Abstract

The exact solution for the scattering from a sphere with a nonconcentric inclusion of arbitrary shape and optical constants is derived in terms of a vector harmonic expansion. The only information required of the inclusion is its freestanding scattering coefficients, so the inclusion can be composed of multiple refractive indices or be a set of separate inclusions. We then compare the absorption of glycerin host spheres containing small oblate and prolate carbon spheroids in different orientations.

© 1995 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).
  5. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
    [Crossref]
  6. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90–92 (1988).
    [Crossref] [PubMed]
  7. K. A. Fuller, G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: Clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [Crossref] [PubMed]
  8. J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [Crossref]
  9. F. Borghese, P. Denti, R. Saija, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [Crossref]
  10. K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
    [Crossref]
  11. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London A 433, 599–614 (1991).
    [Crossref]
  12. M. M. Mazumder, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of the layered T-matrix method and the time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
    [Crossref]
  13. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991);erratum, J. Opt. Soc. Am. A 9, 844–845 (1992).
    [Crossref]
  14. G. Videen, “Light scattering from a sphere behind a surface,” J. Opt. Soc. Am. A 10, 110–117 (1993).
    [Crossref]
  15. G. Videen, D. Ngo, P. Chýlek, “Effective-medium predictions of absorption by graphitic carbon in water droplets,” Opt. Lett. 19, 1675–1677 (1994).
    [Crossref] [PubMed]
  16. S. Stein, “Addition theorems for spherical wave function,” Q. Appl. Math. 19, 15–24 (1961).
  17. O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  18. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).
  19. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Press U., Princeton, N.J., 1957).

1994 (1)

1993 (1)

1992 (2)

1991 (2)

G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991);erratum, J. Opt. Soc. Am. A 9, 844–845 (1992).
[Crossref]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London A 433, 599–614 (1991).
[Crossref]

1988 (2)

1986 (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

1979 (1)

1971 (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[Crossref]

1967 (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

1962 (1)

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

1961 (1)

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

Barber, P. W.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Borghese, F.

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[Crossref]

Chýlek, P.

Cruzan, O. R.

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

Denti, P.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Press U., Princeton, N.J., 1957).

Fikioris, J. G.

Fuller, K. A.

Hill, S. C.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Liang, C.

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[Crossref]

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London A 433, 599–614 (1991).
[Crossref]

Mazumder, M. M.

Ngo, D.

Saija, R.

Stein, S.

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

Uzunoglu, N. K.

van de Hulst, H. C.

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

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

IEEE Trans. Antennas Propag. (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Lett. (3)

Physica (Utrecht) (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–241 (1986).

Proc. R. Soc. London A (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London A 433, 599–614 (1991).
[Crossref]

Q. Appl. Math. (2)

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

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

Radio Sci. (1)

C. Liang, Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481–1495 (1967).

Other (5)

K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
[Crossref]

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Press U., Princeton, N.J., 1957).

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Figures (3)

Fig. 1
Fig. 1

Geometry of the scattering system.

Fig. 2
Fig. 2

Absorption cross section divided by the mass of the carbon inclusion as a function of eccentricity when (a) d = 0 and (b) d = 0.97a1. In both (a) and (b) the host n1 = 1.4746 + 0.0i, a1 = 1.590766 μm, the inclusion n2 = 1.94 + 0.5i, abc = 10−6μm3, λ = 0.5145 μm, and the incident angle α = 0.

Fig. 3
Fig. 3

(a) Absorption cross section divided by the mass of the carbon inclusion as a function of eccentricity for the same system as that in Fig. 2, except that the incident angle α = π/2. (b) Individual polarization components of absorption for the oblate spheroid having the same parameters as those in (a).

Equations (63)

