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Abstract
In this work, we investigate quantitatively the applicability conditions of single scattering approximation (SSA) through direct simulation of electromagnetic scattering by small volume elements filled with randomly distributed spherical particles. The influences of size parameter x, volume fraction fv, complex refractive index m and number N of particles on the nondimensional extinction cross section ηext and absorption cross section ηabs of particle groups are discussed. For non-absorbing particles with small size parameters (x = 0.1 and 0.2 in this study), due to the small phase shift across particles, the particle refractive index has almost no influence on the criteria for SSA. However, when the particle size increases or particle absorption is enhanced, the criteria for SSA will be closely related to the particle complex refractive index. Moreover, when the particle size is small, due to the weak multiple scattering between particles, the criteria for SSA can be regarded as the criteria for independent scattering approximation (ISA). But as the particles increase to relatively large sizes (x = 4.0 in this study), because of the enhancement of multiple scattering, the criteria for SSA and ISA should be treated differently. The widely used criteria obtained for bispheres may not be applicable to particle groups composed of lots of particles, and the optical thickness of dispersed media is not suitable for evaluating the applicability conditions of SSA. For particle groups composed of different particle numbers, due to the differences in dependent scattering and multiple scattering, the criteria for SSA are obviously different and the particle volume fraction should be small enough to make the SSA sufficiently accurate.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light scattering by particles has been studied for decades due to its relevance in a wide array of fundamental and technological applications [1–6]. In the process of research, the fundamental problem under consideration is the scattering, absorption and emission by single particles. However, in natural environments and practical applications, we are usually confronted with collections of large numbers of particles. Even though it is possible to do experiments with single particles, it is more common to make measurements on many particles, due to the cost and difficulties of experimental techniques. Actually, if certain conditions (i.e., the far-field single scattering approximation, SSA) are satisfied, the total scattered field is just the sum of the fields scattered by the individual particles, each of which is acted on by the external field in isolation from the other particles [2,4]. Under the condition of SSA, the extinction, scattering, and absorption cross sections of a fixed tenuous group of particles can be obtained by summing up the corresponding individual-particle cross sections [7]. Therefore, the SSA is widely used in analyses and interpretation of laboratory measurements of light scattering by tenuous collections of natural and artificial particles [8–12].
With the increase of particle number and volume fraction, two main important problems occur and need to be further addressed in many cases. The first is the multiple scattering which is usually accounted for by meanings of radiative transfer theory [13,14]. The second is the dependent scattering which is caused by the near-field inter-particle effect and far-field interference effect [15,16]. Thereby the primary task in the study of light scattering is judging the particle scattering whether belongs to the regimes of single scattering, multiple scattering or dependent scattering. Aiming at this issue, a plenty of research works have been done and some important results and criteria are obtained [7,8,17–26]. The criteria for SSA were generally considered relevant to the optical thickness τ and an optical thickness of τ < 0.1 was recommended by van de Hulst [17]. Through transmission measurements of polymer microsphere suspensions using rectangular spectrometer cells, the single scattering condition was shown to be valid in the optical thickness range from 0.05 to 1.0 [18]. Based on the experimental results about latex particles [19], an optical thickness of τ = 0.3 was regarded as the demarcation boundary between the single versus multiple scattering regimes. Although many experimental researches on the SSA have been carried out, due to the limitation of experimental condition, the obtained criteria for SSA may be applicable only in limited ranges. In addition, some researchers have attempted to assess the qualitative criteria of applicability of SSA from both the theoretical analysis and numerical modeling. By using the T-matrix method to calculate the scattering cross sections of two spherical particles, Quirantes et al. [20] studied the maximum distance required to produce inter-particle light scattering interactions. Mishchenko et al. [7,8] derived the SSA from the exact Foldy-Lax equations and investigated the conditions of applicability of SSA by using the superposition T-matrix method. The results showed that one may need large inter-particle distances and low particle volume fractions in order to make the SSA sufficiently accurate.
