Abstract

An approach to expanding a Gaussian beam in terms of the spheroidal wave functions in spheroidal coordinates is presented. The beam-shape coefficients of the Gaussian beam in spheroidal coordinates can be computed conveniently by use of the known expression for beam-shape coefficients, g n, in spherical coordinates. The unknown expansion coefficients of scattered and internal electromagnetic fields are determined by a system of equations derived from the boundary conditions for continuity of the tangential components of the electric and magnetic vectors across the surface of the spheroid. A solution to the problem of scattering of a Gaussian beam by a homogeneous prolate (or oblate) spheroidal particle is obtained. The numerical values of the expansion coefficients and the scattered intensity distribution for incidence of an on-axis Gaussian beam are given.

© 2001 Optical Society of America

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References

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  1. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  2. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  3. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
    [CrossRef] [PubMed]
  4. Y. Han, Z. Wu, “Discussion of the boundary condition for electromagnetic scattering by spheroidal particles,” Acta Phys. Sin. 49, 57–60 (2000).
  5. B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids,” IEEE Trans. Antennas Propag. 31, 294–304 (1983).
    [CrossRef]
  6. S. Nag, B. P. Sinha, “Electromagnetic plane wave scattering by a system of two uniformly lossy dielectric prolate spheroids in arbitrary orientation,” IEEE Trans. Antennas Propag. 43, 322–327 (1995).
    [CrossRef]
  7. T. G. Tsuei, P. W. Barber, “Information content of the scattering matrix for spheroidal particles,” Appl. Opt. 24, 2391–2396 (1985).
    [CrossRef] [PubMed]
  8. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
    [CrossRef]
  9. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
    [CrossRef] [PubMed]
  10. J. P. Barton, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 8472–8473 (1995).
    [CrossRef] [PubMed]
  11. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
    [CrossRef]
  12. B. Maheu, G. Gouesbet, G. Grehan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  13. G. Gouesbet, B. Maheu, G. Grehan, “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]
  14. A. Doicu, T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [CrossRef] [PubMed]
  15. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. I. On-axis beam,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  16. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beam,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  17. Z. Wu, X. Fu, “Scattering of fundamental Gaussian beam from a multilayered sphere,” Acta Electron. Sin. 23, 32–36 (1995).
  18. G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatterer centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef]
  19. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  20. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  21. G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  22. D. B. Hodge, “Eigenvalues and eigenfunctions of the spheroidal wave equation,” J. Math. Phys. 11, 2380–2392 (1971).
  23. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).

2000

Y. Han, Z. Wu, “Discussion of the boundary condition for electromagnetic scattering by spheroidal particles,” Acta Phys. Sin. 49, 57–60 (2000).

1999

1998

1997

1995

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, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 8472–8473 (1995).
[CrossRef] [PubMed]

S. Nag, B. P. Sinha, “Electromagnetic plane wave scattering by a system of two uniformly lossy dielectric prolate spheroids in arbitrary orientation,” IEEE Trans. Antennas Propag. 43, 322–327 (1995).
[CrossRef]

Z. Wu, X. Fu, “Scattering of fundamental Gaussian beam from a multilayered sphere,” Acta Electron. Sin. 23, 32–36 (1995).

1994

1988

1986

1985

1983

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids,” IEEE Trans. Antennas Propag. 31, 294–304 (1983).
[CrossRef]

1980

1979

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

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
[CrossRef] [PubMed]

1975

1971

D. B. Hodge, “Eigenvalues and eigenfunctions of the spheroidal wave equation,” J. Math. Phys. 11, 2380–2392 (1971).

Asano, S.

Barber, P. W.

Barton, J. P.

Davis, L. W.

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

Doicu, A.

Flammer, C.

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

Fu, X.

Z. Wu, X. Fu, “Scattering of fundamental Gaussian beam from a multilayered sphere,” Acta Electron. Sin. 23, 32–36 (1995).

Gouesbet, G.

Grehan, G.

Han, Y.

Y. Han, Z. Wu, “Discussion of the boundary condition for electromagnetic scattering by spheroidal particles,” Acta Phys. Sin. 49, 57–60 (2000).

Hodge, D. B.

D. B. Hodge, “Eigenvalues and eigenfunctions of the spheroidal wave equation,” J. Math. Phys. 11, 2380–2392 (1971).

Lock, J. A.

MacPhie, R. H.

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids,” IEEE Trans. Antennas Propag. 31, 294–304 (1983).
[CrossRef]

Maheu, B.

Nag, S.

