Abstract

An exact solution to the problem of electromagnetic-wave scattering from a sphere with an arbitrary number of nonoverlapping spherical inclusions is obtained by use of the indirect mode-matching technique. A set of linear equations for the wave amplitudes of the electric field intensity throughout the inhomogeneous sphere and in the surrounding empty space is determined. Numerical results are calculated by truncation and matrix inversion of that set of equations. Specific information about the truncation number pertaining to the multipole expansions of the electric field intensity is given. The theory and the accompanying computer code successfully reproduce the results of other pertinent papers. Some numerical results [Borghese et al., Appl. Opt. 33, 484 (1994)] were not reproduced well, and that discrepancy is discussed. Our numerical investigation is focused on an acrylic sphere with up to four spherical inclusions. This is the first time that numerical results are presented for a sphere with more than two spherical inclusions. Interesting remarks are made about the effect that the look direction and the structure of the inhomogeneity have on backscattering by the acrylic host sphere.

© 2002 Optical Society of America

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References

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  1. J. G. Fikioris, N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [CrossRef]
  2. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [CrossRef]
  3. N. C. Skaropoulos, M. P. Ioannidou, 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]
  4. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953).
  5. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  6. J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Q. Appl. Math. 37, 51–66 (1979).
  7. J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
    [CrossRef]
  8. F. Borghese, P. Denti, R. Saija, “Optical properties of spheres containing several spherical inclusions,” Appl. Opt. 33, 484–493 (1994); errata: 34, 5556 (1995).
    [CrossRef] [PubMed]
  9. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. III. Spheres containing arbitrarily located spherical inhomogeneities,” J. Opt. Soc. Am. A 12, 893–904 (1995).
    [CrossRef]
  10. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles (Academic, San Diego, Calif., 2000).
  11. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  12. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  13. M. P. Ioannidou, N. C. Skaropoulos, D. P. Chrissoulidis, “Study of interactive scattering by clusters of spheres,” J. Opt. Soc. Am. A 12, 1782–1789 (1995).
    [CrossRef]
  14. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  15. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  16. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, San Diego, Calif., 1980).
  17. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
    [CrossRef]

1995 (3)

1994 (2)

1992 (1)

1984 (1)

1979 (2)

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Q. Appl. Math. 37, 51–66 (1979).

J. G. Fikioris, N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
[CrossRef]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Abramowitz, M.

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

Borghese, F.

Chrissoulidis, D. P.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Denti, P.

Feshbach, H.

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

Fikioris, J. G.

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Q. Appl. Math. 37, 51–66 (1979).

J. G. Fikioris, N. K. Uzunoglou, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
[CrossRef]

Fuller, K. A.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

Ioannidou, M. P.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Kakogiannos, N. B.

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Kanellopoulos, J. D.

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Q. Appl. Math. 37, 51–66 (1979).

Morse, P. M.

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

Roumeliotis, J. A.

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

Saija, R.

Sindoni, O. I.

Skaropoulos, N. C.

Stegun, I. A.

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

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Stratton, J. A.

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

Toscano, G.

Uzunoglou, N. K.

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (1)

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Q. Appl. Math. (3)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Q. Appl. Math. 37, 51–66 (1979).

Other (6)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

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

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

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

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

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Figures (5)

Fig. 1
Fig. 1

Geometric configuration.

Fig. 2
Fig. 2

Truncation number M versus size parameter k0αN+1 of host sphere for N=1, k0α1=k0α2/2, n1=1.7, n2=1.3, θ1=0° and N=2, k0α1=k0α2=k0α3/4, n1=n2=1.7, n3=1.3, θ1=0°, θ2=180°; the angle of incidence is θinc=0°.

Fig. 3
Fig. 3

Comparison of IMM solution with the DMM solution of Ref. 8 (N=2, k0α1=k0α2=0.7937, k0α3=3, n1=n2=0.49+i9.9, n3=1.61+i0.004, ι=1, 2): (a) θinc=0° and (b) θinc=90°.

Fig. 4
Fig. 4

Normalized radar cross section σmoι/παN+12 versus angle of incidence θinc (k0αs=N-1/3, ns=1, s=1, 2,, N, k0αN+1=3, nN+1=1.61+i0.004, ι=1, 2): (a) N=1, (b) N=2, (c) N=3, (d) N=4.

