Abstract

It has been shown in the past that the Ewald-Oseen extinction theorem can be generalized into a nonlocal boundary-value problem and that the scattering of an electromagnetic wave on a medium of arbitrary response can be described in terms of this boundary-value problem. The scattered field inside the medium (the interior-scattering problem) is determined, without any reference to the scattered field outside the medium, by solving a set of coupled partial differential equations, subject to certain nonlocal boundary conditions which are generalizations of the Ewald-Oseen extinction theorem. Once the field on the boundary of the medium is known, the exterior field (the exterior-scattering problem) is evaluated by direct integration. Moreover, a hypothesis was put forward according to which these nonlocal boundary conditions completely and uniquely specify the solution of the interior-scattering problem. In this paper it is shown that this hypothesis is true. The equivalence of the interior- and exterior-scattering problems to Maxwell’s equations and boundary conditions is proven. For simplicity, the proof is confined to nonmagnetic media. Thus, if Maxwell’s equations and boundary conditions have a unique solution, the interior-scattering problem will also have a unique solution, namely, that obtained from the former. Some illustrative examples are presented in the special case of a linear medium.

© 1978 Optical Society of America

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References

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  1. P. P. Ewald, Ann. Phys. (Leipz.) 49, 1 (1916).
    [Crossref]
  2. C. W. Oseen, Ann. Phys. (Leipz.) 48, 1 (1915).
    [Crossref]
  3. D. N. Pattanayak and E. Wolf, Opt. Commun. 6, 217 (1972).
    [Crossref]
  4. E. Wolf, “Electromagnetic Scattering as a Non-Local Boundary Value Problem,” Instituto Nazionale di Alta Matematica, Symposia Mathematica, Vol. XVIII (Academic, London and New York, 1976).
  5. Sometimes, the subscript “in” or “out” will be used on a vector field F(R,ω) and it signifies that R lies inside Vin or Vout, respectively.
  6. Equation (3a) was obtained from the relation E=-∇ Φ0(s)+(iω/c)×A0(s), where Φ0(s),A0(s) are the scalar -and vector potentials, respectively. The scalar potential was eliminated with the help of the Lorentz condition Φ(s)= (c/iω)∇·A(s). Finally, use was made of the inhomogeneous Helmholtz equation ∇2A0(s)+(ω/c)2A0(s)=-(4π/c)J0 and also of the identity ∇×∇×A0(s)=∇[∇·A0(s)]-∇2×A0(s). Equation (4) is the solution of the inhomogeneous Helmholtz equation, just given above, subject to the Sommerfeld radiation condition at infinity.
  7. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York and London, 1965), p. 736, Eq. 6.677(5).
  8. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 86, Eq. (3.151).
  9. A. Erderlyi, Asymptotic Expansions (Dover, New York, 1956), p. 51
  10. The superscript (+) will be used for all relevant quantities in this example.
  11. Equation (64) is obtained from Eq. (46) and the formula 6.532(4), p. 678 of the reference given in Ref. 7. After the order of the integrations is inverted, use is made of the formula 3.723(2), p. 406 of the same reference.
  12. The superscript (−) will be used for all relevant quantities in this example.
  13. For the case of beta decay, cf. Sec. 15.7, p. 526 in Ref. 8.

1972 (1)

D. N. Pattanayak and E. Wolf, Opt. Commun. 6, 217 (1972).
[Crossref]

1916 (1)

P. P. Ewald, Ann. Phys. (Leipz.) 49, 1 (1916).
[Crossref]

1915 (1)

C. W. Oseen, Ann. Phys. (Leipz.) 48, 1 (1915).
[Crossref]

Erderlyi, A.

A. Erderlyi, Asymptotic Expansions (Dover, New York, 1956), p. 51

Ewald, P. P.

P. P. Ewald, Ann. Phys. (Leipz.) 49, 1 (1916).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York and London, 1965), p. 736, Eq. 6.677(5).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 86, Eq. (3.151).

Oseen, C. W.

C. W. Oseen, Ann. Phys. (Leipz.) 48, 1 (1915).
[Crossref]

Pattanayak, D. N.

D. N. Pattanayak and E. Wolf, Opt. Commun. 6, 217 (1972).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York and London, 1965), p. 736, Eq. 6.677(5).

Wolf, E.