S. Nag, B. P. Sinha, “Electromagnetic plane wave scattering by a system of two uniformly lossy dielectric prolate spheroids in arbitrary orientation,” IEEE Trans. Antennas Propag. 43, 322–327 (1995).
[CrossRef]

Sato, M.

Schulz, F. M.

Sinha, B. P.

S. Nag, B. P. Sinha, “Electromagnetic plane wave scattering by a system of two uniformly lossy dielectric prolate spheroids in arbitrary orientation,” IEEE Trans. Antennas Propag. 43, 322–327 (1995).
[CrossRef]

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids,” IEEE Trans. Antennas Propag. 31, 294–304 (1983).
[CrossRef]

Stamnes, J. J.

Stamnes, K.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tsuei, T. G.

Wriedt, T.

Wu, Z.

Y. Han, Z. Wu, “Discussion of the boundary condition for electromagnetic scattering by spheroidal particles,” Acta Phys. Sin. 49, 57–60 (2000).

Z. Wu, X. Fu, “Scattering of fundamental Gaussian beam from a multilayered sphere,” Acta Electron. Sin. 23, 32–36 (1995).

Yamamoto, G.

Acta Electron. Sin.

Z. Wu, X. Fu, “Scattering of fundamental Gaussian beam from a multilayered sphere,” Acta Electron. Sin. 23, 32–36 (1995).

Acta Phys. Sin.

Y. Han, Z. Wu, “Discussion of the boundary condition for electromagnetic scattering by spheroidal particles,” Acta Phys. Sin. 49, 57–60 (2000).

Appl. Opt.

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[CrossRef] [PubMed]

S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
[CrossRef] [PubMed]

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

T. G. Tsuei, P. W. Barber, “Information content of the scattering matrix for spheroidal particles,” Appl. Opt. 24, 2391–2396 (1985).
[CrossRef] [PubMed]

A. Doicu, T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
[CrossRef] [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, “Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 8472–8473 (1995).
[CrossRef] [PubMed]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatterer centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag.

B. P. Sinha, R. H. MacPhie, “Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids,” IEEE Trans. Antennas Propag. 31, 294–304 (1983).
[CrossRef]

S. Nag, B. P. Sinha, “Electromagnetic plane wave scattering by a system of two uniformly lossy dielectric prolate spheroids in arbitrary orientation,” IEEE Trans. Antennas Propag. 43, 322–327 (1995).
[CrossRef]

J. Math. Phys.

D. B. Hodge, “Eigenvalues and eigenfunctions of the spheroidal wave equation,” J. Math. Phys. 11, 2380–2392 (1971).

J. Opt. (Paris)

B. Maheu, G. Gouesbet, G. Grehan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

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

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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

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Figures (5)

Fig. 1
Fig. 1

Prolate spheroidal coordinate system.

Fig. 2
Fig. 2

For the plane wave, angular distribution of the intensity for unpolarized incident light as a function of zenith angle θ on the incidence plane for prolate spheroids with ñ = 1.33, α = 2πa/λ = 6, and a/ b = 4, 2, 1.

Fig. 3
Fig. 3

Angular distribution of the intensity functions for incidence of an unpolarized plane wave and a Gaussian beam (w 0 = λ) of prolate spheroids with ñ = 1.33, a/ b = 4, and α = 10.

Fig. 4
Fig. 4

Scattering intensity for TE and TM polarization of an incident Gaussian beam with w 0 = 2λ as a function of scattering angle θ for a prolate spheroid with ñ = 2, a/ b = 4, and α = 12.

Fig. 5
Fig. 5

Angular distributions of intensity for incidence of an unpolarized plane wave and Gaussian beams with waist radii w 0 = 3λ, 6λ. ñ = 1.33, a/ b = 1.1, and α = 60.

Equations (41)

