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Debye-series expansion of T-matrix for light scattering by non-spherical particles computed from Riccati-differential equations

Lei Bi and Gérard Gouesbet

Open Access Open Access

Abstract

A new formulation of the Debye series based on the Riccati-differential equations was developed to compute electromagnetic wave scattering by non-spherical particles. In this formulation, the T-matrix was expanded in terms of the Debye series. The zeroth-order term, which corresponds to a combination of diffraction and external reflection, is given by unity minus the external reflection matrix. The higher-order terms are generated from the transmission matrix from the medium to the particle, the internal reflection matrix within the particle and the transmission matrix from the particle to the medium. We demonstrate that the aforementioned four reflection-transmission matrices satisfy the Riccati-differential equations, which can be numerically solved by the fourth-order Runge-Kutta method. The present algorithm can be applied to generalized convex non-spherical particles. The differential equations were analytically validated in the case of a homogeneous sphere. Representative results were given in the case of spheroids. The impacts of the Debye series with various orders on the optical properties of spheroids were revealed with significant details.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Electromagnetic wave scattering by a three-dimensional macroscopic particle is fundamental to various applications in science and engineering disciplines [1]. Although significant progress has been made on solving Maxwell’s equations for an accurate solution of electromagnetic wave scattering, additional efforts are still in great need if scattering wave characteristics require an accurate interpretation with physical insight. Moreover, in the last several decades, the geometric ray-tracing technique has been extensively used to compute the optical properties of non-spherical particles much larger than the incident wavelength [2–10]. However, due to the semi-empirical nature of geometric-optics, the accuracy of the ray-tracing technique for non-spherical particles often requires assessment or improvement [e.g., 9,10].

One of the most suitable approaches for investigating these issues is the Debye series, which was first proposed for electromagnetic wave scattering by an infinite circular cylinder [11]. In the Debye series, the partial scattering amplitude was expanded into an infinite number of terms and correspondingly related to diffraction and external reflection, and the transmitted wave from the particle to the medium as the waves undergo successive internal reflections [12,13]. With further developments, the approach was extended to electromagnetic wave scattering by several regularly shaped particles such as spheres, coated spheres, and spheroids [e.g., 12–26] which conform with the relevant orthogonal curvilinear coordinate system so that the variable-separation method is applicable. In Bi et al. [27], the T-matrix (rather than the amplitude scattering matrix) that relates the incident and the scattered fields was successfully expanded in terms of the Debye series. The Debye series has exact terms correspondingly associated with the wave reflection and refraction; thus, it can be used to investigate the accuracy of geometric-optics approximation [28,29]. In addition to the plane-wave incidence in the frequency domain, the Debye series was developed for Bessel incident waves and in time-domain analysis [23]. However, little progress has been made toward further generalizing the theory for arbitrarily shaped particles until recently [30–33].

Xu et al. [30,31] applied the extended boundary condition method (EBCM) and formulated the Debye series for more general non-spherical particle shapes. In the traditional application of the EBCM method to calculate scattering by a non-spherical particle, inversion of a transition (T) matrix is often needed to solve for scattering coefficients [1]. When particle size gets larger, the matrix becomes ill-conditioned which leads to instabilities of numerical convergence [1]. The EBCM-based Debye series method shows similar convergence issues as the particle size and non-sphericity (e.g., measured by the axis ratio for a spheroid) increase. In an alternative approach, Bi et al. [32,33] computed the zeroth-order Debye series for solving the Riccati-differential equations in terms of a reflection matrix obtained from the electric-field volume integral equations. Based on the zeroth-order Debye series, the optical tunneling effects on the extinction efficiency and phase matrix of spheroids and faceted particles are examined [32,33]. In addition, an analytical formula of extinction efficiency of spheroids with optimal edge-effect terms was obtained in the framework of complex angular momentum theory and the zeroth- order Debye series [29,32]. However, for transparent particles, high-order terms are essentially required for further investigations.

In this paper, we present a complete algorithm to compute the Debye series with arbitrary orders for non-spherical particles via an invariant imbedding approach. This paper is organized as follows. Section 2 contains the algorithm formulation to compute the Debye series. Representative numerical results for spheroids are given in Section 3. Section 4 summarizes the present study.

2. Method

In the T-matrix method for solving electromagnetic wave scattering by a three-dimensional particle, the incident electric field and the scattered electric field are expanded in terms of suitable vector spherical wave functions [1]:

$${{\textbf E}^{\textrm{inc}}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {a_l}Rg{{\textbf M}_l}({k{\textbf r}} )+ {b_l}Rg{{\textbf N}_l}({k{\textbf r}} ), $$
$${{\textbf E}^{\textrm{sca}}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {p_l}{{\textbf M}_l}({k{\textbf r}} )+ {q_l}{{\textbf N}_l}({k{\textbf r}} ), $$
where k is the wave vector in the medium, $Rg{{\textbf M}_l}$ and $Rg{{\textbf N}_l}$ are the regular vector spherical functions, and ${{\textbf M}_l}$ and ${{\textbf N}_l}$ are the irregular vector spherical functions. The index l is related to the total angular momentum ($n$) and the projected angular momentum ($m, - n \le m \le n$) numbers via $l = n({n + 1} )+ m$. The T-matrix ${\textbf T}$ is defined as a transition matrix that transfers the coefficients of the incident electric field (${a_l},{\; }{b_l}$) to those of the scattered electric field (${p_l},{\; }{q_l}$). The internal field is given by
$${{\textbf E}^{\textrm{int}}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {c_l}Rg{{\textbf M}_l}({\tilde{m}k{\textbf r}} )+ {d_l}Rg{{\textbf N}_l}({\tilde{m}k{\textbf r}} ), $$
$\tilde{m}$ is the complex refractive index (a tilde is included to distinguish the complex refractive index from the projected angular momentum number). The scattered field (Eq. (2)) contains all wave-particle interactions including diffraction, external reflections and various transmissions. To isolate the wave-surface interactions with a similar picture in the framework of geometric- optics, we first consider the reflection and transmission of a radially incident wave given by
$$\tilde{{\textbf E}}_{\textrm{inc}}^{\textrm{ext}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {\tilde{a}_l}{\tilde{{\textbf M}}_l}({k{\textbf r}} )+ {\tilde{b}_l}{\tilde{{\textbf N}}_l}({k{\textbf r}} ). $$

In Eq. (4), ${\tilde{{\textbf M}}_l}$ and ${\tilde{{\textbf N}}_l}$ are similar ${{\textbf M}_l}$ and ${{\textbf N}_l}$ except that vector spherical wave functions are constructed from spherical Hankel functions of the second kind. For convenience, the definitions of all vector spherical functions are given in Appendix A. The reflected field and the transmitted field are given by

$${\textbf E}_{\textrm{ref}}^{\textrm{ext}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {p_{l,ref}}{{\textbf M}_l}({k{\textbf r}} )+ {q_{l,ref}}{{\textbf N}_l}({k{\textbf r}} ), $$
$${\textbf E}_{\textrm{tra}}^{\textrm{ext}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {a_{l,tra}}{\tilde{{\textbf M}}_l}({\tilde{m}k{\textbf r}} )+ {b_{l,tra}}{\tilde{{\textbf N}}_l}({\tilde{m}k{\textbf r}} ). $$

As illustrated in Fig. 1(a), the matrix ${{\textbf R}^{\textrm{ext}}}$ is defined to transfer the coefficients (${\tilde{a}_l}$ and ${\tilde{b}_l}$) into those of the reflected field (${p_{l,ref}}$ and ${q_{l,ref}}$) whereas ${{\textbf T}^{\textrm{ext}}}$ is defined to transfer the coefficients (${\tilde{a}_l}$ and ${\tilde{b}_l}$) into those of the transmitted field (${a_{l,tra}}$ and ${b_{l,tra}}$). Next, we consider the internal reflection and transmission of the outgoing internal wave given by

$$\tilde{{\textbf E}}_{\textrm{out}}^{\textrm{int}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {\tilde{a}_{l,out}}{{\textbf M}_l}({\tilde{m}k{\textbf r}} )+ {\tilde{b}_{l,out}}{{\textbf N}_l}({\tilde{m}k{\textbf r}} ),$$
which are defined by
$${\textbf E}_{\textrm{ref}}^{\textrm{int}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {a_{l,ref}}{\tilde{{\textbf M}}_l}({\tilde{m}k{\textbf r}} )+ {b_{l,ref}}{\tilde{{\textbf N}}_l}({\tilde{m}k{\textbf r}} ), $$
$${\textbf E}_{\textrm{tra}}^{\textrm{int}}({\textbf r} )= \mathop \sum \nolimits_{l = 1}^\infty {p_{l,tra}}{{\textbf M}_l}({k{\textbf r}} )+ {q_{l,tra}}{{\textbf N}_l}({k{\textbf r}} ).$$

 figure: Fig. 1.

