Abstract

Electromagnetic (EM) scattering by particles in an absorbing host medium is frequently encountered in practical applications, which makes the conventional EM scattering theory controversial and most of the theoretical methods for EM scattering inapplicable. Most of the relevant works in literature are confined to spherical particles. In this work, we develop the discrete dipole approximation (DDA) method for EM scattering by an arbitrary particle immersed in an absorbing host medium. We elaborate how the near- and far-field scattering quantities can be calculated by DDA. The accuracy of DDA is validated by comparison with the apparent and inherent scattering quantities of spherical particles computed by exact Mie theory. Then EM extinction by non-absorbing spheroids in absorbing host medium is studied by DDA. We find that particles that are prolonged in the incident direction are more likely to produce a negative apparent extinction, which is also supported by the near-field electric field distribution. The DDA method we develop will be useful and flexible in the study of EM scattering by particles in absorbing host medium.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Electromagnetic (EM) scattering by small particles is a ubiquitous phenomenon that finds applications in a wide range of science and engineering disciplines. The classic EM scattering theories of particles are established upon the basic assumption that the host medium is non-absorbing [14]. The conventional Lorenz-Mie theory and other well-developed numerical methods for nonspherical particles (e.g., the T-matrix method, DDA) typically do not consider the absorption of the host medium. In many situations, however, we encounter scattering problems in a host medium that is absorbing at wavelengths of interest. Examples are infrared light scattering by particles in water and strongly absorbing gases, or light scattering by particles in various absorbing polymers. The absorption of the host medium raises great controversy on the definition and application of the scattering quantities that are well-defined for non-absorbing host medium.

EM scattering by particles in absorbing host medium has been a subject of active research during the past decades [535]. Most of the works studied the EM scattering of spherical particles in absorbing host medium by an extension of the conventional Lorenz-Mie theory, with great controversies on the definition of the scattering, absorption and extinction cross sections. There are mainly two approaches to defining the cross sections, namely the inherent (based on the integration of the near-field EM field over the particle surface) and the apparent (based on the far-field asymptotic form of the EM field) scattering properties [15]. In recent years, using first-principle approach, Mishchenko [2527] elaborated a theoretical framework for single and multiple EM scattering of particles as well as the radiative transfer in absorbing host medium. Several far-field scattering quantities are rigorously derived, which can be applied to the radiative transfer equation. Under this framework, Mishchenko and Yang [36] introduced a far-field Lorenz-Mie theory and computing program for EM scattering by a homogeneous spherical particle in absorbing host medium.

However, for scattering problems in absorbing host medium, very little attention is paid to nonspherical particles that constitute a great portion of both natural and artificial particles. Sun et al. [1719] developed the finite difference time domain method for light scattering by particles in absorbing host medium and studied light scattering by a spheroid and an infinite cylinder, they found that the host medium absorption has an important effect on the phase function. Till now, many aspects of EM scattering by particles (especially nonspherical particles) in absorbing host medium remain unclear, for instance, the far- and near-field scattering properties of nonspherical particles, the effect of host medium absorption on the EM interactions with particles, the energy budget and power flow during EM scattering by particles, etc. A vital factor that limits our understanding of the above aspects is the lack of efficient computing methods for EM scattering by an arbitrary particle in absorbing host medium.

Therefore, it is necessary to develop computing methods for EM scattering by an arbitrary particle in absorbing host medium. Since the EM scattering by an arbitrary particle is frequently described by the volume integral equation (VIE) [25,3739], a natural solution to this problem is the DDA method, which discretizes the target and transforms the VIE into a set of linear equations [4045]. DDA has been extensively studied and developed for particle scattering in non-absorbing host medium [43], but has not been extended to absorbing host medium so far. In this work, we develop DDA for EM scattering by an arbitrary particle in absorbing host medium. In Sec. 2, we elaborate the theoretical framework of DDA for EM scattering in absorbing host medium. In Sec. 3, we validate our DDA code. In Sec. 4, we give some numerical results, and we conclude this article in the last section.

2. DDA for absorbing host medium

In this section, we describe in detail the DDA method for EM scattering by particles in absorbing host medium. For completeness and self-consistency, we start from VIE to give a general description of the EM scattering by an arbitrary particle, then we elaborate how VIE is solved and how the relevant scattering quantities can be calculated by DDA.

2.1 Volume integral equation (VIE)

We consider the EM scattering by an arbitrary particle embedded in a nonmagnetic, infinite, homogenous, linear, isotropic and generally absorbing host medium. VINT denotes the volume occupied by the particle while VEXT denotes the infinite exterior region such that ${V_{\textrm{INT}}} \cup {V_{\textrm{EXT}}} = {{\mathbb R}^3}$. The materials of the particle are assumed to be nonmagnetic, isotropic, linear and possibly inhomogeneous. Assuming the exp(-iωt) time dependence of all the fields, the frequency-domain monochromatic Maxwell equations describing the scattering problem are [25]

$$\left. {\begin{array}{l} {\nabla \times {\textbf E}({\textbf r} )= \textrm{i}\omega {\mu_0}{\textbf H}({\textbf r} )}\\ {\nabla \times {\textbf H}({\textbf r} )={-} \textrm{i}\omega {\varepsilon_\textrm{h}}{\textbf E}({\textbf r} )} \end{array}} \right\}\;\;{\textbf r} \in {V_{\textrm{EXT}}}$$
$$\left. {\begin{array}{l} {\nabla \times {\textbf E}({\textbf r} )= \textrm{i}\omega {\mu_0}{\textbf H}({\textbf r} )}\\ {\nabla \times {\textbf H}({\textbf r} )={-} \textrm{i}\omega {\varepsilon_\textrm{p}}({\textbf r} ){\textbf E}({\textbf r} )} \end{array}} \right\}\;\;{\textbf r} \in {V_{\textrm{INT}}}$$
where εh is the complex permittivity of the host medium, εp(r) is the position-dependent complex permittivity of the scattering particle, μ0 is the vacuum permeability, E and H are the electric and magnetic fields, respectively.

Equations (1) and (2) can be casted into the following vector wave equations [25]:

$$\nabla \times \nabla \times {\textbf E}({\textbf r} )- k_\textrm{h}^2{\textbf E}({\textbf r} )= 0\;\;,\;\;{\textbf r} \in {V_{\textrm{EXT}}}$$
$$\nabla \times \nabla \times {\textbf E}({\textbf r} )- k_\textrm{p}^2({\textbf r} ){\textbf E}({\textbf r} )= 0\;\;,\;\;{\textbf r} \in {V_{\textrm{INT}}}$$
the wave numbers of the host medium and the particle are given by ${k_\textrm{h}} = \omega \sqrt {{\varepsilon _\textrm{h}}{\mu _0}} $ and ${k_\textrm{p}}({\textbf r} )= \omega \sqrt {{\varepsilon _\textrm{p}}({\textbf r} ){\mu _0}} $, respectively, which are generally complex numbers (e.g. ${k_\textrm{h}}\textrm{ = }{k^{\prime}_\textrm{h}}\textrm{ + i}{k^{\prime\prime}_\textrm{h}}$). Eqs. (3) and (4) can be combined into a single inhomogeneous differential equation [25]
$$\nabla \times \nabla \times {\textbf E}({\textbf r} )- k_\textrm{h}^2{\textbf E}({\textbf r} )= \;{\textbf j}({\textbf r} )\;,\;\;{\textbf r} \in {{\mathbb R}^3}$$
where the source function is [25]
$${\textbf j}({\textbf r} )\textrm{ = }k_\textrm{h}^2[{\tilde{\varepsilon }({\textbf r} )- 1} ]{\textbf E}({\textbf r} )$$
$\tilde{\varepsilon }({\textbf r} )$ is the relative electric permittivity given by
$$\tilde{\varepsilon }({\textbf r} )\textrm{ = }\left\{ {\begin{array}{l} {1,\;\;{\textbf r} \in {V_{\textrm{EXT}}}}\\ {{{k_\textrm{p}^2({\textbf r} )} / {k_\textrm{h}^2 = {{{\varepsilon_\textrm{p}}({\textbf r} )} / {{\varepsilon_\textrm{h}}}},\;\;{\textbf r} \in {V_{\textrm{INT}}}}}} \end{array}} \right.$$

The general solution to Eq. (5) can be written in terms of the conventional VIE, i.e. [25,46]

$${\textbf E}({\textbf r} )\textrm{ = }{{\textbf E}^{\textrm{inc}}}({\textbf r} )+ k_\textrm{h}^2\int_{{V_{\textrm{INT}}}}^{} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )\chi ({{\textbf r^{\prime}}} ){\textbf E}({{\textbf r^{\prime}}} ){\textrm{d}^3}r^{\prime}} \;\;, \;\;{\textbf r} \in {{\mathbb R}^3}$$
where Einc(r) and E(r) are the incident and total electric fields, $\chi ({{\textbf r^{\prime}}} )\textrm{ = }\tilde{\varepsilon }({{\textbf r^{\prime}}} )- 1$ is the susceptibility of the medium, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )$ is the free-space dyadic Green’s function given by
$$\begin{aligned}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} ) &= \left( {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textrm I}} } + \frac{1}{{k_\textrm{h}^2}}\nabla \otimes \nabla } \right)\frac{{\exp ({\textrm{i}{k_\textrm{h}}|{{\textbf r} - {\textbf r^{\prime}}} |} )}}{{4\pi |{{\textbf r} - {\textbf r^{\prime}}} |}}\\ &\textrm{ = }\frac{{\exp ({\textrm{i}{k_\textrm{h}}r} )}}{{4\pi r}}\left[ {\left( {1 + \frac{{\textrm{i}{k_\textrm{h}}r - 1}}{{k_\textrm{h}^2{r^2}}}} \right){\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textrm I}} } + \left( {\frac{{\textrm{3} - 3\textrm{i}{k_\textrm{h}}r - k_\textrm{h}^2{r^2}}}{{k_\textrm{h}^2{r^2}}}} \right)\hat{{\textbf r}} \otimes \hat{{\textbf r}}} \right]\end{aligned}$$
where $r = |{{\textbf r} - {\textbf r^{\prime}}} |$, $\hat{{\textbf r}} = {{|{{\textbf r} - {\textbf r^{\prime}}} |} / r}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textrm I}} }$ is a 3×3 unit tensor. Note that the only difference of Eqs. (8) and (9) from those of non-absorbing host medium is that kh is now a complex number. The key to solving Eq. (8) is to obtain the internal electric fields of the scattering particle.

