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

respectively, and the forcing function J by

J = ks2 m2rt − 1 E ,

where

!

1 in Ds ,

mrt = mr in Di ,

we see that the total electric and magnetic fields satisfy the Maxwell curl equations,

j

× E = jk0H , × H = −jk0εsE − k0 J in Ds Di .

By taking the curl of the first equation we obtain an inhomogeneous di erential equation for total electric field

Making use of the di erential equation for the free space dyadic Green function (1.66) and the identity

× G (ks, r, r ) · J (r ) = × G (ks, r, r ) · J (r ) ,

we derive

× × G (ks, r, r ) · J (r )

−ks2 G (ks, r, r ) · J (r ) = I · J (r ) δ (r − r ) .

Integrating this equation over all r and using the identity δ(r−r ) = δ(r −r), gives [131]

40 1 Basic Theory of Electromagnetic Scattering

Because (1.67) and (1.68) have the same right-hand side we deduce that

and, since J = 0 in Ds, we obtain

This vector field is the particular solution to the di erential equation (1.67) that depends on the forcing function. For r Ds, the particular solution satisfies the Silver–M¨uller radiation condition and gives the scattered field. The solution to the homogeneous equation or the complementary solution satisfies the equation

× × Ee − ks2Ee = 0 in Ds Di .

and describes the field that would exist in the absence of the scattering object, i.e., the incident field. Thus, the complete solution to (1.67) can be written as

We note that for a nontrivial magnetic permeability of the particle, a volumesurface integral equation has been derived by Volakis [245], and a “pure” volume-integral equation has been given by Volakis et al. [246].

1.4.2 Far-Field Pattern and Amplitude Matrix

Application of Stratton–Chu representation theorem to the vector fields Es and Ee in the domain Ds together with the boundary conditions es + ee = ei and hs + he = hi, yield

Es(r) = × ei (r ) g (ks, r, r ) dS(r )

S

where Es, Hs and Ei, Hi solve the transmission boundary-value problem. The above equation is known as the Huygens principle and it expresses the field in the domain Ds in terms of the surface fields on the surface S (see, for

example, [229]). Application of Stratton–Chu representation theorem in the domain Di gives the (general) null-field equation or the extinction theorem:

which shows that the radiation of the surface fields into Di extinguishes the incident wave [229]. In the null-field method, the extinction theorem is used to derive a set of integral equations for the surface fields, while the Huygens principle is employed to compute the scattered field.

Every radiating solution Es, Hs to the Maxwell equations has the asymptotic form

uniformly for all directions er = r/r. The vector fields Es∞ and Hs∞ defined on the unit sphere are the electric and magnetic far-field patterns, respectively, and satisfy the relations:

Hs∞ = εs er × Es∞ , µs

er · Es∞ = er · Hs∞ = 0 .

Because Es∞ also depends on the incident direction ek , Es∞ is known as the scattering amplitude from the direction ek into the direction er [229]. Using the Huygens principle and the asymptotic expressions

as r → ∞, we obtain the following integral representations for the far-field patterns [40]

The quantity σd = |Es∞|2 is called the di erential scattering cross-section and describes the angular distribution of the scattered light. The di erential scattering cross-section depends on the polarization state of the incident field and on the incident and scattering directions. The quantities σdp = |Es∞,θ |2 and σds = |Es∞,ϕ|2 are referred to as the di erential scattering cross-sections for parallel and perpendicular polarizations, respectively. The di erential scattering cross-section has the dimension of area, and a dimensionless quantity is the normalized di erential scattering cross-section σdn = σd/πa2c , where ac is a characteristic dimension of the particle.

To introduce the concepts of tensor scattering amplitude and amplitude matrix it is necessary to choose an orthonormal unit system for polarization description. In Sect. 1.2 we chose a global coordinate system and used the vertical and horizontal polarization unit vectors eα and eβ , to describe the polarization state of the incident wave (Fig. 1.9a). For the scattered wave we can proceed analogously by considering the vertical and horizontal polarization unit vectors eϕ and eθ . Essentially, (ek , eβ , eα) are the spherical unit vectors of ke, while (er , eθ , eϕ) are the spherical unit vectors of ks in

Fig. 1.9. Reference frames: (a) global coordinate system and (b) beam coordinate system

the global coordinate system. A second choice is the system based on the scattering plane. In this case we consider the beam coordinate system with the Z-axis directed along the incidence direction, and define the Stokes vectors with respect to the scattering plane, that is, the plane through the direction of incidence and scattering (Fig. 1.9b). For the scattered wave, the polarization description is in terms of the vertical and horizontal polarization unit vectors eϕ and eθ , while for the incident wave, the polarization description is in terms of the unit vectors e = eϕ and e = e × ek . The advantage of this system is that the scattering amplitude can take simple forms for particles with symmetry, and the disadvantage is that e and e depend on the scattering direction. Furthermore, any change in the direction of light incidence also changes the orientation of the particle with respect to the reference frame. In our analysis we will use a fixed global coordinate system to specify both the direction of propagation and the states of polarization of the incident and scattered waves and the particle orientation (see also, [169, 228]).

The tensor scattering amplitude or the scattering dyad is given by [169]

and since er · Es∞ = 0, it follows that:

Because the incident wave is a transverse wave, ek · Ee0 = 0, the dot product A(er , ek ) ·ek is not defined by (1.71), and to complete the definition, we take

Although the scattering dyad describes the scattering of a vector plane wave, it can be used to describe the scattering of any incident field, because any regular solution to the Maxwell equations can be expressed as an integral over vector plane waves. As a consequence of (1.72) and (1.73), only four components of the scattering dyad are independent and it is convenient to introduce the 2 × 2 amplitude matrix S to describe the transformation of the transverse components of the incident wave into the transverse components of the scattered wave in the far-field region. The amplitude matrix is given by [17, 169, 228]

where Ee0,β and Ee0,α do not depend on the incident direction. Essentially, the amplitude matrix is a generalization of the scattering amplitudes including polarization e ects. The amplitude matrix provides a complete description of the far-field patterns and it depends on the incident and scattering directions as well on the size, optical properties and orientation of the particle. The elements of the amplitude matrix