Use of Debye&#x2019;s series to determine the optimal edge-effect terms for computing the extinction efficiencies of spheroids

Wushao Lin; Lei Bi; Dong Liu; Kejun Zhang

doi:10.1364/OE.25.020298

1. Introduction

The extinction efficiency is an important parameter in electromagnetic wave scattering by small particles and is of great interest in many scientific disciplines. In atmospheric sciences, for example, the extinction efficiencies of atmospheric aerosols are fundamentally related to the extinction coefficient (the amount of scattering and absorption per unit path length). The values of the extinction coefficient for aerosols are an important input parameter for global climate models. In addition, the extinction coefficient determines the optical thickness of aerosols and is a critical parameter in the atmospheric radiative transfer and remote sensing simulations [1]. Furthermore, the spectral extinction technique is often used for particle characterization to retrieve the particle size distributions [2–4] or the refractive indices [5]. The physical mechanism underlying the extinction of light by small particles is also an interesting but challenging research topic in mathematical physics [6,7].

Spheroidal shape is a first-order approximation of the overall shape of non-spherical particles. It has been widely used in applications for computing the optical properties of atmospheric aerosols and other particle types [8]. Even though these geometries are simplified as spheroids, efficient and accurate computational tools are unavailable to compute their optical properties across a complete range of size parameters. Practically, if only the extinction efficiency is required, then an alternative approach based on approximate analytical formulas has been developed to economize the computational time [9–13]. Previous studies have shown that most formulae have the following formalism $Q_{e x t} = Q_{g e o} + Q_{e d g e}$ , where $Q_{g e o}$ represents the contribution from diffraction and ray optics, and $Q_{e d g e}$ contains all the edge-effect contributions that are fundamentally related to wave tunneling near the penumbra regions. The edge effect plays a critical role in determining the accuracy of $Q_{e x t}$ for small-to-moderate size parameters, although it decreases as the size parameter increases [14–16].

A thorough understanding of the edge effect is only known for spheres—analytical forms of the edge-effect terms were derived from Maxwell’s equations by Nussenzveig and Wiscombe [17] by applying the Complex Angular Momentum (CAM) theory. For spheroids, however, it is extremely challenging to derive a similar formula. Bi and Yang extended the four edge-effect terms for spheres to spheroids with either fixed or random orientation [18] by using a semi-empirical approach based on Fock’s similarity principle [19,20]. However, an appropriate choice of edge-effect terms is crucial in determining the accuracy of the final results.

In this paper, we apply an advancement of the Debye series for non-spherical particles [21,22] to solve the aforementioned problem associated with the optimal edge-effect terms. More specifically, we employ the Debye series expansion technique for computing the rigorous edge-effect efficiencies for a substantial number of refractive indices. With these rigorous solutions as the reference, new insight is gained to examine the priori number of edge-effect terms. These insights can be further employed to improve the accuracy of the approximate formula for spheroids. For particles with strong absorption, the extinction results from the zeroth-order Debye series is equivalent to their counterparts computed from the T-matrix method. In such scenarios, we directly employ the T-matrix method for computation because it is much more computationally efficient than the Debye series approach. Section 2 describes the formulae associated with the extinction efficiency and the method to determine the optimal edge-effect terms are described. Representative results are given in Section 3, a look-up table approach is discussed for practical computations for improving the numerical accuracy of high-frequency extinction formulae of spheroids. The conclusions are given in Section 4.

2. Computational methods

Figure 1(a) and 1(b) show a prolate spheroid and an oblate spheroid, respectively; the semi-horizontal axis is denoted as a, and the semi-vertical axis is denoted as c. The orientation of the particle with respect to the incident light is specified by the angle θ between the direction of the incident light and the symmetric axis c of the spheroid. Let λ be the wavelength and k be the wavenumber ( $k = 2 π / λ$ ); the size parameter is defined in terms of the maximum of the semi-axes, i.e., $β = k c$ for prolate spheroids and $β = k a$ for oblate spheroids.

Fig. 1 Prolate spheroid (a) and oblate spheroid (b).