2.2 Solution of the VIE by DDA

In the spirit of DDA, the scattering object is discretized into a set of polarizable sub-volumes, i.e. electric dipoles. Then the VIE is transformed into a set of linear equations with unknown polarizations that need to be solved. We first note that the Green’s function ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )$ in VIE is singular when ${\textbf r} \to {\textbf r^{\prime}}$. To exclude the singularity of the Green’s function, the general form of VIE should be expressed as [43,47]

$${\textbf E}({\textbf r} )\textrm{ = }{{\textbf E}^{\textrm{inc}}}({\textbf r} )+ k_\textrm{h}^2\int\limits_{{V_{\textrm{INT}}}\backslash {V_0}} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )\chi ({{\textbf r^{\prime}}} ){\textbf E}({{\textbf r^{\prime}}} ){\textrm{d}^3}r^{\prime}} \textrm{ + }{\textbf M}({{V_0},{\textbf r}} )- {\textbf L}({\partial {V_0},{\textbf r}} )\chi ({\textbf r} ){\textbf E}({\textbf r} )$$
where V0 is the exclusion volume containing the singularity of ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )$, M(V0,r) is given by the following integral associated with the volume V0 [43,47]
$${\textbf M}({{V_0},{\textbf r}} )\textrm{ = }k_\textrm{h}^2\int_{{V_0}}^{} {{\textrm{d}^3}r^{\prime}\left[ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )\chi ({{\textbf r^{\prime}}} ){\textbf E}({{\textbf r^{\prime}}} )- {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }}_\textrm{s}}({{\textbf r},{\textbf r^{\prime}}} )\chi ({{\textbf r^{\prime}}} ){\textbf E}({{\textbf r^{\prime}}} )} \right]}$$
where
$${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }_\textrm{s}}({{\textbf r},{\textbf r^{\prime}}} )\textrm{ = }\frac{1}{{k_\textrm{h}^2}}\nabla \nabla \frac{1}{{4\pi |{{\textbf r} - {\textbf r^{\prime}}} |}}$$

${\textbf L}({\partial {V_0},{\textbf r}} )$ is the self-term dyadic that accounts for the depolarization of the excluded volume V0, which is given by [43]

$${\textbf L}({\partial {V_0},{\textbf r}} )={-} \oint_{\partial {V_0}} {{\textrm{d}^2}r^{\prime}\hat{{\textbf n}}\frac{{{\textbf r} - {\textbf r^{\prime}}}}{{4\pi {{|{{\textbf r} - {\textbf r^{\prime}}} |}^3}}}}$$
where $\hat{{\textbf n}}$ is an external normal to the surface of V0.

Discretizing the particle into a set of polarizable sub-volumes and assuming the size of each sub-volume is much smaller than the wavelength, E and χ in Eqs. (10) and (11) can be regarded as constant inside each sub-volume. Thus, Eq. (10) can be casted into the discretized form [43], i.e.

$${{\textbf E}_i} = {\textbf E}_i^{\textrm{inc}} + k_\textrm{h}^2\sum\limits_{j \ne i}^{} {{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }}_{ij}}{V_j}{\chi _j}{{\textbf E}_j}} + ({{{\textbf M}_i} - {{\textbf L}_i}} ){\chi _i}{{\textbf E}_i}$$
where ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }_{ij}}\textrm{ = }{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{{\textbf r}_i},{{\textbf r}_j}} )$, ${{\textbf L}_i}\textrm{ = }{\textbf L}({\partial {V_i},{{\textbf r}_i}} )$ and Mi is given by [43]
$${{\textbf M}_i}\textrm{ = }k_\textrm{h}^2\int_{{V_i}}^{} {{\textrm{d}^3}r^{\prime}\left[ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }({{\textbf r},{\textbf r^{\prime}}} )- {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }}_\textrm{s}}({{\textbf r},{\textbf r^{\prime}}} )} \right]}$$

Defining the polarizations of the sub-volumes as

$${{\textbf P}_i}\textrm{ = }{\varepsilon _\textrm{h}}{V_i}{\chi _i}{{\textbf E}_i}$$

Eq. (14) can be rewritten as:

$$\frac{1}{{{\varepsilon _\textrm{h}}}}\hat{\alpha }_i^{ - 1}{{\textbf P}_i} - \frac{{k_\textrm{h}^2}}{{{\varepsilon _\textrm{h}}}}\sum\limits_{j \ne i}^{} {{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }}_{ij}}{{\textbf P}_j}} = {\textbf E}_i^{\textrm{inc}}$$
where $\hat{\alpha }_i^{}$ is the so-called electric polarizability tensor defined by
$$\hat{\alpha }_i^{}\textrm{ = }{V_i}{\chi _i}{\left[ {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf I}} } - ({{{\textbf M}_i} - {{\textbf L}_i}} ){\chi_i}} \right]^{ - 1}}$$

Eqs. (17) and (18) are the main formulae of DDA, they represent a system of linear equations with unknown polarizations of sub-volumes that need to be solved. It is noted that the formulae of DDA show no difference from those of non-absorbing host medium except that the kh is now a complex number.

2.3 Dipole polarizability

Before solving the linear equations of DDA, one has to obtain the electric polarizabilities of the sub-volumes given by Eq. (18), the key to which is to determine the tensors Mi and Li. We first consider the self-term dyadic Li. It can be inspected from the integral in Eq. (13) that Li only depends on the geometry of the discretized sub-volume but has nothing relevant with the absorption of the host medium. We only consider cubic sub-volumes. According to previous study, the integral over a cubic sub-volume yields [48]

$${{\textbf L}_i}\textrm{ = }{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf I}} }} / 3}$$

The tensor Mi [Eq. (15)] has been studied by many researchers in the development of DDA for non-absorbing host medium. Various approximations of Mi have been proposed [43]. For absorbing host medium, the absorption of the medium is reflected by the complex wave number in Mi. Note that Mi tends to vanish with decreasing size of the sub-volume. If Mi is neglected, i.e. Mi = 0, we obtain the Clausius-Mossotti (CM) polarizability $\hat{\alpha }_i^{}\textrm{ = 3}{V_i}{{[{\tilde{\varepsilon }({\textbf r} )- 1} ]} / {[{\tilde{\varepsilon }({\textbf r} )+ 2} ]}}$. However, the CM polarizability does not satisfy the optical theorem [41,49]. Many methods have been proposed to correct $\hat{\alpha }_i^{}$ in the development of DDA [43], one of which is based on an exact evaluation of Mi for spherical sub-volumes in non-absorbing host medium [50]. For cubic sub-volumes, it is common place to implement the value of equivalent volume sphere [5153]

$${{\textbf M}_i}\textrm{ = }\frac{2}{3}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf I}} }[{({1\textrm{ - i}{k_\textrm{h}}a} )\exp ({\textrm{i}{k_\textrm{h}}a} )- 1} ]$$
where a = d(3/4π)1/3, d is the size of the cubical sub-volume. It is noted that the integral of Eq. (15) depends only on the geometry of the sub-volumes, thus Mi remains the same form for absorbing host medium except that kh is a complex number.

It is noted that the polarizability form has an important impact on the performance of DDA [43]. For non-absorbing host medium, the C-M polarizability can be corrected by considering the optical theorem of an electric dipole [41,49]. In addition, Draine and Goodman [54] bypassed the integral in Eq. (15) and derived the lattice dispersion relation polarizability, which is widely applied in current popular DDA programs. For absorbing host medium, however, these corrected forms of the polarizability need to be studied in future work. At current state, we only consider the polarizability based on the evaluation of Mi given by Eq. (20).

2.4 Solution of the linear equations

With the electric polarizabilities of the sub-volumes determined, the linear equations [Eq. (17)] need to be solved to obtain the unknown polarizations of the sub-volumes. In compact form, it reads

$${\bar{\textbf D}\bar{\textbf P}} + {\bar{\textbf A}\bar{\textbf P}} = {\varepsilon _\textrm{h}}{\bar{{\textbf E}}^{\textrm{inc}}}$$
where the elements of the matrices are ${\bar{{\textbf D}}_{ij}} = {\delta _{ij}}\hat{\alpha }_i^{ - 1}$, ${\bar{{\textbf A}}_{ij}} ={-} ({1 - {\delta_{ij}}} )k_\textrm{h}^2{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf G}} }_{ij}}$. Note that the incident field can be arbitrary. Einc in each sub-volume of the target is calculated first. For large number of discretized sub-volumes, the linear equations are typically solved by iterative method. Various iterative methods of DDA have been studied and implemented in previous works for non-absorbing host medium [43]. In this work, we implement the BICGSTAB(L) algorithm [5557]. In addition, the translational invariance nature of the Green’s function enables great reduction in the memory storage of matrix $\bar{{\textbf A}}$ [43]. Utilizing the Block-Toeplitz structure of matrix $\bar{{\textbf A}}$, 3D fast Fourier transform is implemented to accelerate the matrix-vector product in the iteration process [58]. After the polarizations of the sub-volumes being solved, the associated scattering quantities can be easily calculated.

2.5 Near-field scattering quantities

For EM scattering of particles in absorbing host medium, one important approach to the definition of the cross sections is to integrate the near-field EM fields over the particle surface, the associated quantities are called inherent scattering properties [1315]. In applications like solar energy utilization, photothermal therapy, etc., knowledge of the near-field EM fields and energy deposition around the particle is necessary [30]. In this sub-section, we show how DDA can be applied to obtain the near-field scattering quantities.

For any points in space, the total EM fields are the superposition of the incident and scattered fields, i.e.

$${\textbf E}({\textbf r} )\textrm{ = }{{\textbf E}^{\textrm{inc}}}({\textbf r} )+ {{\textbf E}^{\textrm{sca}}}({\textbf r} )$$
$${\textbf H}({\textbf r} )= {{\textbf H}^{\textrm{inc}}}({\textbf r} )+ {{\textbf H}^{\textrm{sca}}}({\textbf r} )$$

The total electric fields inside the particle can be easily obtained by Eq. (16). The scattered field at any point r outside the particle is calculated by

$${{\textbf E}^{\textrm{sca}}}({\textbf r} )= \frac{{k_\textrm{h}^2}}{{{\varepsilon _\textrm{h}}}}\sum\limits_{i = 1}^N {{\textbf G}({{\textbf r},{{\textbf r}_i}} ){{\textbf P}_i}}$$

3D fast Fourier transfer can be used to accelerate the calculation of the electric fields on an extended lattice [59]. The relevant magnetic field can then be obtained by the curl equations.

