Abstract

Since 1908, when Mie reported analytical expressions for the fields scattered by a spherical particle upon incidence of plane-waves, generalizing his analysis for the case of an arbitrary incident wave has been an open question because of the cancellation of the prefactor radial spherical Bessel function. This cancellation was obtained before by our own group for a highly focused beam centered in the objective. In this work, however, we show for the first time how these terms can be canceled out for any arbitrary incident field that satisfies Maxwells equations, and obtain analytical expressions for the beam shape coefficients. We show several examples on how to use our method to obtain analytical beam shape coefficients for: Bessel beams, general hollow waveguide modes and specific geometries such as cylindrical and rectangular. Our method uses the vector potential, which shows the interesting characteristic of being gauge invariant. These results are highly relevant for speeding up numerical calculation of light scattering applications such as the radiation forces acting on spherical particles placed in an arbitrary electromagnetic field, as in an optical tweezers system.

© 2016 Optical Society of America

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2014 (2)

G. Gouesbet, “Latest achievements in generalized Lorenz-Mie theories : A commented reference database,” Annalen der Physik 526, 461–489 (2014).
[Crossref]

F. Lanusse, J.-L. Starck, A. Woiselle, and M. J. Fadili, “3D Sparse Representations”, Advances in Imaging and Electron Physics,  183, 99–204, (2014).
[Crossref]

2013 (2)

2009 (2)

2008 (1)

T. J. Dufva, J. Sarvas, and J. C.-E. Sten, “Unified Derivation of the Translational Addition theorems for Spherical Scalar and Vector Wave Functions,” Progress in Electromagnetics Research B,  4, 79–99, (2008).
[Crossref]

2006 (3)

2005 (2)

J. Ng, C. T. Chan, P. Sheng, and Z. Lin, “Strong optical force induced by morphology-dependent resonances,” Opt. Lett. 30, 1956–1958 (2005).
[Crossref] [PubMed]

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]

2003 (1)

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

2002 (3)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

R. Suda and M. Takami, “A fast spherical harmonics transform algorithm,” Math. Comp. 71, 703–715 (2002).
[Crossref]

2001 (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 62–637 (2001).
[Crossref]

2000 (2)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

1999 (1)

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[Crossref]

1998 (2)

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]

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

1997 (1)

1996 (1)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

1994 (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[Crossref] [PubMed]

1993 (2)

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

1992 (1)

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[Crossref] [PubMed]

1988 (1)

1987 (1)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref] [PubMed]

1985 (1)

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” Journal of Optics 16, 83–93 (1985).
[Crossref]

1982 (1)

G. Gouesbet and G. Grehan, “Sur la generalisation de la theorie de Lorenz-Mie,” Journal of Optics 13, 97–103 (1982).
[Crossref]

1979 (2)

G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

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

1974 (1)

J. E. Hansen and J. W. Hovenier, “Interpretation of the Polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974).
[Crossref]

1970 (1)

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

1955 (1)

A. Nisbet, “Hertzian Electromagnetic Potentials and Associated Gauge Transformations”, Proc. R. Soc. A 231, 250–260 (1955).
[Crossref]

1953 (1)

P Moon and D. E. Spencer, “The meaning of the vector Laplacian,” J. Franklin Inst.,  6, 551–558 (1953).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908). See the English translation at: http://diogenes.iwt.uni-bremen.de/vt/laser/papers/SAND78-6018-Mie-1908-translation.pdf
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, 1970).

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]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Arnold, S.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Ashkin, A.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref] [PubMed]

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

Barbosa, L. C.

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] [PubMed]

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: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. 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] [PubMed]

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]

Barends, P.

G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

Betzig, E.

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[Crossref] [PubMed]

Birks, T. A.

Block, S. M.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[Crossref] [PubMed]

Blom, P.

G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

Brakenhoff, G. J.

G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Brune, M.

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Cesar, C. L.

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] [PubMed]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. 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] [PubMed]

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: Math. Gen. 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]

Chan, C. T.