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For a sphere, the edge-effect efficiency from the CAM theory with up to four terms is given by [17]:

Q_{e x t, e d g e} = c_{1} β^{- 2 / 3} + c_{2} β^{- 1} + c_{3} β^{- 4 / 3} + c_{4} β^{- 5 / 3},

where the coefficients

c_{i} (i = 1, 4)

are given as follows:

c_{1} = 1.9924,

c_{2} = 2 Im [(m^{2} + 1) / \sqrt{m^{2} - 1}],

c_{3} = - 0.7154,

c_{4} = - 0.6641 Im [e^{i π / 3} (m^{2} + 1) (2 m^{4} - 6 m^{2} + 3) / {(m^{2} - 1)}^{3 / 2}] .

Here m is the complex refractive index. According to Bi and Yang [18], the edge-effect efficiency for oblique spheroids is written as:

Q_{e x t, e d g e} = q_{1} \frac{c_{1}}{β^{2 / 3}} {(\frac{c}{a})}^{4 / 3} + q_{2} \frac{c_{2}}{β} (\frac{c}{a}) + q_{3} \frac{c_{3}}{β^{4 / 3}} {(\frac{c}{a})}^{2 / 3} + q_{4} \frac{c_{4}}{β^{5 / 3}} {(\frac{c}{a})}^{1 / 3},

where

q_{i} (i = 1, 4)

depends on the particle orientation. For prolate spheroids,

q_{1} = p^{- 2 / 3} {}_{2}F_{1} [- 2 / 3, 1 / 2, 1, 1 - p^{- 2}],

q_{2} = {}_{2}F_{1} [- 1 / 2, 1 / 2, 1, 1 - p^{- 2}],

q_{3} = p^{2 / 3} {}_{2}F_{1} [- 1 / 3, 1 / 2, 1, 1 - p^{- 2}],

q_{4} = p^{4 / 3} {}_{2}F_{1} [- 1 / 6, 1 / 2, 1, 1 - p^{- 2}] .

whereas for oblate spheroids,

q_{1} = p^{- 2} {}_{2}F_{1} [- 2 / 3, 1 / 2, 1, 1 - p^{2}],

q_{2} = p^{- 1} {}_{2}F_{1} [- 1 / 2, 1 / 2, 1, 1 - p^{2}],

q_{3} = {}_{2}F_{1} [- 1 / 3, 1 / 2, 1, 1 - p^{2}],

q_{4} = p {}_{2}F_{1} [- 1 / 6, 1 / 2, 1, 1 - p^{2}] .

In Eq. (7)-Eq. (14),

{}_{2}F_{1}

is the hypergeometric function, and p is defined as follows:

p = {(\cos^{2} θ + \frac{a^{2}}{c^{2}} \sin^{2} θ)}^{1 / 2} .

Note, the first term in Eq. (6) was also obtained in Fournier and Evans [9]. With Eq. (6), the edge-effect efficiency of a randomly oriented spheroid can be computed through a standard numerical averaging procedure. Of the four edge-effect terms, both

c_{1}

and

c_{3}

are constant whereas two terms

c_{2}

and

c_{4}

are dependent of the refractive indices. More specifically, Fig. 2(a) and 2(b) show the values of

c_{2}

and

c_{4}

respectively in the complex refractive index domain. The

c_{2}

and

c_{4}

coefficients can be zero when the real part and the imaginary part have the particular relationship indicated by the black curves. When

c_{2}

is close to zero, then the edge-effect efficiencies with one term or two terms have negligible differences. Similar to

c_{2}

, when

c_{4}

is close to zero, the edge-effect efficiencies with three terms or four terms are nearly identical. Furthermore, the values of

c_{2} β^{- 1} + c_{3} β^{- 4 / 3}

and

c_{2} β^{- 1} + c_{3} β^{- 4 / 3} + c_{4} β^{- 5 / 3}

, shown in Fig. 2(c) and 2(d) have been computed (the value of zero is indicated by black curves). The size parameter β is chosen to be 10. When

c_{2} β^{- 1} + c_{3} β^{- 4 / 3}

is close to zero, the edge-effect efficiencies with one term or three terms are nearly the same. When

c_{2} β^{- 1} + c_{3} β^{- 4 / 3} + c_{4} β^{- 5 / 3}

is close to zero, the edge-effect efficiencies with one term or four terms are nearly identical. Note that the two curves in Fig. 2(c) and 2(d) depend on the size parameter. As will be discussed in Section 3, the black curves in Fig. 2 can partly explain the distribution of the optimal edge-effect terms in the complex refractive index domain.

