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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 10270–10293
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High-frequency extinction efficiencies of spheroids: rigorous T-matrix solutions and semi-empirical approximations

Lei Bi and Ping Yang  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 10270-10293 (2014)
http://dx.doi.org/10.1364/OE.22.010270


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Abstract

A semi-empirical high-frequency formula is developed to efficiently and accurately compute the extinction efficiencies of spheroids in the cases of moderate and large size parameters under either fixed or random orientation condition. The formula incorporates the semi-classical scattering concepts formulated by extending the complex angular momentum approximation of the Lorenz-Mie theory to the spheroid case on the basis of the physical rationales associated with changing the particle morphology from a sphere to a spheroid. The asymptotic edge-effect expansion is truncated with an optimal number of terms based on a priori knowledge obtained from comparing the semi-classical Mie extinction efficiencies with the Lorenz-Mie solutions. The present formula is fully tested in comparison with the T-matrix results for spheroids with the aspect ratios from 0.5 to 2.0, and for various refractive indices mr + imi, with mr from 1.0 to 2.0 and mi from 0 to 0.5.

© 2014 Optical Society of America

1. Introduction

The physics underlying the extinction behavior of a sphere is well understood. Kokhanovsky and Zege [14

14. A. A. Kokhanovsky and E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. 28(1), 1–21 (1997). [CrossRef]

] provided a review of approximate analytical solutions. For large particle sizes, Nussenzveig and Wiscombe [15

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]

,16

16. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).

] derived an expression of the asymptotic extinction efficiency for spheres (hereafter, the NW formula) from the Debye series [17

17. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

] 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

18. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [CrossRef] [PubMed]

]. 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

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

] and geometric optics methods (GOM) [19

19. P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007). [CrossRef]

22

22. K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011). [CrossRef]

]. Fournier and Evans [9

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

] 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

23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

], to spheroids, and includes an edge-effect contribution term following Jones’s assumption [24

24. D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A 239, 338–348 (1957). [CrossRef]

,25

25. D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957). [CrossRef]

] (see Section 4 for details regarding the assumption). In comparison with the extended-boundary condition method (EBCM) [26

26. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971). [CrossRef]

] coded by Hill et al. [27

27. S. C. Hill, A. C. Hill, and P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. 23(7), 1025–1031 (1984). [CrossRef] [PubMed]

], 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

28. T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28(19), 4096–4102 (1989). [CrossRef] [PubMed]

] 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

5. L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012). [CrossRef]

] 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

29. N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993). [CrossRef]

]. With noticeable differences, the GEA captures the main extinction characteristics. Note that, in both studies [5

5. L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012). [CrossRef]

,9

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

], the rigorous results serving as comparison benchmarks have a limited size parameter range (less than approximately 30).

2. T-matrix method

To illustrate the II-TM’s efficiency, Table 1 shows the computational time for obtaining the optical properties of large spheroids.

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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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 T1n1n' is only required for the end-on incidence. As seen from Table 1, for a fixed size parameter, the computational time increases with |1a/c|, 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.

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

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]

] based on the Debye series of the Lorenz-Mie solution and the CAM, can be written as being composed of two parts
Qext=Qext,geo+Qext,edge,
(6)
with
Qext,geo=28βIm{m2(m1)(m+1)2e2i(m1)β×[1i16β(516+12m9mm1)j1(m1)2j+1(m+1)2je4imjβ2jm+1+O(β2)]},
(7)
Qext,edge=c1β2/3+c2β1+c3β4/3+c4β5/3,
(8)
c1=1.9924,
(9)
c2=2Im[(m2+1)/N],
(10)
c3=0.7154,
(11)
c4=0.6641Im[eiπ/3(m2+1)(2m46m2+3)/N3].
(12)
wherem=mr+imiis the complex refractive index; N=m21;β is the size parameter defined as 2πa/λ, where a is the radius of sphere; and, Im indicates the imaginary part of the index of refraction. Equation (7) is associated with the diffraction, blocking of light, and the interference between the diffraction and the forward transmission, and the term i/(16β)[51/6+1/(2m)9m/(m1)]is modified according to Ref. [37

37. W. T. Grandy, Scattering of waves from large spheres (Cambridge University, 2000).

]. Two principal features of the extinction can be explained by Eq. (7): the asymptotic value of 2; and, the macroscopic oscillation of the extinction curve plotted versus the size parameter. The first and third terms in Eq. (8) correspond to the edge-effect contribution in conjunction with the particle geometry, and the second and fourth terms in Eq. (8) are the edge-effect terms dependent on the geometry and the refractive index. Note that some misprints in Eq. (1) of Ref. [15

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]

] have been corrected in Eq. (13.12) of Ref. [16

16. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).