Chen, J. S. Y.

Cheung, G.

Chillce, E.

Collot, L.

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Cremer, C.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Cruz, C. H. B.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. 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] [PubMed]

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: Math. Gen. 39, L293–L296 (2006).
[Crossref]

Davis, L. W.

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

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. 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]

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Dufva, T. J.

T. J. Dufva, J. Sarvas, and J. C.-E. Sten, “Unified Derivation of the Translational Addition theorems for Spherical Scalar and Vector Wave Functions,” Progress in Electromagnetics Research B,  4, 79–99, (2008).
[Crossref]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref] [PubMed]

Euser, T. G.

Fadili, M. J.

F. Lanusse, J.-L. Starck, A. Woiselle, and M. J. Fadili, “3D Sparse Representations”, Advances in Imaging and Electron Physics,  183, 99–204, (2014).
[Crossref]

Fontes, 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: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. 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] [PubMed]

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] [PubMed]

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]

Furusawa, A.

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

Garbos, M. K.

Georgiades, N. Ph.

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

Gouesbet, G.

G. Gouesbet, “Latest achievements in generalized Lorenz-Mie theories : A commented reference database,” Annalen der Physik 526, 461–489 (2014).
[Crossref]

G. Gouesbet and J. A. Lock, “List of problems for future research in generalized Lorenz-Mie theories and related topics, review and prospectus [Invited],” Appl. Opt. 52, 897–916 (2013).
[Crossref] [PubMed]

G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[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]

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” Journal of Optics 16, 83–93 (1985).
[Crossref]

G. Gouesbet and G. Grehan, “Sur la generalisation de la theorie de Lorenz-Mie,” Journal of Optics 13, 97–103 (1982).
[Crossref]

G. Gouesbet and G. Grehan, Generalized Lorenz-Mie theories (Springer, Berlin, 2011).
[Crossref]

Grehan, G.

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” Journal of Optics 16, 83–93 (1985).
[Crossref]

G. Gouesbet and G. Grehan, “Sur la generalisation de la theorie de Lorenz-Mie,” Journal of Optics 13, 97–103 (1982).
[Crossref]

G. Gouesbet and G. Grehan, Generalized Lorenz-Mie theories (Springer, Berlin, 2011).
[Crossref]

Gréhan, G.

Hansen, J. E.

J. E. Hansen and J. W. Hovenier, “Interpretation of the Polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974).
[Crossref]

Haroche, S.

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Heckenberg, N. R.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 62–637 (2001).
[Crossref]

Hell, S.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Hovenier, J. W.

J. E. Hansen and J. W. Hovenier, “Interpretation of the Polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974).
[Crossref]

Ilchenko, V. S.

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

J. D. Jackson, “From Lorenz to Coulomb and other explicit gauge transformations”, arXiv:physics/0204034v2 (2002).

Jaques, F.

Khoshsima, M.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Kimble, H. J.

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Knight, J. C.

Lanusse, F.

F. Lanusse, J.-L. Starck, A. Woiselle, and M. J. Fadili, “3D Sparse Representations”, Advances in Imaging and Electron Physics,  183, 99–204, (2014).
[Crossref]

Lefèvre-Seguin, V.

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Lin, Z.

Lock, J. A.

Lock, J.A.

J.A. Lock, “Angular spectrum and localized model of Davis-type beam,” Journal of the Optical Society of America A 30, 489–500 (2013).
[Crossref]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Maheu, B.

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]

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” Journal of Optics 16, 83–93 (1985).
[Crossref]

McDonald, K. T.

K. T. McDonald, “Bessel Beams,” arXiv:physics/0006046v1 (2000).

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908). See the English translation at: http://diogenes.iwt.uni-bremen.de/vt/laser/papers/SAND78-6018-Mie-1908-translation.pdf
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Moon, P

P Moon and D. E. Spencer, “The meaning of the vector Laplacian,” J. Franklin Inst.,  6, 551–558 (1953).
[Crossref]

Moreira, W.