Fig. 2 The values of $c_{2}$ , $c_{4}$ , $c_{2} β^{- 1} + c_{3} β^{- 4 / 3}$ and $c_{2} β^{- 1} + c_{3} β^{- 4 / 3} + c_{4} β^{- 5 / 3}$ in the complex refractive index domain. The size parameter β is equal to 10. The black curves represent a value of zero.

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The edge-effect efficiency is rigorously encoded in Maxwell’s equations. For a sphere, the optimum number can be determined by comparing the edge-effect efficiencies computed from the approximate formula and their rigorous counterparts obtained from the Debye expansion of the Lorenz-Mie coefficients. In the Debye expansion, the amplitude scattering amplitude of a sphere is composed of the diffraction plus refection term, the direct transmission term, and higher-order terms corresponding to transmission following multiple internal reflections. For a nonspherical particle, similar expansion can be applied to the T-matrix. The edge-effect contribution to the extinction efficiency can be explicitly obtained from the diffraction plus reflection term. Specifically, the edge- effect efficiency is given by [21]:

Q_{e x t, e d g e} = 2 + \frac{1}{k^{2} r^{2}} Re \sum_{l = 1}^{\infty} (2 - a_{l} - b_{l}),

where r is the radius of sphere, and

a_{l}

and

b_{l}

are reflection coefficients in the Debye series, given by [23]

a_{l} = \frac{ς_{n}^{(2)}^{'} (x) ς_{n}^{(2)} (m x) - m ς_{n}^{(2)}^{'} (x) ς_{n}^{(2)}^{'} (m x)}{- ς_{n}^{(1)}^{'} (x) ς_{n}^{(2)}^{'} (m x) + m ς_{n}^{(1)}^{'} (x) ς_{n}^{(2)}^{'} (m x)},

b_{l} = \frac{m ς_{n}^{(2)}^{'} (x) ς_{n}^{(2)} (m x) - ς_{n}^{(2)}^{'} (x) ς_{n}^{(2)}^{'} (m x)}{- m ς_{n}^{(1)}^{'} (x) ς_{n}^{(2)}^{'} (m x) + ς_{n}^{(1)}^{'} (x) ς_{n}^{(2)}^{'} (x)} .

In Eq. (17) and (18),

ς_{n}^{(1)}^{'}

is the Riccati-Bessel function of i-th kind. Similarly, for spheroids, the optimum number of the edge-effect terms can be obtained by minimizing the differences between the results from the formula and those from the Debye series. The general formalism for the edge-effect computation of non-spherical particle is referred to [21]. In contrast to spheres, this requires substantial computational recourses. Note that the extinction efficiency can be accurately obtained from the invariant imbedding T-matrix method (II-TM) [24,25]. Thus, the II-TM can examine the accuracy of the total extinction efficiency. For highly absorptive particles, the II-TM results are the same as the zero-th order Debye series. For non-absorptive and weakly absorptive particles, the use of the II-TM for examining the accuracy of the edge-effect efficiency may incorrectly determine the optimal edge-effect terms because of a ripple structure or subtle uncertainties associated with the geometric-optics term. In this case, we used the Debye series (as will be discussed in Section 3).

3. Results

The extinction efficiencies of spheres and spheroids were computed for a set of refractive indices. Because the real part ( $m_{r}$ ) of the refractive indices ranges from 1.2 to 1.9 for most atmospheric aerosols (e.g., dust, sea salt and soot), $m_{r}$ is constrained from 1.1 to 2.0 with a step of 0.01. The imaginary part ( $m_{i}$ ) is constrained from 0.01 to 1.0 with a step of 0.01 for moderately and strongly absorptive particles. Non-absorptive particles ( $m_{i}$ = 0) and weakly absorptive particles ( $m_{i}$ = 10⁻⁷, 10⁻⁶, 10⁻⁵, 10⁻⁴, and 10⁻³) were also considered. A total of ten aspect ratios (a/c) of randomly oriented spheroids are chosen; 0.5, 0.6, 0.7, 0.8, and 0.9 for prolate spheroids, and 1.2, 1.4, 1.6, 1.8, and 2.0 for oblate spheroids.

