Abstract

Nonradiating sources and the fields that they generate within the source domain are characterized in a novel way, as solutions to an overspecified boundary value problem. This characterization is used to describe a procedure for determining all nonradiating, spherically symmetric sources of a finite radius. An example of a source of this kind is presented and is discussed in detail.

© 1989 Optical Society of America

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References

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  1. For a review of this subject see, for example, B. Hoenders, “The uniqueness of inverse problems,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), Chap. 3, pp. 41–82.
    [CrossRef]
  2. The only investigations concerning fields that nonradiating sources generate within the source region that we are aware of are the following: A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973); Kim and Wolf.3
    [CrossRef]
  3. K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1 (1986).
    [CrossRef]
  4. F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 27, 551–576 (1973).
    [CrossRef]
  5. Cf. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 740, Eq. (16.5) with l= 0.
  6. See, for example, W. H. Carter, “Band-limited angular-spectrum approximation to a spherical scalar wave field,” J. Opt. Soc. Am. 65, 1054–1058 (1975), appendix.
    [CrossRef]
  7. O. D. Kellog, Foundations of Potential Theory (Dover, New York, 1953), Chap. VI, Sec. 3.

1986 (1)

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1 (1986).
[CrossRef]

1975 (1)

1973 (2)

The only investigations concerning fields that nonradiating sources generate within the source region that we are aware of are the following: A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973); Kim and Wolf.3
[CrossRef]

F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 27, 551–576 (1973).
[CrossRef]

Carter, W. H.

Devaney, A. J.

The only investigations concerning fields that nonradiating sources generate within the source region that we are aware of are the following: A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973); Kim and Wolf.3
[CrossRef]

Friedlander, F. G.

F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 27, 551–576 (1973).
[CrossRef]

Hoenders, B.

For a review of this subject see, for example, B. Hoenders, “The uniqueness of inverse problems,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), Chap. 3, pp. 41–82.
[CrossRef]

Jackson, Cf. J. D.

Cf. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 740, Eq. (16.5) with l= 0.

Kellog, O. D.

O. D. Kellog, Foundations of Potential Theory (Dover, New York, 1953), Chap. VI, Sec. 3.

Kim, K.

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1 (1986).
[CrossRef]

Wolf, E.

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1 (1986).
[CrossRef]

The only investigations concerning fields that nonradiating sources generate within the source region that we are aware of are the following: A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973); Kim and Wolf.3
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1 (1986).
[CrossRef]

Phys. Rev. D (1)

The only investigations concerning fields that nonradiating sources generate within the source region that we are aware of are the following: A. J. Devaney, E. Wolf, “Radiating and non-radiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973); Kim and Wolf.3
[CrossRef]

Proc. London Math. Soc. (1)

F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 27, 551–576 (1973).
[CrossRef]

Other (3)

Cf. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 740, Eq. (16.5) with l= 0.

For a review of this subject see, for example, B. Hoenders, “The uniqueness of inverse problems,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1978), Chap. 3, pp. 41–82.
[CrossRef]

O. D. Kellog, Foundations of Potential Theory (Dover, New York, 1953), Chap. VI, Sec. 3.

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Figures (2)

Fig. 1
Fig. 1

Normalized nonradiating source distribution N Q 1 ( r ) ( N = 2 a 10 π a / k 2 ), for different values of the parameter ka = 2πa/λ.

Fig. 2
Fig. 2

Normalized field MV1(r), ( M = a 5 a / 2 π), within the spherical source, generated by the nonradiating distribution Q1(r), as a function of the normalized distance r/a from the center of the source.

Equations (64)

