Vector spherical harmonics

The eigenfunctions of the M operators are the physically more meaningful quantities. They are denoted byYjM and called vector spherical harmonics. In terms of the spin variable a(= x, y, z), and written as functions of the unit vector n = k/k,... 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

It is not possible to ascribe a definite value of the orbital angular momentum to a photon state since the vector spherical harmonic YjM may be a function of different values of . This provides the evidence that, strictly speaking, it... 

If j = 0 there is only one vector spherical harmonic which is identical with the longitudinal harmonic Y 1 = nVoo- From this observation it follows that there are no transverse spherical harmonics for j = 0. It also means that the state with angular momentum zero represents a spherically symmetrical state, but a spherically symmetrical vector field can only be longitudinal. Thus, a photon cannot exist in a state of angular momemtum zero. 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

The vector spherical harmonics generated by the even and odd generating functions defined by Eq. (56) and (57) become... 

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. 

The orthogonality of all the vector spherical harmonics, which was established in the preceding section, implies that the coefficients in the expansion (4.23) are of the form... 

Suppose that a plane jc-polarized wave is incident on a homogeneous, isotropic sphere of radius a (Fig. 4.1). As we showed in the preceding section, the incident electric field may be expanded in an infinite series of vector spherical harmonics. The corresponding incident magnetic field is obtained from the curl of (4.37) ... 

We may also expand the scattered electromagnetic field (E5,H5) and the field (E H,) inside the sphere in vector spherical harmonics. At the boundary between the sphere and the surrounding medium we impose the conditions (3.7) ... 

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 the point matching method (Oguchi, 1973 Bates, 1975) the fields inside and outside a particle are expanded in vector spherical harmonics and the resulting series truncated the tangential field components are required to be continuous at a finite number of points on the particle boundary. Although easy to describe and to understand, the practical usefulness of this method is limited to nearly spherical particles large demands on computer time and uncertain convergence are also drawbacks (Yeh and Mei, 1980). 

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 transverse components of eM are physical but the longitudinal component corresponding to M = 0 is unphysical. This asserts two states of transverse polarization in the vacuum left and right circular. However, this assertion amounts to Cq = e[i = 0, meaning the incorrect disappearance of some vector spherical harmonics that are nonzero from fundamental group theory because some irreducible representations are incorrectly set to zero. 

All three of ea>, e(2 e(3] can be expressed in terms of vector spherical harmonics. Thus, in addition to the nonlinear B cyclic theorem, the following linear relations occur... 

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 ... 

These expansions can always be made, due to the completeness properties of the standard and vector spherical harmonics. The summations in Eqs. 

---