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M n m , j ( p ) = θ ̂ j [ i m sin θ j z n ( p ) ( k r j ) P n m ( cos θ j ) exp ( i m ϕ j ) ] θ ̂ j [ z n ( p ) ( k r j ) d d θ j [ P n m ( cos θ j ) ] exp ( i m ϕ j ) ] , N n m , j ( p ) = r ̂ j [ 1 k r j z n ( p ) ( k r j ) n ( n + 1 ) P n m ( cos θ j ) exp ( i m ϕ j ) ] + θ j { 1 k r j d d r j [ r j z n ( p ) ( k r j ) ] d d θ j × [ P n m ( cos θ j ) ] exp ( i m ϕ j ) } + ϕ ̂ j { 1 k r j d d r j [ r j z n ( p ) ( k r j ) ] × i m sin θ j P n m ( cos θ j ) exp ( i m ϕ j ) } ,
E 1 inc = n = 0 m = n n a n m M n m , 1 ( 1 ) + b n m N n m , 1 ( 1 ) .
E 1 sca = n = 0 m = n n c nm M n m , 1 ( 3 ) + d n m N n m , 1 ( 3 ) .
E 1 sph = n = 0 m = n n e n m M n m , 1 ( 3 ) + f n m N n m , 1 ( 3 ) + g n m M n m , 1 ( 4 ) + h n m N n m , 1 ( 4 ) .
a n m k 1 ψ n ( k a 1 ) + c n m k 1 ξ n ( 1 ) ( k a 1 ) = e n m k ξ n ( 1 ) ( k 1 a 1 ) + g n m k ξ n ( 2 ) ( k 1 a 1 ) ,
a n m ψ n ( k a 1 ) + c n m ξ n ( 1 ) ( k a 1 ) = e n m ξ n ( 1 ) ( k 1 a 1 ) + g n m ξ n ( 2 ) ( k 1 a 1 ) ,
b n m ψ n ( k a 1 ) + d n m ξ n ( 1 ) ( k a 1 ) = f n m ξ n ( 1 ) ( k 1 a 1 ) + h n m ξ n ( 2 ) ( k 1 a 1 ) ,
k 1 b n m ψ n ( k a 1 ) + k 1 d n m ξ n ( 1 ) ( k a 1 ) = k f n m ξ n ( 1 ) ( k 1 a 1 ) + k h n m ξ n ( 2 ) ( k 1 a 1 ) ,
ψ n ( r ) = r j n ( r ) , ξ n ( q ) ( r ) = r h n ( q ) ( r ) ,
E 2 ext = n = 0 m = n n r n m M n m , 2 ( 3 ) + s n m N n m , 2 ( 3 ) + t n m M n m , 2 ( 4 ) + u n m N n m , 2 4 .
E inc = A n m ( 1 ) M n m ( 4 ) + A p q ( 2 ) N p q ( 4 ) ,
E sca = n , m A n m ( 1 ) D n m 1 ( n , m , 1 ) M n m ( 3 ) + A n m ( 1 ) D n m 2 ( n , m , 1 ) N n m ( 3 ) + A p q ( 2 ) D n m 1 ( p , q , 2 ) M n m ( 3 ) + A p q ( 2 ) D n m 2 ( p , q , 2 ) M n m ( 3 ) ,
r n m = n , m t n m D n m 1 ( n , m , 1 ) + u n m D n m 1 ( n , m , 2 ) ,
s n m = n , m t n m D n m 2 ( n , m , 1 ) + u n m D n m 2 ( n , m , 2 ) .
e n m = n = 0 r n m A n ( n , m ) + s n m B n ( n , m ) ,
f n m = n = 0 s n m A n ( n , m ) + r n m B n ( n , m ) ,
g n m = n = 0 t n m A n ( n , m ) + u n m B n ( n , m ) ,
h n m = n = 0 u n m A n ( n , m ) + t n m B n ( n , m ) .
a n m k 1 ψ n ( k a 1 ) / k + c n m k 1 ξ n ( 1 ) ( k a 1 ) / k = n { t n m A n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + u n m B n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + ξ n ( 1 ) ( k 1 a 1 ) p m t p m [ A n ( n , m ) D n m 1 ( p , m , 1 ) + B n ( n , m ) D n m 2 ( p , m , 1 ) ] + u p m [ A n ( n , m ) D n m 1 ( p , m , 2 ) + B n ( n , m ) D n m 2 ( p , m , 2 ) ] } ,
