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 [1–4]. 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 [5–35]. 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 [25–27] 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. [17–19] 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) [25,37–39], a natural solution to this problem is the DDA method, which discretizes the target and transforms the VIE into a set of linear equations [40–45]. 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. V_INT denotes the volume occupied by the particle while V_EXT 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]

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

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

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]

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

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.

Defining the polarizations of the sub-volumes as

Eq. (14) can be rewritten as:

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 k_h 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 M_i and L_i. We first consider the self-term dyadic L_i. It can be inspected from the integral in Eq. (13) that L_i 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]

The tensor M_i [Eq. (15)] has been studied by many researchers in the development of DDA for non-absorbing host medium. Various approximations of M_i have been proposed [43]. For absorbing host medium, the absorption of the medium is reflected by the complex wave number in M_i. Note that M_i tends to vanish with decreasing size of the sub-volume. If M_i is neglected, i.e. M_i= 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 M_i 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 [51–53]

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 M_i 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

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 [13–15]. 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.

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

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

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

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

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

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

The power absorption by the particle can be calculated by

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

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

Thus, W^ext by the particle is calculated in discretized form as

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 W^sca 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

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

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

$\nabla \cdot {\textbf S}_\textrm{h}^{\textrm{ext}}({\textbf r} )$ 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]

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

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

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.

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].

 

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]

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

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

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

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

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 Q^ext=C^ext/(πR²). In DDA, C^ext is calculated by Eq. (56). The refractive index of the particle is assumed to be m_p=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 Q^ext for x approaching 30 but can still predict the interference structure of Q^ext. 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.

 

Fig. 2. Apparent extinction efficiency Q^ext 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 m_p=1.59. N is the number of the discretized sub-volumes.

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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 m_p=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 Q^sca with increasing x, which is more obvious for smaller absorption of the host medium.

 

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 m_p=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.

 

Fig. 5. The apparent extinction efficiency Q^ext 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. Q^ext 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 Q^ext of the sphere, prolate and oblate spheroids as a function of x for different $m_\textrm{h}^{{{\prime\prime}}}$ of the host medium. Q^ext 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, Q^ext 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. Q^ext 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 Q^ext 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 Q^ext 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 Q^ext, which shows very small difference from the apparent Q^ext for small absorption of the host medium. However, the amplitudes of the oscillations decrease with increasing host medium absorption, and Q^ext gradually approaches unity with increasing particle size, which is similar to those observed in literature [13,14]. A special phenomenon is that Q^ext exhibits a large increase for the largest host medium absorption and x>20.

 

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.

 

Fig. 7. The magnitude of the total electric field E^tot 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.