W. Moreira, “rvswf: Vector Spherical Wave Functions in R,” figshare. Retrieved 14:53, Aug20, 2015 (GMT). http://dx.doi.org/10.6084/m9.figshare.1513857 .

Moreira, W. L.

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]

Neves, A. A. R.

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] [PubMed]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. 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] [PubMed]

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: Math. Gen. 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]

Ng, J.

Nieminen, T. A.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 62–637 (2001).
[Crossref]

Nisbet, A.

A. Nisbet, “Hertzian Electromagnetic Potentials and Associated Gauge Transformations”, Proc. R. Soc. A 231, 250–260 (1955).
[Crossref]

Novotny, L.

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[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]

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: Math. Gen. 39, L293–L296 (2006).
[Crossref]

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. 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] [PubMed]

Painter, O.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Pozzo, L. de Y.

Raimond, J. M.

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

Reiner, G.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Rodriguez, E.

Rubinsztein-Dunlop, H.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 62–637 (2001).
[Crossref]

Russell, P. S. J.

Sánchez, E. J.

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[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]

Sarvas, J.

T. J. Dufva, J. Sarvas, and J. C.-E. Sten, “Unified Derivation of the Translational Addition theorems for Spherical Scalar and Vector Wave Functions,” Progress in Electromagnetics Research B,  4, 79–99, (2008).
[Crossref]

Sheng, P.

Spencer, D. E.

P Moon and D. E. Spencer, “The meaning of the vector Laplacian,” J. Franklin Inst.,  6, 551–558 (1953).
[Crossref]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Starck, J.-L.

F. Lanusse, J.-L. Starck, A. Woiselle, and M. J. Fadili, “3D Sparse Representations”, Advances in Imaging and Electron Physics,  183, 99–204, (2014).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, 1970).

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Sten, J. C.-E.

T. J. Dufva, J. Sarvas, and J. C.-E. Sten, “Unified Derivation of the Translational Addition theorems for Spherical Scalar and Vector Wave Functions,” Progress in Electromagnetics Research B,  4, 79–99, (2008).
[Crossref]

Suda, R.

R. Suda and M. Takami, “A fast spherical harmonics transform algorithm,” Math. Comp. 71, 703–715 (2002).
[Crossref]

Svoboda, K.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[Crossref] [PubMed]

Takami, M.

R. Suda and M. Takami, “A fast spherical harmonics transform algorithm,” Math. Comp. 71, 703–715 (2002).
[Crossref]

Teraoka, I.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Thomaz, A. A. de

Trautman, J. K.

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[Crossref] [PubMed]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Vernooy, D. W.

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

Vollmer, F.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).

Woiselle, A.

F. Lanusse, J.-L. Starck, A. Woiselle, and M. J. Fadili, “3D Sparse Representations”, Advances in Imaging and Electron Physics,  183, 99–204, (2014).
[Crossref]

Xie, X. S.

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[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]

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref] [PubMed]

Advances in Imaging and Electron Physics (1)

F. Lanusse, J.-L. Starck, A. Woiselle, and M. J. Fadili, “3D Sparse Representations”, Advances in Imaging and Electron Physics,  183, 99–204, (2014).
[Crossref]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908). See the English translation at: http://diogenes.iwt.uni-bremen.de/vt/laser/papers/SAND78-6018-Mie-1908-translation.pdf
[Crossref]

Annalen der Physik (1)

G. Gouesbet, “Latest achievements in generalized Lorenz-Mie theories : A commented reference database,” Annalen der Physik 526, 461–489 (2014).
[Crossref]

Annu. Rev. Biophys. Biomol. Struct. (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Europhys. Lett. (1)

L. Collot, V. Lefèvre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high–Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
[Crossref]

J. Atmos. Sci. (1)

J. E. Hansen and J. W. Hovenier, “Interpretation of the Polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974).
[Crossref]

J. Franklin Inst. (1)

P Moon and D. E. Spencer, “The meaning of the vector Laplacian,” J. Franklin Inst.,  6, 551–558 (1953).
[Crossref]