Fig. 1. (a) Reflection (${{\textbf R}^{\textrm{ext}}}$) and transmission (${{\textbf T}^{\textrm{ext}}}$) of an incoming spherical wave. (b) Internal reflection (${{\textbf R}^{\textrm{int}}}$) and transmission (${{\textbf T}^{\textrm{int}}}$) of an outgoing spherical wave.

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As illustrated in Fig. 1(b), the matrix ${{\textbf R}^{\textrm{int}}}$ transfers the coefficients (${\tilde{a}_{l,out}}$ and ${\tilde{b}_{l,out}}$) into those of the reflected field (${a_{l,ref}}$ and ${b_{l,ref}}$), whereas ${{\textbf T}^{\textrm{int}}}$ transfers the coefficients (${\tilde{a}_{l,out}}$ and ${\tilde{b}_{l,out}}$) into those of the transmitted field (${p_{l,tra}}$ and ${q_{l,tra}}$). Figure 1 is a generalization of Fresnel reflection and transmission upon a planar surface to a three-dimensional closed surface. In the case of a spherical surface, the reflection and transmission matrices can be analytically obtained and have been shown to approach Fresnel formulas as the radius of sphere approach infinity [34]. In the Debye series formalism, a superposition of outgoing internal waves (Eq. (7)), reflected ingoing internal waves (Eq. 8) and the transmitted ingoing internal wave (Eq. 6) is the total internal field, which should be proportional to $\textrm{Rg}{{\textbf M}_l}({\tilde{m}k{\textbf r}} )$ and $\textrm{Rg}{{\textbf N}_l}({\tilde{m}k{\textbf r}} )$ that are used to represent the internal field for the original scattering problem (see Eq. (3)). That means, the coefficients before ${{\textbf M}_l}({\tilde{m}k{\textbf r}} )$ and ${{\textbf N}_l}({\tilde{m}k{\textbf r}} )$ that represent outgoing internal waves should be identical to their counterparts before ${\tilde{{\textbf M}}_l}({\tilde{m}k{\textbf r}} )$ and ${\tilde{{\textbf N}}_l}({\tilde{m}k{\textbf r}} )$ that represent ingoing internal waves so that irregular parts in the vector spherical functions are cancelled in the summation. Thus, we have the following relation,

$$\left[ {\begin{array}{{c}} {{{\tilde{{\boldsymbol a}}}_{{\textbf out}}}}\\ {{{\tilde{{\boldsymbol b}}}_{{\textbf out}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol a}_{{\boldsymbol ref}}}}\\ {{{\boldsymbol b}_{{\boldsymbol ref}}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol a}_{{\boldsymbol tra}}}}\\ {{{\boldsymbol b}_{{\boldsymbol tra}}}} \end{array}} \right] = {{\textbf R}^{{\textbf int}}}\left[ {\begin{array}{*{20}{c}} {{{\tilde{{\boldsymbol a}}}_{{\textbf out}}}}\\ {{{\tilde{{\boldsymbol b}}}_{{\textbf out}}}} \end{array}} \right] + {{\textbf T}^{{\textbf ext}}}\left[ {\begin{array}{*{20}{c}} {\tilde{{\boldsymbol a}}}\\ {\tilde{{\boldsymbol b}}} \end{array}} \right]. $$

The transmitted field from the particle to the medium is given by

$$\left[ {\begin{array}{{c}} {{{\boldsymbol p}_{{\textbf tra}}}}\\ {{{\boldsymbol q}_{{\textbf tra}}}} \end{array}} \right] = {{\textbf T}^{\textrm{int}}}\left[ {\begin{array}{*{20}{c}} {{{\tilde{{\boldsymbol a}}}_{\textrm{out}}}}\\ {{{\tilde{{\boldsymbol b}}}_{\textrm{out}}}} \end{array}} \right] = {{\textbf T}^{\textrm{int}}}{({1 - {{\textbf R}^{\textrm{int}}}} )^{ - 1}}{{\textbf T}^{\textrm{ext}}}\left[ {\begin{array}{*{20}{c}} {\tilde{{\boldsymbol a}}}\\ {\tilde{{\boldsymbol b}}} \end{array}} \right]. $$

In Eq. (10) and Eq. (11), $\tilde{{\boldsymbol a}}$, $\tilde{{\boldsymbol b}}$, ${\tilde{{\boldsymbol a}}_{\textrm{out}}}$, ${\tilde{{\boldsymbol b}}_{\textrm{out}}}$, ${{\boldsymbol p}_{{\textbf tra}}}$, and ${{\boldsymbol q}_{{\textbf tra}}}$ are column vectors with elements ${\tilde{a}_l}$, ${\tilde{b}_l},{\; }{\tilde{a}_{l,out}}$, ${\tilde{b}_{l,out}},{\; }{p_{l,tra}}$ and ${q_{l,tra}}({l = 0,\infty } )$, respectively. Note, we used Eq. (10) to derive Eq. (11). With the aforementioned definitions, the total scattered field can be represented as a superposition of the external reflection and various transmitted waves from the particle to the medium. The T-matrix can be explicitly written as an infinite series (first given in Ref. [27]),

$${\textbf T} ={-} \frac{1}{2}\left[ {1 - {{\textbf R}^{\textrm{ext}}} - {{\textbf T}^{\textrm{int}}}\left( {\mathop \sum \nolimits_{p = 0}^\infty {{({{{\textbf R}^{\textrm{int}}}} )}^p}} \right){{\textbf T}^{\textrm{ext}}}} \right] ={-} \frac{1}{2}\left[ {1 - {{\textbf R}^{\textrm{ext}}} - {{\textbf T}^{\textrm{int}}}\frac{1}{{1 - {{\textbf R}^{\textrm{int}}}}}{{\textbf T}^{\textrm{ext}}}} \right]. $$

Equation (12) is obtained by demanding that the total field in the medium (Eq. (1)+Eq. (2)) is equivalent to the summation of the incident field of Eq. (4), the reflection field of Eq. (5), and various orders of the transmitted field of Eq. (9); note that $({{{\tilde{a}}_l},{{\tilde{b}}_l}} )= ({{a_l},{b_l}} )/2$ should be satisfied so as to match the total field represented from the two formalisms. In this study, we show that the four matrices shown in the right side of Eq. (12) satisfies the following first-order ordinary differential equations,