] and Ref. [37

37. W. T. Grandy, Scattering of waves from large spheres (Cambridge University, 2000).

]. However, e4ijβin Eq. (13.11) of Ref. [16

16. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).

] should be e4imjβ, and N2 in Eq. (6.3) of Ref. [37

37. W. T. Grandy, Scattering of waves from large spheres (Cambridge University, 2000).

] should be N3. Because of the existence of the edge effect, the extinction efficiency usually approaches two from above (>2). As the particle size parameter increases, the edge effect contribution decreases. In particular, the first term in Eq. (8) dominates the edge-effect contribution for large size parameters (>~100).

The accuracy of Eq. (6) can be assessed in comparison with the Lorenz-Mie theory. The accuracy generally depends on (m1)β. For example, for optically soft particles (m = 1.01), Eq. (6) is reasonably accurate when β is larger than approximately 200; however, when m = 1.1, the results from Eq. (6) have reasonable accuracy when βis larger than approximately 20. In addition, Eq. (7) does not include the surface wave contribution, i.e., the ripple structure of the extinction curve. Qext,geois invalid at m=2j+1 because the denominator 2jm+1 is zero, and the summation terms associated with higher-order central rays in Eq. (7) are obtained under the condition that mr is smaller than 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

15. H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]

], 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 mr+imi, with mr from 1.0 to 2.0 with the interval of 0.5 and mi 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 whenmris less than 1.45 but the accuracy deteriorates by the third or the fourth term for a larger mr. Note that because miis small, c2is nearly zero.

When miincreases, c2β1 generally becomes nontrivial. However, the constant c2is zero for absorptive particles with the refractive indicesmr,0+imi,0, satisfying the following conditions:
mr,0=(1mi,02)+21mi,02,0<mi,0<1.
(13)
c2is negative formr+imi,0when mr<mr,0 and becomes positive when mr>mr,0. For complex refractive indices with mi1,c2is always positive. When the imaginary part is fixed, c2increases with respect to the real part mrand approaches to the asymptotic value (2mi). Similar to the weakly absorptive particles, we tested the accuracy of Eq. (6) for refractive indices mr+imi, with mr from 1.0 to 2.0. The corresponding imaginary part of the refractive index, mi, varies from 0.01 to 0.1 with an interval of 0.01 and from 0.1 to 0.5 with an interval of 0.1. As an example, Fig. 2 shows the comparison of the extinction efficiencies for representative refractive indices.
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.
For the sake of comparison, two curves, with one corresponding to the optimal number of expansion terms, are included. As is evident, the optimal number depends on the refractive index. The same optimal number will be considered in the computation of the extinction efficiencies of spheroids in Section 4.

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

23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

,38

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35(3), 515–531 (1996). [CrossRef] [PubMed]

40

40. F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45(20), 5000–5009 (2006). [CrossRef] [PubMed]

] is given by
S(Θ)=ka(ηΘηsinΘ)1/2ε,
(14)
where Θis the scattering angle, η is a defined variable such that the impact factor,ηa, specifies the distance between an incident ray and the central ray. Note, η=0corresponds to the central ray, whereas η=1 indicates the ray of glancing incidence. In Eq. (14), the coefficient ε is associated with the Fresnel reflection and refraction and is independent of the aspect ratio for the central ray. According to the optical theorem, the extinction cross section is associated with the amplitude scattering matrix in the forward direction. Here, we are interested in the transmitted rays without undergoing internal reflections because high order transmitted rays are less important (see relevant discussion in [23

23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

] for a sphere). Thus, at Θ=0, Eq. (14) is simplified as
S=kalimη0ηΘε,
(15)
and
ε=[1(1m1+m)2].
(16)
In the case of a sphere, van de Hulst [23

23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

] proved the following relation:
Ssphere=kam2(m1)ε=ka2m2(m+1)(m21).
(17)
For a spheroid, after some mathematical analysis (see Appendix), we obtain
Sspheroid=kam2(m1)ε|mc/a(m1)c3/a3|.
(18)
When the sphere is modified to be a spheroid, a change in the phase delay is obvious, given by ρ=2(m1)βc, where βc=kc. Based on Eq. (18), the change of the phase delay and the cross-sectional area of πa2, Qext,geois modified to be
Qext,geo=28βc(ca)Im{m2(m1)(m+1)2e2i(m1)βcmc/a(m1)c3/a3×[1i16βc(516+12m9mm1)j1(m1)2j+1(m+1)2je4imjβc2jm+1+O(βc2)].
(19)
For optically soft particles (i.e.,m1), Eq. (19) reduces to the following form:
Qext,geo=24Im[eiρρ(1+i98ρ)].
(20)
According to the ADT, the extinction efficiency is given in the form [9

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

]
Qext,geo=24Im[eiρρ+ieiρ1ρ2].
(21)
Equations (20) and (21) are independent of the aspect ratio and have the same lower order terms but different higher order terms. Note that the FE formula directly extends Eq. (21), valid for optically soft particles, to optically hard particles without modifications except for a change in the phase delay. In the present formalism, Eq. (19) is essentially valid for all the refractive indices.