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x=f1-η21/2ζ2-11/2 cos ϕ, y=f1-η21/2ζ2-11/2 sin ϕ, z=fηζ,
x=f1-η21/2ζ2+11/2 cos ϕ, y=f1-η21/2ζ2+11/2 sin ϕ, z=fηζ,
2E+k2E=0,  2H+k2H=0.
2φ+k2φ=0
φemnojc; ζ, η, ϕ=Smnc; ηRmnjc; ζsincos mϕ.
Smnc, η=r=0,1 drmncPm+rmη,
ddη1-η2dSmnc, ηdη+λmnc-c2η2-m21-η2Smnc, η=0.
ddζζ2-1dRmnc, ζdζ-λmnc-c2ζ2-m21-ζ2Rmnc, ζ=0,
Mmn=×aφmn,  Nmn=k-1·×Mmn.
Ei=n=1 inGncMo1nr1c, ζ, η, ϕ-iFncNe1nr1×c, ζ, η, ϕ.
Ei=n=1gncMo1nr1r, θ, ϕ+fncNe1nr1r, θ, ϕ
Ei=n=1 gncMo1nr1r, θ, ϕ-iNe1nr1r, θ, ϕ,
gn=Enk2n+1in-1πnn+10π0 rFjnkrP1ncos θ×sin θdθdkr, F=Ψ0 sin θ1-2ikwo2 r cos θexpikr cos θ, En=E0in2n+1nn+1, Ψo=exp-r2w02,
Mo1nr1r, θ, ϕ=1sin θ jnkrPn1cos θcos ϕeˆθ-jnkrPn1cos θθsin ϕeˆϕ, Ne1nr1r, θ, ϕ=nn+1kr jnkrPn1cos θcos ϕeˆr+1krrrjnkrPn1cos θθ s×cos ϕeˆθ-1kr sin θr×rjnkrPn1cos θsin ϕeˆϕ,
Pn1cos θjnkr=2nn+12n+1l=1,2 il-nN1l×dn-11lS1lc, ηR1l1c, ζ.
eˆr=ζζ2-1ζ2-η2ζ2+η2-11/2aˆζ+η1-η2ζ2-η2ζ2+η2-11/2aˆη, eˆθ=η1-η2ζ2-η2ζ2+η2-11/2aˆζ-ζζ2-1ζ2-η2ζ2+η2-11/2aˆη, eˆϕ=aˆϕ;
r=fζ2+η2-11/2, sin θ=ζ2-11-η2ζ2+η2-11/2, cos θ=ζηζ2+η2-11/2.
Mo1nr1r, θ, ϕ=l=1,2 2nn+12n+1il-nN1l×dn-11lMo1lζc, ζ, η, ϕaˆζ+Mo1lηc, ζ, η, ϕaˆη+Mo1lϕc, ζ, η, ϕaˆϕ=l=1,2 2nn+12n+1il-nN1l×dn-11lMo1lr1c, ζ, η, ϕ.
Ne1nr1r, θ, ϕ=l=1,2 2nn+12n+1il-nN1l×dn-11lNe1lζc, ζ, η, ϕaˆζ+Ne1lηc, ζ, η, ϕaˆη+Ne1lϕc, ζ, η, ϕaˆϕ=l=1,2 2nn+12n+1il-nN1l dn-11l×Ne1lr1c, ζ, η, ϕ.
Ei=n=1 gnMo1nr1r, θ, ϕ-iNe1nr1r, θ, ϕ=n=1 gnl=1,2 2nn+12n+1il-nN1l×dn-11lMo1lr1c, ζ, η, ϕ-iNe1lr1c, ζ, η, ϕ,
Ei=l=1 gln=1,2 2ll+12l+1in-lN1n×dl-11nMo1nr1c, ζ, η, ϕ-i Ne1nr1c, ζ, η, ϕ=n=1 N1n-1l=1,2 gl2ll+12l+1 in-l×dl-11nMo1nr1c, ζ, η, ϕ-iNe1nr1c, ζ, η, ϕ.
Gn=Fn=N1n-1l=1,2 gl2ll+12l+1 i-ldl-11n,
Ei=n=1 inGnMo1nr1c, ζ, η, ϕ-iNe1nr1c, ζ, η, ϕ, Hi=-k1ωμ1n=1 inGnMe1nr1c, ζ, η, ϕ+iNo1nr1c, ζ, η, ϕ.
Ei=n=1 inGnMe1nr1c, ζ, η, ϕ+iNo1nr1c, ζ, η, ϕ, Hi=-k1ωμ1n=1 inGnMo1nr1c, ζ, η, ϕ-iNe1nr1c, ζ, η, ϕ.
Ew=n inγnMo1nr1cII, ζ, η, ϕ-iδn×Ne1nr1cII, ζ, η, ϕ, Hw=k2ωμ2n inδnMe1nr1cII, ζ, η, ϕ-iγn×No1nr1cII, ζ, η, ϕ, Es=n inαnMo1nr3cI, ζ, η, ϕ-iβn×Ne1nr3cI, ζ, η, ϕ, Hs=k1ωμ1n inβnMe1nr3cI, ζ, η, ϕ+iαn×No1nr3cI, ζ, η, ϕ;