$$\frac{{d{{\textbf R}^{\textrm{ext}}}({kr} )}}{{d({kr} )}} = \frac{i}{2}[{{{\tilde{{\textbf H}}}^\textrm{T}}({kr} )+ {{\textbf R}^{\textrm{ext}}}({kr} ){{\textbf H}^\textrm{T}}({kr} )} ]{\textbf U}[{\tilde{{\textbf H}}({kr} )+ {\textbf H}({kr} ){{\textbf R}^{\textrm{ext}}}({kr} )} ], $$
$$\frac{{d{{\textbf R}^{\textrm{int}}}({kr} )}}{{d({kr} )}} = \frac{i}{2}\tilde{m}[{{{\textbf H}^\textrm{T}}({\tilde{m}kr} )+ {{\textbf R}^{\textrm{int}}}({kr} ){{\tilde{{\textbf H}}}^\textrm{T}}({\tilde{m}kr} )} ]{{\textbf U}_2}[{{\textbf H}({\tilde{m}kr} )+ \tilde{{\textbf H}}({\tilde{m}kr} ){{\textbf R}^{\textrm{int}}}({kr} )} ], $$
$$\frac{{d{{\textbf T}^{\textrm{ext}}}({kr} )}}{{d({kr} )}} = \frac{i}{2}\tilde{m}[{{{\textbf H}^\textrm{T}}({\tilde{m}kr} )+ {{\textbf R}^{\textrm{int}}}({kr} ){{\tilde{{\textbf H}}}^\textrm{T}}({\tilde{m}kr} )} ]{{\textbf U}_3}[{\tilde{{\textbf H}}({kr} )+ {\textbf H}({kr} ){{\textbf R}^{\textrm{ext}}}({kr} )} ], $$
$$\frac{{d{{\textbf T}^{\textrm{int}}}({kr} )}}{{d({kr} )}} = \frac{i}{2}[{{{\tilde{{\textbf H}}}^\textrm{T}}({kr} )+ {{\textbf R}^{\textrm{ext}}}({kr} ){{\textbf H}^\textrm{T}}({kr} )} ]{{\textbf U}_3}[{{\textbf H}({\tilde{m}kr} )+ \tilde{{\textbf H}}({\tilde{m}kr} ){{\textbf R}^{\textrm{int}}}({kr} )} ],$$
where super-matrices H are defined as follows
$${{\textbf H}_{ll^{\prime}}}(x )= {\delta _{ll\mathrm{^{\prime}}}}\left[ {\begin{array}{{cc}} {h_n^{(1 )}(x )}&0\\ 0&{\frac{1}{x}\frac{\partial }{{\partial x}}[{xh_n^{(1 )}(x )} ]}\\ 0&{\sqrt {n({n + 1} )} h_n^{(1 )}(x )/x} \end{array}} \right],\;\; n = ceil(\sqrt {l + 1} ),$$
where $h_n^{(1 )}(x )$ is the Hankel function of the first kind and $ceil$ is a function to round its variable to its nearest integer (greater or equal to itself). In Eqs. (13)–(16), $\tilde{{\textbf H}}$ is similar to ${\textbf H}$ except that $h_n^{(1 )}(x )$ is replaced with $h_n^{(2 )}(x )$; the superscript “T” indicates “transpose”. We also have
$${{\textbf U}_{ll^{\prime}}}({kr} )= {f_{ll\mathrm{^{\prime}}}}\mathop \smallint \nolimits_0^{2\pi } d\phi \mathop \smallint \nolimits_0^{2\pi } d\theta sin\theta {\; }{e^{ - i({m - m\mathrm{^{\prime}}} )\phi }}[{{{\tilde{m}}^2}({r,\theta ,\phi } )- 1} ]\left[ {\begin{array}{{ccc}} {U_{ll\mathrm{^{\prime}}}^a}&{ - U_{ll\mathrm{^{\prime}}}^b}&0\\ {U_{ll\mathrm{^{\prime}}}^b}&{U_{ll\mathrm{^{\prime}}}^a}&0\\ 0&0&{U_{ll\mathrm{^{\prime}}}^c} \end{array}} \right], $$
$${[{{\textrm{U}_2}} ]_{ll\mathrm{^{\prime}}}}({kr} )= {f_{ll\mathrm{^{\prime}}}}\mathop \smallint \nolimits_0^{2\pi } d\phi \mathop \smallint \nolimits_0^{2\pi } d\theta sin\theta {\; }{e^{ - i({m - m\mathrm{^{\prime}}} )\phi }}[{{{\tilde{m}}^2}({r,\theta ,\phi } )- 1} ]\left[ {\begin{array}{{ccc}} {U_{ll\mathrm{^{\prime}}}^a}&{ - U_{ll\mathrm{^{\prime}}}^b}&0\\ {U_{ll\mathrm{^{\prime}}}^b}&{U_{ll\mathrm{^{\prime}}}^a}&0\\ 0&0&{U_{ll\mathrm{^{\prime}}}^d} \end{array}} \right], $$
$${[{{\textrm{U}_3}} ]_{ll\mathrm{^{\prime}}}}({kr} )= {f_{ll\mathrm{^{\prime}}}}\mathop \smallint \nolimits_0^{2\pi } d\phi \mathop \smallint \nolimits_0^{2\pi } d\theta sin\theta {\; }{e^{ - i({m - m\mathrm{^{\prime}}} )\phi }}[{{{\tilde{m}}^2}({r,\theta ,\phi } )- 1} ]\left[ {\begin{array}{{ccc}} {U_{ll\mathrm{^{\prime}}}^a}&{ - U_{ll\mathrm{^{\prime}}}^b}&0\\ {U_{ll\mathrm{^{\prime}}}^b}&{U_{ll\mathrm{^{\prime}}}^a}&0\\ 0&0&{U_{ll\mathrm{^{\prime}}}^e} \end{array}} \right], $$
where ${f_{ll\mathrm{^{\prime}}}}$ is defined by
$${f_{ll\mathrm{^{\prime}}}} = {k^2}{r^2}{({ - 1} )^{m + m\mathrm{^{\prime}}}}{\left[ {\frac{{2n + 1}}{{4\pi n({n + 1} )}}} \right]^{1/2}}{\left[ {\frac{{2n\mathrm{^{\prime}} + 1}}{{4\pi n\mathrm{^{\prime}}({n\mathrm{^{\prime}} + 1} )}}} \right]^{1/2}}, $$
and the elements in the bracket are given as follows
$$U_{ll^{\prime}}^a(\theta )= {\pi _{mn}}(\theta ){\pi _{m\mathrm{^{\prime}}n\mathrm{^{\prime}}}}(\theta )+ {\tau _{mn}}(\theta ){\tau _{m\mathrm{^{\prime}}n\mathrm{^{\prime}}}}(\theta ), $$
$$U_{ll^{\prime}}^b(\theta )= i[{{\pi_{mn}}(\theta ){\tau_{m\mathrm{^{\prime}}n\mathrm{^{\prime}}}}(\theta )+ {\tau_{mn}}(\theta ){\pi_{m\mathrm{^{\prime}}n\mathrm{^{\prime}}}}(\theta )} ], $$
$$U_{ll\mathrm{^{\prime}}}^c(\theta )= \sqrt {n({n + 1} )n^{\prime}({n^{\prime} + 1} )} d_{0m}^n(\theta )d_{0m^{\prime}}^{n^{\prime}}(\theta )/{\tilde{m}^2}, $$
$$U_{ll\mathrm{^{\prime}}}^d(\theta )= \sqrt {n({n + 1} )n^{\prime}({n^{\prime} + 1} )} d_{0m}^n(\theta )d_{0m^{\prime}}^{n^{\prime}}(\theta ){\tilde{m}^2}, $$
$$U_{ll^{\prime}}^e(\theta )= \sqrt {n({n + 1} )n^{\prime}({n^{\prime} + 1} )} d_{0m}^n(\theta )d_{0m^{\prime}}^{n^{\prime}}(\theta ). $$

The definition of angular function (${\pi _{mn}}$, ${\tau _{mn}}$) and Wigner d function $d_{0m}^n{\; }$ in Eqs. (22)–(26) can be seen in Appendix A. Note that ${\textbf U}$, ${\textbf{U}_2}$ and ${\textbf{U}_3}$ are similar except that the 33 terms have a difference related to the complex refractive index. Equation (13) has been obtained in the previous studies [32,33]. Equations (14)–(16) are new equations, which can be obtained via a similar approach presented in Appendix A of Ref. [32]. The structure symmetry involved in Eqs. (13)–(16) is obvious, which is fundamentally related to the expansion of the Green function with respect to suitable vector spherical functions. In Appendix B of the present paper, we validated Eqs. (14) and (16) for the case of a homogeneous sphere, which has an analytical solution for the Debye series. Therefore, Eqs. (23)–(26) provide a complete system of equations to compute the Debye series expansion of T-matrix. The algorithm is in principle applied to generalized convext non-spherical particle geometries; the only shape-dependent part in the algorithm is U matrices (Eqs. 18–20), in which the complex refractive index $\tilde{m}({r,\theta ,\phi } )$ is unity when the position ($r,\theta ,\phi $) is outside of the particle. For axially symmetric particles, Eqs. (18)–(20) can be analytically obtained without performing numerical integration (refer to the Appendix in Ref. [33]). In the present study, we used the fourth-order Runge-Kutta method for solving the differential equations. Although the initial position of $kr$ can be chosen at origin for solving the differential equations, the availability of analytical solution of the Debye series for a sphere provides a natural choice of initial position at the radius of the inscribed sphere. It is interesting to note that the derivative of transmission matrices (${{\textbf T}^{\textrm{ext}}}$, ${{\textbf T}^{\textrm{int}}}$) can be expressed with only reflection matrices (${{\textbf R}^{\textrm{ext}}}$, ${{\textbf R}^{\textrm{int}}}$).

All the optical properties including the extinction efficiency, single-scattering albedo and scattering matrix can be directedly computed from the T-matrix. The invariant imbedding T-matrix method (IITM) directly compute the T-matrix [35], whereas in the Debye series framework, we compute the T-matrix by using the series given in Eq. (12). As such, the impact of different order of scattering to the total scattering can be examined. The concept of the T-matrix formulation and the differences of related algorithms (IITM, EBCM) to compute the T-matrix are not reiterated in the present context and can be found in previous studies [36–39]. Representative results are given in the next section.

