Abstract

The translational addition theorem for the spherical vector wave functions (SVWFs) of the first kind is derived in an integral form by the use of the relations between the SVWFs and cylindrical vector wave functions. The integral representation provides a theoretical procedure for the calculation of the beam shape coefficients in the generalized Lorenz–Mie theory. The beam shape coefficients in the cylindrical or spheroidal coordinates, which correspond to an arbitrarily oriented infinite cylinder or spheroid, can be obtained conveniently by using the addition theorem for the SVWF under coordinate rotations and the expansions of the SVWF in terms of the cylindrical or spheroidal vector wave functions.

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References

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  1. L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  2. J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  3. G. Gouesbet, B. Maheu, and G. Gréhan, "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]
  4. G. Gouesbet, G. Gréhan, and 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]
  5. G. Gouesbet, G. Gréhan, and B. Maheu, "Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory," J. Opt. Soc. Am. A 7, 998-1003 (1990).
    [CrossRef]
  6. B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).
  7. S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).
  8. O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).
  9. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, 1957).
  10. J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres part I -- multipole expansion and ray-optical solution," IEEE Trans. Antennas Propag. 19, 378-390 (1971).
    [CrossRef]
  11. A. Doicu and 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]
  12. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997).
    [CrossRef]
  13. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  14. H. Y. Zhang, Y. P. Han, and G. X. Han, "Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions," J. Opt. Soc. Am. A 24, 1383-1391 (2007).
  15. Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
    [CrossRef]
  16. Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express 15, 735-746 (2007).
    [PubMed]
  17. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  18. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).

2007 (2)

H. Y. Zhang, Y. P. Han, and G. X. Han, "Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions," J. Opt. Soc. Am. A 24, 1383-1391 (2007).

Y. P. Han, H. Y. Zhang, and G. X. Han, "Expansion of shaped beam with respect to an arbitrarily oriented spheroidal particle," Opt. Express 15, 735-746 (2007).
[PubMed]

2006 (1)

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

1997 (2)

1990 (1)

1989 (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

1988 (2)

1979 (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1971 (1)

J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres part I -- multipole expansion and ray-optical solution," IEEE Trans. Antennas Propag. 19, 378-390 (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 wave functions," Q. Appl. Math. 19, 15-24 (1961).

1954 (1)

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Appl. Opt. (2)

Appl. Phys. B (1)

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B 84, 485-492 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

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

Opt. Commun. (1)

A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Opt. Commun. 136, 114-124 (1997).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Q. Appl. Math. (3)

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

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).

Other (4)

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

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

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

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

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Figures (1)

Fig. 1
Fig. 1

Geometry under study.

Equations (41)