The EM power absorbed by the particle equals to the net rate at which the EM energy crosses the enclosure surface A of the particle, i.e.

$${W^{\textrm{abs}}} ={-} \oint_A {{\textbf S}({\textbf r} )\cdot \hat{{\textbf n}}\textrm{d}A}$$
where
$${\textbf S}({\textbf r} )= \frac{1}{2}{\textrm{Re}} [{{\textbf E}({\textbf r} )\times {\textbf H}{{({\textbf r} )}^ \ast }} ]$$
is the time-averaged Poynting vector, $\hat{{\textbf n}}$ denotes the unit vector normal to the particle surface A. It is common place to separate the Poynting vector into its incident and scattered components. Inserting Eqs. (22) and (23) into Eqs. (25) and (26) yields the well-known relation
$${W^{\textrm{ext}}} = {W^{\textrm{abs}}}\textrm{ + }{W^{\textrm{sca}}} - {W^{\textrm{inc}}}$$

Wsca is the EM power scattered out of surface A of the particle, which is given by

$${W^{\textrm{sca}}} = \oint_A {{{\textbf S}^{\textrm{sca}}}({\textbf r} )\cdot \hat{{\textbf n}}\textrm{d}A} \textrm{ = }\oint_A {\frac{1}{2}{\textrm{Re}} [{{{\textbf E}^{\textrm{sca}}}({\textbf r} )\times {{\textbf H}^{\textrm{sca}}}{{({\textbf r} )}^ \ast }} ]\cdot \hat{{\textbf n}}\textrm{d}A}$$

Winc is the absorption of the incident wave in a host medium volume enclosed by A, which is given by

$${W^{\textrm{inc}}} ={-} \oint_A {{{\textbf S}^{\textrm{inc}}}({\textbf r} )\cdot \hat{{\textbf n}}\textrm{d}A} \textrm{ = } - \oint_A {\frac{1}{2}{\textrm{Re}} [{{{\textbf E}^{\textrm{inc}}}({\textbf r} )\times {{\textbf H}^{\textrm{inc}}}{{({\textbf r} )}^ \ast }} ]\cdot \hat{{\textbf n}}\textrm{d}A}$$
and Wext is the net power flow crossing surface A due to the interaction between the incident and the scattered fields, which is given by
$${W^{\textrm{ext}}} ={-} \oint_A {{{\textbf S}^{\textrm{ext}}}({\textbf r} )\cdot \hat{{\textbf n}}\textrm{d}A} \textrm{ = } - \oint_A {\frac{1}{2}{\textrm{Re}} [{{{\textbf E}^{\textrm{inc}}}({\textbf r} )\times {{\textbf H}^{\textrm{sca}}}{{({\textbf r} )}^ \ast }\textrm{ + }{{\textbf E}^{\textrm{sca}}}({\textbf r} )\times {{\textbf H}^{\textrm{inc}}}{{({\textbf r} )}^ \ast }} ]\cdot \hat{{\textbf n}}\textrm{d}A}$$

To calculate the above powers by DDA, it is convenient to transform the surface integrals into volume integrals using the divergence theorem, i.e.

$$\oint_A {{\textbf S}({\textbf r} )\cdot \hat{{\textbf n}}\textrm{d}A} \textrm{ = }\int_V {\nabla \cdot {\textbf S}({\textbf r} )} \textrm{d}V$$

$\nabla \cdot {\textbf S}({\textbf r} )$ can be viewed as the power absorption per unit volume at position r. Thus, Eq. (27) is transformed into the relation

$$\nabla \cdot {\textbf S}({\textbf r} )\textrm{ = }\nabla \cdot {{\textbf S}^{\textrm{sca}}}({\textbf r} )\textrm{ + }\nabla \cdot {{\textbf S}^{\textrm{ext}}}({\textbf r} )\textrm{ + }\nabla \cdot {{\textbf S}^{\textrm{inc}}}({\textbf r} )$$

In DDA, we first discretize the target volume into a set of sub-volumes, then the volume integration over the target is transformed into the summation of the divergence of the corresponding Poynting vector in each sub-volume. Using Eqs. (1), (2), and (25)–(32), we obtain

$$\nabla \cdot {\textbf S}({\textbf r} )\textrm{ = } - \frac{\omega }{2}{|{{\textbf E}({\textbf r} )} |^2}{\textrm{Im}}[{\varepsilon_\textrm{p}^{}({\textbf r} )} ]\;,\;{\textbf r} \in {V_{\textrm{INT}}}$$

The power absorption by the particle can be calculated by

$$W_{}^{\textrm{abs}}\textrm{ = }\frac{\omega }{2}\int_{{V_\textrm{p}}} {{{|{{\textbf E}({\textbf r} )} |}^2}{\textrm{Im}}[{\varepsilon_\textrm{p}^{}({\textbf r} )} ]\textrm{d}V} \textrm{ = }\frac{\omega }{2}\sum\limits_{i = 1}^N {{{|{{{\textbf E}_i}} |}^2}{\textrm{Im}}({\varepsilon_{\textrm{p},i}^{}} ){V_i}}$$
where Ei is the total electric field in each sub-volume. Similarly, for the incident wave we have
$$\nabla \cdot {\textbf S}_{}^{\textrm{inc}}({\textbf r} )\textrm{ = } - \frac{\omega }{2}{|{{{\textbf E}^{\textrm{inc}}}({\textbf r} )} |^2}{\textrm{Im}}({\varepsilon_\textrm{h}^{}} )\;,\;{\textbf r} \in {V_{\textrm{INT}}}$$

The power absorption by the host medium in a volume corresponding to the particle is calculated by

$$W_{}^{\textrm{inc}}\textrm{ = }\frac{\omega }{2}\sum\limits_{i = 1}^N {{{|{{\textbf E}_i^{\textrm{inc}}} |}^2}\textrm{Im}({\varepsilon_\textrm{h}^{}} ){V_i}}$$

$\nabla \cdot {\textbf S}_{}^{\textrm{ext}}$, after some algebra, can be expressed as

$$\nabla \cdot {\textbf S}_{}^{\textrm{ext}}\textrm{ = }\frac{\omega }{2}{\mathop{\rm Im}\nolimits} [{{\textbf E}_{}^{\textrm{inc}}({\textbf r} )\cdot {{\textbf p}^ \ast }({\textbf r} )} ]+ \omega {\mathop{\rm Im}\nolimits} ({\varepsilon_\textrm{h}^ \ast } ){\textrm{Re}} [{{\textbf E}_{}^{\textrm{inc}}({\textbf r} )\cdot {\textbf E}_{}^ \ast ({\textbf r} )- {{|{{\textbf E}_{}^{\textrm{inc}}({\textbf r} )} |}^2}} ]\;,\;{\textbf r} \in {V_{\textrm{INT}}}$$

Thus, Wext by the particle is calculated in discretized form as

$$W_{}^{\textrm{ext}}\textrm{ = }\frac{\omega }{2}\sum\limits_{i = 1}^N {{\mathop{\rm Im}\nolimits} ({{{\textbf p}_i} \cdot {\textbf E}{{_i^{\textrm{inc}}}^ \ast }} )} {V_i} + \omega \sum\limits_{i = 1}^N {{\mathop{\rm Im}\nolimits} ({\varepsilon_\textrm{h}^ \ast } ){\textrm{Re}} ({{{|{{\textbf E}_i^{\textrm{inc}}} |}^2} - {\textbf E}_i^{\textrm{inc}} \cdot {\textbf E}_i^ \ast } )} {V_i}$$

Note that for non-absorbing host medium, the second term in Eq. (38) disappears, we obtain the formula that is frequently used in DDA for non-absorbing host medium. With the above power quantities calculated, the scattered power out of the particle surface Wsca can be calculated via the relation given by Eq. (27).

2.5.1 Inherent scattering, absorption, and extinction cross sections

According to the definition of the inherent scattering properties [14], the inherent absorption power $W_{\textrm{inh} }^\textrm{a}$ is calculated by Eq. (34); the inherent scattered power is calculated by

$$W_{\textrm{inh} }^{\textrm{sca}} = {W^{\textrm{sca}}} = {W^{\textrm{ext}}}\textrm{ + }{W^{\textrm{inc}}} - {W^{\textrm{abs}}}$$
and the inherent extinction power is the sum of the inherent absorption and scattering, i.e.
$$W_{\textrm{inh} }^{\textrm{ext}} = W_{\textrm{inh} }^{\textrm{sca}}\textrm{ + }W_{\textrm{inh} }^{\textrm{abs}}\textrm{ = }W_{}^{\textrm{ext}}\textrm{ + }W_{}^{\textrm{inc}}$$

To define the inherent cross sections, we have to divide the corresponding power by the intercepted power of the incident energy. For instance, the inherent absorption cross section $C_{\textrm{inh}}^{\textrm{abs}}$ can be defined as

$$C_{\textrm{inh} }^{\textrm{abs}}\textrm{ = }{{W_{\textrm{inh}}^{\textrm{abs}}} / {I_{}^{\textrm{inc}}}}$$
the intercepted power Iinc is calculated by [5]
$${I^{\textrm{inc}}} = {{2I_0^{\textrm{inc}}[{1 + ({\eta - 1} ){e^\eta }} ]} / {{\eta ^2}}}$$
where $\eta = 2{k^{\prime\prime}_\textrm{h}}{r_{\textrm{eff}}}$, and $I_0^{\textrm{inc}}\textrm{ = }{{{{k^{\prime}}_\textrm{h}}{{|{{{\textbf E}_0}} |}^2}} / {({2{\mu_0}\omega } )}}$ is the incident intensity at origin. Since the host medium is absorbing, Iinc considers the variation of the wave intensity on the half spherical surface facing the incident wave. For nonspherical particles, especially concave particles, it is difficult to accurately compute the intercepted power. Since the size of nonspherical particles is frequently measured by the equivalent volume or surface sphere, it is reasonable to choose reff as the radius of the equivalent volume or surface sphere.

2.5.2 Energy deposition near the particle

In absorbing host medium, the presence of the particle also affects the power absorption and scattering by the host medium near the particle. The divergence of the Poynting vectors at the host medium outside the particle is

$$\nabla \cdot {{\textbf S}_\textrm{h}}({\textbf r} )\textrm{ = } - \frac{\omega }{2}{|{{\textbf E}({\textbf r} )} |^2}\textrm{Im}({\varepsilon_\textrm{h}^{}} )\;,\;{\textbf r} \in {V_{\textrm{EXT}}}$$
where E(r)=Esca(r)+Einc(r). Similarly, the divergence of the incident wave is
$$\nabla \cdot {\textbf S}_\textrm{h}^{\textrm{inc}}({\textbf r} )\textrm{ = } - \frac{\omega }{2}{|{{{\textbf E}^{\textrm{inc}}}({\textbf r} )} |^2}\textrm{Im}({\varepsilon_\textrm{h}^{}} )\;,\;{\textbf r} \in {V_{\textrm{EXT}}}$$
and that of the scattered wave is
$$\nabla \cdot {\textbf S}_\textrm{h}^{\textrm{sca}}({\textbf r} )\textrm{ = } - \frac{\omega }{2}{|{{{\textbf E}^{\textrm{sca}}}({\textbf r} )} |^2}\textrm{Im}({\varepsilon_\textrm{h}^{}} )\;,\;{\textbf r} \in {V_{\textrm{EXT}}}$$

$\nabla \cdot {\textbf S}_\textrm{h}^{\textrm{ext}}({\textbf r} )$ can be calculated by the relation of Eq. (32) with the above quantities determined. For a given volume of the host medium outside the particle, we can discretize the volume into sub-volumes and calculate the various power components in discretized form as Eq. (34).