D. N. Pattanayak and E. Wolf, Opt. Commun. 6, 217 (1972).
[Crossref]

E. Wolf, “Electromagnetic Scattering as a Non-Local Boundary Value Problem,” Instituto Nazionale di Alta Matematica, Symposia Mathematica, Vol. XVIII (Academic, London and New York, 1976).

Ann. Phys. (Leipz.) (2)

P. P. Ewald, Ann. Phys. (Leipz.) 49, 1 (1916).
[Crossref]

C. W. Oseen, Ann. Phys. (Leipz.) 48, 1 (1915).
[Crossref]

Opt. Commun. (1)

D. N. Pattanayak and E. Wolf, Opt. Commun. 6, 217 (1972).
[Crossref]

Other (10)

E. Wolf, “Electromagnetic Scattering as a Non-Local Boundary Value Problem,” Instituto Nazionale di Alta Matematica, Symposia Mathematica, Vol. XVIII (Academic, London and New York, 1976).

Sometimes, the subscript “in” or “out” will be used on a vector field F(R,ω) and it signifies that R lies inside Vin or Vout, respectively.

Equation (3a) was obtained from the relation E=-∇ Φ0(s)+(iω/c)×A0(s), where Φ0(s),A0(s) are the scalar -and vector potentials, respectively. The scalar potential was eliminated with the help of the Lorentz condition Φ(s)= (c/iω)∇·A(s). Finally, use was made of the inhomogeneous Helmholtz equation ∇2A0(s)+(ω/c)2A0(s)=-(4π/c)J0 and also of the identity ∇×∇×A0(s)=∇[∇·A0(s)]-∇2×A0(s). Equation (4) is the solution of the inhomogeneous Helmholtz equation, just given above, subject to the Sommerfeld radiation condition at infinity.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York and London, 1965), p. 736, Eq. 6.677(5).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 86, Eq. (3.151).

A. Erderlyi, Asymptotic Expansions (Dover, New York, 1956), p. 51

The superscript (+) will be used for all relevant quantities in this example.

Equation (64) is obtained from Eq. (46) and the formula 6.532(4), p. 678 of the reference given in Ref. 7. After the order of the integrations is inverted, use is made of the formula 3.723(2), p. 406 of the same reference.

The superscript (−) will be used for all relevant quantities in this example.

For the case of beta decay, cf. Sec. 15.7, p. 526 in Ref. 8.

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Equations (131)