Similar to the SSA, lots of research works on the regime of independent scattering approximation (ISA) have been done, but some disputed and unanswered questions still exist [16,27–38]. According to first-principles analysis of Mishchenko [16], the scattering by a multiple-particle group can be called independent if certain optical observables for the entire group can be expressed in appropriate single-particle observables. Based on the previous studies, the criteria for ISA were found to be correlated with the clearance-to-wavelength ratio c/λ, inter-particle distance-to-wavelength ratio d/λ, or inter-particle distance-to-radius ratio d/r, etc [20,29–32]. A review of the transition criteria previous proposed and the corresponding range of parameters was present recently by Galy et al. [31]. Moreover, by using the discrete-dipole approximation (DDA) algorithm, the authors assessed the validity of the transition criteria between the independent and dependent scattering regimes proposed in previous literature, and derived a new criteria based on electromagnetic scattering calculation of bispheres, disordered and ordered suspensions and aggregates. However, the research only focused on the non-absorbing particles and the multiple scattering effects were neglected.
Although the SSA has been studied for decades from different perspectives, it is still difficult to state precise general conditions under which the single scattering criterion is satisfied. Meanwhile, due to the limits of material type of standard particles, previous experimental researches mainly focused on the non-absorbing and weak absorbing particles. In this work, similar to the previous modeling approaches proposed by Mishchenko et al. [8], extensive computations of electromagnetic scattering by small volume elements filled with randomly distributed spherical particles are conducted by using the numerically exact superposition T-matrix method (STMM) [39,40]. By taking into account of the number, size parameter, volume fraction and complex refractive indexes of particles, we investigate the transition relation between the single scattering, multiple scattering and dependent scattering, and further make a quantitative analysis of the applicability conditions of SSA.
2. Modeling methodology
We consider light scattering and absorption in an imaginary spherical volume V with a radius R filled with N randomly distributed, identical spherical particles with radius r and complex refractive indexes m = n + iκ, as shown in Fig. 1. The background medium is assumed to be vacuum, and the size parameter of the particle is denoted as x = 2πr/λ, where λ is the incident light wavelength. By varying the radius R of the imaginary spherical volume in a certain interval, the particle groups with different particle volume fractions fv = N(r/R)3 can be obtained. The scattering volume is illuminated by a parallel quasi-monochromatic beam of light and the observation point is located in the far-field zone of the entire volume. Electromagnetic interactions of light with the particles are solved by the highly efficient and accurate STMM program (i.e., MSTM code [41], which is developed by Mackowski and Mishchenko) [39,40]. This program renders a complete description of the electromagnetic fields, in both the near and far-field regions that result from the excitation of a target of multiple spheres with a time harmonic field. Besides, the program can be applied to arbitrary configurations of spheres located internally or externally to other spheres, and the spheres taken in this description should to be homogeneous and isotropic. The only restriction of this program is that the surfaces of the spheres are not allowed to be overlapped. To model statistical randomness of particle positions within the imaginary spherical volume, we use at least 40 randomly configured N-particle groups and average over all possible orientations of each configuration with respect to the laboratory coordinate system by using the highly efficient STMM orientation averaging procedure [39,40].
Fig. 1. Spherical volume elements filled with N randomly distributed, identical spherical particles. (a) N = 8, (b) N = 16 and (c) N = 32.
