Abstract

The analytical formulation of electromagnetic wave scattering from an eccentrically stratified dielectric sphere is greatly simplified through the indirect mode-matching technique. The resulting exact solution is the most compact available and hence the least prone to analytical or numerical errors. After some checks we present a comparison between our solution and two previous solutions that were obtained through direct mode matching. Our numerical investigation is focused on an acrylic sphere with an eccentric cavity. All four elements of the scattering matrix are available, and specific information about the possibility of detecting the scatterer’s internal asymmetry is given.

© 1994 Optical Society of America

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References

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  1. G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metalloesungen,” Ann. Phys. 25, 377 (1908).
    [Crossref]
  2. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  3. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  4. A. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [Crossref]
  5. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
  6. K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
    [Crossref] [PubMed]
  7. T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
    [Crossref]
  8. C. C. Johnson, A. W. Guy, “Nonionizing electromagnetic wave effects in biological materials and systems,” Proc. IEEE 60, 692–718 (1972).
    [Crossref]
  9. J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1365 (1979).
    [Crossref]
  10. 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]
  11. J. A. Roumeliotis, J. G. Fikioris, “Scattering of plane waves from an eccentrically coated metallic sphere” J. Franklin Inst. 312, 41–59 (1981).
    [Crossref]
  12. J. D. Kanellopoulos, J. G. Fikioris, “Resonant frequencies in an electromagnetic eccentric spherical cavity,” Quart. Appl. Math. 37, 51–66 (1979).
  13. R. Mittra, Computer Techniques for Electromagnetics (Pergamon, New York, 1973).
  14. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, New York, 1953).
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. I (Academic, New York, 1978).
  17. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–40 (1962).
  18. S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

1992 (1)

1991 (1)

1983 (1)

T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
[Crossref]

1981 (1)

J. A. Roumeliotis, J. G. Fikioris, “Scattering of plane waves from an eccentrically coated metallic sphere” J. Franklin Inst. 312, 41–59 (1981).
[Crossref]

1979 (2)

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

J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1365 (1979).
[Crossref]

1972 (1)

C. C. Johnson, A. W. Guy, “Nonionizing electromagnetic wave effects in biological materials and systems,” Proc. IEEE 60, 692–718 (1972).
[Crossref]

1962 (1)

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

1961 (1)

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

1951 (1)

A. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

1908 (1)

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

Abramowitz, M.

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

Aden, A.

A. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

Barrick, D. E.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Borghese, F.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. 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. A. Roumeliotis, J. G. Fikioris, “Scattering of plane waves from an eccentrically coated metallic sphere” J. Franklin Inst. 312, 41–59 (1981).
[Crossref]

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

J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1365 (1979).
[Crossref]

Fuller, K. A.

Guy, A. W.

C. C. Johnson, A. W. Guy, “Nonionizing electromagnetic wave effects in biological materials and systems,” Proc. IEEE 60, 692–718 (1972).
[Crossref]

Ishimaru, A.

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

Johnson, C. C.

C. C. Johnson, A. W. Guy, “Nonionizing electromagnetic wave effects in biological materials and systems,” Proc. IEEE 60, 692–718 (1972).
[Crossref]

Kanellopoulos, J. D.

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

Kerker, M.

A. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

Krichbaum, C. K.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Mie, G.

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

Mittra, R.

R. Mittra, Computer Techniques for Electromagnetics (Pergamon, New York, 1973).

Morse, P. M.

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

Oguchi, T.

T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
[Crossref]

Roumeliotis, J. A.

J. A. Roumeliotis, J. G. Fikioris, “Scattering of plane waves from an eccentrically coated metallic sphere” J. Franklin Inst. 312, 41–59 (1981).
[Crossref]

Ruck, G. T.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Saija, R.

Sindoni, O. I.

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,” Quart. Appl. Math. 19, 15–24 (1961).

Stratton, J. A.