a n m ψ n ( k a 1 ) + c n m ξ n ( 1 ) ( k a 1 ) = n { t n m A n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + u n m B n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + ξ n ( 1 ) ( k 1 a 1 ) p m t p m [ A n ( n , m ) D n m 1 ( p , m , 1 ) + B n ( n , m ) D n m 2 ( p , m , 1 ) ] + u p m [ A n ( n , m ) D n m 1 ( p , m , 2 ) + B n ( n , m ) D n m 2 ( p , m , 2 ) ] } ,
b n m ψ n ( k a 1 ) + d n m ξ n ( 1 ) ( k a 1 ) = n { t n m B n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + u n m A n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + ξ n ( 1 ) ( k 1 a 1 ) p m t p m [ B n ( n , m ) D n m 1 ( p , m , 1 ) + A n ( n , m ) D n m 2 ( p , m , 1 ) ] + u p m [ B n ( n , m ) D n m 1 ( p , m , 2 ) + A n ( n , m ) D n m 2 ( p , m , 2 ) ] } ,
b n m k 1 ψ n ( k a 1 ) / k + d n m k 1 ξ n ( 1 ) ( k a 1 ) / k = n { t n m B n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + u n m A n ( n , m ) ξ n ( 2 ) ( k 1 a 1 ) + ξ n ( 1 ) ( k 1 a 1 ) p m t p m [ B n ( n , m ) D n m 1 ( p , m , 1 ) + A n ( n , m ) D n m 2 ( p , m , 1 ) ] + u p m [ B n ( n , m ) D n m 1 ( p , m , 2 ) + A n ( n , m ) D n m 2 ( p , m , 2 ) ] } .
a n m γ n = n { t n m A n ( n , m ) X n ( 2 ) + u n m B n ( n , m ) X n ( 2 ) + X n ( 1 ) p m t p m [ A n ( n , m ) D n m 1 ( p , m , 1 ) + B n ( n , m ) D n m 2 ( p , m , 1 ) ] + u p m [ A n ( n , m ) D n m 1 ( p , m , 2 ) + B n ( n , m ) D n m 2 ( p , m , 2 ) ] } ,
b n m γ n = n { t n m B n ( n , m ) Y n ( 2 ) + u n m A n ( n m ) Y n ( 2 ) + Y n ( 1 ) p m t p m [ B n ( n , m ) D n m 1 ( p , m , 1 ) + A n ( n , m ) D n m 2 ( p , m , 1 ) ] + u p m [ B n ( n , m ) D n m 1 ( p , m , 2 ) + A n ( n , m ) D n m 2 ( p , m , 2 ) ] } ,
γ n = k 1 [ ξ n ( 1 ) ( k a 1 ) ψ n ( k a 1 ) ψ n ( k a 1 ) ξ n ( 1 ) ( k a 1 ) ] ,
X n ( q ) = k ξ n ( 1 ) ( k a 1 ) ξ n ( q ) ( k 1 a 1 ) k 1 ξ n ( 1 ) ( k a 1 ) ξ n ( q ) ( k 1 a 1 ) ,
Y n ( q ) = k 1 ξ n ( 1 ) ( k a 1 ) ξ n ( q ) ( k 1 a 1 ) k ξ n ( 1 ) ( k a 1 ) ξ n ( q ) ( k 1 a 1 ) ,
A n ( n , m ) = δ n n , B n ( n , m ) = 0 .
D n m p ( n , m , q ) = δ m m D n m p ( n , m , q ) ,
a n m = a n m TE = 2 i n + 2 n ( n + 1 ) d d α P n m ( cos α ) ,
b n m = b n m TE = 2 i n + 2 n ( n + 1 ) m P n m ( cos α ) sin α .
a n m = a n m TM = i b n m TE ,
b n m = b n m TM = i a n m TE .
h n ( 1 ) ( k r ) ~ ( i ) n i k r exp ( i k r ) .
( E ϕ sca E θ sca ) = exp ( i k r 1 ) i k r 1 [ S 1 S 4 S 3 S 2 ] ( E TE inc E TM inc ) .
S 1 = n = 0 m = n n ( i ) n exp ( i m ϕ 1 ) [ d n m TE m sin θ 1 P n m ( cos θ 1 ) + c n m TE θ 1 P n m ( cos θ 1 ) ] , S 2 = i n = 0 m = n n ( i ) n exp ( i m ϕ 1 ) [ c n m TM m sin θ 1 P n m ( cos θ 1 ) + d n m TE θ 1 P n m ( cos θ 1 ) ] , S 3 = i n = 0 m = n n ( i ) n exp ( i m ϕ 1 ) [ c n m TE m sin θ 1 P n m ( cos θ 1 ) + d n m TE θ 1 P n m ( cos θ 1 ) ] , S 4 = n = 0 m = n n ( i ) n exp ( i m ϕ 1 ) [ d n m TM m sin θ 1 P n m ( cos θ 1 ) + c n m TM θ 1 P n m ( cos θ 1 ) ] .
