Vector plane waves

In this section, we characterize the polarization state of vector plane waves and derive vector spherical wave expansions for the incident field. The first topic is relevant in the analysis of the scattered field, while the second one plays an important role in the derivation of the transition matrix. 

We consider a right-handed Cartesian coordinate system OXYZ with a fixed spatial orientation. This reference frame will be referred to as the global coordinate system or the laboratory coordinate system. The direction of propagation of the vector plane wave is specified by the unit vector e, or equivalently, by the zenith and azimuth angles / and a, respectively (Fig. 1.2). The polarization state of the incident wave will be described in terms of the vertical polarization unit vector Cq, = x e /le x e l and the horizontal polarization vector e/3 = Ca x Bk. Note that other names for vertical polarization are TM polarization, parallel polarization and p polarization, while other names for horizontal polarization are TE polarization, perpendicular polarization and s polarization. 

In the frequency domain, a vector plane wave propagating in a medium... 

There are three ways of describing the polarization state of vector plane waves. 

Taking into account the representation of a vector plane wave in the time domain... 

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 OXYZ through the Euler angles ap, j3p and 7p (Fig. 1.5). In our analysis, vector plane waves and Gaussian beams are considered as external excitations. 

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

The polarization unit vector of a linearly polarized vector plane wave is given by (1.18). If the vector plane wave propagates along the z-axis we have / = a = 0 and for / = 0, the spherical vector harmonics are zero unless m = 1. Using the special values of the angular functions and when (3 = 0,... 

Thus, for a vector plane wave polarized along the a -axis we have in = -a-In = j" V2n+ 1,... 

The Gaussian beam becomes a plane wave if wq tends to infinity and for this specific case, the expressions of the expansion coefficients reduce to those of a vector plane wave. 

The Green second vector theorem applied to the incident fields Eei and E 2 in any bounded domain shows that the vector plane wave terms do not contribute to the integral. Furthermore, using the far-field representation... 

Scpjs = Esoo,cp- Analogously, for a vector plane wave linearly polarized in the... 

Another important scattering characteristic is the angular distribution of the scattered field. For an ensemble of randomly oriented particles illuminated by a vector plane wave of unit amplitude and polarization vector Cpoi = epoi,/3e/3 + epoi,a6a, the differential scattering cross-sections in the scattering plane are given by... 

For a vector plane wave of unit amplitude and wave vector fee, fee = kgCk, we have... 

Fig. 2.14. Geometry of an axisymmetric particle situated near a plane surface. The external excitation is a vector plane wave propagating in the ambient medium...

The geometry of the scattering problem is shown in Fig. 2.14. An axisymmetric particle is situated in the neighborhood of a plane surface so that its axis of symmetry is normal to the plane surface. The 2-axis of the particle coordinate system Oxyz is directed along the axis of symmetry and the origin O is situated at the distance zg below the plane surface. The incident radiation is a linearly polarized vector plane wave propagating in the ambient medium (the medium below the surface S)... 

Fig. 2.15. Reflection and refraction of a vector plane wave (propagating in the ambient medium) at the interface S...

For the incident vector plane wave given by (2.193), the transmitted (or the refracted) vector plane wave is... 

We briefly recall the definitions and the basic properties of scalar and vector plane waves [14]. The scalar plane wave is defined by... 

The incident field is a prescribed field, whose sources are assumed to be situated in Dg. This means that in any case, the sources are below the fictitious plane z = and we can represent the external excitation as an integral over up-going vector plane waves... 

For r D and 2 > 2 >, we use the plane wave expansion of the Green dyad and the orthogonality properties of the vector plane waves to obtain... 

Taking into account that the tangential vector plane waves nxM -, K ) and n X N -, K j ), form a complete system of vector function on S, we represent the surface fields n X Ei and n x (V x E ) as integrals over up-going vector plane waves, i.e.,... 

To compute the scattered field we consider (2.213) for r Dg, expand the free space dyadic Green function in terms of vector plane waves by assuming z < z, and obtain... 

Representation of the incident field as an integral over vector plane waves. 

Returning to the scattering problem of a particle near an arbitrary infinite surface, we see that our analysis is complete if, according to Step 1, we are able to represent the plane electromagnetic wave and the radiating vector spherical wave functions and as integrals over vector plane waves. 

For the complete uniform distribution function, the external excitation is a vector plane wave propagating along the Z-axis of the global coordinate system and the scattering plane is the XZ-plane. The code computes the following orientation-averaged quantities ... 

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