3. Results

Figure 2 shows the extinction and scattering efficiency factors of spheroids computed from the IITM and the Debye series method. The incident light is aligned with the symmetric axis. In this case, the optical properties are determined by the T-matrix with a projected angular momentum of $m = 1$. Two numerical examples ($\textrm{a}/\textrm{c} = \textrm{b}/\textrm{c} = 0.8$, $\textrm{a}/\textrm{c} = \textrm{b}/\textrm{c} = 0.5$; a, b, and c are three semi-axes of ellipsoids aligned with x, y, and z directions, respectively) are considered and the refractive index is 1.3 + i0.01. The excellent agreement between the two methods validates the numerical implementation associated with Eq. (12). The Debye series approach is now applicable to a wide range of size parameters; the size parameter is defined as ${\textrm{x}_\textrm{c}} = k \cdot c$, where k = 2π/λ and λ is the wavelength.

 figure: Fig. 2.

Fig. 2. The extinction and scattering efficiency factors computed from the IITM and the Debye series.

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In the framework of Debye series, the contribution to the extinction efficiency from all orders can be accurately obtained. Figure 3 shows the extinction efficiency computed from the T-matrix from different orders. The refractive index is 1.3 and the aspect ratio is 0.8. The $p = 0$ term is associated with the blocking effect, the Fraunhofer diffraction, and the tunneling/edge effect contribution, which have been justified in Bi et al. [32,33]. For contributions with $p \le 1$, the exact extinction curve can be reasonably recaptured except the fine oscillation feature that is particularly close to the peaks and valleys. If the T-matrix is computed with $p \le 4$, then the extinction curves from the exact method and the Debye series are almost identical. According to the optical theorem, the extinction efficiency is related to the forward scattering amplitude; note, the optical theorem is invalid for structured beams [40,41]. These results show that the forward amplitude computed from the diffraction and first-order transmission can be used to compute the extinction efficiency with reasonable accuracy. It is now also clear that the extinction efficiency computed from the anomalous diffraction theory (ADT) [42,43] can reasonably explain most features of the rigorous results, although the ADT is obtained from an optically soft particle. In the ADT theory, the first order transmission is only considered for deriving an approximate formula. However, we note that the higher-order contributions to the other optical properties (e.g., phase matrix elements) are large, which is particularly true for large scattering angles (as will be shown in the following).

 figure: Fig. 3.

Fig. 3. A comparison of the extinction efficiency computed from the T-matrix (pink line) and the Debye series expansion (blue line, orange dotted line and dots). The p denotes the order of the Debye series.

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Figure 4 shows phase function (${P_{11}}$) of spheroids computed from the Debye series with the order p from 0 to 5. Different from Fig. 2 and Fig. 3, spheroids were assumed to be randomly oriented. The size parameters are xc = 50 and xa = 40, which are defined in terms of semi-axes c and a, respectively. The refractive index is again 1.3. It is evident that the contributions from p = 0, p = 1, p = 4, and p = 5 are dominant at the scattering angle less than 90°, whereas the contributions from p = 2 and p = 3 are important at scattering angles larger than 90°. The relative intensity of the Debye series with different orders is critical to understanding the contribution of the p-th order Debye series to the total phase function. However, the interference among different orders should be taken into account in computing the total scattering phase function.

 figure: Fig. 4.

Fig. 4. Phase function computed from the Debye series with p = 0, p = 1, p = 2, p = 3, p = 4, and p = 5. xc and xa are size parameters defined in terms of semi-axes c and a, respectively. The refractive index is 1.3.

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Figure 5 shows phase matrix elements computed from the Debye series with $p = 0$, $p \le 1$ and $p \le 2$ for randomly oriented spheroids. When $p = 0$, as expected, the phase function (${P_{11}}$) has flat backscattering ($\theta > {90^o}$), which is determined from the external reflection (same as the $p = 0$ result shown in Fig. 4). The forward peak is dominated from the Fraunhofer diffraction with the tunneling/edge effect contribution. The element $- {P_{21}}/{P_{11}}$ is unity at the Brewster angle upon the external reflection. The element ${P_{22}}/{P_{11}}$ is unity, which means the diffraction and the external reflection has no contribution to the depolarization of the incident plane wave. The ${P_{34}}/{P_{11}}$ is close to zero with fine oscillations for the scattering angle (0°, 60°). When $p \le 1$, in addition to the diffraction and external reflection, the contribution to the scattering from the first order transmission is included. From the phase function, it can be seen that forward scattering significantly increases. $- {P_{21}}/{P_{11}}$ turns to be negative when the scattering angle is less than 90°. For the ${P_{22}}/{P_{11}}$ element, a slight difference from unity is observed. Differences in the other elements (${P_{33}}/{P_{11}}$, ${P_{34}}/{P_{11}}$, ${P_{44}}/{P_{11}}$) caused by the $p = 1$ term are also observable. When $p \le 2$, the contribution from the second transmission after one-time internal reflection is included. This term has significant contribution to the backscattering (scattering angle >90°). For example, the phase function increases at the scattering angles between 120° and 180°. The ${P_{22}}/{P_{11}}$ element now obviously deviates from unity, which means that strong depolarization will happen if the incident light is linearly polarized. ${P_{33}}/{P_{11}}$ and ${P_{44}}/{P_{11}}$ elements show a peak around 120°.

 figure: Fig. 5.

Fig. 5. Six phase matrix elements computed from the Debye series expansion ($p = 0$, $p \le 1$, and $p \le 2$). The refractive index is 1.3.

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Figure 6 shows the phase matrix elements by including more Debye series terms ($p \le 3,{\; }p \le 5$). The results by incorporating all Debye series terms are also included for comparison; the second formula $- 1/2[{1 - {{\textbf R}^{\textrm{ext}}} - {{\textbf T}^{\textrm{int}}}{{({1 - {{\textbf R}^{\textrm{int}}}} )}^{ - 1}}{{\textbf T}^{\textrm{ext}}}} ]$ in Eq. (12) was used for computing the T-matrix. All the phase matrix elements are similar to the $p \le 2$ results shown in Fig. 5, although differences are observable. For example, the ${P_{22}}/{P_{11}}$ element in Fig. 6 exhibits oscillation features around 120°. A close examination of the Debye series shows that the ${P_{22}}/{P_{11}}$ element converges slowly at the scattering angles between 60° and 120°. Figure 7 compares the ${P_{22}}/{P_{11}}\; $ element at three selected scattering angles (60°, 100°, and 140°) as function of the maximum order of the Debye series. It is evident that ${P_{22}}/{P_{11}}$ at 60° and 140° quickly converge to the results of the original scattering problem (black dashed lines) by including a few Debye series terms. However, ${P_{22}}/{P_{11}}$ at 140° requires a few tens of the Debye series terms.

 figure: Fig. 6.

Fig. 6. Six phase matrix elements computed from the Debye series expansion. $p \le 3$, $p \le 5$, and all orders of Debye series terms are included here. The refractive index is 1.3.

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 figure: Fig. 7.

Fig. 7. ${P_{22}}/{P_{11}}$ at three selected scattering angles (60°, 100°, and 140°) as function of the maximum order of the Debye series. Dashed black lines are the results with all terms.

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Thus, the results shown in Figs. 4–7 demonstrate the application of the new theoretical framework (Eqs. (12)–(26)) for computing the Debye series of light scattering by non-spherical particles. While the Debye series offer insights into the contributions of order p to total scattering, the sum of the Debye series terms from $p = 0\; $ to $p = {p_{max}}$ eventually converges to total scattering when ${p_{max}}$ is sufficiently large.

4. Summary

In this paper, we developed a new approach to computing the Debye series for electromagnetic wave scattering by non-spherical particles. We demonstrated that all the reflection and transmission matrices (from the medium to particle or the particle to medium) satisfy the Riccati-differential equations. The initial matrices for solving the differential equations can be obtained from the analytical solution of the Debye series for an inscribed sphere. Representative results were given in the case of spheroids and the contributions of different order Debye series to the total scattering were explicitly identified and explained. In principle, this approach can be extended to generalized convex non-spherical particles (e.g., a superegg recently studied in [44]), although the applicability should be carefully examined. The application of the present method (high-order Debye series, in particular) to transparent hexagonal ice crystals would be also interesting. Note that in a previous publication, we investigated the zeroth-order Debye series for hexagonal ice crystals. The size parameter can be up to 50.