2.6 Far-field scattering quantities

In this sub-section, we show how the far-field scattering quantities (i.e. the apparent scattering quantities) introduced by Mishchenko [25] can be calculated by DDA. The far-field scattering quantities are defined in such a way that they either theoretically model what a far-field detector reads or quantify the EM energy budget of a finite volume [36] such that the scattering quantities can be applied with the radiative transfer equation.

In the far-field, the outgoing scattered waves become spherical waves, the electric field of which can be approximated as [25]

$${{\textbf E}^{\textrm{sca}}}({\textbf r} )\approx \frac{{\exp ({\textrm{i}{k_\textrm{h}}r} )}}{r}{\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )= \exp ({ - {{k^{\prime\prime}}_\textrm{h}}r} )\frac{{\exp ({\textrm{i}{{k^{\prime}}_\textrm{h}}r} )}}{r}{\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )$$

${\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )$ is the distance-independent amplitude of the transverse outgoing spherical wave given by [25]

$${\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )\textrm{ = }\frac{{k_\textrm{h}^2}}{{4\pi }}\left( {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf I}} } - {{\hat{{\textbf r}}}^{\textrm{sca}}} \otimes {{\hat{{\textbf r}}}^{\textrm{sca}}}} \right)\int_{{V_{\textrm{INT}}}} {\textrm{d}{\textbf r^{\prime}}\chi ({{\textbf r^{\prime}}} ){\textbf E}({{\textbf r^{\prime}}} )\exp ({ - \textrm{i}{k_\textrm{h}}{{\hat{{\textbf r}}}^{\textrm{sca}}} \cdot {\textbf r^{\prime}}} )}$$
where ${\hat{{\textbf r}}^{\textrm{sca}}}$ denotes the propagation direction of the outgoing scattered wave. With the polarization of the discretized sub-volumes being determined, ${\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )$ can be calculated by
$${\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )\textrm{ = }\frac{{k_\textrm{h}^2}}{{4\pi {\varepsilon _\textrm{h}}}}\left( {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf I}} } - {{\hat{{\textbf r}}}^{\textrm{sca}}} \otimes {{\hat{{\textbf r}}}^{\textrm{sca}}}} \right)\sum\limits_{i = 1}^N {{{\textbf P}_i}\exp ({ - \textrm{i}{k_\textrm{h}}{{\hat{{\textbf r}}}^{\textrm{sca}}} \cdot {{\textbf r}_i}} )}$$

Note that the effects of the particle morphology, particle material and host medium absorption are all contained in the solved polarizations of the sub-volumes.

We consider a homogeneous plane incident wave given by

$${{\textbf E}^{\textrm{inc}}}({\textbf r} )= {\textbf E}_0^{\textrm{inc}}\exp ({\textrm{i}{k_\textrm{h}}{{\hat{{\textbf r}}}^{\textrm{inc}}} \cdot {\textbf r}} )$$
where ${\textbf E}_0^{\textrm{inc}}$ is the incident electric field at origin of the laboratory coordinate system, ${\hat{{\textbf r}}^{\textrm{inc}}}$ is a unit vector in the incident direction. ${\textbf E}_1^{\textrm{sca}}$ and ${\textbf E}_0^{\textrm{inc}}$ can be related by the 3×3 scattering dyadic ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf A}} }$ [25], i.e.
$${\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )= {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf A}} }({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )\cdot {\textbf E}_0^{\textrm{inc}}$$

In spherical coordinate systems, only four out of nine elements of ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf A}} }$ are independent [25]. Thus, we can introduce the 2×2 amplitude scattering matrix S that relates the θ- and φ-components (see Fig. 1) of the far-field scattering electric field to those of the incident electric field [25], i.e.

$$\left[ {\begin{array}{c} {{\textbf E}_\theta^{\textrm{sca}}({r{{\hat{{\textbf r}}}^{\textrm{sca}}}} )}\\ {{\textbf E}_\varphi^{\textrm{sca}}({r{{\hat{{\textbf r}}}^{\textrm{sca}}}} )} \end{array}} \right]\textrm{ = }\frac{{\exp ({\textrm{i}{k_\textrm{h}}r} )}}{r}{\textbf S}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )\left[ {\begin{array}{c} {{\textbf E}_{0\theta }^{\textrm{inc}}({{{\hat{{\textbf r}}}^{\textrm{inc}}}} )}\\ {{\textbf E}_{0\varphi }^{\textrm{inc}}({{{\hat{{\textbf r}}}^{\textrm{inc}}}} )} \end{array}} \right]$$

The elements of the amplitude scattering matrix have dimension of length; its relation with ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\textbf A}} }$ can be found in Ref. [25].

 figure: Fig. 1.

Fig. 1. Schematic of far-field EM scattering by a fixed particle.

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In practical applications, the far-field observables are typically expressed by the Stokes parameters I = [I, Q, U, V]T. The Stokes parameters of the scattered and incident waves are related by the 4×4 scattering matrix ${\textbf F}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )$ [25,36]

$${{\textbf I}^{\textrm{sca}}}\textrm{ = }\frac{{\exp ({ - 2{{k^{\prime\prime}}_\textrm{h}}r} )}}{{{r^2}}}{\textbf F}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} ){{\textbf I}^{\textrm{inc}}}$$

The elements of ${\textbf F}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )$ have dimension of area, which can be calculated by the elements of the amplitude scattering matrix (see Refs. [25,36] for calculation details).

Utilizing the distance-independent amplitude of the transverse outgoing spherical wave ${\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )$, the effective scattering cross section can be defined as

$$C_{\textrm{eff}}^{\textrm{sca}} = \frac{1}{{{{|{{\textbf E}_0^{\textrm{inc}}} |}^2}}}\int_{4\pi }^{} {\textrm{d}\Omega {{|{{\textbf E}_1^{\textrm{sca}}({{{\hat{{\textbf r}}}^{\textrm{sca}}}} )} |}^2}}$$

Then, the normalized scattering matrix is defined as [36]

$$\tilde{{\textbf F}}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )\textrm{ = }\frac{{4\pi }}{{C_{\textrm{eff}}^{\textrm{sca}}}}{\textbf F}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )$$

The elements of $\tilde{{\textbf F}}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )$ is dimensionless. The (1,1) element of $\tilde{{\textbf F}}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )$ is referred to as the phase function that satisfies the normalization condition

$$\frac{1}{{4\pi }}\int_0^{4\pi } {\textrm{d}\Omega {{\tilde{F}}_{11}}({{{\hat{{\textbf r}}}^{\textrm{sca}}},{{\hat{{\textbf r}}}^{\textrm{inc}}}} )} = 1$$

According to derivation by Mishchenko [25], the apparent extinction cross section for unpolarized incident wave can be determined by the amplitude scattering matrix elements

$${C^{\textrm{ext}}} = \frac{{2\pi }}{{k^{\prime}_\textrm{h}}}{\mathop{\rm Im}\nolimits} [{{S_{11}}(0 )+ {S_{22}}(0 )} ]$$
where 0 denotes the forward scattering direction. To obtain the amplitude scattering matrix S, it is convenient to consider two linear polarizations of the incident wave at origin of the laboratory coordinate system. In our DDA program, we assume ${\hat{{\textbf r}}^{\textrm{inc}}}\textrm{ = }\hat{{\textbf x}}$ (See Fig. 1) and consider $\hat{{\textbf y}}$ and $\hat{{\textbf z}}$ linear polarizations of incident wave.

3. Validation of DDA for absorbing host medium

We validate the DDA method from two aspects. First, we compare the apparent scattering quantities between DDA and the Lorenz-Mie theory for EM scattering of spherical particles in absorbing host medium proposed by Mishchenko and Yang [36]. Second, we compare the inherent scattering quantities between DDA and Ref. [14].

Figure 2 compares the apparent extinction efficiency of spherical particles calculated by DDA and the far-field Lorenz-Mie theory [36]. The apparent extinction efficiency is defined as Qext=Cext/(πR2). In DDA, Cext is calculated by Eq. (56). The refractive index of the particle is assumed to be mp=1.59 while that of the host medium is ${m_\textrm{h}}\textrm{ = 1}\textrm{.33 + i}m_\textrm{h}^{{{\prime\prime}}}$. As shown, DDA can well predict the extinction efficiency for x<20 for all the $m_\textrm{h}^{{{\prime\prime}}}$ considered. For x>20, however, the performance of DDA tends to deteriorate with increasing x and $m_\textrm{h}^{{{\prime\prime}}}$. At the largest absorption of the host medium, DDA tends underestimate Qext for x approaching 30 but can still predict the interference structure of Qext. We also increase the total number of the discretized sub-volumes for x>20 in DDA. Clearly, the performance of DDA is improved. Figure 3 further compares the scattering phase function calculated by DDA and the Lorenz-Mie theory [36]. As shown, DDA can predict very well the phase function for all the cases considered, although there are some discrepancies between DDA and Lorenz-Mie theory at backscattering angles for the largest absorption of the host medium.

 figure: Fig. 2.

Fig. 2. Apparent extinction efficiency Qext of spherical particle as a function of size parameter x: comparison between DDA and the far-field Lorenz-Mie theory for absorbing host medium [36]. The refractive index of the host medium is ${m_\textrm{h}}\textrm{ = 1}\textrm{.33 + i}m_\textrm{h}^{{{\prime\prime}}}$, the imaginary part is varied among 0.0, 0.01, 0.02 and 0.05; the refractive index of the particle is mp=1.59. N is the number of the discretized sub-volumes.

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 figure: Fig. 3.

Fig. 3. Scattering phase function p(θ) of spherical particle: comparison between DDA and the far-field Lorenz-Mie theory for absorbing host medium [36]. All conditions are the same to Fig. 2 except that $m_\textrm{h}^{{{\prime\prime}}}$ is chosen to be 0.0, 0.05 and x is chosen to be 10, 20.

