Abstract

Electromagnetic scattering of a Gaussian beam by an off-axis dielectric sphere is treated by the sum-of-waves formulation, which is inherent in Lorenz–Mie theory. Each “wave” is a spherical eigenvector, defined in the natural frame of the scatterer, and the coefficient of that wave is the “wave amplitude.” Decomposition of the beam into homogeneous plane waves lays the ground for a synthesis of the wave amplitudes, which is done by an integration over the polar angle that defines the direction of propagation of the plane-wave constituents of the beam. Concise analytical results are developed for (a) the electric-field intensity in every part of space, (b) the bistatic and monostatic radar cross sections of the scatterer, and (c) the power extracted from the beam by scattering and absorption. Numerical calculations are made for a spherical glycerol droplet of radius 1.5 μm that is excited by an adjacent, infrared, Gaussian beam of wavelength 1.1424 μm and spot size 2 μm. The numerical application manifests (a) how the beam is coupled with the droplet and (b) the effect of the droplet on the power intercepted by a receiver-end fiber placed on the beam axis, beyond the focal plane. Comparisons to numerical results obtained by a finite-element method software (a) validate the sum-of-waves theory, (b) evaluate the performance of the code implementing that theory, and, succinctly, (c) manifest the limits of the plane-wave decomposition of the beam.

© 2017 Optical Society of America

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References

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  1. L. V. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbolger belyst Kugle,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).
  2. L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphere transparente,” in Œuvres Scientifiques de L. Lorenz, Revues et Annotées par H. Valentiner (Lehmann & Stage, 1898), Vol. 1, pp. 403–529.
  3. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [Crossref]
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. I.
  5. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  6. P. W. Barber and S. S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
  7. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, 2000).
  8. W. Hergert and T. Wriedt, eds., The Mie Theory (Springer, 2012).
  9. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Part II.
  10. M. Kahnert, “Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: a tutorial review,” J. Quant. Spectrosc. Radiat. Transfer 178, 22–37 (2016).
    [Crossref]
  11. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
    [Crossref]
  12. M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
    [Crossref]
  13. N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
    [Crossref]
  14. E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [Crossref]
  15. G. Gouesbet and J. A. Lock, “On the electromagnetic scattering of arbitrary shaped beams by arbitrary shaped particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 162, 31–49 (2015).
    [Crossref]
  16. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A 24, 1695–1703 (2007).
    [Crossref]
  17. A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. A 24, 3437–3443 (2007).
    [Crossref]
  18. D. P. Chrissoulidis and J. M. Laheurte, “Dyadic Green’s function of a nonspherical model of the human torso,” IEEE Trans. Microwave Theory Tech. 62, 1265–1274 (2014).
    [Crossref]
  19. D. P. Chrissoulidis and J. M. Laheurte, “Radiation from an encapsulated Hertz dipole implanted in a human torso model,” IEEE Trans. Antennas Propag. 64, 4984–4992 (2016).
    [Crossref]
  20. ANSYS, Inc., “HFSS high frequency structure simulator,” Cecil Township, Washington County, Pennsylvania (2015).
  21. N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
    [Crossref]
  22. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  23. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [Crossref]
  24. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
    [Crossref]
  25. G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
    [Crossref]
  26. J. A. Lock, “Angular spectrum and localized model of Davis-type beam,” J. Opt. Soc. Am. A 30, 489–500 (2013).
    [Crossref]

2016 (2)

M. Kahnert, “Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: a tutorial review,” J. Quant. Spectrosc. Radiat. Transfer 178, 22–37 (2016).
[Crossref]

D. P. Chrissoulidis and J. M. Laheurte, “Radiation from an encapsulated Hertz dipole implanted in a human torso model,” IEEE Trans. Antennas Propag. 64, 4984–4992 (2016).
[Crossref]

2015 (1)

G. Gouesbet and J. A. Lock, “On the electromagnetic scattering of arbitrary shaped beams by arbitrary shaped particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 162, 31–49 (2015).
[Crossref]

2014 (1)

D. P. Chrissoulidis and J. M. Laheurte, “Dyadic Green’s function of a nonspherical model of the human torso,” IEEE Trans. Microwave Theory Tech. 62, 1265–1274 (2014).
[Crossref]

2013 (2)

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

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

2007 (3)

1997 (1)

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[Crossref]

1996 (1)

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[Crossref]

1995 (1)

1994 (1)

1993 (1)

E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

1979 (1)

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

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

1890 (1)

L. V. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbolger belyst Kugle,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Angelescu, D.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Barber, P. W.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

P. W. Barber and S. S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Basset, P.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bourouina, T.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Chrissoulidis, D. P.

