1. Introduction

The extinction of light by small particles is of great interest in many scientific and engineering disciplines. For example, the extinction efficiencies of hydrometeors in the atmosphere are essential to determining the attenuation of a radiation beam by clouds and are fundamentally related to the optical thickness as inputs into atmospheric radiative transfer simulations [1]. Among the particle characterization techniques necessary for application in a variety of fields, the spectral extinction technique is frequently used to retrieve the particle sizes based on comparisons between modeling extinction results and measurements [2–5]. Although significant progress has been achieved in light scattering calculations [6,7], currently sphere is the only geometry for which the optical properties can be exactly obtained. For nonspherical particles, rigorous computational techniques are relatively inefficient and generally have limited capabilities with respect to the size parameter. As an alternative, approximate methods or semi-empirical methods have been developed to efficiently obtain the data [8–13]. Using an approximate approach is often inevitable when rigorous solutions are unavailable. Unlike the exact methods that solve Maxwell’s equations, the development of semi-empirical methods requires well-defined physical concepts involved in the light extinction process. Thus, uncovering the physics underlying the Lorenz-Mie solution is not only motivated by theoretical interest, but also by a practical need of a method to obtain a semi-empirical solution for nonspherical particles. Moreover, because the semi-empirical methods for nonspherical particles lack a priori error estimation, a systematic comparison between semi-empirical results with rigorous counterparts, whenever possible, is always an indispensable part of justifying involved assumptions and quantifying the accuracy.

The physics underlying the extinction behavior of a sphere is well understood. Kokhanovsky and Zege [14] provided a review of approximate analytical solutions. For large particle sizes, Nussenzveig and Wiscombe [15,16] derived an expression of the asymptotic extinction efficiency for spheres (hereafter, the NW formula) from the Debye series [17] by using the complex angular momentum (CAM) method. In the CAM, the contribution to the extinction from the rays that incident beyond the edge of a sphere can be approximately quantified. Applying the CAM to nonspherical particles to obtain an analytical solution is more formidable, although the scattering problem can be solved semi-analytically in the spheroidal coordinate system with the method of separation of variables (SOV) [18]. However, the asymptotic extinction efficiency with the incorporation of the edge effects, obtained from the Lorenz-Mie theory, provides a prototype for understanding the light scattering by nonspherical particles. A number of studies based on semi-empirical principles have been undertaken to incorporate the edge effects into the extinction efficiencies of nonspherical particles predicted from the anomalous diffraction theory (ADT) [9] and geometric optics methods (GOM) [19–22]. Fournier and Evans [9] developed an approximate formula (hereafter, the FE formula) for calculating the extinction efficiency for randomly oriented spheroids. The FE formula is obtained by extending the approximate formula for a sphere derived from the ADT, developed by van de Hulst [23], to spheroids, and includes an edge-effect contribution term following Jones’s assumption [24,25] (see Section 4 for details regarding the assumption). In comparison with the extended-boundary condition method (EBCM) [26] coded by Hill et al. [27], the FE formula can obtain the extinction efficiency for randomly oriented spheroids with relatively reasonable accuracy; however, the FE formula is not always accurate for specifically oriented spheroids (e.g., the end-on incidence for prolate spheroids, and the side-on incidence of oblate spheroids). The physical explanation is that inconsistency is incurred when extending the approximate theory from optically soft particles to optically hard particles. In addition, the absorption-dependent edge effect terms are not considered in the FE formula. Based on a generalized eikonal approximation (GEA), Chen [28] derived an analytical expression of the extinction efficiency of a sphere and proposed two ad hoc parameters to include the edge effects. The GEA has been extended by Wang et al. [5] to calculate the extinction efficiency of randomly oriented spheroids. The GEA accuracy was studied by comparing the results with the counterparts computed with the SOV [29]. With noticeable differences, the GEA captures the main extinction characteristics. Note that, in both studies [5,9], the rigorous results serving as comparison benchmarks have a limited size parameter range (less than approximately 30).