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Figure 4 compares the inherent extinction and scattering efficiencies of spherical particles between DDA and the reference data. In this case, the refractive indices of the particle and the host medium are mp=1.4 + 0.05i and ${m_\textrm{h}}\textrm{ = 1}\textrm{.2 + i}m_\textrm{h}^{{{\prime\prime}}}$. As shown, DDA can well capture the inherent extinction efficiency for all the cases considered. As to the inherent scattering efficiency, DDA tends to underestimate Qsca with increasing x, which is more obvious for smaller absorption of the host medium.

 figure: Fig. 4.

Fig. 4. Inherent extinction and scattering efficiencies of a spherical particle as a function of size parameter x: comparison between the data extracted from Fig. 4 of Ref. [14] and DDA. The number of discretized sub-volumes is N=268096.

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4. Numerical examples on nonspherical particles

In this section, we give some numerical examples on the apparent and inherent extinction efficiencies of sphere and spheroids. The refractive indices of the particle and the host medium are assumed to be mp=1.59 and ${m_\textrm{h}}\textrm{ = 1}\textrm{.33 + i}m_\textrm{h}^{{{\prime\prime}}}$. We consider particles at fixed orientation, the incident wave is applied along + x direction (see Fig. 5). The prolate spheroid has an axis ratio of 2 with its major axis parallel to the x axis while the oblate spheroid has an axis ratio of 0.5. The size parameter of the spheroids is defined using the equivalent volume sphere radius. For x<20, the numbers of the discretized sub-volumes are 226088 and 231912 for prolate and oblate spheroids, while for x>20 the numbers of sub-volumes are 536122 and 549312.

 figure: Fig. 5.

Fig. 5. The apparent extinction efficiency Qext of the sphere, prolate and oblate spheroids as a function of size parameter x for different host medium absorption. The step of x is 0.2. Qext of the sphere is calculated by the far-field Lorenz-Mie theory [36] while those of the spheroids are calculated by DDA.

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Figure 5 shows the apparent extinction efficiency Qext of the sphere, prolate and oblate spheroids as a function of x for different $m_\textrm{h}^{{{\prime\prime}}}$ of the host medium. Qext of the sphere is calculated by the far-field Lorenz-Mie theory [36] while those of the spheroids are calculated by DDA. As shown in Fig. 5, Qext exhibits oscillating structure with increasing x for all the three particles. The longer the particle extends along the incident direction, the more oscillations it has. The oscillating structure can be interpreted as the interference between the EM waves transmitted and diffracted by the particle. For the case of $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.001}$, which shows very small differences from those of non-absorbing host medium, the amplitudes of the oscillations decrease with increasing x. Qext approaches 2 with increasing x, which is known as the extinction paradox [1]. However, the amplitudes of the oscillations of all the three particles increase with increasing absorption of the host medium, which is most remarkable for the prolate spheroid. Since the particles are non-absorbing, the transmitted waves is not attenuated over the pathlength of the particle [28], therefore causes growing amplitude of the interference structure with increasing host medium absorption. As a result, the oscillations of Qext of the prolate spheroid increases dramatically for the largest $m_\textrm{h}^{{{\prime\prime}}}$. A parallel effect of increasing host medium absorption is the negative extinction. A comparison among Qext of the three particles shows that prolate spheroid is more likely to have negative extinction. The oblate spheroid, however, does not exhibit noticeable negative extinction for all the cases considered. Thus, it can be inferred that negative extinction is more likely to occur for particles that are less absorptive than the host medium but are geometrically more prolonged in the incident wave direction. In addition, ripple structures can be observed for the sphere, which are attenuated with increasing host medium absorption. However, ripple structures are not notably observed for the spheroids.

Figure 6 shows the same as Fig. 5 but for the inherent extinction efficiency. The oscillating structure can also be observed for the inherent Qext, which shows very small difference from the apparent Qext for small absorption of the host medium. However, the amplitudes of the oscillations decrease with increasing host medium absorption, and Qext gradually approaches unity with increasing particle size, which is similar to those observed in literature [13,14]. A special phenomenon is that Qext exhibits a large increase for the largest host medium absorption and x>20.

 figure: Fig. 6.

Fig. 6. Same as Fig. 5 but for the inherent extinction efficiency; all the results are calculated by DDA.

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For a deeper understanding of the phenomena observed in Figs. 5 and 6, we depict in Fig. 7 the x-y plane magnitude of the total electric fields near and inside the sphere, the prolate and oblate spheroids, which are calculated by DDA. As shown, the non-absorbing particles serve as optical lenses, focusing EM waves in the forward direction due to the near-field interaction between transmitted waves and diffracted waves through the particles. For the case of $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.001}$, the focus point of the oblate spheroid is located relatively farther from the particle, whereas that of the prolate particle locates close to or even inside the particle. For the case of $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.05}$, however, the absorption of the host medium greatly suppresses the near-field forward scattering intensity of the sphere and the oblate spheroid. On the contrary, the electric field intensity inside and in the near front of the prolate spheroid becomes stronger, which supports the negative apparent extinction and the local minimum of the inherent extinction at x=10 observed in Fig. 5 and Fig. 6, respectively. The non-absorbing prolate spheroid can reserve more energy in the forward direction, thus is more likely to produce negative apparent extinction. In addition, it can be induced that the effect of host medium absorption on the near-field scattering properties highly depends on the particle morphology.

 figure: Fig. 7.

Fig. 7. The magnitude of the total electric field Etot inside and near the particle in the x-y plane, the first row is for imaginary part of the host medium refractive index of $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.001}$, the second row is for $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.05}$. From left to right are for the sphere, prolate and oblate spheroids, respectively. The size parameter is x=10. The incident field is y-polarized and its magnitude at origin is assumed unity. The x and y coordinates denote the position of the discretized sub-volumes in DDA.

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5. Conclusion

In conclusion, we have developed the DDA method for EM scattering by an arbitrary particle immersed in an absorbing host medium. We have elaborated how the near- and far-field scattering quantities can be calculated by DDA. The accuracy of DDA is examined and validated by comparison with the far-field Lorenz-Mie theory and the inherent scattering quantities in literature. DDA can well predict the scattering properties of particles in absorbing host medium, although its performance tends to deteriorate for very large size parameter and host medium absorption. Numerical examples of DDA show that particles that are less absorptive than the host medium and extend longer in the incident direction are more likely to produce negative apparent extinction, which makes it easier for an experimental observation. The region map of the near-field electric field shows that the prolate spheroid can concentrate more EM energy in the forward direction, serving as an evidence of the negative extinction.

The DDA method we develop will be useful and flexible to study both the far- and near-field scattering by particles in absorbing host medium. However, many aspects of the DDA method still need to be investigated. For instance, the convergence and accuracy should be tested for more cases with respect to the refractive indices of both the medium and the particle, the iteration method, and the form of the electric polarizability.

Funding

National Natural Science Foundation of China (51906128, 52076123); Natural Science Foundation of Shandong Province (ZR2019BEE011); China Postdoctoral Science Foundation.

Disclosures

The authors declare no conflicts of interest.

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51. J. I. Hage and J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990). [CrossRef]  

52. J. I. Hage, J. M. Greenberg, and R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30(9), 1141–1152 (1991). [CrossRef]  

53. A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98(6), 699 (1993). [CrossRef]  

54. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti-Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993). [CrossRef]  

55. G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Transactions on Numerical Analysis 1, 11–32 (1993).

56. G. L. G. Sleijpen and H. A. Van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algor. 10(2), 203–223 (1995). [CrossRef]  

57. B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT 7.3,” https://arxiv.org/ 1305, 6497 (2013).

58. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16(15), 1198–1200 (1991). [CrossRef]  

59. P. J. Flatau and B. T. Draine, “Fast near field calculations in the discrete dipole approximation for regular rectilinear grids,” Opt. Express 20(2), 1247–1252 (2012). [CrossRef]  