In the numerical computations, we found that the present approach (based on the Runge-Kutta method) can be unstable with an upper limit on the particle size with a fixed aspect ratio. For example, the algorithm starts to lose accuracy when the size parameter is close to 90 when the aspect ratio is 0.8. The upper size limit will decrease as the aspect ratio increases. The stability is fundamentally associated with the stiff problem [45]. For example, when the size parameter is small, most elements in the reflection matrix (with large angular momentum) are close to unity and vary slowly as the size parameter increases. When the size parameter reaches certain threshold values, these elements rapidly change. This feature is typical characteristics of a stiff problem. By using non-stiff ordinary differential equation solvers, the numerical step should be sufficiently small so that the algorithm is stable when the size parameter is large. However, we found that decreasing the numerical step is not so helpful when the particle aspect ratio is extreme. We believe that a systematic comparison of differential equation solvers is worth studying, but this is beyond the scope of the present paper.

In addition to the stiff problem that remains to be investigated, the intercomparisons with the EBCM regarding their applicability regions (the aspect ratio and size parameters) and the examinations of the geometric optics can be also future topics. However, it should be pointed out that, for practical applications, the T-matrix method (directly computing the T-matrix) is recommended for computing the optical properties. The Debye series approach is only recommended for cases studies by analyzing the scattering mechanism involved in the electromagnetic wave scattering process.

Appendix A

The definitions of vector spherical functions are given as follows [1]:

$$\textrm{Rg}{\textrm{M}_\textrm{l}}({\textrm{k}{\textbf r}} )= {\mathrm{\gamma }_\textrm{l}}{\textrm{j}_\textrm{n}}({\textrm{kr}} ){\textrm{C}_\textrm{l}}({\mathrm{\theta },\mathrm{\varphi }} ), $$
$$\textrm{Rg}{\textrm{N}_\textrm{l}}({\textrm{k}{\textbf r}} )= {{\mathbf \gamma }_\textrm{l}}\left\{ {\frac{{\textrm{n}({\textrm{n} + 1} )}}{{\textrm{kr}}}{\textrm{j}_\textrm{n}}({\textrm{k}{\textbf r}} ){\textrm{P}_\textrm{l}}({\mathrm{\theta },\mathrm{\varphi }} )+ \frac{1}{{{\textrm{kr}}}}\frac{\textrm{d}}{{{\textrm{dkr}}}}[{\textrm{kr} \cdot {\textrm{j}_\textrm{n}}({\textrm{kr}} )} ]{\textrm{B}_\textrm{l}}({\mathrm{\theta },\mathrm{\varphi }} )} \right\},$$
$${\textrm{M}_\textrm{l}}({\textrm{k}{\textbf r}} )= {\mathrm{\gamma }_\textrm{l}}\textrm{h}_\textrm{n}^{(1 )}({\textrm{kr}} ){\textrm{C}_\textrm{l}}({\mathrm{\theta },\mathrm{\varphi }} ),$$
$${\textrm{N}_\textrm{l}}({\textrm{k}{\textbf r}} )= {\mathrm{\gamma }_\textrm{l}}\left\{ {\frac{{\textrm{n}({\textrm{n} + 1} )}}{{{\textrm{kr}}}}\textrm{h}_\textrm{n}^{(1 )}({\textrm{kr}} ){\textrm{P}_\textrm{l}}({\mathrm{\theta },\mathrm{\varphi }} )+ \frac{1}{{{\textrm{kr}}}}\frac{\textrm{d}}{{{\textrm{dkr}}}}[{\textrm{kr} \cdot \textrm{h}_\textrm{n}^{(1 )}({\textrm{kr}} )} ]{\textrm{B}_\textrm{l}}({\mathrm{\theta },\mathrm{\varphi }} )} \right\},$$
$${\mathrm{\tilde{M}}_\textrm{l}}({\textrm{k}{\textbf r}} )= {\mathrm{\gamma }_\textrm{l}}\textrm{h}_\textrm{n}^{(2 )}({\textrm{kr}} ){\textrm{C}_{\textbf l}}({\mathrm{\theta },\mathrm{\varphi }} ),$$
$${\tilde{{\textbf N}}_\textrm{l}}({\textrm{k}{\textbf r}} )= {{\mathbf \gamma }_\textrm{l}}\left\{ {\frac{{\textrm{n}({\textrm{n} + 1} )}}{{\textrm{kr}}}\textrm{h}_\textrm{n}^{(2 )}({\textrm{kr}} ){\textrm{P}_{\textbf l}}({\mathrm{\theta },\mathrm{\varphi }} )+ \frac{1}{{{\textrm{kr}}}}\frac{\textrm{d}}{{{\textrm{dkr}}}}[{\textrm{kr} \cdot \textrm{h}_\textrm{n}^{(2 )}({\textrm{kr}} )} ]{\textrm{B}_{\textbf l}}({\mathrm{\theta },\mathrm{\varphi }} )} \right\},$$
where vector spherical harmonics are given by
$${\textrm{C}_{\textbf l}}({\mathrm{\theta },\mathrm{\varphi }} )= {\mathbf {\hat{\theta }i}}{\mathrm{\pi }_{\textrm{mn}}}(\mathrm{\theta } )- \hat{{\mathbf \varphi }}{\mathrm{\tau }_{\textrm{mn}}}(\mathrm{\theta } ),$$
$${\textrm{P}_{\textbf l}}({\mathrm{\theta },\mathrm{\varphi }} )= \hat{{\textbf r}}{({ - 1} )^\textrm{m}}\sqrt {\frac{{({\textrm{n} + \textrm{m}} )!}}{{({\textrm{n} - \textrm{m}} )!}}} \textrm{d}_{0\textrm{m}}^\textrm{n}(\mathrm{\theta } ){\textrm{e}^{\mathrm{im\varphi }}},$$
$${\textrm{B}_{\textbf l}}({\mathrm{\theta },\mathrm{\varphi }} )= \hat{{\mathbf \theta }}{\mathrm{\tau }_{\textrm{mn}}}(\mathrm{\theta } )+ \hat{{\mathbf \varphi }}\textrm{i}{\mathrm{\pi }_{\textrm{mn}}}(\mathrm{\theta } ),$$
where
$${\mathrm{\pi }_{\textrm{mn}}}(\mathrm{\theta } )= \frac{\textrm{m}}{{\mathrm{sin\theta }}}\textrm{d}_{0\textrm{m}}^\textrm{n}(\mathrm{\theta } ),$$
$${\mathrm{\tau }_{\textrm{mn}}}(\mathrm{\theta } )= \frac{\textrm{d}}{{\mathrm{d\theta }}}\textrm{d}_{0\textrm{m}}^\textrm{n}(\mathrm{\theta } ),$$
$$\textrm{d}_{0\textrm{m}}^\textrm{n}(\mathrm{\theta } )= {({ - 1} )^\textrm{m}}\sqrt {\frac{{({\textrm{n} - \textrm{m}} )!}}{{({\textrm{n} + \textrm{m}} )!}}} \textrm{P}_\textrm{n}^\textrm{m}(\textrm{x} ),x = \mathrm{cos\theta ,}$$
$$\textrm{P}_\textrm{n}^\textrm{m}(\textrm{x} )= {({ - 1} )^\textrm{m}}{({1 - {\textrm{x}^2}} )^{\textrm{m}/2}}\frac{{{\textrm{d}^\textrm{m}}}}{{\textrm{d}{\textrm{x}^\textrm{m}}}}{\textrm{P}_\textrm{n}}(\textrm{x} ),{\; }{\textrm{P}_\textrm{n}}(\textrm{x} )= \frac{1}{{{2^\textrm{n}}\textrm{n}!}}\frac{{{\textrm{d}^\textrm{n}}}}{{\textrm{d}{\textrm{x}^\textrm{n}}}}{({{\textrm{x}^2} - 1} )^\textrm{n}}, $$
$${\mathrm{\gamma }_\textrm{l}} = {\left[ {\frac{{({2\textrm{n} + 1} )({\textrm{n} - \textrm{m}} )!}}{{4\mathrm{\pi n}({\textrm{n} + 1} )({\textrm{n} + \textrm{m}} )!}}} \right]^{1/2}}.$$

