Abstract

A spheroidal coordinate separation-of-variables solution has been developed for the determination of the internal, near-surface, and far-scattered electromagnetic fields for irregularly shaped elongated (prolatelike) and irregularly shaped flattened (oblatelike) particles with arbitrary monochromatic illumination. Calculated results are presented, for both plane-wave and focused-Gaussian-beam illumination, which demonstrate the effects of particle geometry on the internal, near-surface, and far-scattered-field distributions.

© 2002 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. 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]
  3. J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spheroidal particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
    [CrossRef]
  4. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chaps. 1, 2.
  5. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 349–354, 392–399, 414–420.
  6. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chaps. 3, 4, 6.
  7. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  8. J. P. Barton, “Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. I. Internal and near-field effects,” Appl. Opt. 36, 1303–1311 (1997).
    [CrossRef] [PubMed]
  9. J. P. Barton, “Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering,” Appl. Opt. 37, 3339–3344 (1998).
    [CrossRef]

1998 (1)

1997 (1)

1995 (1)

1991 (1)

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spheroidal particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

1989 (1)

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

1988 (1)

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]

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spheroidal particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

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

Barton, J. P.

J. P. Barton, “Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering,” Appl. Opt. 37, 3339–3344 (1998).
[CrossRef]

J. P. Barton, “Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. I. Internal and near-field effects,” Appl. Opt. 36, 1303–1311 (1997).
[CrossRef] [PubMed]

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, “Electromagnetic fields for an irregularly shaped, near-spheroidal particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

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

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chaps. 1, 2.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chaps. 3, 4, 6.

Schaub, S. A.

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]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 349–354, 392–399, 414–420.

Appl. Opt. (3)

J. Appl. Phys. (3)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (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 fields for an irregularly shaped, near-spheroidal particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

Other (3)

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chaps. 1, 2.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 349–354, 392–399, 414–420.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957), Chaps. 3, 4, 6.

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Figures (10)

Fig. 1
Fig. 1

Schematic of the geometrical arrangement. The boundary of the particle rotates about the z axis. For prolatelike particles (as shown), the major axis of the particle is along the z axis. For oblatelike particles, the major axis would be along the x axis. The surface of the particle is located at ξ=ξˆ(η).

Fig. 2
Fig. 2

Surface grid plot of electric field magnitude in the xz plane for a plane wave incident upon an irregularly shaped prolatelike particle (ξˆ(η)=ξ0+Δξ sin[N(1-η)π/2] with ξ0=1.060660, Δξ=+0.094040, and N=1). Incident propagation angle θbd=45°, incident electric field polarization angle ϕbd=90°, spheroidal coordinate size parameter hext=10.0, and relative index of refraction n=1.33. The arrow indicates the propagation direction of the incident plane wave.

Fig. 3
Fig. 3

Gray-level (white implies low, black implies high) plot of electric field magnitude (on the left) and polar plot (log scale, range from 0.001 to 10.0) of the far-field scattering (on the right) in the xz plane for a plane wave incident upon an irregularly shaped prolatelike particle. For the gray-level plot, Emax=3.575738 and Emin=0.025947. For the polar plot, Smax=6.254843 and Smin=0.000178. The conditions are the same as those in Fig. 2. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=1.060660, Δξ=+0.094040, and N=1. The arrows indicate the propagation direction of the incident plane wave.

Fig. 4
Fig. 4

Gray-level (white implies low, black implies high) plot of electric field magnitude (on the left) and polar plot (log scale, range from 0.001 to 10.0) of the far-field scattering (on the right) in the xz plane for a plane wave incident upon an irregularly shaped prolatelike particle For the gray-level plot, Emax=2.699819 and Emin=0.153948. For the polar plot, Smax=3.904914 and Smin=4.79×10-7. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=1.154701, Δξ=-0.094040, and N=1.

Fig. 5
Fig. 5

Gray-level (white implies low, black implies high) plot of electric field magnitude (on the left) and polar plot (log scale, range from 0.001 to 10.0) of the far-field scattering (on the right) in the xz plane for a plane wave incident upon an irregularly shaped prolatelike particle. For the gray-level plot, Emax=3.503359 and Emin=0.019923. For the polar plot, Smax=7.399795 and Smin=1.91×10-4. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=1.114701, Δξ=-0.04, and N=3.

Fig. 6
Fig. 6

Surface grid plot of electric field magnitude in the xz plane for a plane wave incident upon an irregularly shaped oblatelike particle. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=0.577350, Δξ=-0.223797, and N=1.

Fig. 7
Fig. 7

Gray-level (white implies low, black implies high) plot of electric field magnitude (on the left) and polar plot (log scale, range from 0.001 to 10.0) of the far-field scattering (on the right) in the xz plane for a plane wave incident upon an irregularly shaped oblatelike particle. For the gray-level plot, Emax=3.619381 and Emin=0.022880. For the polar plot, Smax=17.75742 and Smin=3.58×10-4. The conditions are the same as those in Fig. 6. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=0.577350, Δξ=-0.223797, and N=1.

Fig. 8
Fig. 8

Gray-level (white implies low, black implies high) plot of electric field magnitude (on the left) and polar plot (log scale, range from 0.001 to 10.0) of the far-field scattering (on the right) in the xz plane for a plane wave incident upon an irregularly shaped oblatelike particle. For the gray-level plot, Emax=2.946149 and Emin=0.021813. For the polar plot, Smax=26.24844 and Smin=3.48×10-6. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=0.353553, Δξ=+0.223797, and N=1.