References

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  1. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).
  2. M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).
  3. H. C. van de Hulst, Light scattering by small particles (John Wiley & Sons, 1957).
  4. M. Wendisch and P. Yang, Theory of atmospheric radiative transfer: a comprehensive introduction (John Wiley & Sons, 2012).
  5. W. C. Mundy, J. A. Roux, and A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64(12), 1593–1597 (1974).
    [Crossref]
  6. P. Chýlekt, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67(4), 561–563 (1977).
    [Crossref]
  7. C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
    [Crossref]
  8. K. Kamiuto, “Near-field scattering by a large spherical particle embedded in a nonabsorbing medium,” J. Opt. Soc. Am. 73(12), 1819–1822 (1983).
    [Crossref]
  9. P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “A note on the definition of scattering cross sections and phase functions for spheres immersed in an absorbing medium,” Wave Random Complex 3(3), 147–156 (1993).
    [Crossref]
  10. M. Quinten and J. Rostalski, “Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13(2), 89–96 (1996).
    [Crossref]
  11. A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
    [Crossref]
  12. I. W. Sudiarta and P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 709–714 (2001).
    [Crossref]
  13. I. W. Sudiarta and P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. 18(6), 1275–1278 (2001).
    [Crossref]
  14. Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
    [Crossref]
  15. P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, and S.-C. Tsay, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41(15), 2740–2759 (2002).
    [Crossref]
  16. G. Videen and W. Sun, “Yet another look at light scattering from particles in absorbing media,” Appl. Opt. 42(33), 6724–6727 (2003).
    [Crossref]
  17. W. Sun, N. G. Loeb, and Q. Fu, “Light scattering by coated sphere immersed in absorbing medium: a comparison between the FDTD and analytic solutions,” J. Quant. Spectrosc. Radiat. Transfer 83(3-4), 483–492 (2004).
    [Crossref]
  18. W. Sun, N. G. Loeb, and Q. Fu, “Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41(27), 5728–5743 (2002).
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  19. W. Sun, N. G. Loeb, and B. Lin, “Light scattering by an infinite circular cylinder immersed in an absorbing medium,” Appl. Opt. 44(12), 2338–2342 (2005).
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  20. Q. Fu and W. Sun, “Apparent optical properties of spherical particles in absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 100(1-3), 137–142 (2006).
    [Crossref]
  21. J. Randrianalisoa, D. Baillis, and L. Pilon, “Modeling radiation characteristics of semitransparent media containing bubbles or particles,” J. Opt. Soc. Am. 23(7), 1645–1656 (2006).
    [Crossref]
  22. J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. 23(11), 2784–2796 (2006).
    [Crossref]
  23. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
    [Crossref]
  24. S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
    [Crossref]
  25. M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007).
    [Crossref]
  26. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008).
    [Crossref]
  27. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109(14), 2386–2390 (2008).
    [Crossref]
  28. M. I. Mishchenko and J. M. Dlugach, “Scattering and extinction by spherical particles immersed in an absorbing host medium,” J. Quant. Spectrosc. Radiat. Transfer 211, 179–187 (2018).
    [Crossref]
  29. M. I. Mishchenko and J. M. Dlugach, “Plasmon resonances of metal nanoparticles in an absorbing medium,” OSA Continuum 2(12), 3415–3421 (2019).
    [Crossref]
  30. R. K. Harrison and A. Ben-Yakar, “Point-by-point near-field optical energy deposition around plasmonic nanospheres in absorbing media,” J. Opt. Soc. Am. 32(8), 1523–1535 (2015).
    [Crossref]
  31. Y. Ivanenko, M. Gustafsson, and S. Nordebo, “Optical theorems and physical bounds on absorption in lossy media,” Opt. Express 27(23), 34323–34342 (2019).
    [Crossref]
  32. R. L. Peck, A. G. Brolo, and R. Gordon, “Absorption leads to narrower plasmonic resonances,” J. Opt. Soc. Am. B 36(8), F117–F122 (2019).
    [Crossref]
  33. S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
    [Crossref]
  34. F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. 33(5), 947–953 (2016).
    [Crossref]
  35. J. R. Frisvad, “Phase function of a spherical particle when scattering an inhomogeneous electromagnetic plane wave,” J. Opt. Soc. Am. 35(4), 669–680 (2018).
    [Crossref]
  36. M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
    [Crossref]
  37. M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transfer 214, 158–167 (2018).
    [Crossref]
  38. A. E. Moskalensky and M. A. Yurkin, “Energy budget and optical theorem for scattering of source-induced fields,” Phys. Rev. A 99(5), 053824 (2019).
    [Crossref]
  39. M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
    [Crossref]
  40. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [Crossref]
  41. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  42. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. 11(4), 1491–1499 (1994).
    [Crossref]
  43. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 558–589 (2007).
    [Crossref]
  44. P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70(3), 036606 (2004).
    [Crossref]
  45. P. C. Chaumet, T. Zhang, and A. Sentenac, “Fast far-field calculation in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transfer 165, 88–92 (2015).
    [Crossref]
  46. M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering (Cambridge University, 2006).
  47. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 775 (2003).
    [Crossref]
  48. A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
    [Crossref]
  49. S. Albaladejo, R. Gómez-Medina, L. S. Froufe-Pérez, H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18(4), 3556–3567 (2010).
    [Crossref]
  50. D. E. Livesay and K.-M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. 22(12), 1273–1280 (1974).
    [Crossref]
  51. J. I. Hage and J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
    [Crossref]
  52. J. I. Hage, J. M. Greenberg, and R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30(9), 1141–1152 (1991).
    [Crossref]
  53. A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98(6), 699 (1993).
    [Crossref]
  54. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti-Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [Crossref]
  55. G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Transactions on Numerical Analysis 1, 11–32 (1993).
  56. G. L. G. Sleijpen and H. A. Van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algor. 10(2), 203–223 (1995).
    [Crossref]
  57. B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT 7.3,” https://arxiv.org/ 1305, 6497 (2013).
  58. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16(15), 1198–1200 (1991).
    [Crossref]
  59. P. J. Flatau and B. T. Draine, “Fast near field calculations in the discrete dipole approximation for regular rectilinear grids,” Opt. Express 20(2), 1247–1252 (2012).
    [Crossref]

2019 (5)

2018 (5)

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

M. I. Mishchenko and J. M. Dlugach, “Scattering and extinction by spherical particles immersed in an absorbing host medium,” J. Quant. Spectrosc. Radiat. Transfer 211, 179–187 (2018).
[Crossref]

J. R. Frisvad, “Phase function of a spherical particle when scattering an inhomogeneous electromagnetic plane wave,” J. Opt. Soc. Am. 35(4), 669–680 (2018).
[Crossref]

M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transfer 214, 158–167 (2018).
[Crossref]

2016 (1)

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. 33(5), 947–953 (2016).
[Crossref]

2015 (2)

R. K. Harrison and A. Ben-Yakar, “Point-by-point near-field optical energy deposition around plasmonic nanospheres in absorbing media,” J. Opt. Soc. Am. 32(8), 1523–1535 (2015).
[Crossref]

P. C. Chaumet, T. Zhang, and A. Sentenac, “Fast far-field calculation in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transfer 165, 88–92 (2015).
[Crossref]

2012 (1)

2010 (1)

2008 (2)

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008).
[Crossref]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109(14), 2386–2390 (2008).
[Crossref]

2007 (4)

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
[Crossref]

M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 558–589 (2007).
[Crossref]

2006 (3)

Q. Fu and W. Sun, “Apparent optical properties of spherical particles in absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 100(1-3), 137–142 (2006).
[Crossref]

J. Randrianalisoa, D. Baillis, and L. Pilon, “Modeling radiation characteristics of semitransparent media containing bubbles or particles,” J. Opt. Soc. Am. 23(7), 1645–1656 (2006).
[Crossref]

J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. 23(11), 2784–2796 (2006).
[Crossref]

2005 (1)

2004 (2)

W. Sun, N. G. Loeb, and Q. Fu, “Light scattering by coated sphere immersed in absorbing medium: a comparison between the FDTD and analytic solutions,” J. Quant. Spectrosc. Radiat. Transfer 83(3-4), 483–492 (2004).
[Crossref]

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70(3), 036606 (2004).
[Crossref]

2003 (2)

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 775 (2003).
[Crossref]

G. Videen and W. Sun, “Yet another look at light scattering from particles in absorbing media,” Appl. Opt. 42(33), 6724–6727 (2003).
[Crossref]

2002 (2)

2001 (3)

I. W. Sudiarta and P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 709–714 (2001).
[Crossref]

I. W. Sudiarta and P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. 18(6), 1275–1278 (2001).
[Crossref]

Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
[Crossref]

1999 (1)

A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

1996 (1)

M. Quinten and J. Rostalski, “Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13(2), 89–96 (1996).
[Crossref]

1995 (1)

G. L. G. Sleijpen and H. A. Van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algor. 10(2), 203–223 (1995).
[Crossref]

1994 (1)

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. 11(4), 1491–1499 (1994).
[Crossref]

1993 (4)

A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98(6), 699 (1993).
[Crossref]

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti-Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Transactions on Numerical Analysis 1, 11–32 (1993).

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “A note on the definition of scattering cross sections and phase functions for spheres immersed in an absorbing medium,” Wave Random Complex 3(3), 147–156 (1993).
[Crossref]

1991 (2)

1990 (1)

J. I. Hage and J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[Crossref]

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1983 (1)

1980 (1)

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[Crossref]

1979 (1)

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

1977 (1)

1974 (2)

W. C. Mundy, J. A. Roux, and A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64(12), 1593–1597 (1974).
[Crossref]

D. E. Livesay and K.-M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. 22(12), 1273–1280 (1974).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Albaladejo, S.

Armelles, G.

Baillis, D.

J. Randrianalisoa, D. Baillis, and L. Pilon, “Modeling radiation characteristics of semitransparent media containing bubbles or particles,” J. Opt. Soc. Am. 23(7), 1645–1656 (2006).
[Crossref]

Baum, B. A.

Ben-Yakar, A.

R. K. Harrison and A. Ben-Yakar, “Point-by-point near-field optical energy deposition around plasmonic nanospheres in absorbing media,” J. Opt. Soc. Am. 32(8), 1523–1535 (2015).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).

Brolo, A. G.

Bruscaglioni, P.

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “A note on the definition of scattering cross sections and phase functions for spheres immersed in an absorbing medium,” Wave Random Complex 3(3), 147–156 (1993).
[Crossref]

Calvo-Perez, O.

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
[Crossref]

Carminati, R.

Chaumet, P. C.

P. C. Chaumet, T. Zhang, and A. Sentenac, “Fast far-field calculation in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transfer 165, 88–92 (2015).
[Crossref]

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70(3), 036606 (2004).
[Crossref]

Chen, K.-M.

D. E. Livesay and K.-M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. 22(12), 1273–1280 (1974).
[Crossref]

Chylek, P.

I. W. Sudiarta and P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 709–714 (2001).
[Crossref]

I. W. Sudiarta and P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. 18(6), 1275–1278 (2001).
[Crossref]

Chýlekt, P.

Dlugach, J. M.

M. I. Mishchenko and J. M. Dlugach, “Plasmon resonances of metal nanoparticles in an absorbing medium,” OSA Continuum 2(12), 3415–3421 (2019).
[Crossref]

M. I. Mishchenko and J. M. Dlugach, “Scattering and extinction by spherical particles immersed in an absorbing host medium,” J. Quant. Spectrosc. Radiat. Transfer 211, 179–187 (2018).
[Crossref]

Draine, B. T.

P. J. Flatau and B. T. Draine, “Fast near field calculations in the discrete dipole approximation for regular rectilinear grids,” Opt. Express 20(2), 1247–1252 (2012).
[Crossref]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. 11(4), 1491–1499 (1994).
[Crossref]

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti-Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16(15), 1198–1200 (1991).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT 7.3,” https://arxiv.org/ 1305, 6497 (2013).

Durant, S.

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
[Crossref]

Flatau, P. J.

P. J. Flatau and B. T. Draine, “Fast near field calculations in the discrete dipole approximation for regular rectilinear grids,” Opt. Express 20(2), 1247–1252 (2012).
[Crossref]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. 11(4), 1491–1499 (1994).
[Crossref]

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16(15), 1198–1200 (1991).
[Crossref]

B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT 7.3,” https://arxiv.org/ 1305, 6497 (2013).

Fokkema, D. R.

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Transactions on Numerical Analysis 1, 11–32 (1993).

Frezza, F.

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. 33(5), 947–953 (2016).
[Crossref]

Frisvad, J. R.

J. R. Frisvad, “Phase function of a spherical particle when scattering an inhomogeneous electromagnetic plane wave,” J. Opt. Soc. Am. 35(4), 669–680 (2018).
[Crossref]

Froufe-Pérez, L. S.

Fu, Q.

Q. Fu and W. Sun, “Apparent optical properties of spherical particles in absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 100(1-3), 137–142 (2006).
[Crossref]

W. Sun, N. G. Loeb, and Q. Fu, “Light scattering by coated sphere immersed in absorbing medium: a comparison between the FDTD and analytic solutions,” J. Quant. Spectrosc. Radiat. Transfer 83(3-4), 483–492 (2004).
[Crossref]

W. Sun, N. G. Loeb, and Q. Fu, “Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41(27), 5728–5743 (2002).
[Crossref]

Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
[Crossref]

Gao, B.-C.