Appendix B

In this appendix, we derive the derivative of the reflection and transmission coefficients with respect to the radius for a sphere. First, we consider the TE mode. When a spherically outgoing wave encounters the particle surface, the reflection and transmission coefficients are given by [27]

$$\textrm{R}_{11}^{\textrm{int}} = \textrm{A}/\textrm{B}, $$
$$\textrm{T}_{11}^{\textrm{int}} ={-} 2\textrm{i}/\textrm{B}, $$
where
$$A(x )= \zeta _n^{(1 )}\mathrm{^{\prime}}(x )\zeta _n^{(1 )}({\tilde{m}x} )- \tilde{m}\zeta _n^{(1 )}(x )\zeta _n^{(1 )\mathrm{^{\prime}}}({\tilde{m}x} ), $$
$$B(x )={-} \zeta _n^{(1 )}\mathrm{^{\prime}}(x )\zeta _n^{(2 )}({\tilde{m}x} )+ \tilde{m}\zeta _n^{(1 )}(x )\zeta _n^{(2 )}\mathrm{^{\prime}}({\tilde{m}x} ). $$

In (43) and (44), $x = kr{\; }$ is the size parameter, and $\zeta _n^{(i )}(x )$ is the Riccati-Hankel function of the i-th kind. Note, the prime indicates the derivative of a function in terms of its argument. The Wronskian identity shows

$$\zeta _n^{(1 )}({\tilde{m}x} )\zeta _n^{(2 )}\mathrm{^{\prime}}({\tilde{m}x} )- \zeta _n^{(1 )}\mathrm{^{\prime}}({\tilde{m}x} )\zeta _n^{(2 )}({\tilde{m}x} )={-} 2i, $$
and we obtain
$$A(x )\zeta _n^{(2 )}({\tilde{m}x} )+ B(x )\zeta _n^{(1 )}({\tilde{m}x} )={-} 2i\tilde{m}\zeta _n^{(1 )}(x ), $$
$$A(x )\zeta _n^{(2 )\mathrm{^{\prime}}}({\tilde{m}x} )+ B(x )\zeta _n^{(1 )\mathrm{^{\prime}}}({\tilde{m}x} )={-} 2i\zeta _n^{(1 )\mathrm{^{\prime}}}(x ). $$

By taking the following operation,

$$\frac{{d[{46} ]}}{{dx}} - \tilde{m},$$
we obtain
$$A^{\prime}(x )\zeta _n^{(2 )}({\tilde{m}x} )+ B^{\prime}(x )\zeta _n^{(1 )}({\tilde{m}x} )= 0.$$
With Eqs. (46) and (48) and taking $A^{\prime}(x )\times [{46} ]- A(x )\times [{48} ]$, we have
$$[{A^{\prime}(x )B(x )- A(x )B\mathrm{^{\prime}}(x )} ]={-} 2i\tilde{m}\frac{{\zeta _n^{(1 )}(x )}}{{\zeta _n^{(1 )}({\tilde{m}x} )}}A\mathrm{^{\prime}}(x ).$$

Based on Eq. (43), we have

$$A^{\prime}(x )= \zeta _n^{{{(1 )}^{\mathrm{^{\prime\prime}}}}}(x )\zeta _n^{(1 )}({\tilde{m}x} )- {\tilde{m}^2}\zeta _n^{(1 )}(x )\zeta _n^{{{(1 )}^{\mathrm{^{\prime\prime}}}}}({\tilde{m}x} ), $$
and using the Riccati-differential equation,
$$\frac{{\zeta _n^{{{(1 )}^{\mathrm{^{\prime\prime}}}}}({\tilde{m}x} )}}{{\zeta _n^{(1 )}({\tilde{m}x} )}} = \left[ {\frac{{n({n + 1} )}}{{{{({\tilde{m}x} )}^2}}} - 1} \right], $$
$$\frac{{\zeta _n^{{{(1 )}^{\mathrm{^{\prime\prime}}}}}(x )}}{{\zeta _n^{(1 )}(x )}} = \left[ {\frac{{n({n + 1} )}}{{{x^2}}} - 1} \right], $$
we found
$$\begin{aligned}{\left[A^{\prime}(x)B(x)-A(x)B^{\prime}(x)\right]=-2i\tilde{m} \frac{\zeta_{n}^{(1)}(x)}{\zeta_{n}^{(1)}(\widetilde{m}x)}} \\ \times\left[\left(\frac{n(n+1)}{x^{2}}-1\right)\zeta_{n}^{(1)}(x)\zeta_{n}^{(1)}(\tilde{m}x)-\tilde{m}^{2}\left(\frac{n(n+1)}{(\tilde{m}x)^{2}}-1\right) \zeta_{n}^{(1)}(x) \zeta_{n}^{(1)}(\tilde{m}x)\right] \\ =-2i\tilde{m}\left[\zeta_{n}^{(1)}(x)\right]^{2}\left[\tilde{m}^{2}-1\right].\end{aligned}$$

Now, we can obtain the derivative of the internal reflection coefficient

$$\begin{aligned} \frac{{dR_{11}^{int}}}{{dx}} &= \frac{{A^{\prime}(x )B(x )- A(x )B^{\prime}(x )}}{{{B^2}(x )}}\\ &={-} 2i\frac{{{{\tilde{m}}^2} - 1}}{{\tilde{m}}}{\left[ {\frac{{\tilde{m}\zeta_n^{(1 )}(x )}}{{B(x )}}} \right]^2}\\ &= \frac{i}{2}\frac{{{{\tilde{m}}^2} - 1}}{{\tilde{m}}}{\left[ {\frac{{A(x )}}{{B(x )}}\zeta_n^{(2 )}({\tilde{m}x} )+ \zeta_n^{(1 )}({\tilde{m}x} )} \right]^2}. \end{aligned}$$

We used Eq. (46) in deriving Eq. (54). By using Eq. (41), Eq. (54) can be simplified as

$$\frac{{dR_{11}^{int}}}{{dx}} = \frac{i}{2}\frac{{{{\tilde{m}}^2} - 1}}{{\tilde{m}}}{[{{R^{\textrm{int}}}\zeta_n^{(2 )}({\tilde{m}x} )+ \zeta_n^{(1 )}({\tilde{m}x} )} ]^2}. $$

The derivative of the transmission matrix with respect to the radius can be obtained similarly.

$$\frac{{dT_{11}^{int}}}{{dx}} = 2i\frac{{B^{\prime}(x )}}{{{B^2}(x )}}. $$

By using (44), (51) and (52), we have

$$\begin{aligned} B^{\prime}(x )&={-} \zeta _n^{{{(1 )}^{\mathrm{^{\prime\prime}}}}}(x )\zeta _n^{(2 )}({\tilde{m}x} )+ {{\tilde{m}}^2}\zeta _n^{(1 )}(x )\zeta _n^{{{(2 )}^{\mathrm{^{\prime\prime}}}}}({\tilde{m}x} )\\ &= \left\{ { - \left[ {\frac{{n({n + 1} )}}{{{x^2}}} - 1} \right] + \left[ {\frac{{n({n + 1} )}}{{{x^2}}} - {m^2}} \right]} \right\}\zeta _n^{(1 )}(x )\zeta _n^{(2 )}({\tilde{m}x} )\\ &= ({1 - {{\tilde{m}}^2}} )\zeta _n^{(1 )}(x )\zeta _n^{(2 )}({\tilde{m}x} )\end{aligned}$$
and then we show that
$$\begin{aligned} \frac{{dT_{11}^{int}}}{{dx}} &= 2i\frac{{B\mathrm{^{\prime}}(x )}}{{{B^2}(x )}} = 2i({1 - {{\tilde{m}}^2}} )\frac{{\zeta _n^{(1 )}(x )\zeta _n^{(2 )}({\tilde{m}x} )}}{{{B^2}(x )}}\\ &= 2i({1 - {{\tilde{m}}^2}} )\frac{{\zeta _n^{(1 )}(x )}}{{B(x )}}\frac{{\zeta _n^{(2 )}({\tilde{m}x} )}}{{B(x )}}\\ &= \frac{i}{{2\tilde{m}}}({{{\tilde{m}}^2} - 1} )[{{R^{\textrm{int}}}\zeta_n^{(2 )}({\tilde{m}x} )+ \zeta_n^{(1 )}({\tilde{m}x} )} ][{{R^{\textrm{ext}}}\zeta_n^{(1 )}(x )+ \zeta_n^{(2 )}(x )} ].\end{aligned}$$

Rext can be found in the Appendix B of Ref. [32].

