Abstract

A spheroidal coordinate separation-of-variables solution has been developed for the determination of the internal, near-surface, and scattered fields of a spheroid (either prolate or oblate) with an embedded source of arbitrary type, location, and orientation. Presented results for (1, 0) and (1, 1) electric multipoles embedded in 2:1 axis ratio prolate and oblate spheroids (equal volume sphere size parameter equal to 20) illustrate that the presence of the spheroid interface can have a profound effect on the corresponding far-field scattering pattern. The calculation procedure could be used, for example, to model the emission of inelastic scattered light (Raman, fluorescence, etc.) from biological particles of appreciably elongated (prolatelike) or appreciably flattened (oblatelike) geometries.

© 2000 Optical Society of America

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References

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  1. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
    [CrossRef] [PubMed]
  2. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [CrossRef]
  3. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J. Opt. Soc. Am. 68, 1676–1689 (1978).
    [CrossRef]
  4. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
    [CrossRef]
  5. S. D. Druger, P. J. McNulty, “Radiation pattern of fluorescence from molecules embedded in small particles: general case,” Appl. Opt. 22, 75–82 (1983).
    [CrossRef] [PubMed]
  6. H. Chew, D.-S. Wang, M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56–66 (1984).
    [CrossRef]
  7. S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
    [CrossRef]
  8. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
    [CrossRef] [PubMed]
  9. J. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).
    [CrossRef]
  10. S. C. Hill, G. Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B 14, 2522–2529 (1997).
    [CrossRef]
  11. J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
    [CrossRef] [PubMed]
  12. J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 31, 7140–7146 (1992).
    [CrossRef] [PubMed]
  13. N. Velesco, G. Schweiger, “Geometrical optics calculation of inelastic scattering on large particles,” Appl. Opt. 38, 1046–1052 (1999).
    [CrossRef]
  14. H. Chew, P. J. McNulty, M. Kerker, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
    [CrossRef]
  15. H. Chew, D. D. Cooke, M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
    [CrossRef] [PubMed]
  16. Z.-L. Wang, W.-G. Lin, “Raman or fluorescent scattering by active molecules or ions embedded in a single-mode optical fiber,” Appl. Opt. 32, 6645–6649 (1993).
    [CrossRef] [PubMed]
  17. D.-S. Wang, M. Kerker, H. W. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
    [CrossRef] [PubMed]
  18. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  19. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  20. J. P. Barton, D. R. Alexander, 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]

1999 (1)

1997 (1)

1995 (1)

1993 (1)

1992 (2)

1989 (1)

J. P. Barton, D. R. Alexander, 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]

1988 (2)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1987 (1)

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

1984 (1)

1983 (1)

1980 (2)

1978 (2)

1976 (2)

H. Chew, P. J. McNulty, M. Kerker, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[CrossRef]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Alexander, D. R.

J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 31, 7132–7139 (1992).
[CrossRef] [PubMed]

J. Zhang, D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 31, 7140–7146 (1992).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, 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, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).
[CrossRef]

Arnold, S.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Barton, J. P.

J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, 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, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).
[CrossRef]

Chew, H.

Chew, H. W.

Cooke, D. D.

Druger, S. D.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

S. D. Druger, P. J. McNulty, “Radiation pattern of fluorescence from molecules embedded in small particles: general case,” Appl. Opt. 22, 75–82 (1983).
[CrossRef] [PubMed]

Folan, L. M.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Hill, S. C.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Kerker, M.

Lin, W.-G.

McNulty, P. J.

Pendleton, J. D.

Schaub, S. A.

J. P. Barton, D. R. Alexander, 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, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Schweiger, G.

Sculley, M.

Velesco, N.

Videen, G.

Wang, D.-S.

Wang, Z.-L.

Zhang, J.

Appl. Opt. (8)

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, 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. Chem. Phys. (1)

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. B (2)

Phys. Rev. A (2)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Other (2)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

J. P. Barton, D. R. Alexander, “Electromagnetic field calculations for a tightly focused laser beam incident upon a microdroplet: applications to nonlinear optics,” in Nonlinear Optics and Materials, C. D. Cantrell, C. M. Bowden, eds., Proc. SPIE1497, 64–77 (1991).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Schematic of geometrical arrangement. The coordinate origin of the embedded source is located at r0. The boundary of the surface of the spheroid (located at ξ=ξ0) rotates about the z axis. For the prolate spheroid (as shown), the major axis of the spheroid is along the z axis. For the oblate spheroid, the major axis of the spheroid is along the x axis.

Fig. 2
Fig. 2

Far-field scattering patterns in the xz plane for an embedded source in a spheroid with n=(1.00, 0.00). (Left) a (1, 0) electric multipole. (Right) a (1, 1) electric multipole.

Fig. 3
Fig. 3

Internal and near-surface electric field magnitude in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. Contour plot (left) and gray-level visualization (right). (1, 0) electric multipole source; source location, (x0, y0, z0)=(0.25, 0.00, 0.50); source orientation angle, θdir=0°; relative index of refraction, n=(1.33, 0.00); and equal volume sphere size parameter, αsp=2πasp/λext=20.

Fig. 4
Fig. 4

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. Same conditions as in Fig. 3. (1, 0) electric multipole source located at (x0, y0, z0)=(0.25, 0.00, 0.50) with θdir=0°.