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J m ( λ r ) exp ( i m ϕ ) exp ( i h z ) = s = J s ( λ r 0 ) exp ( i s θ 0 ) exp ( i h z 0 ) J m + s ( λ r ) exp [ i ( m + s ) ϕ ] exp ( i h z ) ,
r 0 = x 0 2 + y 0 2 , tan θ 0 = y 0 x 0 .
( m m λ ( 1 ) n m λ ( 1 ) ) exp ( i h z ) = s = J s ( λ r 0 ) exp ( i s θ 0 ) exp ( i h z 0 ) ( m ( m + s ) λ ( 1 ) n ( m + s ) λ ( 1 ) ) exp ( i h z ) .
( m m λ ( 1 ) n m λ ( 1 ) ) exp ( i h z ) = m = J m m ( λ r 0 ) exp [ i ( m m ) θ 0 ] exp ( i h z 0 ) ( m m λ ( 1 ) n m λ ( 1 ) ) exp ( i h z ) .
( m m n r ( 1 ) ( k R , θ , ϕ ) n m n r ( 1 ) ( k R , θ , ϕ ) ) = 0 π [ ( c m n ( ζ ) a m n ( ζ ) ) m m λ ( 1 ) + ( a m n ( ζ ) c m n ( ζ ) ) n m λ ( 1 ) ] exp ( i h z ) sin ζ d ζ ,
c m n ( ζ ) = i m n + 1 2 k d P n m ( cos ζ ) d ( cos ζ ) ,
a m n ( ζ ) = m k λ 2 i m n 1 2 P n m ( cos ζ ) .
( m m n r ( 1 ) ( k R , θ , ϕ ) n m n r ( 1 ) ( k R , θ , ϕ ) ) = m = 0 π [ ( C ( m , m , n ) A ( m , m , n ) ) m m λ ( 1 ) + ( A ( m , m , n ) C ( m , m , n ) ) n m λ ( 1 ) ] exp ( i h z ) sin ζ d ζ ,
C ( m , m , n ) = c m n ( ζ ) J m m ( λ r 0 ) exp [ i ( m m ) θ 0 ] exp ( i h z 0 ) = i m n 1 2 k sin ζ d P n m ( cos ζ ) d ζ J m m ( k r 0 sin ζ ) exp [ i ( m m ) θ 0 ] exp ( i k z 0 cos ζ ) ,
A ( m , m , n ) = a m n ( ζ ) J m m ( λ r 0 ) exp [ i ( m m ) θ 0 ] exp ( i h z 0 ) = m k λ 2 i m n 1 2 P n m ( cos ζ ) J m m ( k r 0 sin ζ ) exp [ i ( m m ) θ 0 ] exp ( i k z 0 cos ζ ) .
( m m λ exp ( i h z ) n m λ exp ( i h z ) ) = n = m i n m + 1 k sin ζ 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ d P n m ( cos ζ ) d ζ ( m m n ( k R , θ , ϕ ) n m n ( k R , θ , ϕ ) ) + m P n m ( cos ζ ) sin ζ ( n m n ( k R , θ , ϕ ) m m n ( k R , θ , ϕ ) ) ] .
( m m n r ( 1 ) ( k R , θ , ϕ ) n m n r ( 1 ) ( k R , θ , ϕ ) ) = m = n = m A m n m n [ ( m m n ( k R , θ , ϕ ) n m n ( k R , θ , ϕ ) ) + B m n m n ( n m n ( k R , θ , ϕ ) m m n ( k R , θ , ϕ ) ) ] ,
A m n m n = i n m + 1 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 π k [ C ( m , m , n ) d P n m ( cos ζ ) d ζ + A ( m , m , n ) m P n m ( cos ζ ) sin ζ ] sin 2 ζ d ζ = i n n ( m m ) 2 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 π J m m ( k r 0 sin ζ ) exp [ i ( m m ) θ 0 ] exp ( i k z 0 cos ζ ) [ d P n m ( cos ζ ) d ζ d P n m ( cos ζ ) d ζ + m m P n m ( cos ζ ) sin ζ P n m ( cos ζ ) sin ζ ] sin ζ d ζ ,
B m n m n = i n m + 1 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 π k [ C ( m , m , n ) m P n m ( cos ζ ) sin ζ + A ( m , m , n ) d P n m ( cos ζ ) d ζ ] sin 2 ζ d ζ = i n n ( m m ) 2 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 π J m m ( k r 0 sin ζ ) exp [ i ( m m ) θ 0 ] exp ( i k z 0 cos ζ ) [ m d P n m ( cos ζ ) d ζ P n m ( cos ζ ) sin ζ + m P n m ( cos ζ ) sin ζ d P n m ( cos ζ ) d ζ ] sin ζ d ζ .
E i = E 0 n = 1 m = ± 1 C n m [ i g n , TE m m m n r ( 1 ) ( k R , θ , ϕ ) + g n , TM m n m n r ( 1 ) ( k R , θ , ϕ ) ] ,
C n m = { i n 1 2 n + 1 n ( n + 1 ) , m 0 ( 1 ) m ( n + m ) ! ( n m ) ! i n 1 2 n + 1 n ( n + 1 ) , m < 0 ,
g n , TE 1 = g n , TE 1 = 1 2 g n , g n , TM 1 = g n , TM 1 = i 2 g n ,
g n , TE 1 = g n , TE 1 = i 2 g n , g n , TM 1 = g n , TM 1 = 1 2 g n .
g n = exp [ s 2 ( n + 1 2 ) 2 ] ,