Ew=n inδnMe1nr1cII, ζ, η, ϕ+iγn×No1nr1cII, ζ, η, ϕ, Hw=k2ωμ2n inγnMo1nr1cII, ζ, η, ϕ-iδn×Ne1nr1cII, ζ, η, ϕ, Es=n inβnMe1nr3cI, ζ, η, ϕ+iαn×No1nr3cI, ζ, η, ϕ, Hs=k1ωμ1n inαnMo1nr3cI, ζ, η, ϕ-iβn×Ne1nr3cI, ζ, η, ϕ.
Eηi+Eηs=Eηw, Eϕi+Eϕs=Eϕw, Hηi+Hηs=Hηw, Hϕi+Hϕs=Hϕw
n=m inVmn3,tcIαmn+Umn3,tcIβmn-Vmn1,tcIIγmn-Umn1,tcIIδmn=-n=m ingmnζUmn1,tcI+fmnζVmn1,tcI.
Vmnj,t=ichm2Rmnjch; ζ0ζ02-1ζ02-12Dtmnch+2ζ02-1Ctmnch+Ftmnch-Rmnjch; ζ0λmn-chζ02+m2ζ02-1×ζ02-1Ctmnch+Ftmnch+ζ0ζ02-1ddζ Rmnjch; ζ0ζ02Ctmnchζ0+ζ02-1Gtmnch+Itmnch+Rmnjch; ζ0ζ02-12Gtmnch+3ζ02-1Itmnch
Vmnj,t=ich-m2Rmnj-ich; iζ0ζ02+1ζ02+12Dtmn-ich-2ζ02+1Ctmn-ich+Ftmn-ich+Rmnj-ich; iζ0λmn-chζ02-m2ζ02+1ζ02+1Ctmn-ich-Ftmn-ich+ζ0ζ02+1×ddζRmnj-ich; iζ0ζ0-2Ctmnchζ0+ζ02+1Gtmn-ich-Itmn-ich+Rmnj-ich; iζ0ζ02+12Gtmn-ich-3ζ02+1Itmn-ich
1-η21/2Snη, 1-η2-1/2Snη, η1-η21/2Snη, η1-η2-1/2Snη, 1-η23/2Snη, η1-η23/2Snη, η1-η21/2dSnη/dη, 1-η21/2dSnη/dη, 1-η23/2dSnη/dη,
Atmn=Nm-1,m-1-t-1r=0,1 drmn-1+11-η21/2 Pm+rmηPm-1+tm-1ηdη=0n-m+t oddt+2m-1t+2m2t+2m+1 dtmn-tt-12t+2m-3 dt-2mnn-m+t even,
Htmn=Nm-1,m-1+t-1r=0,1 drmn-1+1 η1-η21/2dPm+rmηdη Pm-1+tm-1dη=0n-m+t odd-tt-1t+m-22t+2m-3dt-2mn+tm+t+12m+t-m+tt+12t+2m-12m+2t+1×dtmn+m2t+2m-1r=t+2 drmnn-m+t even.
c-ic,  ζiζ
q0p0s2q2p2···s2rq2rp2r···d0ncd2nc···d2rnc···=λnd0ncd2nc···d2rnc···  n odd,
q1p1s3q3p3···s2r+1q2r+1p2r+1···d1ncd3nc···d2r+1nc···=λnd1ncd3nc···d2r+1nc···  n even,
pr=r+4r+32r+72r+5 c2, qr=r+1r+2+2r+1r+2-32r+52r+1 c2, sr=rr-12r-12r+1 c2.
Me1n,ηor3-in+1Sncos θsin θ1krexpikrsin-1cos ϕ, Me1n,ϕor3--in+1dSncos θdθ1krexpikrsincos ϕ, Ne1n,ηor3--indSncos θdθ1krexpikrcossin ϕ, Ne1n,ϕor3--inSncos θsin θ1krexpikrsin-1cos ϕ.
-Eηs=Hϕsk1ωμ1=iλI2πrexpi 2πrλI×nαndSncos θdθ+βnSncos θsin θsin ϕ, Eϕs=Hηsk1ωμ1=iλI2πrexpi 2πrλI×nαnSncos θsin θ+βndSncos θdθcos ϕ;
-Eηs=-Hϕsk1ωμ1=iλI2πrexpi 2πrλI×nβndSncos θdθ+αnSncos θsin θcos ϕ, Eϕs=Hηsk1ωμ1=iλI2πrexpi 2πrλI×nβnSncos θsin θ+αndSncos θdθsin ϕ.
I=EsEs*=EηsEηs*+EϕsEϕs*.

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