In this paper, we present the rigorous extinction efficiencies of spheroids computed from the invariant imbedding T-matrix method (II-TM) [30–32] in a relatively wide size parameter range from the Rayleigh regime to the geometric-optics regime. With the T-matrix solution as a reference, new insight is gained to study the extinction efficiency of spheroids in a large size parameter range from the perspective of a semi-empirical approach. Different from previous studies [5,9,10], we develop a new approximate formula to extend the NW formula from spheres to spheroids on the basis of physical rationales associated with the change of the particle morphology. The new high-frequency formula is valid under both random and fixed orientation conditions. In Section 2, we summarize the T-matrix methodology for the extinction efficiency calculation. The NW formula [15] is recaptured in Section 3 and the accuracy is examined in order to obtain an optimal number for the truncation of the edge-effect series. Stemmed from the semi-classical Mie extinction efficiency, a semi-empirical approximation for the spheroid extinction efficiency is developed in Section 4. The physical and mathematical fundamentals for extending the NW formula to spheroids include the end-on incidence, the oblique incidence, and the random orientation scenarios. In Section 5, we present a comprehensive comparison between the extinction efficiency determined from the T-matrix method and the new approximation formula. The summary and conclusions are given in Section 6.

2. T-matrix method

The T-matrix method is a first-principle approach to solving Maxwell’s equations. In the T-matrix method, the incident field and the scattered field are expanded in terms of vector spherical functions, representing a complete basis of the fields satisfying the vector Helmholtz equation. The expansion coefficients of the scattered field and the incident field are related through a matrix, namely the T-matrix, which contains all the light extinction process information. A general framework and technical details associated with the T-matrix are found in Ref. [33]. We use the II-TM computational program developed in Ref. [31] to compute the T-matrix. In this section, we summarize the method of computing the extinction efficiencies from the T-matrix and illustrate the II-TM’s computational capability and efficiency.

Figure 1 is a schematic diagram showing prolate and oblate spheroids; the semi-horizontal axis and semi-vertical axis are denoted as $a$and $c$, respectively. Let $\lambda$be the wavelength of the incident light and $\theta$ be the angle between the direction of the incident light and the symmetric axis $c$ of the spheroid. The size parameters given with respect to two semi-axes are ${\beta }_a=ka$ and ${\beta }_c=kc$, where $k=2\pi /\lambda$. Following the convention of the definition given in Ref. [33], the T-matrix is denoted as $T_{mnm\text{'}n\text{'}}$, where $n(n\text{'})$ is the angular momentum number and $m(m\text{'})$is the projected angular momentum number. For spheroids with the symmetric axis aligned with the z-axis in the laboratory coordinate system, the T-matrix is decoupled in terms of $m$, namely, $T_{mnm\text{'}n\text{'}}=T_{mnm\text{'}n\text{'}}{\delta }_{mm\text{'}}$. For computing the extinction efficiency of a spheroid, different procedures are employed for fixed and random orientations. For the end-on incidence (i.e.,$\theta =0$), the extinction efficiency is obtained from the sub-matrix $T_{1n1n\text{'}}$with $m=m\text{'}=1$,

 

Fig. 1 Prolate spheroid (a) and oblate spheroid (b). The semi-axis of rotation is denoted as $c$. The other axis is denoted as $a$. The size parameters for the two axes are denoted as ${\beta }_c$and ${\beta }_a$.

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To illustrate the II-TM’s efficiency, Table 1 shows the computational time for obtaining the optical properties of large spheroids. All the simulations are performed by using the Texas A&M supercomputing facilities, particularly, on a supercomputer whose single node contains eight 64-bit 2.8 Ghz processors. For the end-on incidence, the program is implemented with Open Multi-Processing (OpenMP) and 8 processors are used in the computation. For randomly oriented spheroids, the program is parallelized based on the Message Passing Interface (MPI) and two nodes with 16 processors are used in the present computation. The computational wall time for the end-on incidence is much less than that of random orientation because $T_{1n1n\text{'}}$ is only required for the end-on incidence. As seen from Table 1, for a fixed size parameter, the computational time increases with $\left|1-a/c\right|$, because the radius of the inscribed sphere is smaller, requiring more iterative steps in the invariant imbedding procedure starting from the inscribed sphere and ending at the circumscribed sphere. For a fixed aspect ratio, the computational time increases significantly with respect to the size parameter because the T-matrix becomes large and the iterative steps increase. Note that, computationally, the EBCM is much more efficient than the II-TM in the EBCM’s applicable domain; however, for consistency, we use the II-TM for all the computational cases in this study.