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( 2 + k 2 ) V ( r ) = - 4 π Q ( r ) ,
V ( r s ^ ) ~ A ( s ^ ) e i k r r
( 4 π ) A ( s ^ ) 2 d Ω = 0 ,
V ( r ) = 0             when             r D ,
V ( r ) 0             when             r D .
V NR ( r ) r S = 0 ,
V NR ( r ) n | r S = 0 ,
( 2 + k 2 ) V NR ( r ) = - 4 π Q NR ( r ) ,
[ 1 r 2 d d r ( r 2 d d r ) + k 2 ] V NR ( r ) = - 4 π Q NR ( r ) ,
V ( a ) = 0 ,
[ d V ( r ) d r ] r = a = 0.
[ 1 r 2 d d r ( r 2 d d r ) + k n 2 ] ψ n ( r ) = 0             ( n = 0 , 1 , 2 ) ,
ψ n ( a ) = 0.
r a ψ n * ( r ) ψ m ( r ) d 3 r = δ n m ,
ψ n ( r ) = k n 2 π a j 0 ( k n r ) k n 2 π a [ sin ( k n r ) k n r ] ,
k n = ( n + 1 ) π a .
V NR ( r ) = n = 0 a n ψ n ( r ) ,
n = 0 α n a n = 0 ,
α n = d d r ψ n ( r ) | r = a .
d ψ n ( r ) d r = k n a 2 π a [ cos ( k n a ) - sin ( k n a ) k n a ] ,
α n d ψ n ( r ) d r | r = a = ( - 1 ) n + 1 ( n + 1 ) π a 2 2 π a .
Q NR ( r ) = 1 4 π n = 0 ( k 2 - k n 2 ) a n ψ n ( r ) .
V 1 ( r ) , V 2 ( r ) , V 3 ( r ) , V m ( r ) , ,
V m ( r ) = v 0 ( m ) ψ 0 ( r ) + v 1 ( m ) ψ 1 ( r ) + + v m ( m ) ψ m ( r )
D V n * ( r ) V m ( r ) d 3 r = δ n m .
Q m ( r ) = - 1 4 π j = 0 m ( k 2 - k j 2 ) v j ( m ) ψ j ( r ) .
V 1 ( r ) = v 0 ( 1 ) ψ 0 ( r ) + v 1 ( 1 ) ψ 1 ( r ) .
α 0 v 0 ( 1 ) + α 1 v 1 ( 1 ) = 0 ,
D V 1 * ( r ) V 1 ( r ) d 3 r = 1 ,
v 0 ( 1 ) 2 + v 1 ( 1 ) 2 = 1.
v 0 ( 1 ) = α 1 α 0 ( 1 + | α 1 α 0 | 2 ) - 1 / 2 exp [ i ϕ ( 1 ) ] ,
v 1 ( 1 ) = ( 1 + | α 1 α 0 | 2 ) - 1 / 2 exp [ i ϕ ( 1 ) ] ,
v 0 ( 1 ) = 2 5 exp [ i ϕ ( 1 ) ] ,             v 1 ( 1 ) = 1 5 exp [ i ϕ ( 1 ) ] .
V 1 ( r ) = 1 a ( 2 π 5 a ) 1 / 2 [ j 0 ( π r a ) + j 0 ( 2 π r a ) ] .
Q 1 ( r ) = - k 2 2 a 10 π a { [ 1 - ( π k a ) 2 ] j 0 ( π r a ) + [ 1 + ( 2 π k a ) 2 ] j 0 ( 2 π r a ) } .
Q 1 ( r ) ~ - k 2 2 a 10 π a [ j 0 ( π r a ) + j 0 ( 2 π r a ) ] ,
V ( r > a ) = A e i k r r ,
A = 4 π 0 a Q ( r ) j 0 ( k r ) r 2 d r .
Q ( r ) Q ( n ) ( r ) = j 0 ( n π r a )             ( n = 1 , 2 , 3 , ) ,
V ( r > a ) V ( n ) ( r > a ) = A ( n ) e i k r r ,
A ( n ) = 4 π 0 a j 0 ( n π r a ) j 0 ( k r ) r 2 d r .
A ( n ) = sin ( k a ) k a 4 π ( - 1 ) n - 1 a 3 ( n π ) 2 - ( k a ) 2 .
Q 1 ( r ) = - k 2 2 a 10 π a { [ 1 - ( π k a ) 2 ] Q ( 1 ) ( r ) + [ 1 - ( 2 π k a ) 2 ] Q ( 2 ) ( r ) } .
V 1 ( r > a ) = - k 2 2 a 10 π a { [ 1 - ( π k a ) 2 ] A ( 1 ) + [ 1 - ( 2 π k a ) 2 ] A ( 2 ) } e i k r r
[ 1 - ( π k a ) 2 ] A ( 1 ) = - 4 π a k 2 [ sin ( k a ) k a ] ,
[ 1 - ( 2 π k a ) 2 ] A ( 2 ) = + 4 π a k 2 [ sin ( k a ) k a ] .
V 1 ( r ) = 0             for all r > a .
V m ( r ) = C m r { 3 2 m sin [ π ( m + 1 ) r / a ] + ( m + 1 ) sin ( π m r / a ) cos 2 ( π r / 2 a ) + m ( 2 m + 1 ) sin [ π ( m + 1 ) r / a ] } ,
C m = 1 [ 2 π a m ( m + 2 ) ( 2 m + 1 ) ( 2 m + 3 ) ] 1 / 2
Q NR ( r ) = m = 1 b m Q m ( r ) ,
Q m ( r ) = - 1 4 π ( 2 + k 2 ) V m ( r )
m = 1 m 4 b m 2 < .
V ( r ) = D exp ( i k r - r ) r - r Q ( r ) d 3 r
V ( r ) M D d 3 r r - r M r R + R 0 d 3 r r = C M ( R + R 0 ) 2             for all r in S R .
| exp ( i k r 1 - r ) r 1 - r - exp ( i k r 2 - r ) r 2 - r | C r 1 - r 2 ( 1 r 1 - r 2 + 1 r 2 - r 2 ) .
V ( r ) = D r ( exp ( i k r - r ) r - r ) Q ( r ) d 3 r .
V ( r ) = ( D S + D \ S ) × [ r - r r - r 3 ( i k r - r - 1 ) exp ( i k r - r ) Q ( r ) ] d 3 r .
I ( κ , k ; a ) = 0 a j 0 ( κ r ) j 0 ( k r ) r 2 d r .
j 0 ( x ) = sin x x ,
I ( κ , k ; a ) = 1 κ k 0 a sin ( κ r ) sin ( k r ) d r .
I ( κ , k ; a ) = a 2 κ k { j 0 [ ( κ - k ) a ] - j 0 [ ( κ + k ) a ] } .
k n = ( n + 1 ) π a             ( n = 0 , 1 , 2 , ) .
j 0 [ ( k n ± k ) a ] = j 0 [ ( n + 1 ) π ± k a ] = sin [ ( n + 1 ) π ± k a ] ( n + 1 ) π ± k a = ( - 1 ) n sin ( k a ) ( n + 1 ) π ± k a ,
I ( k n , k ; a ) = [ sin ( k a ) k a ] ( - 1 ) n a 3 [ ( n + 1 ) π ] 2 - ( k a ) 2 .

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