Abstract

The electric dyadic Green’s function (dGf) of a cluster of spheres is obtained by application of the superposition principle, dyadic algebra, and the indirect mode-matching method. The analysis results in a set of linear equations for the unknown, vector, wave amplitudes of the dGf; that set is solved by truncation and matrix inversion. The theory is exact in the sense that no simplifying assumptions are made in the analytical steps leading to the dGf, and it is general in the sense that any number, position, size and electrical properties can be considered for the spheres that cluster together. The point source can be anywhere, even within one of the spheres. Energy conservation, reciprocity, and other tests prove that this solution is correct. Numerical results are presented for an electric Hertz dipole radiating in the presence of an array of rexolite spheres, which manifests lensing and beam-forming capabilities.

© 2007 Optical Society of America

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References

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2007 (2)

2005 (2)

P. Ghenuche, R. Quidant, and G. Badenes, "Cumulative plasmon field enhancement in finite metal particle chains," Opt. Lett. 30, 1882-1884 (2005).
[CrossRef] [PubMed]

P. Muehlschleger, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant optical antennas," Science 308, 1607-1609 (2005).
[CrossRef]

2004 (1)

E. Hao and G. C. Schatz, "Electromagnetic fields around silver nanoparticles and dimers," J. Chem. Phys. 120, 357-366 (2004).
[CrossRef] [PubMed]

2001 (2)

M. Paulus and O. J. F. Martin, "Light propagation and scattering in stratified media: a Green's tensor approach," J. Opt. Soc. Am. A 18, 854-861 (2001).
[CrossRef]

L. W. Li, M. S. Leong, and Y. Huang, "Electromagnetic radiation of antennas in the presence of an arbitrarily shaped dielectric object: Green dyadics and their application," IEEE Trans. Antennas Propag. 49, 1646-1652 (2001).

1999 (1)

G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theory for assemblies of spheres and aggregates," J. Opt. A, Pure Appl. Opt. 1, 706-712 (1999).
[CrossRef]

1998 (1)

Y. L. Xu, "Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories," J. Comput. Phys. 139, 137-165 (1998).
[CrossRef]

1995 (2)

1994 (1)

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, "Electromagnetic dyadic Green's function in spherically multilayered media," IEEE Trans. Microwave Theory Tech. 42, 2302-2310 (1994).
[CrossRef]

1991 (1)

D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
[CrossRef]

1988 (2)

1987 (1)

M. S. Narasihman and S. Ravishankar, "Multiple scattering of em waves by dielectric spheres located in the near field of a source of radiation," IEEE Trans. Antennas Propag. 35, 399-405 (1987).
[CrossRef]

1979 (1)

1971 (2)

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I--multipole expansion and ray-optical solutions," IEEE Trans. Antennas Propag. 19, 378-390 (1971).
[CrossRef]

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part II--numerical and experimental results," IEEE Trans. Antennas Propag. 19, 391-399 (1971).
[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 vector wave functions," Q. Appl. Math. 19, 15-24 (1961).

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (4)

L. W. Li, M. S. Leong, and Y. Huang, "Electromagnetic radiation of antennas in the presence of an arbitrarily shaped dielectric object: Green dyadics and their application," IEEE Trans. Antennas Propag. 49, 1646-1652 (2001).

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I--multipole expansion and ray-optical solutions," IEEE Trans. Antennas Propag. 19, 378-390 (1971).
[CrossRef]

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part II--numerical and experimental results," IEEE Trans. Antennas Propag. 19, 391-399 (1971).
[CrossRef]

M. S. Narasihman and S. Ravishankar, "Multiple scattering of em waves by dielectric spheres located in the near field of a source of radiation," IEEE Trans. Antennas Propag. 35, 399-405 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, "Electromagnetic dyadic Green's function in spherically multilayered media," IEEE Trans. Microwave Theory Tech. 42, 2302-2310 (1994).
[CrossRef]

J. Chem. Phys. (1)

E. Hao and G. C. Schatz, "Electromagnetic fields around silver nanoparticles and dimers," J. Chem. Phys. 120, 357-366 (2004).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

Y. L. Xu, "Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories," J. Comput. Phys. 139, 137-165 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

G. Gouesbet and G. Grehan, "Generalized Lorenz-Mie theory for assemblies of spheres and aggregates," J. Opt. A, Pure Appl. Opt. 1, 706-712 (1999).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Opt. Lett. (3)

Proc. R. Soc. London, Ser. A (1)

D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London, Ser. A 433, 599-614 (1991).
[CrossRef]

Q. Appl. Math. (2)

S. Stein, "Addition theorems for spherical vector 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).

Science (1)

P. Muehlschleger, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, "Resonant optical antennas," Science 308, 1607-1609 (2005).
[CrossRef]

Other (2)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill, 1864-1891, 1953).

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1993).

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Figures (6)

Fig. 1
Fig. 1

Geometric configuration.

Fig. 2
Fig. 2

Clusters of rexolite spheres used for energy conservation tests: (a) pair of spheres with a z-oriented dipole in the middle (size and separation defined in Table 1); rings of (b) six spheres ( k 0 a = 1 , k 0 r = 2 , n max = 7 ) and (c) nine spheres ( k 0 a = 2 , k 0 r = 8 , n max = 14 ) around the source.

Fig. 3
Fig. 3

Line cluster of rexolite spheres ( N = 3 , k 0 a 1 = 2 , k 0 a 2 = 4 , k 0 a 3 = 6 , k 0 d 12 = 7 , k 0 d 23 = 11 ) used for the reciprocity checks of Table 2. The z-oriented dipole source is shown by black or gray asterisks; cases with f = s involve a black and a gray asterisk; if f s , only black asterisks are used.