Table 1. Computational Wall Time of the II-TM for Spheroids with the End-on Orientation and Random Orientations for Different Aspect Ratios and Size Parameters

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3. Semi-classical Mie extinction efficiency

Semi-empirical theory provides an approach to economizing the numerical simulations for large size parameters while keeping reasonable accuracy. In this section, we recapture the semi-classical formalism of the Mie extinction efficiency, which serves as a fundamental basis for the counterpart for spheroids in Section 4. In addition, a detailed comparison between the approximate results and the exact Lorenz-Mie solutions are conducted. To the best of our knowledge, no studies have been conducted on the optimal number needed to truncate the asymptotic expansion of the edge-effect contribution.

According to the physical interpretation of each term, the extinction efficiency derived by Nussenzveig and Wiscombe [15] based on the Debye series of the Lorenz-Mie solution and the CAM, can be written as being composed of two parts

The accuracy of Eq. (6) can be assessed in comparison with the Lorenz-Mie theory. The accuracy generally depends on $\left(m-1\right)\beta$. For example, for optically soft particles ($m$ = 1.01), Eq. (6) is reasonably accurate when $\beta$ is larger than approximately 200; however, when $m$ = 1.1, the results from Eq. (6) have reasonable accuracy when $\beta$is larger than approximately 20. In addition, Eq. (7) does not include the surface wave contribution, i.e., the ripple structure of the extinction curve. $Q_{ext,geo}$is invalid at $m=2j+1$ because the denominator $2j-m+1$ is zero, and the summation terms associated with higher-order central rays in Eq. (7) are obtained under the condition that m_r is smaller than $\sqrt{2}$. Furthermore, we find that the inclusion of higher-order terms does not necessarily improve the accuracy of the resultant extinction efficiency.

In Ref. [15], the accuracy of Eq. (6) is scrutinized by including the four edge-effect terms. However, we found that the accuracy of Eq. (6) depends on an appropriate number of terms and excessive terms can deteriorate the resultant accuracy for certain values of the refractive indices. The optimal number of terms is more sensitive to the refractive index (in particular, a critical value of the real part for a fixed imaginary part of the refractive index) than to the size parameter. According to the Lorenz-Mie theory, an appropriate number of terms can be determined through the comparison of approximate solutions against the rigorous solutions.

For non-absorptive and weakly absorptive particles, we have tested the accuracy of Eq. (6) for refractive indices $m_r+i{m}_i$, with $m_r$ from 1.0 to 2.0 with the interval of 0.5 and $m_i$ from 10⁻⁷ to 10⁻³. By comparing the trend of the extinction curve with respect to the size parameter, we found that four terms are required when$m_r$is less than 1.45 but the accuracy deteriorates by the third or the fourth term for a larger $m_r$. Note that because $m_i$is small, $c_2$is nearly zero.

When $m_i$increases, $c_2{\beta }^{-1}$ generally becomes nontrivial. However, the constant $c_2$is zero for absorptive particles with the refractive indices$m_{r,0}+i{m}_{i,0}$, satisfying the following conditions:

 

Fig. 2 Comparison of the extinction efficiencies of a sphere computed from the Lorenz-Mie theory and Eq. (6). The accuracy of the inclusion of the edge-effect terms is illustrated.