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

Stuart, W. D.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Uzunoglu, N. K.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Ann. Phys. (1)

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

Appl. Opt. (1)

J. Appl. Phys. (1)

A. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[Crossref]

J. Franklin Inst. (1)

J. A. Roumeliotis, J. G. Fikioris, “Scattering of plane waves from an eccentrically coated metallic sphere” J. Franklin Inst. 312, 41–59 (1981).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Proc. IEEE (2)

T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE 71, 1029–1078 (1983).
[Crossref]

C. C. Johnson, A. W. Guy, “Nonionizing electromagnetic wave effects in biological materials and systems,” Proc. IEEE 60, 692–718 (1972).
[Crossref]

Quart. Appl. Math. (3)

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

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

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

Other (7)

R. Mittra, Computer Techniques for Electromagnetics (Pergamon, New York, 1973).

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

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

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

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (7)

Fig. 1
Fig. 1

Geometric configuration.

Fig. 2
Fig. 2

Truncation number N versus size parameter k0α1 (α2 = α1/2,n1 = 1.3,n2 = 1.7,θi = 0°, polarization 1). The eccentricity is k0d = k0α1k0α2 (maximum) or k0d = 0 (concentric inclusion).

Fig. 3
Fig. 3

Comparison of IMM and previous DMM solutions (k0α1 = 2,k0α2 = l,n1 = 1.3,n2 = 1.7,θi = 0°, 180°).

Fig. 4
Fig. 4

Effect of (a) eccentricity (θi = 0°, 45°, 90°) and (b) viewing direction (k0d = 1,2) on the backscattering cross section of an acrylic sphere (k0α1 = 3,n1 = 1.61 + j0.004) with an eccentric spherical cavity (k0α2 = 1, n2 = 1). Incident polarization is 1,2.

Fig. 5
Fig. 5

(a) Extinction cross section (θi= 0°,45°,90°, polarization 1,2) ϕ and (b) differential scattering cross section (k0d = 0.5, θi = 30°, = 90°) of an acrylic sphere (k0α1= 1.5, n1 = 1.61 + j0.004) with an eccentric spherical cavity (k0α2 = 0.5, n2 = 1).

Fig. 6
Fig. 6

Effect of cavity size k0α2 on the backscattering cross section of an acrylic sphere with an eccentric spherical cavity (k0α1 = 3,k0d = 2,n1 = 1.61 + j0.004,n2 = 1,θi = 0°,45°,90°, polarization 1,2).

Fig. 7
Fig. 7

Spherical interface between dielectrics.

Equations (43)