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D ( R , ω ) = { E out ( R , ω ) , if R V out E in ( R , ω ) + 4 π P ( R , ω ) , if R V in .
E o ( R , ω ) = E o ( f ) ( R , ω ) + E o ( s ) ( R , ω ) ,
B o ( R , ω ) = B o ( f ) ( R , ω ) + B o ( s ) ( R , ω ) ,
E o ( s ) ( R , ω ) = - ( 1 / i k ) [ × × A o ( s ) ( R , ω ) - ( 4 π / c ) J o ( R , ω ) ] ,
B o ( s ) ( R , ω ) = × A o ( s ) ( R , ω ) ,
A o ( s ) ( R , ω ) = 1 c V in + V out G ( R - R ; k ) J o ( R , ω ) d 3 R ,
G ( R ; k ) = e i k R / R ,
k = ω / c .
[ 2 + k 2 ] G ( R - R ; k ) = - 4 π δ ( 3 ) ( R - R ) ,
S o E o · d S = 4 π V o ρ o d V ,
S o B o · d S = 0 ,
C E o · d l = i k S B o · d S ,
C B o · d l = - i k S E o · d S + 4 π c S J o · d S ,
ρ o = ( 1 / i k c ) · J o ,
[ 2 + k 2 ] E o ( R , ω ) = ( 4 π / i k c ) [ k 2 J o ( R , ω ) + ( · J o ( R , ω ) ] ,
· E o ( R , ω ) = ( 4 π / i k c ) · J o ( R , ω ) .
E in ( R , ω ) = - ( 1 / i k ) [ × × A in ( R , ω ) - ( 4 π / c ) J t ( R , ω ) ] ,
[ 2 + k 2 ] A in ( R , ω ) = - ( 4 π / c ) J t ( R , ω ) ,
J t ( R , ω ) = J o in ( R , ω ) - i ω P ( R , ω ) .
E in ( R , ω ) = E o in ( R , ω ) + × × Q ( R , ω ) - 4 π P ( R , ω ) ,
E out ( R , ω ) = E o out ( R , ω ) + × × Q ( R , ω ) ,
Q ( R , ω ) = V in G ( R - R ; k ) P ( R , ω ) d 3 R ,
B ( R , ω ) = ( 1 / i k ) × E ( R , ω ) .
B ( R , ω ) = B o ( R , ω ) - i k × Q ( R , ω ) ,
× × F = ( · F ) - 2 F .
S o D · d S = S o E o · d S ,
S o B · d S = S o B o · d S ,
C E · d l = C E o · d l + k 2 C Q · d l ,
S B · d S = S B o · d S - i k C Q · d l ,
C B · d l = C B o · d l - i k C ( × Q ) · d l ,
S D · d S = S E o · d S + C ( × Q ) · d l .
[ 2 + k 2 ] E in ( R , ω ) = ( 4 π / i k c ) { k 2 J t ( R , ω ) + [ · J t ( R , ω ) ] } ,
· E in ( R , ω ) = ( 4 π / i k c ) · J t ( R , ω ) ,
× × Q ( R , ω ) = × × [ in ( R , ω ) + in ( R , ω ) ] + E in ( R , ω ) + 4 π P ( R , ω ) - E o in ( i s ) ( R , ω ) ,
( R , ω ) = 1 4 π k 2 S - [ E in ( R , ω ) G ( R - R ; k ) n - G ( R - R ; k ) E in ( R , ω ) n ] d S ,
( R , ω ) = 1 i k 3 c S - G ( R - R ; k ) · ( J o ( R , ω ) - i ω P ( R , ω ) ) n d S ,
E o in ( i s ) ( R , ω ) = - ( 1 / i k ) [ × × A o ( i s ) ( R , ω ) - ( 4 π / c ) J o in ( R , ω ) ] ,
A o ( i s ) ( R , ω ) = 1 c V in G ( R - R ; k ) J o ( R , ω ) d 3 R .
× × Q ( R , ω ) = × × [ out ( R , ω ) + out ( R , ω ) ] - E o out ( i s ) ( R , ω ) ,
E o out ( i s ) ( R , ω ) = - ( 1 / i k ) × × A o ( i s ) ( R , ω ) .
E o in ( R , ω ) + × × [ in ( R , ω ) + in ( R , ω ) ] = 0.
E o in ( R , ω ) = E o in ( R , ω ) - E o in ( i s ) ( R , ω ) ,