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4. Semi-empirical approach for spheroids

4.1 End-on Incidence

In this section, we focus on the extinction efficiency of dielectric spheroids with the propagation direction of an incident plane wave aligned with the symmetric axis of the particle. The end-on incidence is the simplest scattering case for nonspherical particles and the scattering process is similar to that of sphere. For example, the extinction is independent of the polarization state of the incident light, the forward transmission is associated primarily with the central rays, and the boundary separating the illuminated side and the shadow side is a circle. Based on the similarities, we modify Eq. (6) by including the particle aspect ratio based on ray optics and Jones’s explanation of the edge effect. The validity of the modified equation will be assessed in comparison with the rigorous results calculated from Eq. (1).

We first look at the term associated with the interference between the diffraction and the forward transmission. In ray optics, the amplitude of a scattered ray [23,38–40] is given by

When the particle becomes optically harder, Eq. (19) diverges at the aspect ratio of $\sqrt{m/(m-1)}$. Figure 3 displays a defined modified factor $f=S_{spheroid}/S_{sphere}$ as a function of the particle aspect ratio for different refractive indices. As can be seen from Fig. 3, the modified formula Eq. (19) is valid for all oblate spheroid cases, but only valid for a prolate spheroid with a particle aspect ratio larger than a critical value $\sqrt{m/(m-1)}$ where the denominator of Eq. (18) is zero. At the critical aspect ratio, the ray has no divergence in the forward scattering direction and the forward glory occurs [38].

 

Fig. 3 Shape factor associated with the divergence of the first order transmitted ray. (a) prolate spheroid. (b) oblate spheroid. Note that, for a prolate spheroid, the shape factor has delta peaks, which correspond to the forward glory.

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Now we turn to the modification of $Q_{ext,edge}$. According to van de Hulst [23] and Jones’s explanation [24,25], the edge-effect contribution to the extinction cross section is associated with the curvature of a profile at the glancing point. In the case of the end-on incidence, the profile is an ellipse and the radius of the curvature is the same at each glancing point and equals $c^2/a$. The first order edge-effect contribution to the extinction cross section at one glancing point within a differential length $ds$ of the boundary separating the illuminated and the shadow sides is given by

4.2 Oblique incidence

To extend the formula to the oblique incidence case, for convenience, Eq. (6) is rewritten as:

Now let us consider the oblique incidence case. Figure 4 shows a schematic diagram of light scattering by a spheroid. The red curve is a boundary formed by glancing points. At each glancing point, the profile is defined to be the curve on the spheroid surface with the incident direction as the tangent vector. The position of the initial point of a ray is denoted as ($\eta$,$\xi$), which means,

 

Fig. 4 A schematic diagram to illustrate the oblique incidence. The wave front blocked by the geometry is an ellipse. The red curve is a boundary separating the illuminated and shadow side. The location within the ellipse is denoted as ($\eta$,$\xi$).

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Table 2. Coefficients for Edge-effect Terms in Eq. (45)^a

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4.3 Random orientation

The semi-empirical formula for an arbitrary orientation of a spheroid is given in the previous section, and the average extinction cross section can be calculated through a numerical averaging procedure,

The accuracy of $〈Q_{ext}〉$ depends on the accuracy of $Q_{ext}(\mu )$. However, if $Q_{ext}(\mu )$ is relatively inaccurate at the orientation angles where the weighting function has small values, the averaged extinction efficiency may remain satisfactorily accurate. For the prolate spheroid, the weight is a minimum for the end-on incidence ($\mu =1$) and reaches a maximum for the side-on incidence ($\mu =0$). Ref. [9] demonstrates that for prolate spheroids, the inaccuracy of the FE extinction efficiency at the end-on incidence has negligible effect on the final averaged extinction efficiency because the weight is quite small. The same reasoning is applied to oblate spheroids, for which the FE extinction efficiency at the side-on incidence is relatively inaccurate.

For transparent and semi-transparent particles, $Q_{ext}(\mu )$ tends to be an oscillating function. The induced error may be cancelled in the numerical averaging process. Note that, for each orientation, a numerical procedure is required to compute the divergence factor. However, we find that, for random orientations, the assumption that the divergence factor is the same as that of a sphere has little effect on the averaged extinction efficiency. Therefore, when the random orientation scenario is considered, we assume the convergence correction factor to be unity to avoid tedious numerical computations.