García-Martín, A.

Gartz, M.

A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

Gilra, D. P.

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

Gómez-Medina, R.

Goodman, J.

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti-Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

Goodman, J. J.

Gordon, R.

Greenberg, J. M.

J. I. Hage, J. M. Greenberg, and R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30(9), 1141–1152 (1991).
[Crossref]

J. I. Hage and J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[Crossref]

Greffet, J.-J.

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
[Crossref]

Gustafsson, M.

Hage, J. I.

J. I. Hage, J. M. Greenberg, and R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30(9), 1141–1152 (1991).
[Crossref]

J. I. Hage and J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[Crossref]

Harrison, R. K.

R. K. Harrison and A. Ben-Yakar, “Point-by-point near-field optical energy deposition around plasmonic nanospheres in absorbing media,” J. Opt. Soc. Am. 32(8), 1523–1535 (2015).
[Crossref]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 558–589 (2007).
[Crossref]

Hu, Y. X.

Huang, H.-L.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).

Ismaelli, A.

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “A note on the definition of scattering cross sections and phase functions for spheres immersed in an absorbing medium,” Wave Random Complex 3(3), 147–156 (1993).
[Crossref]

Ivanenko, Y.

Kahnert, F. M.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 775 (2003).
[Crossref]

Kamiuto, K.

Kreibig, U.

A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

Kristensson, G.

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

Lacis, A. A. X.

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering (Cambridge University, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Lakhtakia, A.

A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98(6), 699 (1993).
[Crossref]

Lebedev, A. N.

A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

Lin, B.

Livesay, D. E.

D. E. Livesay and K.-M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. 22(12), 1273–1280 (1974).
[Crossref]

Loeb, N. G.

Mangini, F.

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. 33(5), 947–953 (2016).
[Crossref]

Marinchio, H.

Mirmoosa, M.

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko and J. M. Dlugach, “Plasmon resonances of metal nanoparticles in an absorbing medium,” OSA Continuum 2(12), 3415–3421 (2019).
[Crossref]

M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transfer 214, 158–167 (2018).
[Crossref]

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

M. I. Mishchenko and J. M. Dlugach, “Scattering and extinction by spherical particles immersed in an absorbing host medium,” J. Quant. Spectrosc. Radiat. Transfer 211, 179–187 (2018).
[Crossref]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008).
[Crossref]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109(14), 2386–2390 (2008).
[Crossref]

M. I. Mishchenko, “Electromagnetic scattering by a fixed finite object embedded in an absorbing medium,” Opt. Express 15(20), 13188–13202 (2007).
[Crossref]

P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, and S.-C. Tsay, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41(15), 2740–2759 (2002).
[Crossref]

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering (Cambridge University, 2006).

Moskalensky, A. E.

A. E. Moskalensky and M. A. Yurkin, “Energy budget and optical theorem for scattering of source-induced fields,” Phys. Rev. A 99(5), 053824 (2019).
[Crossref]

Mulholland, G. W.

A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98(6), 699 (1993).
[Crossref]

Mundy, W. C.

Nordebo, S.

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

Y. Ivanenko, M. Gustafsson, and S. Nordebo, “Optical theorems and physical bounds on absorption in lossy media,” Opt. Express 27(23), 34323–34342 (2019).
[Crossref]

Peck, R. L.

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Pilon, L.

J. Randrianalisoa, D. Baillis, and L. Pilon, “Modeling radiation characteristics of semitransparent media containing bubbles or particles,” J. Opt. Soc. Am. 23(7), 1645–1656 (2006).
[Crossref]

J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. 23(11), 2784–2796 (2006).
[Crossref]

Platnick, S. E.

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Quinten, M.

M. Quinten and J. Rostalski, “Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13(2), 89–96 (1996).
[Crossref]

Rahmani, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70(3), 036606 (2004).
[Crossref]

Randrianalisoa, J.

J. Randrianalisoa, D. Baillis, and L. Pilon, “Modeling radiation characteristics of semitransparent media containing bubbles or particles,” J. Opt. Soc. Am. 23(7), 1645–1656 (2006).
[Crossref]

Rostalski, J.

M. Quinten and J. Rostalski, “Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13(2), 89–96 (1996).
[Crossref]

Roux, J. A.

Sáenz, J. J.

Sentenac, A.

P. C. Chaumet, T. Zhang, and A. Sentenac, “Fast far-field calculation in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transfer 165, 88–92 (2015).
[Crossref]

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70(3), 036606 (2004).
[Crossref]

Sleijpen, G. L. G.

G. L. G. Sleijpen and H. A. Van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algor. 10(2), 203–223 (1995).
[Crossref]

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Transactions on Numerical Analysis 1, 11–32 (1993).

Smith, A. M.

Stenzel, O.

A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

Sudiarta, I. W.

I. W. Sudiarta and P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 709–714 (2001).
[Crossref]

I. W. Sudiarta and P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. 18(6), 1275–1278 (2001).
[Crossref]

Sun, W.

Torrado, J. F.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering (Cambridge University, 2006).

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Tretyakov, S.

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

Tsay, S.-C.

van de Hulst, H. C.

H. C. van de Hulst, Light scattering by small particles (John Wiley & Sons, 1957).

Van der Vorst, H. A.

G. L. G. Sleijpen and H. A. Van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algor. 10(2), 203–223 (1995).
[Crossref]

Videen, G.

Vukadinovic, N.

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
[Crossref]

Wang, R. T.

Wendisch, M.

M. Wendisch and P. Yang, Theory of atmospheric radiative transfer: a comprehensive introduction (John Wiley & Sons, 2012).

Winker, D. M.

Wiscombe, W. J.

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[Crossref]

Yang, P.

M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
[Crossref]

P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, and S.-C. Tsay, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41(15), 2740–2759 (2002).
[Crossref]

M. Wendisch and P. Yang, Theory of atmospheric radiative transfer: a comprehensive introduction (John Wiley & Sons, 2012).

Yin, J.

J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. 23(11), 2784–2796 (2006).
[Crossref]

Yurkin, M. A.

A. E. Moskalensky and M. A. Yurkin, “Energy budget and optical theorem for scattering of source-induced fields,” Phys. Rev. A 99(5), 053824 (2019).
[Crossref]

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transfer 214, 158–167 (2018).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 558–589 (2007).
[Crossref]

Zaccanti, G.

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “A note on the definition of scattering cross sections and phase functions for spheres immersed in an absorbing medium,” Wave Random Complex 3(3), 147–156 (1993).
[Crossref]

Zhang, T.

P. C. Chaumet, T. Zhang, and A. Sentenac, “Fast far-field calculation in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transfer 165, 88–92 (2015).
[Crossref]

Appl. Opt. (6)

Astrophys. J. (4)

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti-Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

J. I. Hage and J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[Crossref]

Electronic Transactions on Numerical Analysis (1)

G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Transactions on Numerical Analysis 1, 11–32 (1993).

Eur. Phys. J. C (1)

A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. C 6(2), 365–369 (1999).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

D. E. Livesay and K.-M. Chen, “Electromagnetic fields induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. 22(12), 1273–1280 (1974).
[Crossref]

J. Colloid Interface Sci. (1)

C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979).
[Crossref]

J. Opt. Soc. Am. (12)

K. Kamiuto, “Near-field scattering by a large spherical particle embedded in a nonabsorbing medium,” J. Opt. Soc. Am. 73(12), 1819–1822 (1983).
[Crossref]

W. C. Mundy, J. A. Roux, and A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64(12), 1593–1597 (1974).
[Crossref]

P. Chýlekt, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67(4), 561–563 (1977).
[Crossref]

I. W. Sudiarta and P. Chylek, “Mie-scattering formalism for spherical particles embedded in an absorbing medium,” J. Opt. Soc. Am. 18(6), 1275–1278 (2001).
[Crossref]

J. Randrianalisoa, D. Baillis, and L. Pilon, “Modeling radiation characteristics of semitransparent media containing bubbles or particles,” J. Opt. Soc. Am. 23(7), 1645–1656 (2006).
[Crossref]

J. Yin and L. Pilon, “Efficiency factors and radiation characteristics of spherical scatterers in an absorbing medium,” J. Opt. Soc. Am. 23(11), 2784–2796 (2006).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: diagrammatic expansion of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2943–2952 (2007).
[Crossref]

S. Durant, O. Calvo-Perez, N. Vukadinovic, and J.-J. Greffet, “Light scattering by a random distribution of particles embedded in absorbing media: full-wave Monte Carlo solutions of the extinction coefficient,” J. Opt. Soc. Am. 24(9), 2953–2962 (2007).
[Crossref]

R. K. Harrison and A. Ben-Yakar, “Point-by-point near-field optical energy deposition around plasmonic nanospheres in absorbing media,” J. Opt. Soc. Am. 32(8), 1523–1535 (2015).
[Crossref]

F. Frezza and F. Mangini, “Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method,” J. Opt. Soc. Am. 33(5), 947–953 (2016).
[Crossref]

J. R. Frisvad, “Phase function of a spherical particle when scattering an inhomogeneous electromagnetic plane wave,” J. Opt. Soc. Am. 35(4), 669–680 (2018).
[Crossref]

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. 11(4), 1491–1499 (1994).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transfer (10)

M. I. Mishchenko and P. Yang, “Far-field Lorenz–Mie scattering in an absorbing host medium: theoretical formalism and FORTRAN program,” J. Quant. Spectrosc. Radiat. Transfer 205, 241–252 (2018).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transfer 214, 158–167 (2018).
[Crossref]

Q. Fu and W. Sun, “Apparent optical properties of spherical particles in absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 100(1-3), 137–142 (2006).
[Crossref]

M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109(14), 2386–2390 (2008).
[Crossref]

M. I. Mishchenko and J. M. Dlugach, “Scattering and extinction by spherical particles immersed in an absorbing host medium,” J. Quant. Spectrosc. Radiat. Transfer 211, 179–187 (2018).
[Crossref]

I. W. Sudiarta and P. Chylek, “Mie scattering efficiency of a large spherical particle embedded in an absorbing medium,” J. Quant. Spectrosc. Radiat. Transfer 70(4-6), 709–714 (2001).
[Crossref]

W. Sun, N. G. Loeb, and Q. Fu, “Light scattering by coated sphere immersed in absorbing medium: a comparison between the FDTD and analytic solutions,” J. Quant. Spectrosc. Radiat. Transfer 83(3-4), 483–492 (2004).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 558–589 (2007).
[Crossref]