E out ( R , ω ) = E o out ( R , ω ) + × × [ out ( R , ω ) + out ( R , ω ) ] ,
E o out ( R , ω ) = E o out ( R , ω ) - E o out ( i s ) ( R , ω ) ,
P ( R , ω ) = ( - 1 ) / 4 π E in ( R , ω ) ,
· E in ( R , ω ) = ( 4 π / i k c ) · J o in ( R , ω ) ,
( R , ω ) = 1 i k k 2 c S - G ( R - R ; k ) × [ · J o ( R , ω ) ] n d S ,
k = ( ω / c ) .
A in ( R , ω ) = A in ( R , ω ) + χ ( R , ω )
[ 2 + k 2 ] χ ( R , ω ) = ( - 1 ) · A in ( R , ω ) ,
E in ( R , ω ) = - ( 1 / i k ) × [ × × A in ( R , ω ) - ( 4 π / c ) J o in ( R , ω ) ] ,
[ 2 + k 2 ] A in ( R , ω ) = - ( 4 π / c ) J o in ( R , ω ) .
J o ( R , ω ) = z ˆ q δ ( x ) δ ( y ) e i k o z Θ [ ( L / 2 ) - z ] ,
k o = ω / β c .
× × in ( R , ω ) = 0 ,
E out ( R , ω ) = × × out ( R , ω ) .
E ρ in ( R , ω ) = 2 q ω - σ λ [ K 1 ( λ ρ ) - A ( σ ) I 1 ( λ ρ ) ] f ( σ ) e i σ z d σ ,
E z in ( R , ω ) = - i 2 q ω - λ 2 [ K o ( λ ρ ) + A ( σ ) I 0 ( λ ρ ) ] f ( σ ) e i σ z d σ ,
f ( σ ) = ( 1 / π ) [ sin ( σ - k o ) ( L / 2 ) / ( σ - k o ) ] ,
λ 2 = σ 2 - k 2 ,
G ( R - R ; k ) = 1 π - K o ( λ ρ - ρ ) e i σ ( z - z ) d σ ,
ρ in ( R , ω ) = 2 q ω k 2 - σ λ M 1 ( λ ρ o , λ ρ o , σ ) × I 1 ( λ ρ ) f ( σ ) e i σ z d σ ,
z in ( R , ω ) = - i 2 q ω k 2 - λ 2 M 2 ( λ ρ o , λ ρ o , σ ) × I o ( λ ρ ) f ( σ ) e i σ z d σ ,
ρ out ( R , ω ) = 2 q ω k 2 - σ λ N 1 ( λ ρ o , λ ρ o , σ ) × K 1 ( λ ρ ) f ( σ ) e i σ z d σ ,
z out ( R , ω ) = - i 2 q ω k 2 - λ 2 N 2 ( λ ρ o , λ ρ o , σ ) × K o ( λ ρ ) f ( σ ) e i σ z d σ ,
M 1 ( z , z , σ ) = - z K o ( z ) K 1 ( z ) + z K 1 ( z ) K o ( z ) + A ( σ ) [ z K o ( z ) I 1 ( z ) + z K 1 ( z ) I o ( z ) ] ,
M 2 ( z , z , σ ) = - z K o ( z ) K 1 ( z ) + z K 1 ( z ) K o ( z ) + A ( σ ) [ - z K 1 ( z ) I o ( z ) - z K o ( z ) I 1 ( z ) ] ,
N 1 ( z , z , σ ) = z I o ( z ) K 1 ( z ) + z I 1 ( z ) K o ( z ) + A ( σ ) [ - z I o ( z ) I 1 ( z ) + z I 1 ( z ) I o ( z ) ] ,
N 2 ( z , z , σ ) = z I 1 ( z ) K o ( z ) + z I o ( z ) K 1 ( z ) + A ( σ ) [ z I 1 ( z ) I o ( z ) - z I o ( z ) I 1 ( z ) ] ,
λ 2 = σ 2 - k 2 .
K o ( λ ρ - ρ ) = m = - I m ( λ ρ < ) K m ( λ ρ > ) e i m ( ϕ - ϕ ) ,
( × × in ) ρ = - i - σ D ( σ , ω ) I 1 ( λ ρ ) e i σ z d σ ,
( × × in ) z = - λ D ( σ , ω ) I o ( λ ρ ) e i σ z d σ ,
D ( σ , ω ) = i ( 2 q ρ o / ω ) λ 2 f ( σ ) × [ ( K 1 ( λ ρ o ) K o ( λ ρ o ) - λ λ K o ( λ ρ o ) K 1 ( λ ρ o ) ) + A ( σ ) ( K 1 ( λ ρ o ) I o ( λ ρ o ) + λ λ K o ( λ ρ o ) I 1 ( λ ρ o ) ) ] .
A ( σ ) = - K 1 ( λ ρ o ) K o ( λ ρ o ) - ( λ / λ ) K o ( λ ρ o ) K 1 ( λ ρ o ) K 1 ( λ ρ o ) I o ( λ ρ o ) + ( λ / λ ) K o ( λ ρ o ) I 1 ( λ ρ o ) .