Figure 3 shows the edge-effect efficiencies (in Fig. 3(a) and 3(b)) and the total extinction efficiencies (in Fig. 3(c) and 3(d)) for two representative refractive indices. The results with the optimal number of edge-effect terms and a total of four edge-effect terms are compared to their rigorous counterparts obtained from the Debye series or the Mie theory. Notice that for weakly absorptive particles (m = 1.6 + i10⁻⁷), the total extinction efficiency curves have an oscillation structure due to the geometrical optics term in the formula. In addition to this macroscopic oscillation, the rigorous extinction has a ripple structure associated with the morphology-dependent resonances [6,26]. Under these circumstances, it is difficult to determine the optimal number because it could change with different size parameters. This feature does not occur for highly absorptive particles (m = 1.75 + i0.3). However, this problem can be avoided, if the edge-effect efficiencies are directly computed from the Debye solution. The results from a formula with an appropriate number of terms are consistent with the Debye solutions. The improvement is particularly obvious for smaller size parameters. As the size parameter increases, the differences among the three results decrease substantially. When the size parameter is larger than 50, the improved accuracy by truncating the edge-effect expansion is small because the first-order term dominates the edge-effect contribution, and all higher-order contributions could be reasonably neglected. Therefore, it is no longer necessary to identify an optimal number to achieve the best accuracy. Importantly, as the size parameter increases from 5 to 60, the optimal numbers of edge-effect terms for m = 1.6 + i10⁻⁷ and m = 1.75 + i0.3 remain 2 and 3, respectively. This means that the optimal number is insensitive to the size parameters in these two cases.

Fig. 3 Comparison of the results of a sphere computed from the rigorous solutions and the formulae.

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To examine the dependence of the optimal numbers on the refractive index, Fig. 4(a) shows the optimal number of edge-effect terms for a sphere in a complex refractive index domain with a size parameter of 10. The results of non-absorptive particles ( $m_{i}$ = 0) and weakly absorptive particles ( $m_{i}$ = 10⁻⁷, 10⁻⁶, 10⁻⁵, 10⁻⁴, 10⁻³) are shown in the lower part of the figure. Figure 4(b) shows the absolute error of the approximate results with optimal edge-effect terms; the errors corresponding to the four terms for all refractive indices are shown in Fig. 4(c).

Fig. 4 The optimal number of edge-effect terms (a); the errors of the approximate results with optimal edge-effect terms (b); similar to (b), (c) is for four edge-effect terms;the accuracy improved by truncating the edge-effect terms (d). The lower parts of the figures show the results of non-absorptive particles ( $m_{i}$ = 0) and weakly absorptive particles ( $m_{i}$ = 10⁻⁷, 10⁻⁶, 10⁻⁵, 10⁻⁴, and 10⁻³). The size parameter is 10.

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The optimal number depends on the refractive index. In the shaded red region, the approximate results with four terms offer the best accuracy. However, for a majority of refractive indices, the four edge-effect terms have poor accuracy. In particular, the optimal number of terms significantly improves the accuracy of the approximate formula for the refractive indices with large real parts. More explicitly, Fig. 4(d) shows that the accuracy improved by truncating the edge-effect expansion with an optimal number of edge-effect terms. The improved accuracy is evaluated via Eq. (19):

\frac{| Q_{e d g e, 4 t e r m s} - Q_{e d g e, d e b y e} | - | Q_{e d g e, o p t i m a l} - Q_{e d g e, d e b y e} |}{Q_{e d g e, d e b y e}} \times 100 % .

The accuracy of the results can be improved by up to 32%. As mentioned previously, the distribution of the optimal numbers is partly explained by Fig. 2. The boundaries between different colored regions indicated by the black curves numbered from 1 to 5 are closely related to the curves in Fig. 2 (i.e., a certain edge-effect term or a summation of different edge-effect terms is zero). For the other boundaries (e.g., the boundary between the red and yellow region in the central part), the edge efficiency values with four or three terms have nearly identical absolute errors (one is larger than the true value and the other is smaller than the true value).

To further examine the size-dependence, the optimal number of edge-effect terms are obtained for spherical particles with different size parameters (see Fig. 5). The overall patterns of optimal numbers in terms of refractive indices are similar for different size parameters. Similar to Fig. 3, the optimal numbers do not change with the size parameter for most refractive indices. The red shaded region indicates that the optimal number is 4. This region expands as the size parameter increases. Major differences are found in the upper part of the figures where the imaginary parts of the refractive indices are quite large. In this region, the optimal number is either 1 or 4.

Fig. 5 The optimal numbers of edge-effect terms for four spheres with size parameters of 5, 15, 20, and 25 (subplot (a), (b), (c), and (d), respectively).