P. C. Chaumet, T. Zhang, and A. Sentenac, “Fast far-field calculation in the discrete dipole approximation,” J. Quant. Spectrosc. Radiat. Transfer 165, 88–92 (2015).
[Crossref]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 775 (2003).
[Crossref]

J. Res. Natl. Inst. Stand. Technol. (1)

A. Lakhtakia and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98(6), 699 (1993).
[Crossref]

Numer. Algor. (1)

G. L. G. Sleijpen and H. A. Van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algor. 10(2), 203–223 (1995).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

OSA Continuum (1)

Part. Part. Syst. Charact. (1)

M. Quinten and J. Rostalski, “Lorenz-Mie Theory for Spheres Immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13(2), 89–96 (1996).
[Crossref]

Phys. Rev. A (2)

A. E. Moskalensky and M. A. Yurkin, “Energy budget and optical theorem for scattering of source-induced fields,” Phys. Rev. A 99(5), 053824 (2019).
[Crossref]

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

Phys. Rev. B (1)

S. Nordebo, G. Kristensson, M. Mirmoosa, and S. Tretyakov, “Optimal plasmonic multipole resonances of a sphere in lossy media,” Phys. Rev. B 99(5), 054301 (2019).
[Crossref]

Phys. Rev. E (1)

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70(3), 036606 (2004).
[Crossref]

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[Crossref]

Wave Random Complex (1)

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “A note on the definition of scattering cross sections and phase functions for spheres immersed in an absorbing medium,” Wave Random Complex 3(3), 147–156 (1993).
[Crossref]

Other (6)

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

H. C. van de Hulst, Light scattering by small particles (John Wiley & Sons, 1957).

M. Wendisch and P. Yang, Theory of atmospheric radiative transfer: a comprehensive introduction (John Wiley & Sons, 2012).

M. I. Mishchenko, L. D. Travis, and A. A. X. Lacis, Multiple scattering of light by particles: radiative transfer and coherent backscattering (Cambridge University, 2006).

B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT 7.3,” https://arxiv.org/ 1305, 6497 (2013).

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Figures (7)

Fig. 1.
Fig. 1. Schematic of far-field EM scattering by a fixed particle.
Fig. 2.
Fig. 2. Apparent extinction efficiency Qext of spherical particle as a function of size parameter x: comparison between DDA and the far-field Lorenz-Mie theory for absorbing host medium [36]. The refractive index of the host medium is ${m_\textrm{h}}\textrm{ = 1}\textrm{.33 + i}m_\textrm{h}^{{{\prime\prime}}}$, the imaginary part is varied among 0.0, 0.01, 0.02 and 0.05; the refractive index of the particle is mp=1.59. N is the number of the discretized sub-volumes.
Fig. 3.
Fig. 3. Scattering phase function p(θ) of spherical particle: comparison between DDA and the far-field Lorenz-Mie theory for absorbing host medium [36]. All conditions are the same to Fig. 2 except that $m_\textrm{h}^{{{\prime\prime}}}$ is chosen to be 0.0, 0.05 and x is chosen to be 10, 20.
Fig. 4.
Fig. 4. Inherent extinction and scattering efficiencies of a spherical particle as a function of size parameter x: comparison between the data extracted from Fig. 4 of Ref. [14] and DDA. The number of discretized sub-volumes is N=268096.
Fig. 5.
Fig. 5. The apparent extinction efficiency Qext of the sphere, prolate and oblate spheroids as a function of size parameter x for different host medium absorption. The step of x is 0.2. Qext of the sphere is calculated by the far-field Lorenz-Mie theory [36] while those of the spheroids are calculated by DDA.
Fig. 6.
Fig. 6. Same as Fig. 5 but for the inherent extinction efficiency; all the results are calculated by DDA.
Fig. 7.
Fig. 7. The magnitude of the total electric field Etot inside and near the particle in the x-y plane, the first row is for imaginary part of the host medium refractive index of $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.001}$, the second row is for $m_\textrm{h}^{{{\prime\prime}}}\textrm{ = 0}\textrm{.05}$. From left to right are for the sphere, prolate and oblate spheroids, respectively. The size parameter is x=10. The incident field is y-polarized and its magnitude at origin is assumed unity. The x and y coordinates denote the position of the discretized sub-volumes in DDA.

Equations (56)

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× E ( r ) = i ω μ 0 H ( r ) × H ( r ) = i ω ε h E ( r ) } r V EXT
× E ( r ) = i ω μ 0 H ( r ) × H ( r ) = i ω ε p ( r ) E ( r ) } r V INT
× × E ( r ) k h 2 E ( r ) = 0 , r V EXT
× × E ( r ) k p 2 ( r ) E ( r ) = 0 , r V INT
× × E ( r ) k h 2 E ( r ) = j ( r ) , r R 3
j ( r )  =  k h 2 [ ε ~ ( r ) 1 ] E ( r )
ε ~ ( r )  =  { 1 , r V EXT k p 2 ( r ) / k h 2 = ε p ( r ) / ε h , r V INT
E ( r )  =  E inc ( r ) + k h 2 V INT G ( r , r ) χ ( r ) E ( r ) d 3 r , r R 3
G ( r , r ) = ( I + 1 k h 2 ) exp ( i k h | r r | ) 4 π | r r |  =  exp ( i k h r ) 4 π r [ ( 1 + i k h r 1 k h 2 r 2 ) I + ( 3 3 i k h r k h 2 r 2 k h 2 r 2 ) r ^ r ^ ]
E ( r )  =  E inc ( r ) + k h 2 V INT V 0 G ( r , r ) χ ( r ) E ( r ) d 3 r  +  M ( V 0 , r ) L ( V 0 , r ) χ ( r ) E ( r )
M ( V 0 , r )  =  k h 2 V 0 d 3 r [ G ( r , r ) χ ( r ) E ( r ) G s ( r , r ) χ ( r ) E ( r ) ]
G s ( r , r )  =  1 k h 2 1 4 π | r r |
L ( V 0 , r ) = V 0 d 2 r n ^ r r 4 π | r r | 3
E i = E i inc + k h 2 j i G i j V j χ j E j + ( M i L i ) χ i E i
M i  =  k h 2 V i d 3 r [ G ( r , r ) G s ( r , r ) ]
P i  =  ε h V i χ i E i
1 ε h α ^ i 1 P i k h 2 ε h j i G i j P j = E i inc
α ^ i  =  V i χ i [ I ( M i L i ) χ i ] 1
L i  =  I / 3
M i  =  2 3 I [ ( 1  - i k h a ) exp ( i k h a ) 1 ]
D ¯ P ¯ + A ¯ P ¯ = ε h E ¯ inc
E ( r )  =  E inc ( r ) + E sca ( r )
H ( r ) = H inc ( r ) + H sca ( r )
E sca ( r ) = k h 2 ε h i = 1 N G ( r , r i ) P i
W abs = A S ( r ) n ^ d A
S ( r ) = 1 2 Re [ E ( r ) × H ( r ) ]
W ext = W abs  +  W sca W inc
W sca = A S sca ( r ) n ^ d A  =  A 1 2 Re [ E sca ( r ) × H sca ( r ) ] n ^ d A
W inc = A S inc ( r ) n ^ d A  =  A 1 2 Re [ E inc ( r ) × H inc ( r ) ] n ^ d A
W ext = A S ext ( r ) n ^ d A  =  A 1 2 Re [ E inc ( r ) × H sca ( r )  +  E sca ( r ) × H inc ( r ) ] n ^ d A
A S ( r ) n ^ d A  =  V S ( r ) d V
S ( r )  =  S sca ( r )  +  S ext ( r )  +  S inc ( r )
S ( r )  =  ω 2 | E ( r ) | 2 Im [ ε p ( r ) ] , r V INT
W abs  =  ω 2 V p | E ( r ) | 2 Im [ ε p ( r ) ] d V  =  ω 2 i = 1 N | E i | 2 Im ( ε p , i ) V i
S inc ( r )  =  ω 2 | E inc ( r ) | 2 Im ( ε h ) , r V INT
W inc  =  ω 2 i = 1 N | E i inc | 2 Im ( ε h ) V i
S ext  =  ω 2 Im [ E inc ( r ) p ( r ) ] + ω Im ( ε h ) Re [ E inc ( r ) E ( r ) | E inc ( r ) | 2 ] , r V INT
W ext  =  ω 2 i = 1 N Im ( p i E i inc ) V i + ω i = 1 N Im ( ε h ) Re ( | E i inc | 2 E i inc E i ) V i
W inh sca = W sca = W ext  +  W inc W abs
W inh ext = W inh sca  +  W inh abs  =  W ext  +  W inc
C inh abs  =  W inh abs / I inc
I inc = 2 I 0 inc [ 1 + ( η 1 ) e η ] / η 2
S h ( r )  =  ω 2 | E ( r ) | 2 Im ( ε h ) , r V EXT
S h inc ( r )  =  ω 2 | E inc ( r ) | 2 Im ( ε h ) , r V EXT
S h sca ( r )  =  ω 2 | E sca ( r ) | 2 Im ( ε h ) , r V EXT
E sca ( r ) exp ( i k h r ) r E 1 sca ( r ^ sca ) = exp ( k h r ) exp ( i k h r ) r E 1 sca ( r ^ sca )
E 1 sca ( r ^ sca )  =  k h 2 4 π ( I r ^ sca r ^ sca ) V INT d r χ ( r ) E ( r ) exp ( i k h r ^ sca r )
E 1 sca ( r ^ sca )  =  k h 2 4 π ε h ( I r ^ sca r ^ sca ) i = 1 N P i exp ( i k h r ^ sca r i )
E inc ( r ) = E 0 inc exp ( i k h r ^ inc r )
E 1 sca ( r ^ sca ) = A ( r ^ sca , r ^ inc ) E 0 inc
[ E θ sca ( r r ^ sca ) E φ sca ( r r ^ sca ) ]  =  exp ( i k h r ) r S ( r ^ sca , r ^ inc ) [ E 0 θ inc ( r ^ inc ) E 0 φ inc ( r ^ inc ) ]
I sca  =  exp ( 2 k h r ) r 2 F ( r ^ sca , r ^ inc ) I inc
C eff sca = 1 | E 0 inc | 2 4 π d Ω | E 1 sca ( r ^ sca ) | 2
F ~ ( r ^ sca , r ^ inc )  =  4 π C eff sca F ( r ^ sca , r ^ inc )
1 4 π 0 4 π d Ω F ~ 11 ( r ^ sca , r ^ inc ) = 1
C ext = 2 π k h Im [ S 11 ( 0 ) + S 22 ( 0 ) ]

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