Q sca = 2 ( k a 1 ) 2 [ n = 1 n ( n + 1 ) m = n n ( | c n m TE | 2 + | d n m TE | 2 + | c n m TM | 2 + | d n m TM | 2 ) ] ,
Q ext = 2 ( k a 1 ) 2 Re [ n = 1 n ( n + 1 ) m = n n ( c n m TE a n m TE * + d n m TE b n m TE * + c n m TM a n m TM * + d n m TM b n m TM * ) ] ,
g = 4 π k 2 C sca n , m n ( n + 2 ) [ ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ] ½ × Re ( i d n m TE d n + 1 , m TE * + i c n m TE c n + 1 , m TE * + i d n m TM d n + 1 , m TM * + i c n m TM c n + 1 , m TM * ) + m Re ( c n m TE d n m TE * + c n m TM d n m TM * )
E 102 ( 1 , 0 , 2 ) = 2 i k 1 3 a b c 3 ( k 2 / K 1 ) 2 1 3 + 3 L z [ ( k 2 / k 1 ) 1 ] ,
E 112 ( 1 , 1 , 2 ) = E 1 , 1 , 2 ( 1 , 1 , 2 ) = 2 i k 1 3 a b c 3 ( k 2 / k 1 ) 2 1 3 + 3 L x [ ( k 2 / k 1 ) 2 1 ] ,
L z = g 2 ( e ) [ 1 2 e ln ( 1 + e 1 e ) 1 ] ,
L x = 1 L z 2 ,
g ( e ) = ( e 2 1 ) ½ ,
e 2 = 1 a 2 c 2 .
L x = g ( e ) 2 e 2 { π 2 tan 1 [ g ( e ) ] } g 2 ( e ) 2 ,
L z = 1 2 L x ,
e 2 = 1 c 2 a 2 .
D n m j ( n , m , k ) = 1 + 2 E n m j ( n , m , k ) .
M n m , 2 ( k ) = n = 0 A n ( n , m ) M n m , 1 ( k ) + B n ( n , m ) N n m , 1 ( k ) ,
N n m , 2 ( k ) = n = 0 B n ( n , m ) M n m , 1 ( k ) + A n ( n , m ) N n m , 1 ( k )
A n ( n , m ) = C n ( n , m ) k d n + 1 [ ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ] ½ C n + 1 ( n , m ) k d n [ ( n m ) ( n + m ) ( 2 n + 1 ) ( 2 n 1 ) ] ½ C n 1 ( n , m ) ,
B n ( n , m ) = i k d n ( n + 1 ) m C n ( n , m ) ,
C n ( 0 , 0 ) = ( 2 n + 1 ) ½ j n ( k d ) ,
C n ( 1 , 0 ) = ( 2 n + 1 ) ½ j n ( k d ) ,
[ ( n m + 1 ) ( n + m ) ( 2 n + 1 ) ] ½ C n ( n , m ) = [ ( n m + 1 ) ( n + m ) ( 2 n + 1 ) ] ½ C n ( n , m 1 ) k d [ ( n m + 2 ) ( n + m 1 ) ( 2 n + 3 ) ] ½ C n + 1 ( n , m 1 ) k d [ ( n + m ) ( n + m 1 ) ( 2 n 1 ) ] ½ C n + 1 ( n , m 1 ) ,
C n ( n , m ) = C n ( n , m ) ,
n C n ( n 1 , 0 ) ( 2 n + 1 2 n 1 ) ½ ( n + 1 ) C n ( n + 1 , 0 ) ( 2 n + 1 2 n + 3 ) ½ = ( n + 1 ) C n + 1 ( n , 0 ) ( 2 n + 1 2 n + 3 ) ½ n C n + 1 ( n , 0 ) ( 2 n + 1 2 n 1 ) ½ .
M n m , 2 ( k ) = m D m ( n , m ) M n m , 1 ( k ) ,
N n m , 2 ( k ) = m D m ( n , m ) N n m , 1 ( k ) ,
D m ( n , m ) = exp [ i ( m α + m γ ) ] [ ( n + m ) ! ( n m ) ! ( n + m ) ! ( n m ) ! ] ½ × σ ( n + m n m σ ) ( n m σ ) ( 1 ) n + m σ × [ cos ( β / 2 ) ] 2 σ + m + m [ sin ( β / 2 ) ] 2 n 2 σ m m
c n m , 2 = m D m ( n , m ) c n m , 1 ,
d n m , 2 = m D m ( n , m ) d n m , 1 .

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