Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

36 1 Basic Theory of Electromagnetic Scattering

arbitrary constant unit vector u and apply Green’s second vector theorem (1.64) to a(r ) = Ei(r ) and b(r ) = g(ki, r , r)u, for r Di. Recalling that

= − u · Ei (r ) δ (r − r) dV (r ) = −u · Ei(r) .

Di

Taking into account that for r = r,

× × g (ki, r , r) u = ki2g (ki, r , r) u + · g (ki, r , r) u , we rewrite the right-hand side of (1.64) as

{n · Ei [ · g (ki, ·r) u] + n · [Ei × ( × g (ki, ·r) u)

S

+ ( × Ei) × g (ki, ·r) u]} dS

={n · Ei [ · g (ki, ·r) u] + n · [Ei × ( × g (ki, ·r) u)]

S

+ 12 n · [( × Ei) × ( × × g (ki, ·r) u) ki

− ( × Ei) × · g (ki, ·r) u]} dS .

From Stokes theorem we have

n · { × [Hi · g (ki, ·r) u]} dS = 0 ,

S

whence, using the vector identity ×(αb) = α×b+α ×b and the Maxwell equations, we obtain

Finally, using the vector identity a ·(b ×c) = (a ×b) ·c, the symmetry relationg(ki, r , r) = − g(ki, r , r), the Maxwell equations and the identities

[ × g (ki, r , r) u] · [n (r ) × Ei (r )]

= { × g (ki, r , r) [n (r ) × Ei (r )]} · u

and

[ × × g (ki, r , r) u] · [n (r ) × Hi (r )]

= { × × g (ki, r , r) [n (r ) × Hi (r )]} · u ,

Since u is arbitrary, we have established the Stratton–Chu formula for r Di. If r Ds, we have

u · Ei (r ) δ (r − r) dV (r ) = 0 ,

Di

and the proof follows in a similar manner. For radiating solutions to the Maxwell equations, we see that the proof is established if we can show that

as R → ∞, where SR is a spherical surface situated in the far-field region. To prove this assertion we use the Silver–M¨uller radiation condition and the general assumption Im{ks} ≥ 0.

Alternative representations for Stratton–Chu formulas involve the free space dyadic Green function G instead of the fundamental solution g [228]. A dyad D serves as a linear mapping from one vector to another vector, and in general, D can be introduced as the dyadic product of two vectors: D = a b. The dot product of a dyad with a vector is another vector: D · c = (a b) · c = a(b · c) and c · D = c · (a b) = (c · a)b, while the cross product of a dyad with a vector is another dyad: D×c = (a b)×c = a (b×c) and c × D = c × (a b) = (c × a) b. The free space dyadic Green function

38 1 Basic Theory of Electromagnetic Scattering

where I is the identity dyad (D · I = I · D = D). Multiplying the di erential equation (1.65) by I and using the identities

× × Ig = g − I g ,

× × ( g) = 0 ,

gives the di erential equation for the free space dyadic Green function

The Stratton–Chu formula for vector fields satisfying the Maxwell equations in bounded domains read as

and this integral representation follows from the second vector-dyadic Green theorem [220]:

applied to a(r ) = Ei(r ) and D(r ) = G(ki, r , r), the di erential equation

for the free space dyadic Green function, and the identity a·(b×D) = (a×b)· D. For radiating solutions to the Maxwell equations, we use the asymptotic behavior of the free space dyadic Green function in the far-field region

to show that the integral over the spherical surface vanishes at infinity.

Remark. The Stratton–Chu formulas are surface-integral representations for the electromagnetic fields and are valid for homogeneous particles. For inhomogeneous particles, a volume-integral representation for the electric field can be derived. For this purpose, we consider the nonmagnetic domains Ds and Di (µs = µi = 1), rewrite the Maxwell equations as

× Et = jk0Ht , × Ht = −jk0εtEt in Dt , t = s, i ,

and assume that the domain Di is isotropic, linear and inhomogeneous, i.e.,

εi = εi(r). The Maxwell curl equation for the magnetic field Hi can be written as