The TM mode can be obtained by following a similar procedure. We have

$$\frac{{dR_{22}^{int}}}{{dx}} = \frac{i}{2}\frac{{{{\tilde{m}}^2} - 1}}{{\tilde{m}}}\left\{ {\frac{{n({n + 1} )}}{{{{(x )}^2}}}{{[{{R^{\textrm{int}}}\zeta_n^{(2 )}({\tilde{m}x} )+ \zeta_n^{(1 )}({\tilde{m}x} )} ]}^2} + {{[{{R^{\textrm{int}}}\zeta_n^{(2 )}\mathrm{^{\prime}}({\tilde{m}x} )+ \zeta_n^{(1 )}\mathrm{^{\prime}}({\tilde{m}x} )} ]}^2}} \right\}, $$
$$\begin{aligned} \frac{{dT_{22}^{int}}}{{dx}} &= \frac{i}{2}\frac{{({{{\tilde{m}}^2} - 1} )}}{{\tilde{m}}}\left\{ {\frac{{n({n + 1} )}}{{\tilde{m}{x^2}}}} \right.[{{R^{\textrm{int}}}\zeta_n^{(2 )}({\tilde{m}x} )+ \zeta_n^{(1 )}({\tilde{m}x} )} ][{{R^{\textrm{ext}}}\zeta_n^{(1 )}(x )+ \zeta_n^{(2 )}(x )} ]\\ &+ {[{{R^{\textrm{int}}}\zeta_n^{(2 )}\mathrm{^{\prime}}({\tilde{m}x} )+ \zeta_n^{(1 )}\mathrm{^{\prime}}({\tilde{m}x} )} ][{{R^{\textrm{ext}}}\zeta_n^{(1 )}\mathrm{^{\prime}}(x )+ \zeta_n^{(2 )}\mathrm{^{\prime}}(x )} ]} \}.\end{aligned}$$

For a homogeneous sphere, the expansion of the right hands of Eqs. (14) and (16) contains both the TE and TM modes. Equations (55), (58), (59) and (60) can now be directly compared with the Riccati-differential equations (Eqs. (14) and (16)) for validation. Thus, the correctness of the algorithm was analytically confirmed in the case of a homogeneous sphere.

Funding

National Natural Science Foundation of China (42022038).

Acknowledgments

Lei Bi was supported by the National Natural Science Foundation of China (42022038). We acknowledge discussions with Feng Xu (The University of Oklahoma, USA).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. (a) Reflection (${{\textbf R}^{\textrm{ext}}}$) and transmission (${{\textbf T}^{\textrm{ext}}}$) of an incoming spherical wave. (b) Internal reflection (${{\textbf R}^{\textrm{int}}}$) and transmission (${{\textbf T}^{\textrm{int}}}$) of an outgoing spherical wave.

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Fig. 2.
Fig. 2. The extinction and scattering efficiency factors computed from the IITM and the Debye series.

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Fig. 3.
Fig. 3. A comparison of the extinction efficiency computed from the T-matrix (pink line) and the Debye series expansion (blue line, orange dotted line and dots). The p denotes the order of the Debye series.

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Fig. 4.
Fig. 4. Phase function computed from the Debye series with p = 0, p = 1, p = 2, p = 3, p = 4, and p = 5. xc and xa are size parameters defined in terms of semi-axes c and a, respectively. The refractive index is 1.3.

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Fig. 5.
Fig. 5. Six phase matrix elements computed from the Debye series expansion ($p = 0$, $p \le 1$, and $p \le 2$). The refractive index is 1.3.

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Fig. 6.
Fig. 6. Six phase matrix elements computed from the Debye series expansion. $p \le 3$, $p \le 5$, and all orders of Debye series terms are included here. The refractive index is 1.3.

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Fig. 7.
Fig. 7. ${P_{22}}/{P_{11}}$ at three selected scattering angles (60°, 100°, and 140°) as function of the maximum order of the Debye series. Dashed black lines are the results with all terms.

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Equations (61)