J. Microsc. (2)

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

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

J. Phys. A: Math. Gen. (1)

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: Math. Gen. 39, L293–L296 (2006).
[Crossref]

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

G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[Crossref]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 62–637 (2001).
[Crossref]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[Crossref]

Journal of Optics (2)

G. Gouesbet and G. Grehan, “Sur la generalisation de la theorie de Lorenz-Mie,” Journal of Optics 13, 97–103 (1982).
[Crossref]

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” Journal of Optics 16, 83–93 (1985).
[Crossref]

Journal of the Optical Society of America A (1)

J.A. Lock, “Angular spectrum and localized model of Davis-type beam,” Journal of the Optical Society of America A 30, 489–500 (2013).
[Crossref]

Math. Comp. (1)

R. Suda and M. Takami, “A fast spherical harmonics transform algorithm,” Math. Comp. 71, 703–715 (2002).
[Crossref]

Nature (2)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[Crossref] [PubMed]

Opt. Commun. (1)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (2)

D. W. Vernooy, A. Furusawa, N. Ph. Georgiades, V. S. Ilchenko, and H. J. Kimble, “Cavity QED with high–Q whispering gallery modes,” Phys. Rev. A 57, R2293–R2296 (1998).
[Crossref]

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

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (3)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[Crossref]

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

Proc. R. Soc. A (1)

A. Nisbet, “Hertzian Electromagnetic Potentials and Associated Gauge Transformations”, Proc. R. Soc. A 231, 250–260 (1955).
[Crossref]

Progress in Electromagnetics Research B (1)

T. J. Dufva, J. Sarvas, and J. C.-E. Sten, “Unified Derivation of the Translational Addition theorems for Spherical Scalar and Vector Wave Functions,” Progress in Electromagnetics Research B,  4, 79–99, (2008).
[Crossref]

Science (2)

P. S. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

E. Betzig and J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[Crossref] [PubMed]

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 (8)

G. Gouesbet and G. Grehan, Generalized Lorenz-Mie theories (Springer, Berlin, 2011).
[Crossref]

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).

K. T. McDonald, “Bessel Beams,” arXiv:physics/0006046v1 (2000).

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, 1970).

R Core Team, “R: A Language and Environment for Statistical Computing,” http://www.R-project.org , (2013).

W. Moreira, “rvswf: Vector Spherical Wave Functions in R,” figshare. Retrieved 14:53, Aug20, 2015 (GMT). http://dx.doi.org/10.6084/m9.figshare.1513857 .

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Figures (2)

Fig. 1
Fig. 1 z-Component of the Poynting vectors with respect to truncation in lMAX along the diagonal (y = x).
Fig. 2
Fig. 2 Plots of the components of the Poynting vector of the rectangular and cylindrical TM waveguides and the two cases Bessel Beams: in density is the real part of the z component (normalized); the vector field in blue is the imaginary part of transverse component and in magenta its the real part. If we treat the TE case, the vectors represent the imaginary part (magenta) will invert its directions. All the other values remain unchanged, the same happens if we change p→−p. The green lines indicate the origin of the expansion.

Equations (37)