E ρ out ( R , ω ) = 2 q ω - σ B ( σ ) K 1 ( λ ρ ) f ( σ ) e i σ z d σ ,
E z out ( R , ω ) = - i 2 q ω - λ B ( σ ) K o ( λ ρ ) f ( σ ) e i σ z d σ ,
B ( σ ) = 1 ρ o 1 K 1 ( λ ρ o ) I o ( λ ρ o ) + ( λ / λ ) K o ( λ ρ o ) I 1 ( λ ρ o ) .
B ϕ in ( R , ω ) = 2 q c - λ [ K 1 ( λ ρ ) - A ( σ ) I 1 ( λ ρ ) ] f ( σ ) e i σ z d z ,
B ϕ out ( R , ω ) = 2 q c - B ( σ ) K 1 ( λ ρ ) f ( σ ) e i σ z d σ ,
J o ( + ) ( R , ω ) = z ˆ q δ ( x ) δ ( y ) e i k o z Θ ( z ) ,
· J o ( + ) ( R , ω ) z = 0 + = i q k o δ ( x ) δ ( y ) ,
× × [ in ( + ) ( R , ω ) + in ( + ) ( R , ω ) ] = 0 ,
E out ( + ) ( R , ω ) = × × [ out ( R , ω ) + out ( R , ω ) ] .
E ρ in ( + ) ( R , ω ) = - i q ω 0 σ ( 2 i k o e i k o z λ 2 + k o 2 r + e - λ z λ + i k o - λ D ( + ) ( σ ) e - λ z ) J 1 ( σ ρ ) σ d σ ,
E z in ( + ) ( R , ω ) = i q ω 0 [ - 2 ( 1 - σ 2 λ 2 + k 0 2 ) e i k o z - σ 2 e - λ z λ ( λ + i k o ) + σ 2 D ( + ) ( σ ) e - λ z ] J o ( σ ρ ) σ d σ ,
G ( R - R ; k ) = 0 1 λ e - λ z - z J o ( σ ρ - ρ ) σ d σ .
{ × × [ in ( + ) + in ( + ) ] } ρ = 0 T ( + ) ( σ , ω ) e - λ z J 1 ( σ ρ ) σ d σ ,
{ × × [ in ( + ) + in ( + ) ] } z = 0 σ λ T ( + ) ( σ , ω ) e - λ z J o ( σ ρ ) σ d σ ,
T ( + ) ( σ , ω ) = i q 2 ω ( 1 λ ( λ / - λ ) 1 λ - i k o - ( λ / + λ ) D ( + ) ( σ ) ) .
D ( + ) ( σ ) = 1 λ ( λ / ) - λ ( λ / ) + λ 1 λ - i k o .
{ × × [ out ( + ) + out ( + ) ] } ρ = 0 S ( + ) ( σ , ω ) e λ z J 1 ( σ ρ ) σ d σ ,
{ × × [ out ( + ) + out ( + ) ] } z = - 0 k λ S ( + ) ( σ , ω ) e λ z J o ( σ ρ ) σ d σ ,
S ( + ) ( σ , ω ) = i q 2 ω σ λ [ - ( λ + λ ) + λ ( λ - λ ) D ( + ) ( σ ) ] .
E ρ out ( + ) ( R , ω ) = - i q ω 0 λ σ C ( + ) ( σ ) e λ z J 1 ( σ ρ ) σ d σ ,
E z out ( + ) ( R , ω ) = i q ω 0 σ 2 C ( + ) ( σ ) e λ z J o ( σ ρ ) σ d σ ,
C ( + ) ( σ ) = 2 1 ( λ / ) + λ 1 λ - i k o .
B ϕ in ( + ) ( R , ω ) = q c 0 σ ( 2 e i k o z λ 2 + k o 2 - e - λ z λ ( λ + i k o ) + D ( + ) ( σ ) e - λ z ) J 1 ( σ ρ ) σ d σ ,
B ϕ out ( + ) ( R , ω ) = q c 0 σ C ( + ) ( σ ) e λ z J 1 ( σ ρ ) σ d σ .
J o ( - ) ( R , ω ) = z ˆ q δ ( x ) δ ( y ) e i k o z Θ ( - z ) .
E o in ( - ) ( R , ω ) + × × in ( - ) ( R , ω ) = 0 ,
E out ( - ) ( R , ω ) = E o out ( - ) ( R , ω ) + × × out ( - ) ( R , ω ) .
E ρ in ( - ) ( R , ω ) = i q ω 0 σ λ D ( - ) ( σ ) e - λ z J 1 ( σ ρ ) σ d σ ,
E z in ( - ) ( R , ω ) = i q ω 0 σ 2 D ( - ) ( σ ) e - λ z J o ( σ ρ ) σ d σ ,
[ × × in ( - ) ] ρ = 0 T ˆ ( - ) ( σ , ω ) e - λ z J 1 ( σ ρ ) σ d σ ,
[ × × in ( - ) ] z = 0 σ λ T ˆ ( - ) ( σ , ω ) e - λ z J o ( σ ρ ) σ d σ ,
T ˆ ( - ) ( σ , ω ) = - i ( q / 2 ω ) σ ( λ / + λ ) D ( - ) ( σ ) .