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To evaluate whether these changes are important, Fig. 6 shows the edge-effect extinction efficiencies computed from both the formula and the Debye solution for four representative refractive indices. Here the optimal numbers change with the size parameter. The optimal number changes from 4 to 1 for m = 1.5 + i0.85 (see Fig. 6(a)), from 1 to 4 for m = 1.9 + i1.0 (see Fig. 6(b)), from 1 to 3 for m = 1.75 + i10⁻⁵ (see Fig. 6(c) and 6(d)), and from 2 to 3 for m = 1.6 + i0.15 (see Fig. 6(e) and 6(f)). It is clear that applying the optimal number results in better accuracy at small size parameters. The optimal number likely changes from one to the other at size parameters larger than 20. However, we can see that the improved accuracy resulting from this change is relatively tiny—applying the previous optimal number still results in reasonable accuracy. Thus, we conclude that the optimal numbers obtained at size parameter of 10 are practical for size parameters ranging from 5 to 50. When the size parameter is larger than 50, the approximate edge-effect efficiencies with four terms have reasonable accuracy.

Fig. 6 The edge-effect efficiencies with respect to size parameter: a comparison of the results with four edge-effect terms and their counterparts with an optimal number of edge-effect terms.

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We now turn to spheroids. Figure 7 displays the extinction efficiencies of moderately absorptive particles (m = 2.0 + i0.01) and highly absorptive particles (m = 1.75 + i0.44, the typical refractive index of soot at 532nm) with different aspect ratios as a function of the size parameter. For all spheroids, incorporating four edge-effect terms leads to a non-negligible error for small size parameters. However, the formula is quite accurate when the optimal number of edge-effect terms is applied. As the size parameter increases, the difference between results that include different number of terms decreases, and reasonable precision can be obtained by including all four edge-effect terms into the results. Note that with a fixed refractive index, prolate spheroids and oblate spheroids have the same optimal numbers. In addition, the optimal numbers remain the same for size parameters ranging from 5 to 60. This finding suggests that the optimal number of edge-effect terms for a single size parameter can also be generalized to other spheroid size parameters.

Fig. 7 Comparison of the extinction efficiencies of spheroids computed from the T-matrix solution and the extinction formula.

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Similar to spherical objects, the optimal numbers of edge-effect terms are obtained for spheroids with the size parameter of 10 in a complex refractive index domain. Figure 8 shows the results of the spheroids with different aspect ratios (a/c = 0.5, 0.6, 0.7, 0.8, 0.9, 1.2, 1.4, 1.6, and 1.8). Similarly, we can see that the approximate results with four edge-effect terms do not always guarantee the best accuracy for different refractive indices. In general, the change of optimal numbers with respect to refractive index is similar for all aspect ratios. This indicates that the optimal number is slightly sensitive to the shape of the particles. As the aspect ratio approaches unity, the optimal numbers of spheroids and those of a sphere (see Fig. 4(a)) converge. A portion of the optimal numbers (with the imaginary part of the refractive index ranging from 0.2 to 0.9) is stable for all ten aspect ratios. Note that as the aspect ratio approaches 0.5 or 1.8, the overall pattern becomes similar to that of a sphere with a size parameter of 5 (as shown in Fig. 5(a)). For aspect ratios of 0.5 and 1.8, the boundary between the red and yellow region is not smooth—most likely due to the inaccuracy of the Debye solution upon numerical instability. Actually, there is a slight difference in the Debye solution that could change the optimal edge-effect term.

Fig. 8 The optimal numbers of edge-effect terms for spheroids. The aspect ratios are 0.5 (a), 0.6 (b), 0.7 (c), 0.8 (d), 0.9 (e), 1.2 (f), 1.4 (g), 1.6 (h) and 1.8 (i). The size parameter is 10.

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Comparing Fig. 4(a) and Fig. 8 confirms that the optimal numbers for spheres might also be applied to spheroids with reasonable accuracy. To test this hypothesis, we investigated the improved accuracy in Fig. 9 via an approximate extinction efficiency formula of spheroids with the optimal edge-effect terms determined for spheres and evaluated through the following equation:

\frac{| Q_{e x t, 4 t e r m s} - Q_{e x t, I I T M} | - | Q_{e x t, o p t i m a l} - Q_{e x t, I I T M} |}{Q_{e x t, I I T M}} \times 100 %,

The optimal edge-effect terms are applicative to most refractive indices. Applying the optimal numbers determined for spheres leads to worse accuracy (blue color) only in a small portion of refractive indices at extreme aspect ratios. Importantly, for extreme aspect ratios, the characteristic dimensions for some orientations are too small and high-frequency approximation becomes problematic.