Fig. 9
Fig. 9

Gray-level (white implies low, black implies high) plot of electric field magnitude (on the left) and polar plot (log scale, range from 0.001 to 10.0) of the far-field scattering (on the right) in the xz plane for a plane wave incident upon an irregularly shaped oblatelike particle. For the gray-level plot, Emax=3.194559 and Emin=0.024453. For the polar plot, Smax=19.12084 and Smin=4.47×10-7. hext=10.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=0.477350, Δξ=-0.10, and N=3.

Fig. 10
Fig. 10

Gray-level (white implies low, black implies high) plot of electric field magnitude in the xz plane for a focused beam incident upon an irregularly shaped oblatelike particle. The focusing is on center with a beam waist radius (w0) of 0.50. Emax=1.172466 and Emin=0.000262. hext=20.0, n=1.33, θbd=45°, ϕbd=90°, ξ0=0.353553, Δξ=+0.223797, and N=1.

Equations (48)

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x=[(ξ2±1)(1-η2)]1/2sin ϕ,
y=[(ξ2±1)(1-η2)]1/2cos ϕ,
z=ξη,
a/b=ξ0/ξ02-1
a/b=1+1/ξ02
(2+h2)E=0,
(2+h2)H=0,
Mlm=×(r Πlm),
Nlm=1h ×Mlm,
(2+h2)Π=0,
Πlm=Slm(h, η)Rlm(β)(h, ξ)exp(imϕ),
E(s)=l,m[almNlm(s)+blmMlm(s)],
H(s)=-iextl,m[almMlm(s)+blmNlm(s)],
Πlm(s)=Slm(hext, η)Rlm(3)(hext, ξ)exp(imϕ).
E(w)=l,m[clmNlm(w)+dlmMlm(w)],
H(w)=-iextnl,m[clmMlm(w)+dlmNlm(w)],
Πlm(w)=Slm(hint, η)Rlm(1)(hint, ξ)exp(imϕ).
As=sin θˆAξ+cos θˆAη,
An=cos θˆAξ-sin θˆAη,
θˆ=tan-1[(1-η2)/(ξ2±1)]1/2ξη.
-Es(s)+Es(w)=Es(i),
-Eϕ(s)+Eϕ(w)=Eϕ(i),
-Hs(s)+Hs(w)=Hs(i),
-Hϕ(s)+Hϕ(w)=Hϕ(i).
l=|m|L(-Ilml1alm-Ilml2blm+Ilml3clm+Ilml4dlm)
=12π Alms,
l=|m|L(-Ilml5alm-Ilml6blm+Ilml7clm+Ilml8dlm)
=12π Almϕ,
l=|m|L(-Ilml2alm-Ilml1blm+nIlml4clm+nIlml3dlm)
=i2πext Blms,
l=|m|L(-Ilml6alm-Ilml5blm+nIlml8clm+nIlml7dlm)
=i2πext Blmϕ
Alms=02π-11[sin θˆEξ(i)(ξˆ, η, ϕ)+cos θˆEη(i)(ξˆ, η, ϕ)]×Slm(hext, η)exp(-imϕ)dηdϕ,
Almϕ=02π-11Eϕ(i)(ξˆ, η, ϕ)×Slm(hext, η)exp(-imϕ)dηdϕ,
Blms=02π-11[sin θˆHξ(i)(ξˆ, η, ϕ)+cos θˆHη(i)(ξˆ, η, ϕ)]×Slm(hext, η)exp(-imϕ)dηdϕ,
Blmϕ=02π-11Hϕ(i)(ξˆ, η, ϕ)×Slm(hext, η)exp(-imϕ)dηdϕ.
En(i)+En(s)=n2En(w),
Hn(i)+Hn(s)=Hn(w).
ξˆ(η)=ξ0+Δξ sin[N(1-η)π/2].
Sr(θ, ϕ)=limrr2Sr(c/8π)E02(πf2)=limξξ2πRe[Eϕ(s)Hη(s)*-Eη(s)Hϕ(s)*].
Ilml1=-11[sin θˆNlm,ξ(s)(ξˆ, η, 0)+cos θˆNlm,η(s)×(ξˆ,η,0)]Slm(hext, η)dη,
Ilml2=-11[sin θˆMlm,ξ(s)(ξˆ, η, 0)+cos θˆMlm,η(s)×(ξˆ, η, 0)]Slm(hext, η)dη,
Ilml3=-11[sin θˆNlm,ξ(w)(ξˆ, η, 0)+cos θˆNlm,η(w)×(ξˆ, η, 0)]Slm(hext, η)dη,
Ilml4=-11[sin θˆMlm,ξ(w)(ξˆ, η, 0)+cos θˆMlm,η(w)×(ξˆ, η, 0)]Slm(hext, η)dη,
Ilml5=-11Nlm,ϕ(s)(ξˆ, η, 0)Slm(hext, η)dη,
Ilml6=-11Mlm,ϕ(s)(ξˆ, η, 0)Slm(hext, η)dη,
Ilml7=-11Nlm,ϕ(w)(ξˆ, η, 0)Slm(hext, η)dη,
Ilml8=-11Mlm,ϕ(w)(ξˆ, η, 0)Slm(hext, η)dη.

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