a p * 2 b l a p * ) ] X l m · ( r ^ × X p q * ) r ^ d Ω } .
sin θ X l m · X p q * e i ϕ d Ω = δ q , m + 1 δ l , p + 1 p + 1 p ( p + 2 ) ( p m + 1 ) ( p m ) ( 2 p + 3 ) ( 2 p + 1 ) δ q , m + 1 δ l + 1 , p l + 1 l ( l + 2 ) ( l + m + 2 ) ( l + m + 1 ) ( 2 l + 3 ) ( 2 l + 1 ) ,
sin θ X l m · ( r ^ × X p q * ) e i ϕ d Ω = δ q , m + 1 δ l , p i ( l m ) ( l + m + 1 ) l ( l + 1 ) ,
cos θ X l m · X p q * d Ω = δ q , m δ l , p + 1 p + 1 p ( p + 2 ) ( p + m + 1 ) ( p m + 1 ) ( 2 p + 3 ) ( 2 p + 1 ) + δ q , m δ l + 1 , p l + 1 l ( l + 2 ) ( l + m + 1 ) ( l m + 1 ) ( 2 l + 3 ) ( 2 l + 1 ) ,
cos θ X l m · ( r ^ × X p q * ) d Ω = δ q , m δ l , p i m l ( l + 1 ) .
[ F x F y ] = ϵ | E 0 | 2 ( 2 k ) 2 [ R I ] p = 1 q = p p i p + 1 { p ( p + 2 ) ( p + q + 2 ) ( p + q + 1 ) ( 2 p + 3 ) ( 2 p + 1 ) × [ A p G p + 1 , ( q + 1 ) TM G p , q TM * + A p * G p q TM G p + 1 , q + 1 TM * + B p G p + 1 , ( q + 1 ) TE G p , q TE * + B p * G p q TE G p + 1 , q + 1 TE * ] ( p q ) ( p + q + 1 ) p [ C p G p q TM G p , q + 1 TE * C p * G p q TE G p , q + 1 TM * ] } ,
F z = ϵ | E 0 | 2 2 k 2 R p = 1 q = p p i p + 1 { p ( p + 2 ) ( p + q + 1 ) ( p q + 1 ) ( 2 p + 1 ) ( 2 p + 3 ) × [ A p G p + 1 , q TM G p q TM * + B p G p + 1 , q TE G p q TE * ] q p C p G p q TM G p q TE * } ,
A p = a p + 1 + a p * 2 a p + 1 a p * , B p = b p + 1 + b p * 2 b p + 1 b p * , C p = a p + b p * 2 a p b p * .
N max = 0.76 ϵ 2 / 3 x 1 / 3 4.1 ,
κ q = ( F q q ) r = r eq ,