5. Computational results

The spheroid extinction efficiencies with the II-TM and the approximate formula for a set of optical parameters are computed. The aspect ratio ($a/c$) is constrained from 0.5 to 2.0 with steps of 0.05. The real parts of the refractive indices are chosen to be 1.1, 1.3, 1.5, 1.7, and 2.0. For each real part of the refractive index, five imaginary part values are chosen for the calculations; 10⁻⁷, 10⁻³, 10⁻², 10⁻¹, and 0.5. In the II-TM simulations, the extinction efficiencies for randomly oriented spheroids and for 91 fixed orientations specified with $\theta =0^{\text{o}},1^{\text{o}},2^{\text{o}}\cdots {90}^{\text{o}}$ are computed. In the following discussions, some representative results are presented for illustrating the high-frequency extinction characteristics and the accuracy of the developed semi-empirical calculations.

Figure 5 shows the comparison between the extinction efficiencies of five prolate spheroids ($a/c<1$) and five oblate spheroids ($a/c>1$) as a function of the size parameter. The incident light is aligned with the symmetric axis (i.e., the end-on incidence), and the refractive index is 1.1 + i10⁻⁷. A small imaginary part of the refractive index corresponds to negligible absorption, and the interference between the diffraction and the transmission is evident. For prolate spheroids, because the phase shift is the same for all particle aspect ratios, the peaks and troughs are located at the same size parameter. However, the divergence of the central ray and the edge effect are different and the values of the extinction efficiencies differ. From the comparison, the II-TM agrees excellently with the approximations when the size parameter is larger than 10. For oblate spheroids, the phase shift varies for different aspect ratios because the size parameter is defined in terms of the larger semi-axis (namely, ${\beta }_a$ for oblate spheroids). The peak shift for different aspect ratios is obvious, and the size parameter at which the II-TM and the approximate results converge is relatively larger than those of prolate spheroids. For small size parameters, the preset formula fails, because the semi-classical approximation breaks down. In this region, the semi-empirical formula in Fournier and Evans [9], incorporating a Rayleigh formula, the ADT, and a bridge function, might be a better choice.

 

Fig. 5 Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formula. The incident light is aligned with the symmetric axis. 10 aspect ratios are selected for illustration. The refractive index is assumed to be 1.1 + i10⁻⁷.

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Figure 6 displays the results for spheroids with the refractive index of 1.1 + i0.5. The particles are now highly absorptive. The contribution part of the extinction efficiency from the diffraction and blocking can be well approximated by 2, indicating that the difference between the total efficiency and 2 stems essentially from the edge effect. Figures 6(a) and (b) are plots of one prolate spheroid and one oblate spheroid with three size parameters: 20, 50, and 100. We can see that the II-TM and the present approximation agree for all the orientations from the end-on incidence (θ = 0°) to the side-on incidence (θ = 90°). The differences for the size parameter of 20 are relatively larger, because the semi-classical approximation is valid for large size parameters and does not always yield better results for smaller size parameters. To illustrate the comparison of the results for all the aspect ratios simulated, Fig. 6(c) plots the extinction efficiencies as a function of the aspect ratio for three selected orientations (θ = 0°, 45°, and 90°). Figure 6(d) is the same as Fig. 6(c) except the size parameter is 100. The II-TM and the approximate results agree very well. The three curves have a common point at the unity aspect ratio, because the extinction efficiency is the same for different orientations of a sphere.

 

Fig. 6 Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formula. The refractive index is 1.1 + i0.5. (a) The extinction efficiency is plotted against the particle orientation. The aspect ratio is 0.8. Three size parameters (20, 50, and 100) are selected. (b) Similar to (a) except that the aspect ratio is 1.6. (c) The extinction efficiency is plotted against the particle aspect ratio. The size parameter is 50. Three orientations (θ = 0°, 45 ^o, 90 ^o) are selected. (d) Similar to (c) except that the size parameter is 100.

