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

12 1 Basic Theory of Electromagnetic Scattering

where

Ee(r, t) = Ee,β (r, t)eβ + Ee,α(r, t)eα , has the components

Ee0,β = −jE0 sin χ ,

Ee0,α = E0 cos χ .

Using the transformation rule for rotation of a two-dimensional coordinate system we obtain the desired relations

If b = 0, the ellipse degenerates into a straight line and the wave is linearly polarized. In this specific case χ = 0 and

In view of (1.16) and (1.17) it is apparent that the polarization unit vector is real and is given by

If a = b, the ellipse is a circle and the wave is circularly polarized. We have tan χ = ±1, which implies χ = ±π/4, and choosing ψ = π/2, we obtain

√

2

Ee0,β = 2 E0 ,

√

2

Ee0,α = ±j 2 E0 .

The polarization unit vectors of rightand left-circularly polarized waves then become

and we see that these basis vectors are orthonormal in the sense that eR · eR = 1, eL · eL = 1 and eR · eL = 0.

3. The polarization characteristics of the incident field can also be described by the coherency and Stokes vectors. Although the ellipsometric parameters completely specify the polarization state of a monochromatic wave, they are di cult to measure directly (with the exception of the intensity E02). In contrast, the Stokes parameters are measurable quantities and are of greater usefulness in scattering problems. The coherency vector is defined as

and in terms of E0, ψ and χ, as

Ie = E02 ,

Qe = −E02 cos 2χ cos 2ψ ,

Ue = −E02 cos 2χ sin 2ψ ,

Ve = −E02 sin 2χ .

The above relations show that the Stokes parameters carry information about the amplitudes and the phase di erence, and are operationally defined in terms of measurable quantities (intensities). For a linearly polarized plane wave, χ = 0 and Ve = 0, while for a circularly polarized plane wave, χ = ±π/4 and Qe = Ue = 0. Thus, the Stokes vector of a linearly polarized wave of unit amplitude is given by Ie = [1, cos 2αpol, − sin 2αpol, 0]T, while the Stokes vector of a circularly polarized wave of unit amplitude is Ie = [1, 0, 0, 1]T.

The Stokes parameters of a monochromatic plane wave are not independent since

and we may conclude that only three parameters are required to characterize the state of polarization. For quasi-monochromatic light, the amplitude of the electric field fluctuate in time and the Stokes parameters are expressed in terms of the time-averaged quantities Ee0,pEe0,q , where p and q stand for β and α. In this case, the equality in (1.24) is replaced by the inequality

Ie2 ≥ Q2e + Ue2 + Ve2 ,

and the quantity

is known as the degree of polarization of the quasi-monochromatic beam. For natural (unpolarized) light, P = 0, while for fully polarized light P = 1. The Stokes vector defined by (1.20) is one possible representation of polarization. Other representations are discussed by Hovenier and van der Mee [101], while a detailed discussion of the polarimetric definitions can be found in [17,169,171].

1.2.2 Vector Spherical Wave Expansion

The derivation of the transition matrix in the framework of the null-field method requires the expansion of the incident field in terms of (localized) vector spherical wave functions. This expansion must be provided in the particle coordinate system, where in general, the particle coordinate system Oxyz is obtained by rotating the global coordinate system OXY Z through the Euler angles αp, βp and γp (Fig. 1.5). In our analysis, vector plane waves and Gaussian beams are considered as external excitations.

16 1 Basic Theory of Electromagnetic Scattering

Z, z1

z, z2

βp

Fig. 1.5. Euler angles αp, βp and γp specifying the orientation of the particle coordinate system Oxyz with respect to the global coordinate system OXY Z. The transformation OXY Z → Oxyz is achieved by means of three successive rotations:

(1) rotation about the Z-axis through αp, OXY Z → Ox1y1z1, (2) rotation about the y1-axis through βp, Ox1y1z1 → Ox2y2z2 and (3) rotation about the z2-axis through γp, Ox2y2z2 → Oxyz

Vector Plane Wave

We consider a vector plane wave of unit amplitude propagating in the direction (βg, αg) with respect to the global coordinate system. Passing from spherical coordinates to Cartesian coordinates and using the transformation rules under coordinate rotations we may compute the spherical angles β and α of the wave vector in the particle coordinate system. Thus, in the particle coordinate system we have the representation

Ee(r) = epolejke·r , epol · ek = 0 ,

where as before, ke = ksek .

The vector spherical waves expansion of the incident field reads as