The random orientation cross sections (extinction, scattering and absorption) of individual particles and total volume element can be directly obtained by using the STMM program. To quantitative analyze the applicability conditions of SSA, the non-dimensional cross section η is defined as the ratio of the total optical cross section (extinction or absorption) of the entire ensemble and a single particle in the same environment
where ηext and ηabs represent the non-dimensional extinction and absorption cross section, Cext,Mie and Cabs,Mie represent the extinction and absorption cross section of a single sphere which are calculated using the Mie theory [4]. The random orientation extinction cross section Cext,total and absorption cross section Cabs,total of the entire ensemble are obtained from the sum of the individual sphere cross sections Cext,i and Cabs,i represent the random orientation extinction and absorption cross section of sphere i which can be obtained as [39,40].We focus on the spherical volume elements composed of randomly distributed, identical spherical particles with different particle numbers (N = 8, 16 and 32), particle size parameters (x = 0.1, 0.2, 0.5, 1.0, 2.0 and 4.0), and complex refractive indexes (n = 0.75, 1.3, 1.5, 2.0, 3.0 and 5.0, κ = 0, 0.0001, 0.001, 0.01, 0.1, 1.0, 2.0 and 3.0). These cases can be considered as the situation of air bubbles and metallic particles embedded in water, liquid and ice particles suspended in air, etc. The range of particle volume fraction fv is set to 2.68 × 10−6∼0.296, which represents the main distribution range of fv in practical applications. To generate ensembles of randomly distributed spheres in an imaginary spherical volume, the Metropolis shuffling algorithm is used [43]. Note that the spheres in the simulation volume are not allowed to be overlapped and cross the volume’s outer boundary. For each volume fraction, at least 40 randomly configured N-particle groups are used to obtain the convergence results. For the SSA to be applicable, the particle group must be optically thin and the inter-particle distance between different particles must be sufficiently large. In this study, the SSA is considered to be reached when the results of ηext and ηabs locate between 0.95 and 1.05 [8]. Note that the multiple scattering effect (which can be accounted by the radiative transfer theory) can be neglected to some extent when the optical thickness of particle group is sufficiently thin. Therefore, the criteria for SSA can be also regarded as the criteria for ISA for an optically thin particle group.
3. Results and discussion
3.1 Effect of the particle refractive index
Figure 2 presents the nondimensional extinction cross section ηext of particle groups composed of N = 8 non-absorbing particles as functions of particle volume fraction fv with particle size parameters x = 0.1, 0.2, 0.5, 1.0, 2.0 and 4.0, respectively. For each size of particles, six kinds of particle refractive indexes with n = 0.75, 1.3, 1.5, 2.0, 3.0 and 5.0 are studied. For the convenience of comparative analysis, two dotted lines corresponding to the results of ηext = 0.95 and 1.05 are also illustrated in the figures. As shown in Figs. 2(a) and 2(b), for small particles with x = 0.1 and 0.2, the ηext for different particle refractive indexes are almost the same, which indicates that the particle refractive index has almost no influence on the criteria for SSA under the current circumstances. The main reason for this phenomenon is that the phase shift (ϕ = 2x|m−1|) is negligible for particles with small size parameters. As the particle size parameter increases to x = 0.5, as shown in Fig. 2(c), differences in the ηext between different particle refractive indexes are observed. For instance, the ηext for n = 5.0 is obviously larger than that of others for most cases of particle volume fractions. Meanwhile, it is found that the ηext shows an increasing trend with the increase of particle volume fraction. This result indicates that the inter-particle effect (multiple scattering or dependent scattering) enhances the particle scattering when the particles are close to each other.
Fig. 2. The nondimensional extinction cross section ηext of particle groups composed of N = 8 non-absorbing spherical particles as functions of particle volume fraction fv for particle size parameter of x = 0.1, 0.2, 0.5, 1.0, 2.0 and 4.0, and particle refractive index of n = 0.75, 1.3, 1.5, 2.0, 3.0 and 5.0.
As the particle size parameter continues to increase, different variation tendencies of ηext with the particle volume fraction are observed, as shown in Figs. 2(d)–2(f). Unlike the previous cases of small size particles, the values of ηext can be larger or smaller than 1.0 depending on the particle refractive indexes, which indicates that the inter-particle effect can either enhance or attenuate the particle scattering. Meanwhile, it can be clearly seen that the maximum/critical particle volume fractions fv which make the SSA applicable are obviously different for particles with different refractive indexes. This result verifies that the criteria for SSA have remarkable connections with the particle complex refractive index for particles whose sizes are not very small [35]. Moreover, we observe that the changes of nondimensional extinction cross section ηext with particle volume fraction are not strictly monotonous. This phenomenon is particularly prominent for the case of x = 1.0 and n = 5.0. As shown in Fig. 2(d), with the increase of particle volume fraction, the value of ηext slightly increases to about 1.09 then decreases to about 0.83. This result may be mainly caused by the combined interaction of multiple scattering and dependent scattering between particles. Furthermore, it is also worth noting that the nondimensional extinction cross section does not change monotonically with the particle refractive index. For instance, as for the particles of x = 1.0, 2.0 and 4.0, the particle refractive indexes corresponding to the smallest values of ηext are n = 3.0, 2.0, and 1.5, respectively. This phenomenon may be attributed to the nonlinear variations of extinction cross section of single particles with the particle refractive index.