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When the particle becomes optically harder, comparing the results computed by the FE and the present formulas is interesting because the FE formula is obtained from the ADT approximation. As an example, Fig. 7 displays the results for spheroids with the refractive index of 1.3 + i10⁻⁷. As in Fig. 5, the incident light is aligned with the symmetric axis. For oblate spheroids and prolate spheroids with the aspect ratio close to unity, the ripple structure of the extinction curve is obvious. However, for prolate spheroids with $a/c$ much smaller than unity, the ripple structure disappears. Similar findings have been reported in [43], where the extinction efficiency is computed from the method of separation of variables with the size parameter less than 30. From the comparison, the present formula demonstrates better accuracy than the FE formula, in particular, for elongated prolate spheroids. The differences between the two approximations at the end-on incidence become much smaller for oblate spheroids. However, for oblate spheroids, the accuracy of the FE extinction efficiency for the side-on incidence becomes worse, similar to that of prolate spheroids with the end-on incidence (the results are not shown).

 

Fig. 7 Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formulas. The incident light is aligned with the symmetric axis. The refractive index is 1.3 + i10⁻⁷.

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Table 3 contains the extinction efficiencies for spheroids with excessively large size parameters. As can be seen from the table, the present formula has a precision up to three to four significant digits. The FE(a) includes all edge effect terms similar to the present formula. The same as Ref. [9], the FE(b) includes only the leading edge-effect term, namely, $\text{1}\text{.9924}{\beta }^{-2/3}$ (the FE formula mentioned in all other places is the FE(b) defined here). For the refractive indices, 1.3 + i10⁻⁷ and 1.3 + i0.001, the FE(a) and the FE(b) have close results because the contributions from higher-order terms are small. The accuracy of the present formula is better than the FE(a) and the FE(b) in comparison with the II-TM because the geometric optics term Eq. (19) is more accurate than the ADT formula Eq. (21). For the refractive index, 1.3 + i0.5, Eq. (19) and Eq. (21) are equal to two because the particle is highly absorptive such that the interference between diffraction and transmission is suppressed. However, the higher-order terms (in particular, $c_2{\beta }^{-1}$) have observable effects on the resultant accuracy because $c_2$ is a relatively large number for absorptive particles. Consequently, the differences between the FE(a) and the FE(b) results become larger, and the differences between the FE(a), the present formula, and the II-TM become smaller. From the comparison of decimal points for “asymptotic” extinction efficiencies, the present formula is well validated regarding the modifications of the geometric optics terms and the edge-effect terms. Note that, to make the comparison of decimal points meaningful, we have excluded the cases where the ripple structures are pronounced, particularly, in the case of transparent oblate spheroids.

Table 3. Extinction Efficiencies of Spheroids Computed from the II-TM, the Present Formula, and the FE Formula

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Figure 8 illustrates the accuracy of the new approximation in comparison with the FE approximation for spheroids with 91 fixed orientations. Assuming $f$to be unity in Eq. (41) slightly affects the resultant accuracy. The new approximation is evidently better than the FE formula for spheroids with fixed orientations. The FE formula overestimates or underestimates the extinction efficiency depending on the orientation angles. In the case of random orientations, the extinction cross-sections for individual orientations are averaged. The accuracy of the FE results may be improved due to the error cancellation. The values of the extinction efficiencies of randomly oriented spheroids are given in Table 4. The results computed from the three approximations have comparable accuracy.

 

Fig. 8 Comparison between the extinction efficiencies of spheroids as a function of particle orientation computed from the II-TM, the FE approximation, and the present new approximate formula. The refractive index is 1.3 + i10⁻⁷. The aspect ratio and the size parameters are indicated in the figure.