Fig. 9 Similar to Fig. 4 (d), this shows the percentage improvement in accuracy by truncating edge-effect terms of spheroids. The aspect ratios are 0.5 (a), 0.6 (b), 0.7 (c), 0.8 (d), 0.9 (e), 1.2 (f), 1.4 (g), 1.6 (h) and 1.8 (i). The size parameter is 10.

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Next, to confirm whether it is reasonable to apply the table of the optimal numbers of edge-effect terms for a sphere with size parameter of 10 to spheroids with different aspect ratios and different size parameters, we evaluated the relative error of the approximate results with optimal edge-effect terms obtained for spheres and those with all four edge-effect terms via the following equation:

\frac{| Q_{e x t, o p t i m a l} - Q_{e x t, I I T M} |}{Q_{e x t, I I T M}} \times 100 %,

\frac{| Q_{e x t, 4 t e r m s} - Q_{e x t, I I T M} |}{Q_{e x t, I I T M}} \times 100 % .

Figure 10 shows the results of two refractive indices with a size parameter ranging from 10 to 40. After applying the optimal numbers, the formula offers excellent accuracy. Unlike strongly absorptive particles (m = 2.0 + i0.1), the optimal number from spherical cases is not practical near the small size parameters and moderately absorptive particles (m = 1.7 + i0.01) with extreme aspect ratios. However, as the size parameter becomes larger, applying the optimal number yields results in better accuracy. Therefore, we conclude that it is reasonable to use the table with optimal numbers of edge-effect terms for spheres to spheroids as well. For size parameters ranging from 10 to 50, the optimal number can be efficiently obtained from the table by searching the given refractive index. For size parameters larger than 50, the optimal number can be set to be 4 because the approximate results with four terms included have high accuracy. These can be improved by truncating the edge-effect series, which is a fairly minor processing step.

Fig. 10 The errors of the extinction efficiency computed with different numbers of edge-effect terms.

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4. Summary and conclusion

The accuracy of the high-frequency extinction formulas was systematically examined in computing the extinction efficiencies of spheroids by comparing the approximate results with their rigorous counterparts. The use of Debye’s series allows for a direct examination of the edge-effect efficiency to circumvent the uncertainties associated with geometric-optics terms in the case of particles with weak and zero absorption. The formulas have been fully tested for randomly oriented spheroids with 10 aspect ratios ranging from 0.5 to 2.0 as well as refractive indices with the real part ranging from 1.1 to 2.0 and the imaginary part ranging from 0.0 to 1.0. The accuracy of extinction efficiencies computed from the high-frequency formula depends strongly on an appropriate choice of edge-effect terms. Interestingly, for a given refractive index, the optimal number of edge-effect terms almost insensitive to the aspect ratio (from 0.5 to 2.0) and sizes, although slight changes have been observed for a few refractive indices. Therefore, for practical applications, the optimal numbers for a sphere can be applied to spheroids. With a table of optimal edge-effect terms with respect to different refractive indices, the current approach permits accurate and real-time computing of extinction efficiencies. Needless to say, there remains little incentive to further develop formulae for small size parameters because the T-matrix method is already computationally efficient. It should be noted that the fundamental reason that leads to the existence of the optimal edge-effect remains unclear. The present study does not circumvent the possibility that including even higher-order (>4) terms will improve the accuracy for small size parameters without the truncation given in Fig. 5 and Fig. 8. The fundamental nature of CAM asymptotic series requires further investigation for theorists.

Funding

National Key Research and Development Program of China (2016YFC0200700); National Natural Science Foundation of China (41675025); Fundamental Research Funds for the Central Universities (2017QNA3017).

Acknowledgments

We acknowledge Ms. Rui Liu from the Training Center of Atmospheric Sciences of Zhejiang University for her effort related to computing resources. A portion of computations was performed on the cluster at State Key Lab of CAD&CG at Zhejiang University. The computation was also supported by National Supercomputer Center in Guangzhou (NSCC-GZ).

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Use of Debye’s series to determine the optimal edge-effect terms for computing the extinction efficiencies of spheroids

Abstract

1. Introduction

2. Computational methods

3. Results

4. Summary and conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Equations (22)

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