Null-Field Method

Essentially, the null-field method involves the following steps  

Derivation of an infinite system of integral equations for the surface fields by using the general null-field equation. 

Derivation of integral representations for the scattered field coefficients in terms of surface fields by using the Huygens principle. 

Approximation of the surface fields by a complete and linearly independent system of (tangential) vector functions. 

Truncation of the infinite system of null-field equations and of the scattered-field expansion. 

---

The Mie theory [1] and the T-matrix method [4] are very efficient for (multilayered) spheres and axisymmetric particles (with moderate aspect ratios), respectively. Several methods, applicable to particles of arbitrary shapes, have been used in plasmonic simulations the boundary element method (BEM) [5, 6], the DDA [7-9], the finite-difference time-domain method (FDTD) [10, 11], the finite element method (FEM] [12,13], the finite integration technique (FIT) [14] and the null-field method with discrete sources (NFM-DS) [15,16]. There is also quasi-static approximation for spheroids [12], but it is not discussed here. 

Wriedt, T. (2007) Review of the null-field method with discrete sources,/ Quant Spectrosc. Radiat Transfer, 106,535-545. 

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. 

To solve the scattering problem in the framework of the null-field method it is necessary to approximate the internal field by a suitable system of vector functions. For isotropic particles, regular vector spherical wave functions of the interior wave equation are used for internal field approximations. In this section we derive new systems of vector functions for anisotropic and chiral particles by representing the electromagnetic fields (propagating in anisotropic... 

The transition matrix relates the expansion coefficients of the incident and scattered fields. The existence of the transition matrix is postulated by the T-Matrix Ansatz and is a consequence of the series expansions of the incident and scattered fields and the linearity of the Maxwell equations. Historically, the transition matrix has been introduced within the null-field method formalism (see [253,256]), and for this reason, the null-field method has often been referred to as the T-matrix method. However, the null-field method is only one among many methods that can be used to compute the transition matrix. The transition matrix can also be derived in the framework of the method of moments [88], the separation of variables method [208], the discrete dipole approximation [151] and the point matching method [181]. Rother et al. [205] foimd a general relation between the surface Green function and the transition matrix for the exterior Maxwell problem, which in principle, allows to compute the transition matrix with the finite-difference technique. 

Several studies have addressed the convergence of the null-field method. Ramm [200,201] and Kristensson et al. [126] have derived sufficiency criteria for the convergence of the null-field method, but this criteria are not satisfied for nonspherical surfaces. A major progress has been achieved by Dallas [44], who showed that for ellipsoidal surfaces, the far-field pattern converges in the least-squares sense on the unit sphere. 

The standard scheme for computing the transition matrix in the framework of the null-field method relies on the solution of the general null-field equation... 

In the following analysis we summarize the basic concepts of the null-field method with distributed sources. The distributed vector spherical wave functions are defined as... 

To ameliorate the numerical instability of the null-field method for nonabsorbing particles, Waterman [256,257] and Lakhtakia et al. [133,134] proposed to exploit the unitarity property of the transition matrix. To smnmarize this technique we consider nonabsorbing particles and use the identity = Re Q to rewrite the T-matrix equation... 

The null-field method is a general technique and is applicable for arbitrarily shaped particles. However, for nonaxisymmetric particles, a semi-convergent... 

The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the difference between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundary-value problem. We consider the vector potential Aa with density a... 

An interesting feature of the null-field method is that all matrix equations become considerably simpler and reduce to the corresponding equations of the Lorenz-Mie theory when the particle is spherically. For a spherical particle of radius R, the orthogonality relations of the vector spherical harmonics show that the QP matrices are diagonal... 

The use of distributed vector spherical wave functions improves the numerical stability of the null-field method for highly elongated and flattened layered particles. Although the above formalism is valid for nonaxisymmetric particles, the method is most effective for axisymmetric particles, in which case the 2 -axis of the particle coordinate system is the axis of rotation. Applications of the null-field method with distributed sources to axisymmetric layered spheroids with large aspect ratios have been given by Doicu and Wriedt [50]. 

In this section we extend the null-field method to the case of an arbitrary nrnnber of particles by using the translation properties of the vector spherical wave functions. Taking into account the geometric restriction that the particles do not overlap in space, we derive the expression of the transition matrix for restricted values of translations. Our treatment closely follows the original derivation given by Peterson and Strom [187,188]. 

---