D. P. Chrissoulidis and J. M. Laheurte, “Radiation from an encapsulated Hertz dipole implanted in a human torso model,” IEEE Trans. Antennas Propag. 64, 4984–4992 (2016).
[Crossref]

D. P. Chrissoulidis and J. M. Laheurte, “Dyadic Green’s function of a nonspherical model of the human torso,” IEEE Trans. Microwave Theory Tech. 62, 1265–1274 (2014).
[Crossref]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a sphere with an eccentric spherical inclusion,” J. Opt. Soc. Am. A 24, 1695–1703 (2007).
[Crossref]

A. P. Moneda and D. P. Chrissoulidis, “Dyadic Green’s function of a cluster of spheres,” J. Opt. Soc. Am. A 24, 3437–3443 (2007).
[Crossref]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[Crossref]

M. P. Ioannidou, N. C. Skaropoulos, and D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
[Crossref]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
[Crossref]

Davis, L. W.

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

Doicu, A.

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Part II.

Gaber, N.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Gouesbet, G.

G. Gouesbet and J. A. Lock, “On the electromagnetic scattering of arbitrary shaped beams by arbitrary shaped particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 162, 31–49 (2015).
[Crossref]

Hill, S. C.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

Hill, S. S.

P. W. Barber and S. S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Ioannidou, M. P.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. I.

Kahnert, M.

M. Kahnert, “Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: a tutorial review,” J. Quant. Spectrosc. Radiat. Transfer 178, 22–37 (2016).
[Crossref]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

Laheurte, J. M.

D. P. Chrissoulidis and J. M. Laheurte, “Radiation from an encapsulated Hertz dipole implanted in a human torso model,” IEEE Trans. Antennas Propag. 64, 4984–4992 (2016).
[Crossref]

D. P. Chrissoulidis and J. M. Laheurte, “Dyadic Green’s function of a nonspherical model of the human torso,” IEEE Trans. Microwave Theory Tech. 62, 1265–1274 (2014).
[Crossref]

Lock, J. A.

G. Gouesbet and J. A. Lock, “On the electromagnetic scattering of arbitrary shaped beams by arbitrary shaped particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 162, 31–49 (2015).
[Crossref]

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

Lorenz, L. V.

L. V. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbolger belyst Kugle,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphere transparente,” in Œuvres Scientifiques de L. Lorenz, Revues et Annotées par H. Valentiner (Lehmann & Stage, 1898), Vol. 1, pp. 403–529.

Malak, M.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Mengué, S.

G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
[Crossref]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Moneda, A. P.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Part II.

Nguyen, K. N.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Orjubin, G.

G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
[Crossref]

Picon, O.

G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
[Crossref]

Richalot, E.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
[Crossref]

Skaropoulos, N. C.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Wong, M. F.

G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
[Crossref]

Wriedt, T.

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[Crossref]

Yuan, X.

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

IEEE Trans. Antennas Propag. (2)

E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[Crossref]

D. P. Chrissoulidis and J. M. Laheurte, “Radiation from an encapsulated Hertz dipole implanted in a human torso model,” IEEE Trans. Antennas Propag. 64, 4984–4992 (2016).
[Crossref]

IEEE Trans. Electromagn. Compat. (1)

G. Orjubin, E. Richalot, S. Mengué, M. F. Wong, and O. Picon, “On the FEM modal approach for a reverberation chamber analysis,” IEEE Trans. Electromagn. Compat. 49, 76–85 (2007).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

D. P. Chrissoulidis and J. M. Laheurte, “Dyadic Green’s function of a nonspherical model of the human torso,” IEEE Trans. Microwave Theory Tech. 62, 1265–1274 (2014).
[Crossref]

N. C. Skaropoulos, M. P. Ioannidou, and D. P. Chrissoulidis, “Induced EM field in a layered eccentric spheres model of the head: plane-wave and localized source exposure,” IEEE Trans. Microwave Theory Tech. 44, 1963–1973 (1996).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Quant. Spectrosc. Radiat. Transfer (2)

M. Kahnert, “Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: a tutorial review,” J. Quant. Spectrosc. Radiat. Transfer 178, 22–37 (2016).
[Crossref]

G. Gouesbet and J. A. Lock, “On the electromagnetic scattering of arbitrary shaped beams by arbitrary shaped particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 162, 31–49 (2015).
[Crossref]

Lab Chip (1)

N. Gaber, M. Malak, X. Yuan, K. N. Nguyen, P. Basset, E. Richalot, D. Angelescu, and T. Bourouina, “On the free-space Gaussian beam coupling to droplet optical resonators,” Lab Chip 13, 826–833 (2013).
[Crossref]

Opt. Commun. (1)

A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[Crossref]

Phys. Rev. A (1)

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

Vidensk. Selsk. Skrifter (1)

L. V. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbolger belyst Kugle,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).