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Table 4. Extinction Efficiencies of Randomly Oriented Spheroids for Four Size Parameters Shown in Fig. 8

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Figure 9 shows the extinction efficiency of a randomly oriented spheroid for two larger different refractive indices computed from both the II-TM and the approximate formula. As is evident, the approximate formula captures the main physics associated with the amplitude of the oscillation and the phase shift. The subtle structure associated with the surface waves cannot be taken into account in the semi-empirical formula, and for the comparison purposes, the FE results are included. The present formula is slightly more accurate than the FE formula regarding the amplitude of oscillation. Here, the approximate solutions by the FE method and the present approach are shown only for size parameters with ${\beta }_c>5$, a physically reasonable range of semi-empirical approximation. Note that one edge-effect term is included in Fig. 9(a) and three edge-effect terms are included in Fig. 9(b).

 

Fig. 9 Comparison between the extinction efficiencies of spheroids computed from the II-TM and the approximate formula. The aspect ratio is 0.8.

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Figure 10 illustrates the error of the semi-empirical approach with comparisons made between the T-matrix results as a function of the size parameter and the aspect ratio for four refractive indices. The particle is assumed to be randomly oriented. The difference is quantified through the ratio $Q_{ext,approximation}/Q_{ext,II-TM}$. As can be seen from the error display, the semi-empirical formula yields results with excellent accuracy. A relatively large error is found near the small size parameters and extreme aspect ratios, most likely because the radius of curvature at the particle surface is relatively small causing the inaccuracy of the semi-classical approximation. When the size parameter is larger than 20, the differences become negligible. The present results are slightly more accurate than the FE results. For a concise presentation, the FE results are not shown here, because Figs. 7–9 illustrate the differences.

 

Fig. 10 The ratios of the extinction efficiencies computed from the present formula and the II-TM.

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6. Summary and conclusion

We investigate the extinction efficiencies of dielectric spheroids using the invariant imbedding T-matrix method in a wide size parameter range. The accuracy of the semi-classical Mie extinction efficiency is assessed and the NW formula is extended to spheroids with the modifications based on ray optics and Jones’s explanation of the edge effect. The semi-empirical approximation yields results close to the rigorous solution but is computationally more efficient (e.g., the computational time for approximate methods is on the order 10⁻⁴ seconds).

For transparent spheroids with end-on incidence and aspect ratios close to that associated with the forward glory, the semi-empirical formula fails because of the inapplicability of ray optics. In this scenario, Fresnel and Kirchhoff diffraction laws should be adopted for semi-empirical calculations. However, for randomly oriented spheroids, the drawback does not influence the accuracy of the final results, because of averaging over particle orientations. Therefore, the semi-empirical calculation applies to all particle aspect ratios.

An inconvenience in the present study is that a priori knowledge needs be obtained from the Lorenz-Mie theory regarding whether the higher-order edge effect terms should be included. To remove this inconvenience, further theoretical investigations on the semi-classical Mie extinction efficiency are most likely required or, practically, a table of optimal number of terms can be created with respect to typical refractive indices, although the operational principle behind this truncation is not clear. However, if no treatments are made (i.e., all higher-order edge-effect terms are included for higher real part of refractive indices), the additional resultant errors may not exceed ~5% (see Fig. 2) for moderate size parameters.

We do not consider the calculation of the extinction efficiency based on simplified formulas for small size parameters. Note, little incentive exits for using an approximate method for small size parameters because the II-TM (or, EBCM) is efficient for these cases. Moreover, employing the bridge function developed by Fournier and Evans [9] in transition regions from Rayleigh scattering to the semi-classical domains is theoretically straightforward.

For spheroids, the radius of curvature on the particle surface is finite, and Jones’s approximation is valid. For cylinders with the end-on incidence or for faceted particles, the radius of curvature is infinite and Jones’s assumption breaks down because of the infinite edge-effect contribution [44]. However, from the comparison between rigorous solutions and the geometric-optics results, the edge-effect contribution is finite and obvious. Thus, an accurate quantification of the edge-effect contribution is an open question and will require further investigation.