Fig. 4
Fig. 4

Director effect of rexolite sphere on an x-oriented dipole. The gain in the direction of the z axis, shown in decibels relative to the gain of a single source, is 5.3 dB in case A ( k 0 a 1 = 2 , r 1 a 1 = 2.9 ) and 8 dB in case B ( k 0 a 1 = 3.6 , r 1 a 1 = 2.1 ) .

Fig. 5
Fig. 5

Radiation patterns of source–sphere system corresponding to cases A (solid curve), B (dashed curve), and C (dotted curve) of Fig. 4; the sphere acts as a director (cases A and B) or a reflector (case C).

Fig. 6
Fig. 6

Radiation patterns of system comprising the source and a case A, uniform, line array of spheres ( k 0 a 1 = = k 0 a N = 2 = k 0 a A , d 12 = = d N 1 , N = 2.9 a A , r 1 = 2.9 a A ) ; N = 2 (solid), N = 3 (long-dashed), N = 4 (short-dashed) or N = 5 (dotted).

Tables (3)

Tables Icon

Table 1 Energy Conservation Tests

Tables Icon

Table 2 Numerical Results for Both Sides of the Reciprocity Formula as Applied to the Cluster of Fig. 3

Tables Icon

Table 3 Radiation Characteristics in the Presence of a Case A, Uniform, Line Array of N Spheres a

Equations (11)

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G e ( s ) ( r i , r i ) = 1 k s 2 r ̂ i r ̂ i δ ( r i r i ) + j k s 4 π ν m n c m n F ν , m n ( 1 ̃ ) ( k s r i ) F ν , m n ( 3 ̃ ) ( k s r i ) .
G e , i ( 0 s ) ( r i , r ) = j k s 4 π ν m n c m n F ν , m n ( 3 ) ( k 0 r i ) A ν , m n ( i ) ( r ) ,
G e , i ( i s ) ( r i , r ) = j k s 4 π ν m n c m n F ν , m n ( 1 ) ( k i r i ) C ν , m n ( i ) ( r ) .
G e ( 0 s ) ( r , r ) = δ s 0 1 k s 2 r ̂ r ̂ δ ( r r ) + j k s 4 π ν m n c m n [ δ s 0 F ν , m n ( 1 ̃ ) ( k 0 r ) F ν , m n ( 3 ̃ ) ( k 0 r ) + i F ν , m n ( 3 ) ( k 0 r i ) A ν , m n ( i ) ( r ) ] ,
G e ( i s ) ( r i , r ) = δ s i 1 k s 2 r ̂ i r ̂ i δ ( r i r i ) + j k s 4 π ν m n c m n [ δ s i F ν , m n ( 1 ̃ ) ( k i r i ) F ν , m n ( 3 ̃ ) ( k i r i ) + F ν , m n ( 1 ) ( k i r i ) C ν , m n ( i ) ( r ) ] ,
S F ν , m n ( ι ) ( k r ) × F ν , m n ( ι ) ( k r ) r ̂ d s = 0 ,
S F M , m n ( ι ) ( k r ) × F N , m n ( ι ) ( k r ) r ̂ d s = S F N , m n ( ι ) ( k r ) × F M , m n ( ι ) ( k r ) r ̂ d s = 4 π a 2 ( 1 ) m n ( n + 1 ) 2 n + 1 z n ( ι ) ( k a ) η n ( ι ) ( k a ) δ m , m δ n n ,
C ν , k l ( i ) ( r ) = I ν , l ( 3 , 1 , 1 ) ( k 0 , k i , k 0 , a i ) A ν , k l ( i ) ( r ) δ s i I ν , l ( 3 , 1 , 1 ) ( k i , k i , k 0 , a i ) F ν , k l ( 1 ) ( k i r i ) ,
I M N , l ( 3 , 1 , 1 ) ( k 0 , k 0 , k i , a i ) A M N , k l ( i ) ( r ) + i m n i i [ A m n , 3 k l ( k 0 d i i ) A M N , m n ( i ) ( r ) + B m n , 3 k l ( k 0 d i i ) A N M , m n ( i ) ( r ) ] = δ s i I M N , l ( 3 , 1 , 1 ) ( k i , k 0 , k i , a i ) F M N , k l ( 1 ) ( k i r i ) δ s 0 F M N , k l ( 3 ) ( k 0 r i ) ,
I M , n ( ι 1 , ι 2 , ι 3 ) ( k 1 , k 2 , k 3 , a ) = k 3 z n ( ι 1 ) ( k 1 a ) η n ( ι 3 ) ( k 3 a ) k 1 η n ( ι 1 ) ( k 1 a ) z n ( ι 3 ) ( k 3 a ) k 3 z n ( ι 2 ) ( k 2 a ) η n ( ι 3 ) ( k 3 a ) k 2 η n ( ι 2 ) ( k 2 a ) z n ( ι 3 ) ( k 3 a ) ,
I N , n ( ι 1 , ι 2 , ι 3 ) ( k 1 , k 2 , k 3 , a ) = k 1 z n ( ι 1 ) ( k 1 a ) η n ( ι 3 ) ( k 3 a ) k 3 η n ( ι 1 ) ( k 1 a ) z n ( ι 3 ) ( k 3 a ) k 2 z n ( ι 2 ) ( k 2 a ) η n ( ι 3 ) ( k 3 a ) k 3 η n ( ι 2 ) ( k 2 a ) z n ( ι 3 ) ( k 3 a )

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