E o ρ in ( - ) ( R , ω ) = i q ω 0 σ λ + i k o e - λ z J 1 ( σ ρ ) σ d σ ,
E o z in ( - ) ( R , ω ) = i q ω 0 σ 2 λ ( λ + i k o ) e - λ z J o ( σ ρ ) σ d σ ,
E o ρ in ( - ) ( R , ω ) + [ × × in ( - ) ( R , ω ) ] ρ = o T ( - ) ( σ , ω ) e - λ z J 1 ( σ ρ ) σ d σ ,
E o z in ( - ) ( R , ω ) + [ × × in ( - ) ( R , ω ) ] z = 0 σ λ T ( - ) ( σ , ω ) e - λ z J o ( σ ρ ) σ d σ ,
T ( - ) ( σ , ω ) = i q ω σ [ 1 λ + i k o - 1 2 ( λ + λ ) D ( - ) ( σ ) ] .
D ( - ) ( σ ) = 2 ( λ / ) + λ 1 λ + i k o .
[ × × out ( - ) ] ρ = - i q ω 0 σ λ C ( - ) ( σ ) e λ z J 1 ( σ ρ ) σ d σ ,
[ × × out ( - ) ] z = i q ω 0 σ 2 C ( - ) ( σ ) e λ z J o ( σ ρ ) σ d σ ,
C ( - ) ( σ ) = - ( 1 / 2 λ ) [ ( λ / ) - λ ] D ( - ) ( σ ) .
E o ρ out ( - ) ( R , ω ) = - i q ω 0 σ ( 2 i k o e i k o z λ 2 + k o 2 - e λ z λ - i k o ) J 1 ( σ ρ ) σ d σ ,
E o z out ( - ) ( R , ω ) = i q ω 0 [ - 2 ( 1 - σ 2 λ 2 + k o 2 ) e i k o z - σ 2 e λ z λ ( λ - i k o ) ] J o ( σ ρ ) σ d σ .
E ρ out ( - ) ( R , ω ) = - i q ω 0 σ ( 2 i k o e i k o z λ 2 + k o 2 - e λ z λ - i k o + λ C ( - ) ( σ ) e λ z ) J 1 ( σ λ ) σ d σ ,
E z out ( - ) ( R , ω ) = i q ω 0 [ - 2 ( 1 - σ 2 λ 2 + k 0 2 ) e i k o z - σ 2 e λ z λ ( λ - i k o ) + σ 2 C ( - ) ( σ ) e λ z ] J o ( σ λ ) σ d σ ,
C ( - ) ( σ ) = - 1 λ ( λ / ) - λ ( λ / ) + λ 1 λ + i k o .
B ϕ in ( - ) ( R , ω ) = q c 0 σ D ( - ) ( σ ) e - λ z J 1 ( σ ρ ) σ d σ ,
B ϕ out ( - ) ( R , ω ) = q c 0 σ ( 2 e i k o z λ 2 + k o 2 - e λ z λ ( λ - i k o ) + C ( - ) ( σ ) e λ z ) J 1 ( σ ρ ) σ d σ .
P ( R , ω ) = - 1 4 π k 2 [ 2 + k 2 ] E in ( R , ω ) + 1 i k c × ( J o in ( R , ω ) + 1 k 2 [ · J t ( R , ω ) ] ) ,
G ( R - R ; k ) P ( R , ω ) = 1 k 2 E in ( R , ω ) δ ( 3 ) ( R - R ) + 1 4 π k 2 [ E in ( R , ω ) 2 G ( R - R ; k ) - G ( R - R ; k ) 2 E in ( R , ω ) ] + 1 i k c G ( R - R ; k ) ( J o in ( R , ω ) + 1 k 2 [ · J t ( R , ω ) ] ) .
× × Q ( R , ω ) = 1 k 2 × × E in ( R , ω ) V in δ ( 3 ) ( R - R ) d 3 R + × × ( R , ω ) + 1 i k × × A o ( i s ) ( R , ω ) + 1 i k 3 c × × V in G ( R - R ; k ) [ · J t ( R , ω ) ] d 3 R ,
[ G ( R - R ; k ) · J t ( R , ω ) ] = [ G ( R - R ; k ) ] · J t ( R , ω ) + G ( R - R ; k ) [ · J t ( R , ω ) ] ,
V in [ G ( R - R ; k ) · J t ( R , ω ) ] d 3 R = S - G ( R - R ; k ) · J t ( R , ω ) n d S ,
× [ G ( R - R ; k ) ] = - × [ G ( R - R ; k ) ] = 0 ,
× × Q ( R , ω ) = × × [ ( R , ω ) + ( R , ω ) ] + 1 i k × × A o ( i s ) ( R , ω ) + 1 k 2 × × [ E in ( R , ω ) V in δ ( 3 ) ( R - R ) d 3 R ] .
1 k 2 × × E in ( R , ω ) = 1 k 2 { [ · E in ( R , ω ) ] - 2 E in ( R , ω ) } = E in ( R , ω ) + 4 π P ( R , ω ) - 4 π i k c J o in ( R , ω ) .