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[ E ( r ) Z H ( r ) ] = E 0 l = 1 m = 1 l ( [ G l m T E G l m T M ] M l m ( r ) + [ G l m T M G l m T E ] N l m ( r ) ) ,
E 0 j l ( k r ) [ G l m T E G l m T M ] = d Ω ( r ^ ) X l m ( r ^ ) [ E ( r ) Z H ( r ) ] ,
[ E ( r ) H ( r ) ] = 1 ( 2 π ) 3 / 2 d 3 k [ ε ( k ) ( k ) ] e i k r .
L e i k r = ( i r × d d r ) e i k r = ( i r × i k ) e i k r = ( i k × i r ) e i k r ( i k × d d k ) e i k r .
d 3 k Ψ ( k ) ( i k × d d k ) e i k r = i ε l m n e ^ l d k x d k y d k z Ψ k m k n e i k x x e i k y y e i k z z .
d k z Ψ k z e i k z z = d k z e i k z z k z Ψ .
[ L ψ ( r ) ] = { ψ ( r ) } = Ψ ( k ) .
E 0 j l ( k r ) [ G l m T E G l m T M ] = i l 2 π d 3 k l m ( k ) [ ε ( k ) Z ( k ) ] .
[ ε ( k ) ( k ) ] = δ ( k k ) k 2 [ ε k ( k ^ ) k ( k ^ ) ] .
[ G l m T E G l m T M ] = i l E 0 2 π d Ω ( k ^ ) X l m ( k ^ ) [ ε k ( k ^ ) Z k ( k ^ ) ] .
[ G l m T E G l m T M ] = i l E 0 2 π p , q [ E p , q H p , q ] l , m | | p , q l ( l + 1 ) .
E = Φ + i k c A 0 A
Z H = c A 0 × A ,
ε k = i k Φ ˜ + i E 0 A k
Z k = i E 0 k ^ × A k .
[ G l m T E G l m T M ] = i l + 1 2 π d Ω ( k ^ ) X l m ( k ^ ) [ A k ( k ^ ) k ^ × A k ( k ^ ) ] .
[ E ( r ) Z H ( r ) ] = E 0 e i k r [ ε ^ k ^ × ε ^ ]
[ G l m T E G l m T M ] = i l 4 π X l m * ( k ^ ) [ ε ^ k ^ × ε ^ ] .
[ G l m T E G l m T M ] = i l 2 π ( 2 l + 1 ) δ l p [ 1 i p ] ,
[ d 2 d ρ 2 + γ 2 ] g γ ( ρ ) = 0 ,
[ E T M Z E T E ] = k z k z ^ × [ Z H T M E T E ] = E 0 [ z ^ + i k z γ 2 d d ρ ] g ( r ) ,
[ T M Z T E ] = E 0 G ( k ) [ z ^ k z γ 2 γ ]
[ T E Z T m ] = E 0 k z γ 2 G ( k ) ζ ^ .
[ G l m T E [ T M ] G l m T M [ T E ] ] = 2 i l m l ( l + 1 ) k 2 γ 2 Q l m ( k z k ) G γ m
[ G l m T M [ T M ] G l m T E [ T E ] ] = 2 i l 1 l ( l + 1 ) Q l m ( k z k ) G γ m ,
G l m = d ζ G γ ( ζ ) e i m ζ .
G γ m = α ( e i m ζ e { i m e i ϕ } ± e i m ζ e { i m e i ϕ + } )
ψ m ( k ; r ) = J m ( γ ρ ) e i s m ϕ e i k z z .
Ψ m ( k ; k ^ ) = 2 π ( i ) m e i s m ζ δ ( ξ ξ ) sin ξ .
ψ m ( k ; r + r 0 ) = j = ψ m j ( k ; r 0 ) ψ j ( k ; r ) .
G γ ( ζ ) = 2 π j = ψ M j ( k ; ρ 0 ) ( i ) j e i s j ζ
G γ m = 2 π ( i ) s m ψ M s m ( k ; ρ 0 ) .
G γ m = 2 π ( i ) M δ m , s M .
[ E Z H ] z = i E 0 [ ψ M z ^ + [ ψ M z ^ ] / k 2 i × [ ψ M z ^ ] / k ]
[ E z H ] p = i E 0 [ ψ M e ^ p + [ ψ M e ^ p ] / k 2 i × [ ψ M e ^ p ] / k ] .
[ G l m T E G l m T M ] z = W l , s , m ψ M s m ( k ; r 0 ) [ m Q l m ( k z / k ) i ( γ / k ) 2 Q l m ( k z / k ) ]
[ G l m T E G l m T M + i p ( k z / k ) G l m T E ] p = W l , s , m 2 ψ M s ( m p ) ( k ; r 0 ) [ c p l m Q l m p i p m ( γ / k ) Q l m ]

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