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E III ( r 1 ) = m n C m n III W m n T ( r 1 , k 0 ) ,
C m n III = [ ξ m n λ m n ι a m n ι j ξ m n μ m n ι b m n ι ] ,
W m n ( r , k ) = [ M m n ( 1 ) ( r , k ) M m n ( 3 ) ( r , k ) N m n ( 1 ) ( r , k ) N m n ( 3 ) ( r , k ) ] .
E II ( r 2 ) = m n C m n II W m n T ( r 2 , k 2 ) ,
s 1 ( E I × × Q Q × × E I ) · N ̂ 1 d s + s 2 ( E I × × Q Q × × E I ) · N ̂ 2 d s = V ( Q · × × E I E I · × × Q ) d υ .
s 1 ( E III × × Q Q × × E III ) · r ̂ 1 d s = s 2 ( E II × × Q Q × × E II ) · r ̂ 2 d s ,
ν [ ξ k ν λ k ν ι U ν ( 1 , 1 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l j ξ k ν μ k ν ι V ν ( 1 , 1 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l + a k ν ι U ν ( 3 , 1 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l + b k ν ι V ν ( 3 , 1 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l ] = ( α 2 / α 1 ) 2 c k l ι U l ( 1 , 1 ) ( k 2 , k 1 , α 2 ) ,
ν [ ξ k ν λ k ν ι U ν ( 1 , 3 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l j ξ k ν μ k ν ι V ν ( 1 , 3 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l + a k ν ι U ν ( 3 , 3 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l + b k ν ι V ν ( 3 , 3 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l ] = ( α 2 / α 1 ) 2 c k l ι U l ( 1 , 3 ) ( k 2 , k 1 , α 2 ) ,
ν [ ξ k ν λ k ν ι U ν ( 1 , 1 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l j ξ k ν μ k ν ι V ν ( 1 , 1 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l + a k ν ι U ν ( 3 , 1 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l + b k ν ι V ν ( 3 , 1 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l ] = ( α 2 / α 1 ) 2 d k l ι V l ( 1 , 1 ) ( k 2 , k 1 , α 2 ) ,
ν [ ξ k ν λ k ν ι U ν ( 1 , 3 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l j ξ k ν μ k ν ι V ν ( 1 , 3 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l + a k ν ι U ν ( 3 , 3 ) ( k 0 , k 1 , α 1 ) B k ν , 1 k l + b k ν ι V ν ( 3 , 3 ) ( k 0 , k 1 , α 1 ) A k ν , 1 k l ] = ( α 2 / α 1 ) 2 d k l ι V l ( 1 , 3 ) ( k 2 , k 1 , α 2 ) .
U n ( i 1 , i 2 ) ( u , υ , r ) = 2 n ( n + 1 ) 2 n + 1 × [ υ z n ( i 1 ) ( u r ) η n ( i 2 ) ( υ r ) u η n ( i 1 ) ( u r ) z n ( i 2 ) ( υ r ) ] ,
V n ( i 1 , i 2 ) ( u , υ , r ) = 2 n ( n + 1 ) 2 n + 1 × [ u z n ( i 1 ) ( u r ) η n ( i 2 ) ( υ r ) υ η n ( i 1 ) ( u r ) z n ( i 2 ) ( υ r ) ]
ν = | k | C k ν III O 1 T = 0 ,
ν = | k | C k ν III O 2 T = 0 ,
O 1 = [ Ω k ( 1 ) ( U ν , U l ) Ω k ( 3 ) ( U ν , U l ) Ω k ( 1 ) ( V ν , U l ) Ω k ( 3 ) ( V ν , U l ) ] ,
O 2 = [ Ω k ( 1 ) ( U ν , V l ) Ω k ( 3 ) ( U ν , V l ) Ω k ( 1 ) ( V ν , V l ) Ω k ( 3 ) ( V ν , V l ) ] .
Ω k ( i ) ( X μ , Ψ ν ) = | X μ ( i , 1 ) ( k 0 , k 1 , α 1 ) X μ ( i , 3 ) ( k 0 , k 1 , α 1 ) Ψ ν ( 1 , 1 ) ( k 2 , k 1 , α 2 ) Ψ ν ( 1 , 3 ) ( k 2 , k 1 , α 2 ) | F k μ , 1 k ν ,
F μ ν , 1 m n = { A μ ν , 1 m n B μ ν , 1 m n if ( X μ , Ψ ν ) = { ( U μ , U ν ) or ( V μ , V ν ) ( U μ , V ν ) or ( V μ , U ν ) .