Due to the irregular changes of ηext with the particle refractive index for different particle sizes, it is difficult to exactly determine the criteria for SSA. From another perspective, it can be seen from Fig. 2 that for each size of particles, there exists a critical particle volume fraction fv,n which makes the SSA applicable for different particle refractive indexes. Table 1 lists the critical particle volume fraction fv,n and the corresponding characteristic parameters of the particle groups including the clearance-to-wavelength ratio c/λ, inter-particle distance-to-wavelength ratio d/λ and maximum optical thickness τn. Note that the parameters c/λ, d/λ and τ are calculated as [28,29]
Table 1. Characteristic parameters of particle groups composed of non-absorbing particles (N = 8)
As shown in Table 1, the critical fv,n increases with the increase of particle size, but the ratios c/λ and d/λ have no obvious change regular pattern. Meanwhile, obvious differences in the maximum optical thicknesses τn for different particle refractive indexes are observed. For small size particles with x = 0.1, 0.2 and 0.5, the values of τn are very small which indicates that the multiple scattering effects between particles are very weak. In this case, the critical fv,n for SSA can be also regarded as the critical condition for ISA. As the particle size increases to x = 4.0, due to the strong multiple scattering effect, the τn for different particle refractive indexes all increase to high values. Then, the critical fv for SSA cannot be viewed as the criteria for ISA without taking into account the effect of multiple scattering on particle scattering. For more discussions on the problems with considering the multiple and dependent scattering, please refer to our previous work [35,36].
3.2 Effect of the particle absorption index
Figures 3–6 present the nondimensional extinction cross section ηext and absorption cross section ηabs of particle groups composed of N = 8 spherical particles as functions of particle volume fraction fv with particle size parameters x = 0.1, 0.5, 1.0 and 4.0, respectively. For each size of particles, three kinds of particle refractive indexes with n = 0.75, 1.3 and 2.0, and eight kinds of particle absorption indexes with κ = 0, 0.0001, 0.001, 0.01, 0.1, 1.0, 2.0 and 3.0 are studied. As shown in the figures, both the ηext and ηabs are significantly influenced by the particle absorption index. For small particles with x = 0.1, as shown in Fig. 3, a small change in the particle absorption index may cause a remarkable change in the nondimensional extinction cross section. This result is significantly different with those presented in previous section on non-absorbing particles, which suggests that the variation characteristic of phase shift (ϕ = 2x|m−1|) is not suitable to explain the relationships between the ηext and κ. Meanwhile, because the values of ηext change significantly with the increasing of particle absorption index, it will inevitably have an impact on the criteria for SSA. For example, when n = 1.3 [as shown in Fig. 3(b)], the critical fv for κ = 0 is about 4.3 × 10−6. As the absorption index κ increases to 0.0001 and 0.001, the critical fv increases to 2.4 × 10−5 and 5.7 × 10−4, respectively. For the cases of κ = 0.01, 0.1 and 1.0, the values of ηext always locate between 0.95 and 1.05, which results in that the critical fv could be as high as 0.296. Additionally, as the particle absorption index increases to κ = 2.0 and 3.0, instead of maintaining around 1.0, the values of ηext are large than or equal to 1.0. This indicates that the inter-particle effect enhances the particle extinction when the particle volume fraction is very high. By comparison, the effect of particle absorption index on the nondimensional absorption cross section ηabs is relatively weak, as shown in Figs. 3(d)–3(f). Except for strong absorbing particles with κ = 1.0, 2.0 and 3.0, the values of ηabs always locate between 0.95 and 1.05. On the whole, the criteria for SSA for the extinction and absorption cross section are obviously different.
Fig. 3. The nondimensional extinction cross section ηext and absorption cross section ηabs of particle groups composed of N = 8 spherical particles as functions of particle volume fraction fv for particle size parameter of x = 0.1, particle refractive index of n = 0.75, 1.3 and 2.0, and particle absorption index of κ = 0, 0.0001, 0.001, 0.01, 0.1, 1.0, 2.0 and 3.0.
