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 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 vector spherical waves expansion of the incident field reads as 

To give a justification of the above expansion we consider the integral representation 

Using the orthogonality relations of vector spherical harmonics we see that the expansion coefficients amn and bmn are given by 

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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. 

Considering the scattered field representation (2.5), we replace the surface fields by their approximations and use the vector spherical wave expansion of the dyad gl on a sphere enclosing D to obtain... 

Thus, the derivation of the vector spherical wave expansion of the electromagnetic field scattered by an arbitrary infinite surface involves the following... 

Computation of the vector spherical wave expansion of the scattered field... 

As in the scalar case, integral and series representations for the translation coefficients can be obtained by using the integral representations for the vector spherical wave functions. First we consider the case of regular vector spherical wave functions. Using the integral representation (B.26), the relation r = 0 + T" ) and the vector spherical wave expansion... 

Restricting r to lies inside a sphere enclosed in S2 and using the vector spherical wave expansion of the dyad gl, we see that the first set of closure relations gives 5 = 0 in D fi. Analogously, but restricting r to lies outside a sphere enclosing Si, we deduce that the second set of closure relations yields = 0 in )<,. Theorem 1 can now be used to conclude. ... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

In (1.38)-(1,39), the electromagnetic fields are expressed in terms of the unknown scalar functions T>a and V/3, while in (1.41) and (1.42), the electromagnetic fields are expressed in terms of the unknown expansion coefficients Cmn and dmn These unknowns will be determined from the boundary conditions for each specific scattering problem. The vector functions and can be regarded as a generalization of the regular vector spherical wave functions and For isotropic media, we have eXfSfs = 1, = 0 and... 

As a result, we obtain the familiar expansions of the electromagnetic fields in terms of vector spherical wave functions of the interior wave equation ... 

TV) (hr ), respectively, while Q (kg, kg) contains as rows and columns the vectors M hr ), M (ksr ) and Mj (ksr ), Nj (ksr ), respectively. To compute the scattered field we proceed as in the case of locahzed sources. Application of the Huygens principle yields the expansion of the scattered field in terms of localized vector spherical wave functions as in (2.15) and (2.16). Inserting (2.22) into (2.16) gives... 

To compute the T matrix of the two-particles system and to derive a scattered-field expansion centered at the origin O of the global coordinate system we use the Stratton-Chu representation theorem for the scattered field Eg in Dg. In the exterior of a sphere enclosing the particles, the expansion of the approximate scattered field in terms of radiating vector spherical wave functions reads as... 

The surface fields ei i, /ti,i and e, 2, /ti,2 are approximated by finite expansions in terms of regular vector spherical wave functions as in (2.132) and (2.133) respectively. Inserting these expansions into the null-field equations (2.159) and (2.160), jdelds the system of matrix equations... 

In this case the interacting held is the image of the scattered field and the expansion (2.198) can be derived by using the addition theorem for vector spherical wave functions. The elements of the reflection matrix are the translation coefficients, and as a result, the amount of computer time required to solve the scattering problem is significantly reduced. In this regard it should... 

Relying on these expansions and using the Stratton-Chu representation theorem, orthogonality relations of vector spherical wave functions on arbitrarily closed surfaces can be derived. Let be a bounded domain with boundary S and exterior Ds, and let n be the unit normal vector to S directed into Ds- The wave number in the domain is denoted by k, while the wave number in the domain D is denoted by k. For r Di, application of Stratton-Chu representation theorem to the vector fields Es r) = M ksr) and Hs(r) = -j V s/MsiV (A sr) gives... 

EXPANSION OF A PLANE WAVE IN VECTOR SPHERICAL HARMONICS... 

Expansion of a plane wave in vector spherical harmonics is a lengthy, although straightforward, procedure. In this section we outline how one goes about determining the coefficients in such an expansion. 

In Chapter 4 a plane wave incident on a sphere was expanded in an infinite series of vector spherical harmonics as were the scattered and internal fields. Such expansions, however, are possible for arbitrary particles and incident fields. It is the scattered field that is of primary interest because from it various observable quantities can be obtained. Linearity of the Maxwell equations and the boundary conditions (3.7) implies that the coefficients of the scattered field are linearly related to those of the incident field. The linear transformation connecting these two sets of coefficients is called the T (for transition) matrix. I f the particle is spherical, then the T matrix is diagonal. 

On the U(l) level, the plane wave is subjected to a multipole expansion in terms of the vector spherical harmonics, in which only two physically significant values of M in Eq. (761) are assumed to exist, corresponding to M = +1 and — 1, which translates into our notation as follows ... 

Now consider the measurement of monochromatic plane photons by a photodetector shown in Fig. 14. At far distances, the photons are specified by a unique wave vector k. Mandel s localization of photons in the vicinity of the sensitive area cr assumes that the wave converges to a. This means that there is a variety of directions of the wavevectors near a (Fig. 16). This picture can be described by a proper expansion over spherical waves. In view of the discussion... 

In the above analysis, and Y 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 (e 7 y z) by considering the expansion of the tangential vector function T>c (3,a)Va + T>is j3, a)vjj in terms of vector spherical harmonics. 

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