E sca ι = f ι ( θ , ϕ ) exp ( j k 0 r 1 ) / r 1 ,
f θ ι ( θ , ϕ ) = 1 k 0 m n j n [ m a m n ι P n m ( cos θ ) sin θ + b m n ι d P n m ( cos θ ) d θ ] e j m ϕ ,
f ϕ ι ( θ , ϕ ) = 1 k 0 m n j n + 1 [ m b m n ι P n m ( cos θ ) sin θ + a m n ι d P n m ( cos θ ) d θ ] e j m ϕ ,
I = [ σ 11 σ 12 σ 21 σ 22 ] = 4 π [ | f ϕ 1 ( θ , ϕ ) | 2 | f ϕ 2 ( θ , ϕ ) | 2 | f θ 1 ( θ , ϕ ) | 2 | f θ 2 ( θ , ϕ ) | 2 ] .
E I ( r 2 ) = m n C m n I W m n T ( r 2 , k 1 ) ,
s 2 ( E I × × Q Q × × E I ) · r ̂ 2 d s = s 2 ( E II × × Q Q × × E II ) · r ̂ 2 d s .
e m n ι = U n ( 1 , 3 ) ( k 2 , k 1 , α 2 ) U n ( 1 , 3 ) ( k 1 , k 1 , α 2 ) c m n ι ,
f m n ι = U n ( 1 , 1 ) ( k 2 , k 1 , α 2 ) U n ( 3 , 1 ) ( k 1 , k 1 , α 2 ) c m n ι ,
g m n ι = V n ( 1 , 3 ) ( k 2 , k 1 , α 2 ) V n ( 1 , 3 ) ( k 1 , k 1 , α 2 ) d m n ι ,
h m n ι = V n ( 1 , 1 ) ( k 2 , k 1 , α 2 ) V n ( 3 , 1 ) ( k 1 , k 1 , α 2 ) d m n ι .
a m n ι = ξ m n λ m n ι × U n ( 1 , 1 ) ( k 0 , k 1 , a 1 ) U n ( 1 , 3 ) ( k 2 , k 1 , a 2 ) U n ( 1 , 3 ) ( k 0 , k 1 , a 1 ) U n ( 1 , 1 ) ( k 2 , k 1 , a 2 ) U n ( 3 , 1 ) ( k 0 , k 1 , a 1 ) U n ( 1 , 3 ) ( k 2 , k 1 , a 2 ) U n ( 3 , 3 ) ( k 0 , k 1 , a 1 ) U n ( 1 , 1 ) ( k 2 , k 1 , a 2 ) ,
b m n ι = j ξ m n μ m n ι × V n ( 1 , 1 ) ( k 0 , k 1 , a 1 ) V n ( 1 , 3 ) ( k 2 , k 1 , a 2 ) V n ( 1 , 3 ) ( k 0 , k 1 , a 1 ) V n ( 1 , 1 ) ( k 2 , k 1 , a 2 ) V n ( 3 , 1 ) ( k 0 , k 1 , a 1 ) V n ( 1 , 3 ) ( k 2 , k 1 , a 2 ) V n ( 3 , 3 ) ( k 0 , k 1 , a 1 ) V n ( 1 , 1 ) ( k 2 , k 1 , a 2 ) .
r 1 d M m n ( i ) ( r 2 , k ) = ν = | m | [ A m ν m n , M i m ν ( 1 ) ( r 1 , k ) + B m ν m n , N i m ν ( 1 ) ( r 1 , k ) ] ,
N m n ( i ) ( r 2 , k ) = ν = | m | [ A m ν m n , N i m ν ( 1 ) ( r 1 , k ) + B m ν m n , M i m ν ( 1 ) ( r 1 , k ) ] ;
r 1 d M m n ( i ) ( r 2 , k ) = ν = | m | [ A m ν m n , M 1 m ν ( i ) ( r 1 , k ) + B m ν m n , N 1 m ν ( i ) ( r 1 , k ) ] ,
N m n ( i ) ( r 2 , k ) = ν = | m | [ A m ν m n , N 1 m ν ( i ) ( r 1 , k ) + B m ν m n , M 1 m ν ( i ) ( r 1 , k ) ]
A m ν , i m n = ( 1 ) n + ν + m p a ( m , n | m , ν | p ) a ( n , ν , p ) z p ( i ) ( k d ) ,
B m ν , i m n = ( 1 ) n + ν + m p a ( m , n | m , ν | p + 1 , p ) × b ( n , ν , p + 1 ) z p + 1 ( i ) ( k d ) ,
I = s ( E × × Q Q × × E ) · r ̂ d s ,
E ( r , k o ) = m n C m n W m n T ( r , k o ) ,
I = 2 π α 2 ν = | k | C k ν ν T .
f ν γ = ( 1 ) k A k ν , 1 k l U ν ( ζ , ξ ) ( k o , k i , α ) if w m n γ = M m n ( ζ ) ( r , k o ) , Q = M k l ( ξ ) ( r , k i ) ,
f ν γ = ( 1 ) k A k ν , 1 k l V ν ( ζ , ξ ) ( k o , k i , α ) if w m n γ = N m n ( ζ ) ( r , k o ) , Q = N k l ( ξ ) ( r , k i ) ,
f ν γ = ( 1 ) k B k ν , 1 k l U ν ( ζ , ξ ) ( k o , k i , α ) if w m n γ = M m n ( ζ ) ( r , k o ) , Q = N k l ( ξ ) ( r , k i ) ,
f ν γ = ( 1 ) k B k ν , 1 k l V ν ( ζ , ξ ) ( k o , k i , α ) if w m n γ = N m n ( ζ ) ( r , k o ) , Q = M k l ( ξ ) ( r , k i ) ,

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