When x = 0.5 and 1.0, as shown in Figs. 4 and 5, some phenomena about the ηext different from that of x = 0.1 are observed. The differences in the ηext between non-absorbing particles and weak absorbing particles are very small. For example, when x = 1.0, as shown in Figs. 5(a)–5(c), the ηext for κ = 0, 0.0001 and 0.001 are almost identical. As the particle absorption index κ increases from 0 to 3.0, the values of ηext can decrease from larger than 1.0 to smaller than 1.0, which corresponds to inter-particle effect on particle extinction that changes from enhance to attenuate. Accordingly, the critical fv presents the increasing tendency and then decreasing tendency with the increase of particle absorption index. In addition, we observed that the ηext does not decrease monotonically with the increase of particle absorption index. For example, the ηext for κ = 2.0 is larger than or equal to that for κ = 3.0. As for the nondimensional absorption cross section ηabs, it can be seen from Figs. 4(d)–4(f) and Figs. 5(d)–5(f) that the effect of particle absorption index on the ηabs is more obvious, compared to the case of x = 0.1. For strong absorbing particles with κ = 1.0, 2.0 and 3.0, the critical fv is obviously smaller than that of weak absorbing particles. Besides, for particles with the same absorption index κ, the particle refractive index n can also produce varied effects on the ηabs.
As the particles increase to relatively large sizes with x = 4.0, as shown in Figs. 6(a)–6(c), the differences in the ηext between κ = 0, 0.0001, 0.001 and 0.01 are very small. This result indicates that weak absorbing particles have the same critical fv as that of non-absorbing particles. For particles with n = 0.75, the values of ηext decrease from larger than 1.0 to smaller than 1.0 with the increase of particle absorption index. However, when n = 1.3 and 2.0, all the values of ηext are smaller than or equal to 1.0. Moreover, it is found that the values of ηext for κ = 1.0, 2.0 and 3.0 are always smaller than 1.0 and change little for the three different cases of particle refractive indexes. As for the ηabs, it can be seen from Figs. 6(d)–6(f) that the variation tendency of ηabs with the particle volume fraction is similar to that of x = 1.0. For weak absorbing particles with κ = 0.0001, 0.001 and 0.01, the critical fv can be as high as 0.1∼0.296. However, as the particle absorption index increases to κ = 0.1, the critical fv will be reduced to 0.02∼0.07. Moreover, with the particle absorption index κ increases from 0.1 to 3.0, the values of ηabs for high volume fractions tend to decrease firstly and then increase.
On the whole, due to the complex influences of particle complex refractive index on the nondimensional cross section, it is difficult to establish an accurate criterion or model for SSA based only on limited results. Similar to the proposed method in previous section, through searching the critical/maximum fv,n which makes the SSA applicable for different complex refractive indexes, the applicability of the proposed criteria can be enhanced to some extent. Because only three kinds of particle refractive indexes are considered in this section, more calculation work still needs to be conducted.
3.3 Effect of the particle number
In order to make the SSA accurate, the particle group must be optically thin, which means that the particle number should be small enough. However, it is often difficult or impractical to do experiments with single particles or very few particles. As the particle number increases, the multiple scattering and dependent scattering may be enhanced, which will have an impact on the radiative properties of particle groups. Figures 7–10 present the nondimensional extinction cross section ηext and maximum optical thickness τ [=1.5Cext,totalRfv/(πr3)] of particle groups composed of N = 8, 16 and 32 particles as functions of particle volume fraction fv with particle size parameters x = 0.1, 0.5, 1.0 and 4.0, respectively. For each size of particles, three kinds of particle complex refractive indexes with n = 1.3 and κ = 0, 0.01 and 1.0 are studied.
Fig. 7. The nondimensional extinction cross section ηext and the maximum optical thickness τmax of particle systems composed of N = 2, 8, 16 and 32 spherical particles as functions of particle volume fraction fv for particle size parameter of x = 0.1, particle refractive index of n = 1.3, and particle absorption index of k = 0, 0.01 and 1.0.