( g n , TM m , l o c i g n , TE m , l o c ) = 1 2 i m 1 exp ( i k z 0 ) K n m ψ ¯ 0 0 × { J m 1 ( 2 Q ¯ r 0 ρ n w 0 2 ) exp [ i ( m 1 ) θ 0 ] J m + 1 ( 2 Q ¯ r 0 ρ n w 0 2 ) exp [ i ( m + 1 ) θ 0 ] } ,
ψ ¯ 0 0 = i Q ¯ exp ( i Q ¯ r 0 2 w 0 2 ) exp [ i Q ¯ ( n + 0.5 ) 2 ( k 2 w 0 2 ) ] ,
K n m = { ( i ) m i ( n + 0.5 ) m 1 , m 0 n ( n + 1 ) ( n + 0.5 ) , m = 0 ,
ρ n = ( n + 0.5 ) k , Q ¯ = 1 i + 2 z 0 ( k w 0 2 ) .
w m n r ( 1 ) ( k R , θ , ϕ ) = s = n n ρ ( m , s , n ) w s n r ( 1 ) ( k R , θ , ϕ ) ,
ρ ( m , s , n ) = ( 1 ) s + m e i s γ [ ( n + m ) ! ( n s ) ! ( n m ) ! ( n + s ) ! ] 1 2 u s m ( n ) ( β ) e i m α ,
u s m ( n ) ( β ) = [ ( n + s ) ! ( n s ) ! ( n + m ) ! ( n m ) ! ] 1 2 σ ( n + m n s σ ) ( n m σ ) ( 1 ) n s σ ( cos β 2 ) 2 σ + s + m ( sin β 2 ) 2 n 2 σ s m .
E i = E 0 m = n = m [ i G n , TE m m m n r ( 1 ) ( k R , θ , ϕ ) + G n , TM m n m n r ( 1 ) ( k R , θ , ϕ ) ] ,
( G n , TE m G n , TM m ) = s = n n ρ ( s , m , n ) C n s ( g n , TE s g n , TM s ) .
w e o m n r ( 1 ) ( k R , θ , ϕ ) = l = m , m + 1 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! i l n N m l d n m m l ( c ) W e o m l r ( 1 ) ( c , ζ , η , ϕ ) ,
e ̑ y exp [ i k ( x sin ζ + z cos ζ ) ] = m = i m + 1 k sin ζ m m λ ( 1 ) e i h z .
e ̑ y exp ( i k z ) = n = 0 i n 2 n + 1 n ( n + 1 ) [ m e 1 n r ( 1 ) ( k R , θ , ϕ ) + i n o 1 n r ( 1 ) ( k R , θ , ϕ ) ] .
e ̑ y exp ( i k z ) = n = 0 i n 2 n + 1 n ( n + 1 ) [ 1 2 m 1 n r ( 1 ) + 1 2 ( 1 ) ( n + 1 ) ! ( n 1 ) ! m ( 1 ) n r ( 1 ) + 1 2 n 1 n r ( 1 ) 1 2 ( 1 ) ( n + 1 ) ! ( n 1 ) ! n ( 1 ) n r ( 1 ) ] .
( ρ ( 1 , m , n ) ρ ( 1 , m , n ) ) = σ = 0 n m ( n m ) ! ( n 1 σ ) ! ( m + 1 + σ ) ! ( n m σ ) ! σ ! ( ( n + 1 ) ! ( 1 ) n m σ ( cos ζ 2 ) 2 σ + 1 + m ( sin ζ 2 ) 2 n 2 σ 1 m ( n 1 ) ! ( 1 ) σ ( sin ζ 2 ) 2 σ + 1 + m ( cos ζ 2 ) 2 n 2 σ 1 m ) .
P n m ( cos ζ ) = r = 0 n m ( n + m ) ! n ! ( n r ) ! ( r + m ) ! ( n m r ) ! r ! ( 1 ) n m r sin 2 n m 2 r ζ 2 cos 2 r + m ζ 2 = r = 0 n m ( n + m ) ! n ! ( n r ) ! ( r + m ) ! ( n m r ) ! r ! ( 1 ) r cos 2 n m 2 r ζ 2 sin 2 r + m ζ 2 .
2 m P n m ( cos ζ ) sin ζ = [ m t g ζ 2 P n m ( cos ζ ) + P n m + 1 ( cos θ ) ] + [ m c t g ζ 2 P n m ( cos ζ ) P n m + 1 ( cos ζ ) ] .
2 m P n m ( cos ζ ) sin ζ = σ = 0 n m ( n + m ) ! ( n + 1 ) ! ( n m σ ) ! ( m + 1 + σ ) ! ( n 1 σ ) ! σ ! [ ( 1 ) n m σ ( cos ζ 2 ) 2 σ + m + 1 ( sin ζ 2 ) 2 n 2 σ m 1 + ( 1 ) σ ( sin ζ 2 ) 2 σ + m + 1 ( cos ζ 2 ) 2 n 2 σ m 1 ] .
d d ζ P n m ( cos ζ ) = m P n m ( cos ζ ) sin ζ m t g ζ 2 P n m ( cos ζ ) P n m + 1 ( cos ζ ) .
2 d P n m ( cos ζ ) d ζ = σ = 0 n m ( n + m ) ! ( n + 1 ) ! ( n m σ ) ! ( m + 1 + σ ) ! ( n 1 σ ) ! σ ! [ ( 1 ) n m σ ( cos ζ 2 ) 2 σ + m + 1 ( sin ζ 2 ) 2 n 2 σ m 1 ( 1 ) σ ( sin ζ 2 ) 2 σ + m + 1 ( cos ζ 2 ) 2 n 2 σ m 1 ] .
( ρ ( 1 , m , n ) n ( n + 1 ) ρ ( 1 , m , n ) ) = ( n m ) ! ( n + m ) ! [ m P n m ( cos ζ ) sin ζ + ( 1 1 ) d P n m ( cos ζ ) d ζ ] .
e ̑ y exp [ i k ( x sin ζ + z cos ζ ) ] = m = n = m i n 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ d P n m ( cos ζ ) d ζ m m n r ( 1 ) + m P n m ( cos ζ ) sin ζ n m n r ( 1 ) ] .
m = i m + 1 k sin ζ m m λ ( 1 ) e i h z = m = n = m i n 2 n + 1 n ( n + 1 ) ( n m ) ! ( n + m ) ! [ d P n m ( cos ζ ) d ζ m m n r ( 1 ) + m P n m ( cos ζ ) sin ζ n m n r ( 1 ) ] .

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