Other (9)

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphere transparente,” in Œuvres Scientifiques de L. Lorenz, Revues et Annotées par H. Valentiner (Lehmann & Stage, 1898), Vol. 1, pp. 403–529.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. I.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

P. W. Barber and S. S. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, 2000).

W. Hergert and T. Wriedt, eds., The Mie Theory (Springer, 2012).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Part II.

ANSYS, Inc., “HFSS high frequency structure simulator,” Cecil Township, Washington County, Pennsylvania (2015).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (10)

Fig. 1.
Fig. 1.

Electromagnetic Gaussian beam and off-axis dielectric sphere (not in scale).

Fig. 2.
Fig. 2.

Electric-field intensity of Gaussian beam in xOz plane, as calculated by HFSS (top), the sum-of-waves formulation of Eq. (27) with νmax=30 (bottom, left), and the integral formulation of Eq. (29) (bottom, right).

Fig. 3.
Fig. 3.

Electric-field intensity of Gaussian beam, |Einc| (V/m), along the z-axis, as calculated by the integral formulation (continuous line), the sum-of-waves formulation (dashed line), and HFSS (dots).

Fig. 4.
Fig. 4.

Electric-field intensity in xOz plane and in the presence of on-axis sphere (d=0), as calculated by HFSS (top) and the sum-of-waves code (bottom, νmax=30). Incident, scattered, and total electric-field intensity is shown from left to right in each row. CPU time: 208 (HFSS), 1 (sum-of-waves).

Fig. 5.
Fig. 5.

Electric-field intensity in xOz plane and in the presence of off-axis sphere (d=2a, Θ=90°, ϕ=180°), as calculated by HFSS (top) and the sum-of-waves code with νmax=30 (middle) or νmax=50 (bottom). Incident, scattered, and total electric-field intensity is shown from left to right in each row. CPU time: 233 (HFSS), 23 (sum-of-waves, νmax=30), and 62 (sum-of-waves, νmax=50).

Fig. 6.
Fig. 6.

Electric-field intensity maps in xOz plane, centered on the sphere and calculated by sum-of-waves code. The sphere is on the focal plane, either on the axis (left, νmax=30, CPU time: 1) or displaced from the axis by d=2a (right, νmax=50, CPU time: 39). The top-row maps correspond to non-resonant excitation (λ=1.1424  μm, x0=8.25), whereas those on the bottom row correspond to a frequency that excites a morphology-dependent resonance (λ=0.3880  μm, x0=24.3).

Fig. 7.
Fig. 7.

Intercepted power Pint (dB re Pb) in the absence of the sphere versus the radius af (μm) of a receiver-end fiber placed at distance L=20  μm (continuous line), 150 μm (dashed line), 450 μm (dotted line) from the beam focus.

Fig. 8.
Fig. 8.

Receiver-end power Pr (dB re Pb) versus the (normalized, lateral) displacement d/α of the sphere from the beam axis; Θ=90° and ϕ=0° (continuous line), 45° (dotted line), 90° (dashed line).

Fig. 9.
Fig. 9.

CPU time versus truncation number for sum-of-waves results of Tables 1 and 2.

Fig. 10.
Fig. 10.

Truncation number νmax versus lateral displacement d of sphere for sum-of-waves results of Table 1.

Tables (3)

Tables Icon

Table 1. Intercepted Power (af=2  μm, L=20  μm), Pint (dB re Pb), in the Presence of Spherea

Tables Icon

Table 2. Non-Convergent Sum-of-Waves Results at L=150  μma

Tables Icon

Table 3. Intercepted Power (af=2  μm), Pint (dB re Pb), in the Absence of Spherea

Equations (36)