When the particle size increases to x = 4.0, as shown in Fig. 10, due to the enhancement of particle scattering, the optical thicknesses of particle groups increase to the order of magnitude of 0.1∼4.0, which indicates that the effects of multiple scattering between particles are very strong. In these circumstances, due to the strong multiple scattering between particles, the deviation of ηext from 1.0 may be caused by the coupled effects of dependent scattering and multiple scattering. Therefore, the critical fv for SSA cannot be viewed as the criteria for ISA without taking into account the effect of multiple scattering on particle scattering. For all the three kinds of particle absorption index, the ηext shows a decreasing tendency with the increase of particle volume fraction. This result indicates that the inter-particle effects weaken the particle extinction. Meanwhile, we observe that the critical fv are quite different for particle groups composed of different particle numbers. As shown in Fig. 10(a), with the particle number N increases from 8 to 32, the corresponding critical fv decreases from 0.21 to 0.037.
To summarize, for particle groups composed of different particle numbers, due to the differences in dependent scattering and multiple scattering, the critical fv for SSA are obviously different and it seems that there is no unified criterion to evaluate the SSA. For small size particles, as the particle volume fraction increases, the enhanced effects of dependent scattering lead to the deviation of the ηext and thus the critical fv for SSA can be regarded as the criteria of the ISA. As the particles increase to relatively large sizes, due to the enhancement of multiple scattering, the proposed method can only be used to evaluate the criteria for SSA, which means the criteria for SSA and ISA should be treated differently.
4. Conclusion
In this work, we focus on electromagnetic scattering by spherical volume elements composed of randomly distributed, identical spherical particles with different particle size parameters, complex refractive indexes and particle numbers. Through comparing the nondimensional extinction cross section ηext and absorption cross section ηabs of the particle groups, the transition relations between the single scattering, multiple scattering and dependent scattering are systemically investigated.
The results show that the effects of particle size, complex refractive index and particle number have different degrees influence on the ηext and ηabs, which suggests that the impacts of these factors should be carefully considered in the analysis and evaluation of the criteria for SSA and ISA. For non-absorbing particles with small size parameters (x = 0.1 and 0.2 in this study), due to the small phase shift across particles, the particle refractive index has almost no influence on the criteria for SSA. However, when the particle size increases or particle absorption is enhanced, the criteria of SSA will be closely related to the particle complex refractive index. Moreover, when the particle size is small, due to the weak multiple scattering between particles, the deviation of ηext from 1.0 is caused mainly by the dependent scattering, and thus the critical particle volume fraction fv for SSA can be regarded as the criteria for ISA. But as the particles increases to relatively large sizes (x = 4.0 in this study), because of the enhancement of multiple scattering, the criteria for SSA and ISA should be treated differently. For particle groups composed of different particle numbers, due to the differences in dependent scattering and multiple scattering, the critical fv for SSA are obviously different and the particle volume fraction should be small enough to make the SSA sufficiently accurate. Meanwhile, the widely used criteria obtained for bispheres may not be applicable to particle groups composed of lots of particles, and the optical thickness of dispersed media is not suitable for evaluating the applicability conditions of SSA. Another conclusion is that the criteria for SSA for the extinction and absorption cross section are obviously different, which verifies that the applicability of SSA ultimately depends on the applications.
Due to the complex influences of various factors, it is difficult to establish an accurate and comprehensive criterion or model of SSA based only on limited calculation results. With regards to future research, it is desirable to extend the calculation to groups composed of more different particle size parameters, complex refractive indexes and particle numbers. Meanwhile, by using some new analysis methods, such as the machine-learning method, a regime map demarcating single scattering, multiple scattering and dependent scattering considering the influences of size parameter, volume fraction and complex refractive index of particles will be researched and constructed.
Funding
National Natural Science Foundation of China (51806124, 51906127); China Postdoctoral Science Foundation (2019M662353, 2019M662354, 2020T130365, 2021T140401).
Acknowledgments
We sincerely thank D. W. Mackowski and M. I. Mishchenko for the multiple sphere T-matrix code. Lanxin Ma gratefully acknowledges support from the Young Scholars Program of Shandong University.
Disclosures
The authors declare no conflicts of interest.
Data availability
The MSTM code used in this work is available in Ref. [41]. Other data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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