Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf
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30 1 Basic Theory of Electromagnetic Scattering
and as before, the components of the vector functions Y hmn are given by (1.51)–(1.53) with mπn|m| and τn|m| interchanged.
In the above analysis, Xemn,h and Y emn,h are expressed in the principal coordinate system, but in general, it is necessary to transform these vector functions from the principal coordinate system to the particle coordinate system through a rotation. The vector quasi-spherical wave functions can also be defined for biaxial media (εx = εy = εz ) by considering the expansion of the tangential vector function Dα(β, α)vα + Dβ (β, α)vβ in terms of vector spherical harmonics.
1.3.2 Chiral Media
For a source-free, isotropic, chiral medium, the Maxwell equations are given by (1.10), with the K matrix defined by (1.11). Following Bohren [16], the electromagnetic field is transformed to
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where A is a transformation matrix and |
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A = |
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−j |
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The transformed fields L and R are the leftand right-handed circularly polarized waves, or simply the waves of leftand right-handed types. Explicitly, the electromagnetic field transformation is
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E = L − j |
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R , |
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H = −j |
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and note that this linear transformation diagonalizes the matrix K,
Λ = A−1KA = |
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Defining the wave numbers |
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kL = |
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1 − βk |
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kR = |
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1.3 Internal Field |
31 |
we see that the waves of leftand right-handed types satisfy the equations
× L = kLL , |
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(1.54) |
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× R = −kRR , |
(1.55) |
respectively.
For chiral media, we use the same technique as for anisotropic media and express the leftand right-handed circularly polarized waves as integrals over plane waves. For the Fourier transform corresponding to waves of left-handed type,
L(r) = |
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(k) ejk·r dV (k) , |
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the di erential equations (1.54) yield
Lβ = −j kkL Lα ,
Lα = j kkL Lβ ,
and Lk = 0. The above set of equations form a system of homogeneous equations and setting the determinant equal to zero, gives the dispersion relation for the waves of left-handed type
k2 = kL2 .
Choosing Lβ as an independent scalar function, we express L as
2π π
L(r) = (eβ + jeα) Lβ (β, α)ejkL(β,α)·r sin β dβ dα , (1.56)
00
where kL(β, α) = kLek (β, α). The tangential field (eβ + jeα)Lβ is orthogonal to the vector spherical harmonics of right-handed type with respect to the
scalar product in L2tan(Ω) (cf. (B.16) and (B.17)), and as result, (eβ + jeα)Lβ possesses an expansion in terms of vector spherical harmonics of left-handed
type (cf. (B.14))
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cmn [mmn(β, α) + jnmn(β, α)] . |
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32 1 Basic Theory of Electromagnetic Scattering
Inserting (1.57) into (1.56), yields
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L(r) = − |
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+ nmn(β, α)] ejkL(β,α)·r sin β dβ dα ,
whence, using the integral representation for the vector spherical wave functions (cf. (B.26) and (B.27)), gives
∞n
L(r) = cmnLmn (kLr)
n=1 m=−n
∞n
=cmn M 1mn (kLr) + N 1mn (kLr) ,
n=1 m=−n
where the vector spherical wave functions of left-handed type Lmn are defined as
Lmn = M mn1 + N mn1 . |
(1.58) |
For the waves of right-handed type we proceed analogously. We obtain the integral representation
R(r) = |
2π π (e |
β − |
je ) |
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(β, α)ejkR(β,α)·r sin β dβ dα |
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with Rβ being an independent scalar function, and the expansion
∞n
R(r) = dmnRmn (kRr)
n=1 m=−n
∞n
=dmn M 1mn (kRr) − N 1mn (kRr)
n=1 m=−n
with the vector spherical wave functions of right-handed type Rmn being defined as
Rmn = M mn1 − N mn1 . |
(1.59) |
In conclusion, the electric and magnetic fields propagating in isotropic, chiral media possess the expansions [16, 17, 135]
∞ n
E(r) =
n=1 m=−n
∞ n
H(r) =
n=1 m=−n
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cmnLmn (kLr) − j |
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dmnRmn (kRr) , |
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cmnLmn (kLr) + dmnRmn (kRr) . |
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1.4 Scattered Field |
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An exhaustive treatment of electromagnetic wave propagation in isotropic, chiral media has been given by Lakhtakia et al. [136]. This analysis deals with the conservation of energy and momentum, properties of the infinite-medium Green’s function and the mathematical expression of Huygens’s principle.
1.4 Scattered Field
In this section we consider the basic properties of the scattered field as they are determined by energy conservation and by the propagation properties of the fields in source-free regions. The results are presented for electromagnetic scattering by dielectric particles, which is modeled by the transmission boundary-value problem. To formulate the transmission boundary-value problem we consider a bounded domain Di (of class C2) with boundary S and exterior Ds, and denote by n the unit normal vector to S directed into Ds (Fig. 1.8). The relative permittivity and relative permeability of the domain Dt are εt and µt, where t = s, i, and the wave number in the domain Dt is kt = k0√εtµt, where k0 is the wave number in free space. The unbounded domain Ds is assumed to be lossless, i.e., εs > 0 and µs > 0, and the external excitation is considered to be a vector plane wave
Ee(r) = Ee0ejke·r , He(r) = εs ek × Ee0ejke·r ,
µs
where Ee0 is the complex amplitude vector and ek is the unit vector in the direction of the wave vector ke. The transmission boundary-value problem has the following formulation.
Given Ee, He as an entire solution to the Maxwell equations representing the external excitation, find the vector fields Es, Hs C1(Ds) ∩ C(Ds) and Ei, Hi C1(Di) ∩ C(Di) satisfying the Maxwell equations
× Et = jk0µtHt, × Ht = −jk0εtEt, |
(1.60) |
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ε i, µi, k i |
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Fig. 1.8. The domain Di with boundary S and exterior Ds
34 1 Basic Theory of Electromagnetic Scattering
in Dt, t = s, i, and the two transmission conditions
n × Ei − n × Es = n × Ee ,
n × Hi − n × Hs = n × He , |
(1.61) |
on S. In addition, the scattered field Es, Hs must satisfy the Silver–M¨uller radiation condition
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uniformly for all directions r/r.
It should be emphasized that for the assumed smoothness conditions, the transmission boundary-value problem possesses an unique solution [177].
Our presentation is focused on the analysis of the scattered field in the far-field region. We begin with a basic representation theorem for electromagnetic scattering and then introduce the primary quantities which define the single-scattering law: the far-field patterns and the amplitude matrix. Because the measurement of the amplitude matrix is a complicated experimental problem, we characterize the scattering process by other measurable quantities as for instance the optical cross-sections and the phase and extinction matrices.
In our analysis, we will frequently use the Green second vector theorem
[a · ( × × b) − b · ( × × a)] dV
D
=n · [b × ( × a) − a × ( × b)] dS ,
S
where D is a bounded domain with boundary S and n is the outward unit normal vector to S.
1.4.1 Stratton–Chu Formulas
Representation theorems for electromagnetic fields have been given by Stratton and Chu [216]. If Es, Hs is a radiating solution to Maxwell’s equations in Ds, then we have the Stratton–Chu formulas
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S es (r ) g (ks, r, r ) dS(r ) |
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× × hs (r ) g (ks, r, r ) dS(r ) , |
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