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E inc ( r ) = ∑ l = 1 ∞ ⁡ a l R g M l ( k r ) + b l R g N l ( k r ) ,
E sca ( r ) = ∑ l = 1 ∞ ⁡ p l M l ( k r ) + q l N l ( k r ) ,
E int ( r ) = ∑ l = 1 ∞ ⁡ c l R g M l ( m ~ k r ) + d l R g N l ( m ~ k r ) ,
E ~ inc ext ( r ) = ∑ l = 1 ∞ ⁡ a ~ l M ~ l ( k r ) + b ~ l N ~ l ( k r ) .
E ref ext ( r ) = ∑ l = 1 ∞ ⁡ p l , r e f M l ( k r ) + q l , r e f N l ( k r ) ,
E tra ext ( r ) = ∑ l = 1 ∞ ⁡ a l , t r a M ~ l ( m ~ k r ) + b l , t r a N ~ l ( m ~ k r ) .
E ~ out int ( r ) = ∑ l = 1 ∞ ⁡ a ~ l , o u t M l ( m ~ k r ) + b ~ l , o u t N l ( m ~ k r ) ,
E ref int ( r ) = ∑ l = 1 ∞ ⁡ a l , r e f M ~ l ( m ~ k r ) + b l , r e f N ~ l ( m ~ k r ) ,
E tra int ( r ) = ∑ l = 1 ∞ ⁡ p l , t r a M l ( k r ) + q l , t r a N l ( k r ) .
[ a ~ o u t b ~ o u t ] = [ a r e f b r e f ] + [ a t r a b t r a ] = R i n t [ a ~ o u t b ~ o u t ] + T e x t [ a ~ b ~ ] .
[ p t r a q t r a ] = T int [ a ~ out b ~ out ] = T int ( 1 − R int ) − 1 T ext [ a ~ b ~ ] .
T = − 1 2 [ 1 − R ext − T int ( ∑ p = 0 ∞ ⁡ ( R int ) p ) T ext ] = − 1 2 [ 1 − R ext − T int 1 1 − R int T ext ] .
d R ext ( k r ) d ( k r ) = i 2 [ H ~ T ( k r ) + R ext ( k r ) H T ( k r ) ] U [ H ~ ( k r ) + H ( k r ) R ext ( k r ) ] ,
d R int ( k r ) d ( k r ) = i 2 m ~ [ H T ( m ~ k r ) + R int ( k r ) H ~ T ( m ~ k r ) ] U 2 [ H ( m ~ k r ) + H ~ ( m ~ k r ) R int ( k r ) ] ,
d T ext ( k r ) d ( k r ) = i 2 m ~ [ H T ( m ~ k r ) + R int ( k r ) H ~ T ( m ~ k r ) ] U 3 [ H ~ ( k r ) + H ( k r ) R ext ( k r ) ] ,
d T int ( k r ) d ( k r ) = i 2 [ H ~ T ( k r ) + R ext ( k r ) H T ( k r ) ] U 3 [ H ( m ~ k r ) + H ~ ( m ~ k r ) R int ( k r ) ] ,
H l l ′ ( x ) = δ l l ′ [ h n ( 1 ) ( x ) 0 0 1 x ∂ ∂ x [ x h n ( 1 ) ( x ) ] 0 n ( n + 1 ) h n ( 1 ) ( x ) / x ] , n = c e i l ( l + 1 ) ,
U l l ′ ( k r ) = f l l ′ ∫ 0 2 π ⁡ d ϕ ∫ 0 2 π ⁡ d θ s i n θ e − i ( m − m ′ ) ϕ [ m ~ 2 ( r , θ , ϕ ) − 1 ] [ U l l ′ a − U l l ′ b 0 U l l ′ b U l l ′ a 0 0 0 U l l ′ c ] ,
[ U 2 ] l l ′ ( k r ) = f l l ′ ∫ 0 2 π ⁡ d ϕ ∫ 0 2 π ⁡ d θ s i n θ e − i ( m − m ′ ) ϕ [ m ~ 2 ( r , θ , ϕ ) − 1 ] [ U l l ′ a − U l l ′ b 0 U l l ′ b U l l ′ a 0 0 0 U l l ′ d ] ,
[ U 3 ] l l ′ ( k r ) = f l l ′ ∫ 0 2 π ⁡ d ϕ ∫ 0 2 π ⁡ d θ s i n θ e − i ( m − m ′ ) ϕ [ m ~ 2 ( r , θ , ϕ ) − 1 ] [ U l l ′ a − U l l ′ b 0 U l l ′ b U l l ′ a 0 0 0 U l l ′ e ] ,
f l l ′ = k 2 r 2 ( − 1 ) m + m ′ [ 2 n + 1 4 π n ( n + 1 ) ] 1 / 2 [ 2 n ′ + 1 4 π n ′ ( n ′ + 1 ) ] 1 / 2 ,
U l l ′ a ( θ ) = π m n ( θ ) π m ′ n ′ ( θ ) + τ m n ( θ ) τ m ′ n ′ ( θ ) ,
U l l ′ b ( θ ) = i [ π m n ( θ ) τ m ′ n ′ ( θ ) + τ m n ( θ ) π m ′ n ′ ( θ ) ] ,
U l l ′ c ( θ ) = n ( n + 1 ) n ′ ( n ′ + 1 ) d 0 m n ( θ ) d 0 m ′ n ′ ( θ ) / m ~ 2 ,
U l l ′ d ( θ ) = n ( n + 1 ) n ′ ( n ′ + 1 ) d 0 m n ( θ ) d 0 m ′ n ′ ( θ ) m ~ 2 ,
U l l ′ e ( θ ) = n ( n + 1 ) n ′ ( n ′ + 1 ) d 0 m n ( θ ) d 0 m ′ n ′ ( θ ) .
Rg M l ( k r ) = γ l j n ( kr ) C l ( θ , φ ) ,
Rg N l ( k r ) = γ l { n ( n + 1 ) kr j n ( k r ) P l ( θ , φ ) + 1 kr d dkr [ kr ⋅ j n ( kr ) ] B l ( θ , φ ) } ,
M l ( k r ) = γ l h n ( 1 ) ( kr ) C l ( θ , φ ) ,
N l ( k r ) = γ l { n ( n + 1 ) kr h n ( 1 ) ( kr ) P l ( θ , φ ) + 1 kr d dkr [ kr ⋅ h n ( 1 ) ( kr ) ] B l ( θ , φ ) } ,
M ~ l ( k r ) = γ l h n ( 2 ) ( kr ) C l ( θ , φ ) ,
N ~ l ( k r ) = γ l { n ( n + 1 ) kr h n ( 2 ) ( kr ) P l ( θ , φ ) + 1 kr d dkr [ kr ⋅ h n ( 2 ) ( kr ) ] B l ( θ , φ ) } ,
C l ( θ , φ ) = θ ^ i π mn ( θ ) − φ ^ τ mn ( θ ) ,
P l ( θ , φ ) = r ^ ( − 1 ) m ( n + m ) ! ( n − m ) ! d 0 m n ( θ ) e i m φ ,
B l ( θ , φ ) = θ ^ τ mn ( θ ) + φ ^ i π mn ( θ ) ,
π mn ( θ ) = m s i n θ d 0 m n ( θ ) ,
τ mn ( θ ) = d d θ d 0 m n ( θ ) ,
d 0 m n ( θ ) = ( − 1 ) m ( n − m ) ! ( n + m ) ! P n m ( x ) , x = c o s θ ,
P n m ( x ) = ( − 1 ) m ( 1 − x 2 ) m / 2 d m d x m P n ( x ) , P n ( x ) = 1 2 n n ! d n d x n ( x 2 − 1 ) n ,
γ l = [ ( 2 n + 1 ) ( n − m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 / 2 .
R 11 int = A / B ,
T 11 int = − 2 i / B ,
A ( x ) = ζ n ( 1 ) ′ ( x ) ζ n ( 1 ) ( m ~ x ) − m ~ ζ n ( 1 ) ( x ) ζ n ( 1 ) ′ ( m ~ x ) ,
B ( x ) = − ζ n ( 1 ) ′ ( x ) ζ n ( 2 ) ( m ~ x ) + m ~ ζ n ( 1 ) ( x ) ζ n ( 2 ) ′ ( m ~ x ) .
ζ n ( 1 ) ( m ~ x ) ζ n ( 2 ) ′ ( m ~ x ) − ζ n ( 1 ) ′ ( m ~ x ) ζ n ( 2 ) ( m ~ x ) = − 2 i ,
A ( x ) ζ n ( 2 ) ( m ~ x ) + B ( x ) ζ n ( 1 ) ( m ~ x ) = − 2 i m ~ ζ n ( 1 ) ( x ) ,
A ( x ) ζ n ( 2 ) ′ ( m ~ x ) + B ( x ) ζ n ( 1 ) ′ ( m ~ x ) = − 2 i ζ n ( 1 ) ′ ( x ) .
d [ 46 ] d x − m ~ ,
A ′ ( x ) ζ n ( 2 ) ( m ~ x ) + B ′ ( x ) ζ n ( 1 ) ( m ~ x ) = 0.
[ A ′ ( x ) B ( x ) − A ( x ) B ′ ( x ) ] = − 2 i m ~ ζ n ( 1 ) ( x ) ζ n ( 1 ) ( m ~ x ) A ′ ( x ) .
A ′ ( x ) = ζ n ( 1 ) ′ ′ ( x ) ζ n ( 1 ) ( m ~ x ) − m ~ 2 ζ n ( 1 ) ( x ) ζ n ( 1 ) ′ ′ ( m ~ x ) ,
ζ n ( 1 ) ′ ′ ( m ~ x ) ζ n ( 1 ) ( m ~ x ) = [ n ( n + 1 ) ( m ~ x ) 2 − 1 ] ,
ζ n ( 1 ) ′ ′ ( x ) ζ n ( 1 ) ( x ) = [ n ( n + 1 ) x 2 − 1 ] ,
[ A ′ ( x ) B ( x ) − A ( x ) B ′ ( x ) ] = − 2 i m ~ ζ n ( 1 ) ( x ) ζ n ( 1 ) ( m ~ x ) × [ ( n ( n + 1 ) x 2 − 1 ) ζ n ( 1 ) ( x ) ζ n ( 1 ) ( m ~ x ) − m ~ 2 ( n ( n + 1 ) ( m ~ x ) 2 − 1 ) ζ n ( 1 ) ( x ) ζ n ( 1 ) ( m ~ x ) ] = − 2 i m ~ [ ζ n ( 1 ) ( x ) ] 2 [ m ~ 2 − 1 ] .
d R 11 i n t d x = A ′ ( x ) B ( x ) − A ( x ) B ′ ( x ) B 2 ( x ) = − 2 i m ~ 2 − 1 m ~ [ m ~ ζ n ( 1 ) ( x ) B ( x ) ] 2 = i 2 m ~ 2 − 1 m ~ [ A ( x ) B ( x ) ζ n ( 2 ) ( m ~ x ) + ζ n ( 1 ) ( m ~ x ) ] 2 .
d R 11 i n t d x = i 2 m ~ 2 − 1 m ~ [ R int ζ n ( 2 ) ( m ~ x ) + ζ n ( 1 ) ( m ~ x ) ] 2 .
d T 11 i n t d x = 2 i B ′ ( x ) B 2 ( x ) .
B ′ ( x ) = − ζ n ( 1 ) ′ ′ ( x ) ζ n ( 2 ) ( m ~ x ) + m ~ 2 ζ n ( 1 ) ( x ) ζ n ( 2 ) ′ ′ ( m ~ x ) = { − [ n ( n + 1 ) x 2 − 1 ] + [ n ( n + 1 ) x 2 − m 2 ] } ζ n ( 1 ) ( x ) ζ n ( 2 ) ( m ~ x ) = ( 1 − m ~ 2 ) ζ n ( 1 ) ( x ) ζ n ( 2 ) ( m ~ x )
d T 11 i n t d x = 2 i B ′ ( x ) B 2 ( x ) = 2 i ( 1 − m ~ 2 ) ζ n ( 1 ) ( x ) ζ n ( 2 ) ( m ~ x ) B 2 ( x ) = 2 i ( 1 − m ~ 2 ) ζ n ( 1 ) ( x ) B ( x ) ζ n ( 2 ) ( m ~ x ) B ( x ) = i 2 m ~ ( m ~ 2 − 1 ) [ R int ζ n ( 2 ) ( m ~ x ) + ζ n ( 1 ) ( m ~ x ) ] [ R ext ζ n ( 1 ) ( x ) + ζ n ( 2 ) ( x ) ] .
d R 22 i n t d x = i 2 m ~ 2 − 1 m ~ { n ( n + 1 ) ( x ) 2 [ R int ζ n ( 2 ) ( m ~ x ) + ζ n ( 1 ) ( m ~ x ) ] 2 + [ R int ζ n ( 2 ) ′ ( m ~ x ) + ζ n ( 1 ) ′ ( m ~ x ) ] 2 } ,
d T 22 i n t d x = i 2 ( m ~ 2 − 1 ) m ~ { n ( n + 1 ) m ~ x 2 [ R int ζ n ( 2 ) ( m ~ x ) + ζ n ( 1 ) ( m ~ x ) ] [ R ext ζ n ( 1 ) ( x ) + ζ n ( 2 ) ( x ) ] + [ R int ζ n ( 2 ) ′ ( m ~ x ) + ζ n ( 1 ) ′ ( m ~ x ) ] [ R ext ζ n ( 1 ) ′ ( x ) + ζ n ( 2 ) ′ ( x ) ] } .
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