Fig. 5
Fig. 5

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.25, 0.00, 0.50) with θdir=45°.

Fig. 6
Fig. 6

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.25, 0.00, 0.50) with θdir=90°.

Fig. 7
Fig. 7

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. (1, 1) electric multipole source located at (x0, y0, z0)=(0.25, 0.00, 0.50) with θdir=0°.

Fig. 8
Fig. 8

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.10, 0.00, 0.90) with θdir=0°.

Fig. 9
Fig. 9

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio prolate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.10, 0.00, 0.90) with θdir=90°.

Fig. 10
Fig. 10

Internal and near-surface electric field magnitude in the xz plane for a 2:1 axis ratio oblate spheroid with an embedded source. Contour plot (left) and gray-level visualization (right). (1, 0) electric multipole source located at (x0, y0, z0)=(0.50, 0.00, 0.25) with θdir=0°.

Fig. 11
Fig. 11

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio oblate spheroid with an embedded source. Same conditions as in Fig. 10. (1, 0) electric multipole source located at (x0, y0, z0)=(0.50, 0.00, 0.25) with θdir=0°.

Fig. 12
Fig. 12

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio oblate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.50, 0.00, 0.25) with θdir=90°.

Fig. 13
Fig. 13

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio oblate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.90, 0.00, 0.10) with θdir=0°.

Fig. 14
Fig. 14

Internal and near-surface electric field magnitude (left) and far-field scattering (right) in the xz plane for a 2:1 axis ratio oblate spheroid with an embedded source. (1, 0) electric multipole source located at (x0, y0, z0)=(0.90, 0.00, 0.10) with θdir=90°.

Equations (32)

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f=a[1-(b/a)2]1/2,
E(s)=l,m[almNlm(s)+blmMlm(s)],
H(s)=-iext l,m[almMlm(s)+blmNlm(s)],
E(w)=l,m[clmNlm(w)+dlmMlm(w)],
H(w)=-iextnl,m[clmMlm(w)+dlmNlm(w)].
Mlm=×(rlm),
Nlm=1h×Mlm,
(2+h2)Π=0.
Πlm(s)=Slm(hext, η)Rlm(3)(hext, ξ)exp(imϕ),
Πlm(w)=Slm(hint, η)Rlm(1)(hint, ξ)exp(imϕ),
Eη(s)=Eη(w)+Eη(e),
Eϕ(s)=Eϕ(w)+Eϕ(e),
Hη(s)=Hη(w)+Hη(e),
Hϕ(s)=Hϕ(w)+Hϕ(e).
l=|m|L(+Ilml1alm+Ilml2blm-Ilml3clm-Ilml4dlm)=12πClmη,
l=|m|L(+Ilml5alm+Ilml6blm-Ilml7clm-Ilml8dlm)=12πClmϕ,
l=|m|L(+Ilml2alm+Ilml1blm-nIlml4clm-nIlml3dlm)=i2πextDlmη,
l=|m|L(+Ilml6alm+Ilml5blm-nIlml8clm-nIlml7dlm)=i2πextDlmϕ,
Clmη=02π-11Eη(e)(ξ0, η, ϕ)×Slm(hext, η)exp(-imϕ)dηdϕ,
Clmϕ=02π-11Eϕ(e)(ξ0, η, ϕ)×Slm(hext, η)exp(-imϕ)dηdϕ,
Dlmη=02π-11Hη(e)(ξ0, η, ϕ)×Slm(hext, η)exp(-imϕ)dηdϕ,
Dlmϕ=02π-11Hϕ(e)(ξ0, η, ϕ)×Slm(hext, η)exp(-imϕ)dηdϕ.
Sr(θ, ϕ)=limrr2Src8πE02(πf2)=limξξ2πRe[Eϕ(s)Hη(s)*-Eη(s)Hϕ(s)*].
Qscat=Wscatc8πE02(πf2)=02π0πSr(θ, ϕ)sin θ dθdϕ.
Er(e)=1hint2re2l,m[l(l+1)Clmξl(1)(hintre)Ylm(θe, ϕe)],
Eθ(e)=1hintrel,mClmξl(1)(hintre)Ylm(θe, ϕe)θe+imDlmξl(1)(hintre)Ylm(θe, ϕe)sin θe,
Eϕ(e)=1hintrel,mimClmξl(1)(hintre)Ylm(θe, ϕe)sin θe-Dlmξl(1)(hintre)Ylm(θe, ϕe)θe,
Hr(e)=-iexthinthextre2l,m[l(l+1)Dlmξl(1)(hintre)Ylm(θe, ϕe)],
Hθ(e)=-iexthextrel,mimClmξl(1)(hintre)Ylm(θe, ϕe)sin θe+Dlmξl(1)(hintre)Ylm(θe, ϕe)θe,
Hϕ(e)=-iexthextrel,m-Clmξl(1)(hintre)Ylm(θe, ϕe)θe+imDlmξl(1)(hintre)Ylm(θe, ϕe)sin θe.
Clm=hint2re,02l(l+1)ψl(hintre,0)02π0πEr(e)re,0, θe, ϕe)×Ylm*(θe, ϕe)sin θedθedϕe,
Dlm=ihinthextre,02extl(l+1)ψl(hintre,0)02π0πHr(e)(re,0, θe, ϕe)×Ylm*(θe, ϕe)sin θedθedϕe.

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