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

Einc=e^ejk0i^·r=e^·νμ,α(1)νcμνfα,μν(i^)Fα,μν(1)(k0r)=νμ,αAα,μνFα,μν(1)(k0r).
fM,μν(r^)=jν1[jθ^μτμν(1)(θ)ϕ^τμν(2)(θ)]ejμϕ,fN,μν(r^)=jν[θ^τμν(2)(θ)+jϕ^μτμν(1)(θ)]ejμϕ,
FM,μν(ι)(kr)=(j)ν1zM,ν(ι)(kr)fM,μν(r^),FN,μν(ι)(kr)=r^ν(ν+1)krzM,ν(ι)(kr)Pνμ(cosθ)ejμϕ+(j)νzN,ν(ι)(kr)fN,μν(r^),
Einc=νμ,αAα,μνFα,μν(1)(k0r1),
Aα,μν=ejk0i^·dAα,μν,
E0=Einc+Esca=νμ,αAα,μνFα,μν(1)(k0r1)+νμ,αBα,μνFα,μν(3)(k0r1),E1=νμ,αCα,μνFα,μν(1)(k1r1),
bα,ν=x1zα,ν(1)(x0)zβ,ν(1)(x1)x0zβ,ν(1)(x0)zα,ν(1)(x1)x1zα,ν(3)(x0)zβ,ν(1)(x1)x0zβ,ν(3)(x0)zα,ν(1)(x1),cα,ν=x0zα,ν(3)(x0)zβ,ν(1)(x0)x0zβ,ν(3)(x0)zα,ν(1)(x0)x1zα,ν(3)(x0)zβ,ν(1)(x1)x0zβ,ν(3)(x0)zα,ν(1)(x1),
E0=νμ,αAα,μν[Fα,μν(1)(k0r1)+bα,νFα,μν(3)(k0r1)],E1=νμ,αAα,μνcα,νFα,μν(1)(k1r1).
Fα,μν(3)(k0r1)ejk0rrejk0dcos(θΘ)(1)νk0fα,μν(s^),
Escaejk0rrejk0dcos(θΘ)1k0νμ,α(1)νbα,νAα,μνfα,μν(s^).
f(i^,s^)=ejk0dcos(θΘ)1k0νμ,α(1)νbα,νAα,μνfα,μν(s^).
σbi=4π|f(i^,s^)|2πa2=4x02[νμ,α(1)νbα,νAα,μνfα,μν(s^)]2.
σbi=4x02[νμ,α(1)νbα,νAα,μνfα,μν(s^)]2,
Aα,μν=Aα,1ν(δμ,1(1)δαMδμ1ν(ν+1)),
fα,μν(z^)=12(j)ν[θ^+jϕ^]θ=πejϕδμ,1(1)δαM12(j)νν(ν+1)[θ^jϕ^]θ=πejϕδμ1,
σmo=4x02[ν,αjνAα,1νbα,ν]2.
σmo=1x02[ν(1)ν(2ν+1)(bM,νbN,ν)]2.
ψ(0)=jQejq,ψ(2)=jQ(2+q2)ψ(0),ψ(4)=Q2(6+3q22jq3+12q4)ψ(0),
Einc(r)=jω[x^(1+1k022x2)+y^1k022xy+z^1k022xz]A(r).
A˜0(kx,ky)=A0(x,y)ej(kxx+kyy)dxdy,
Aeq(r)=14π2A˜0(kx,ky)ej(kxx+kyy+kzz)dkxdky.
Einckx2+ky2k02E(e^ejk0i^·r)dkxdky,
Aα,μν=(1)νcμνw024π0k002πe(k)·fα,μν(k01k)×[1+s2(w0κ)2(1(w0κ)216)+]×ejk·d(w0κ)24κdκdγ.
Aα,μν=(j)νμ1cμνHα,μν,
Hα,μν=14  s210[1+(1v2)(1(1v2)16  s2)+]×(vμδαMτμν(2δαM)(v)Gμ(1)+μδαNτμν(2δαN)(v)Gμ(2))×ejxl1v24  s2vdv,Gμ(1)=Jμ1(xt)ej(μ1)ϕJμ+1(xt)ej(μ+1)ϕ,Gμ(2)=Jμ1(xt)ej(μ1)ϕ+Jμ+1(xt)ej(μ+1)ϕ,
Einc=νμ,α[Aα,μν]d=0Fα,μν(1)(k0r).
Eincν,α(j)νc1νIα,ν×[Fα,1ν(1)(k0r)(1)δαMν(ν+1)Fα,1ν(1)(k0r)],Iα,ν=14  s210[vτ1ν(2δαM)(v)+τ1ν(2δαN)(v)]e1v24  s2vdv.
Eincw024πκk0[x^(1kx2k02)y^kxkyk02z^kxkzk02]×ejk·rw02κ24dkxdky.
Eincb01{x^[(1+v2)J0(b0ρ)+(1v2)J2(b0ρ)cos2ϕ]+y^(1v2)J2(b0ρ)sin2ϕ2jz^k01vb0J1(b0ρ)cosϕ}×eb(1v2)+jk0vzvdv,
[Einc]on-axisx^b01(1+v2)eb(1v2)+jk0vzvdv.
[Einc]originx^12[1+(11b)(1eb)],
Pb=12Re{Einc×Hinc*}z=0·z^dxdy=12k0Z0w0440k002π|e|2ew02κ22k02κ2κdκdγ12Z0πw024[212bF(2b)].
Pabs=12SRe{E1×H1*}·r^1ds.
Pabs=4πa22Z0Re{jnνμ,α(1)δαN(ν+μ)!(νμ)!ν(ν+1)2ν+1×|Aα,μν|2|cα,ν|2zα,ν(1)(nk0a)zβ,ν(1)*(nk0a)}.
Psca=12Z04πk02νμ,α(ν+μ)!(νμ)!ν(ν+1)2ν+1|Aα,μν|2|bα,ν|2.
Pint=x2+y2af[S]z=L·z^dxdy.

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