Appendix

Figure 11 shows a case with the end-on incidence. Analytical formulas required to calculate the scattering angle $\Theta$ for a general entry point can be found in Lock and McCollum [45]. Here, we derive the derivative of $\eta$ with respect to $\Theta$ when $\eta \to 0$. For small $\eta$, the entry point coordinates expressed in terms of $\delta$are given by

(50)$$x_0=\frac{c\mathrm{si}n\delta }{\sqrt{{\cos }^2\delta +c^2/a^2{\sin }^2\delta }}\approx c\delta \left(1-\frac{1}{2}\frac{c^2}{a^2}{\delta }^2\right),$$

(51)$$y_0=\frac{-c\cos \delta }{\sqrt{{\cos }^2\delta +c^2/a^2{\sin }^2\delta }}\approx -c\left(1-\frac{1}{2}\frac{c^2}{a^2}{\delta }^2\right).$$

The incident angle is proportional to $\delta$in the form

(52)$${\theta }_i=\text{arctan}\left(\frac{c^2}{a^2}\tan \delta \right)\approx \frac{c^2}{a^2}\delta .$$

Based on Snell’s law, the deflection angle after the first-order refraction can be written as

(53)$${\Delta }_d={\theta }_i-{\theta }_t\approx \left(1-\frac{1}{m}\right)\frac{c^2}{a^2}\delta .$$

Using the Taylor expansion, we have

(54)$$\cos {\Delta }_d\approx 1-\frac{1}{2}{\left(1-\frac{1}{m}\right)}^2\frac{c^4}{a^4}{\delta }^2,$$

(55)$$\sin {\Delta }_d\approx \left(1-\frac{1}{m}\right)\frac{c^2}{a^2}\delta .$$

The length of the trajectory within the particle is approximately

(56)$$L\approx 2c+{\rm O}(\delta ).$$

In the limiting process, the coordinates of the next intersection point are

(57)$$x_1=x_0-L\sin {\Delta }_d\approx c\delta \left[1-2\left(1-\frac{1}{m}\right)\frac{c^2}{a^2}\right]+{\rm O}({\delta }^2),$$

(58)$$y_1=y_0-L\cos {\Delta }_d\approx c+{\rm O}({\delta }^2).$$

The incident angle upon the second-order refraction is given by

(59)$${\overline{\theta }}_i=\arctan \left(\frac{c^2}{a^2}\frac{x_1}{y_1}\right)+{\Delta }_d\approx \frac{c^2}{a^2}\left[1-2\left(1-\frac{1}{m}\right)\frac{c^2}{a^2}\right]\delta +{\Delta }_d.$$

The scattering angle can now be expressed in terms of $\delta$as follows:

(60)$$\Theta ={\overline{\theta }}_t-{\overline{\theta }}_i+{\Delta }_d=\left[2\left(m-1\right)\frac{c^2}{a^2}-\frac{2{(m-1)}^2}{m}\frac{c^4}{a^4}\right]\delta .$$

When $\eta$is small, we have

(61)$$\eta \approx \frac{c}{a}\delta .$$

Therefore,

(62)$$\Theta =\frac{2\left(m-1\right)}{m}\left[m\frac{c}{a}-\left(m-1\right)\frac{c^3}{a^3}\right]\eta .$$

The derivative of interest is given by

 

Fig. 11 A schematic diagram of the trajectory of the first-order refracted ray.

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(63)$$\underset{\eta \to 0}{\lim }\frac{\partial \eta }{\partial \Theta }=\frac{m}{2(m-1)}\frac{1}{\left[mc/a-(m-1)c^3/a^3\right]}.$$

Acknowledgments

This study was supported by a NSF grant (AGS-1338440), a NASA grant (NNX11AK37G), and the endowment funds related to the David Bullock Harris Chair in Geosciences at the College of Geosciences, Texas A&M University. A major portion of the simulations was carried out at the Texas A&M University Supercomputing Facilities, and the authors gratefully acknowledge Facility staff for their help and assistance. The authors also gratefully acknowledge the effort by two anonymous reviewers to improve the manuscript.

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