Fig. 5
Fig. 5

Normalized radar cross section σmoι/παN+12, versus angle of incidence θinc (k0αs=N-1/3, ns=0.49+i9.9, s=1, 2,, N, k0αN+1=3, nN+1=1.61+i0.004, ι=2, N=1, 2, 3, 4).

Equations (32)

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E0(r)=Einc(r)+Esca(r)=n=1m=-nnCmn0WmnT(k0r).
Es(rs)=n=1m=-nnCmnsWmnT(ksrs).
EN+1=s=0Nn=1m=-nnCmnsN+1WmnT(kN+1rs).
SN+1[E0××Q-Q××E0]·rˆ dsN+1
=s=1N Ss[Es××Q-Q××Es]·rˆs dss.
Sq[EN+1××Q-Q××EN+1]·rˆq dsq
=Sq[Eq××Q-Q××Eq]·rˆq dsq.
s=1spNn=1m=-nnαs2(-1)mCmnsN+1Jn1(kN+1, kN+1, αs)
×A-mn,ikl(kN+1dps)-αN+12n=1m=-nn(-1)mCmn0
×Jni(k0, kN+1, αN+1)A-mn,1kl(-kN+1dp)
=-αp2(-1)kg-klpιUl(3, 1)(kN+1, kN+1, αq)
×Ul(1, i)(kp, kN+1, αp)Ul(1,1)(kp, kN+1, αp),
s=1spNn=1m=-nnαs2(-1)mCmnsN+1Jn1(kN+1, kN+1, αs)
×B-mn,ikl(kN+1dps)-αN+12n=1m=-nn(-1)mCmn0
×Jni(k0, kN+1, αN+1)B-mn,1kl(-kN+1dp)
=-αp2(-1)kh-klpιVl(3,1)(kN+1, kN+1, αp)
×Vl(1, i)(kp, kN+1, αp)Vl(1,1)(kp, kN+1, αp),
s=0Nn=1m=-nnCmnsN+1U-klqmns(kN+1)=0,
s=0Nn=1m=-nnCmnsN+1V-klqmns(kN+1)=0,
cmnsι=Un(3,1)(kN+1, kN+1, αs)Un(1,1)(ks, kN+1, αs)gmnsι,
dmnsι=Vn(3,1)(kN+1, kN+1, αs)Vn(1,1)(ks, kN+1, αs)hmnsι,
Un(i, j)(u, v, r)=2n(n+1)2n+1[vzn(i)(ur)ηn(j)(vr)-uηn(i)(ur)zn(j)(vr)],
Vn(i, j)(u, v, r)=2n(n+1)2n+1[uzn(i)(ur)ηn(j)(vr)-vηn(i)(ur)zn(j)(vr)],
Jni(u, v, r)=Un(1, i)(u, v, r)0000Un(3, i)(u, v, r)0000Vn(1, i)(u, v, r)0000Vn(3, i)(u, v, r),
Uni(u, v, r)=Un(1, i)(u, v, r)Un(3, i)(u, v, r)00,
Vni(u, v, r)=00Vn(1, i)(u, v, r)Vn(3, i)(u, v, r),
Amn,iμν(κ)=Amn,iμν(κ)Amn,iμν(κ)Bmn,iμν(κ)Bmn,iμν(κ),Amnμν(κ)=Amn,1μν(κ)Amn,3μν(κ)Bmn,1μν(κ)Bmn,3μν(κ),
Bmn,iμν(κ)=Bmn,iμν(κ)Bmn,iμν(κ)Amn,iμν(κ)Amn,iμν(κ),Bmnμν(κ)=Bmn,1μν(κ)Bmn,3μν(κ)Amn,1μν(κ)Amn,3μν(κ),
Uklqmns(k)=(1-δsq)Ul(1,1)(k, kq, αq)Aklmn(kdsq)+δm,-kδnlδsqUl1(k, kq, αq),
Vklqmns(k)=(1-δsq)Vl(1,1)(k, kq, αq)Bklmn(kdsq)+δm,-kδnlδsqUl1(k, kq, αq).
Escaι=fι(iˆ, sˆ)rexp(jk0r).
fι(iˆ, sˆ)=1k0n=1Mm=-nnj-nmamnιPnm(cos θ)sin θ+bmnιdPnm(cos θ)dθθˆ+jmbmnιPnm(cos θ)sin θ+amnιdPnm(cos θ)dθϕˆexp(jmϕ),

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