When the particle becomes optically harder, Eq. (19) diverges at the aspect ratio of m/(m1). Figure 3 displays a defined modified factor f=Sspheroid/Ssphere as a function of the particle aspect ratio for different refractive indices.
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.
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 m/(m1) 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

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35(3), 515–531 (1996). [CrossRef] [PubMed]

].

Now we turn to the modification of Qext,edge. According to van de Hulst [23

23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

] and Jones’s explanation [24

24. D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A 239, 338–348 (1957). [CrossRef]

,25

25. D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957). [CrossRef]

], 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 c2/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
C1ext,edge=(0.9962k2/3(c2a)1/3)ds.
(22)
Equation (22) is independent of the location of the glancing point. Multiplying Eq. (22) by 2πa/(πa2)yields the efficiency,
Q1ext,edge=1.9924βc2/3(ca)4/3.
(23)
Following the same reasoning and including higher order terms, we have
Qext,edge=c1βc2/3(ca)4/3+c2βc(ca)+c3βc4/3(ca)2/3+c4βc5/3(ca)1/3.
(24)
In contrast to studies in the literature [22

22. K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011). [CrossRef]

] where the edge-effect contribution for nonspherical particles is assumed to be proportional to that of a sphere, the present study finds that each edge-effect term adopts different coefficients associated with the particle geometry rather than a common coefficient. The FE formula only considers the first term in Eq. (24). The accuracy of the numerical results will be illustrated in Section 5.

4.2 Oblique incidence

To extend the formula to the oblique incidence case, for convenience, Eq. (6) is rewritten as:
Qext=24Im[εfFmexp(ρ)ρ(1+iρ(1+m2+116m))]+Qext,edge.
(25)
Note that the contribution terms from higher order central rays are discarded because they no longer exist for the oblique incidence. In terms of the physical implications, ρ is the phase shift factor, ε is the transmission factor, f is the divergence correction factor, and Fis the distance scaling factor. In the case of a sphere,
ρ=2k(m1)a,ε=1(1m1+m)2,f=1,F=1.
(26)
In the case of the end-on incidence of a spheroid,

ρ=2k(m1)c,ε=1(1m1+m)2f=[mc/a(m1)c3/a3]1,F=c/a.
(27)

Now let us consider the oblique incidence case. Figure 4 shows a schematic diagram of light scattering by a spheroid.
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 (η,ξ).
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 (η,ξ), which means,
x=a¯ηcosξ,
(28)
y=c¯ηsinξ,
(29)
where a¯ and c¯ are two semi-axes of the projected ellipse, given by
a¯=a,
(30)
c¯=c2sin2θ+a2cos2θ.
(31)
The direction of an arbitrary scattered ray is denoted as (Θ,Φ), where Θ is the scattering zenith angle and Φ is scattering azimuthal angle. Based on the central ray approximation, the phase correction [10

10. B. T. N. Evans and G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33(24), 5796–5804 (1994). [CrossRef] [PubMed]

] is written as
ρ=2a(mcosϕ){cap[p2cosϕ+ssinϕp2cos2ϕ+q2sin2ϕ+ssin(2ϕ)]},
(32)
where
ϕ=θiθt,
(33)
cosϕ=s2+p2Δm(p4+s2),
(34)
sinϕ=s(Δp2)m(p4+s2),
(35)
Δ=[m2(p4+s2)s2]1/2,
(36)
s=(p2q2c2a2)1/2,
(37)
q=(c2a2cos2(θ)+sin2(θ))1/2,
(38)
p=(cos2(θ)+c2a2sin2(θ))1/2.
(39)
In Eq. (33), θi and θt are the angle of incidence and the angle of refraction of the central ray, as depicted in Fig. 4. The transmission factor is given by
ε=112(r12+r22)112(r'12+r'22),
(40)
where r1 and r2 are the Fresnel reflection coefficients associated with the two polarized directions [41

41. M. Born and E. Wolf, Principles of Optics (Pergamon, 2001).

],
r1=mcosθicosθtmcosθi+cosθt,r2=cosθimcosθtcosθi+mcosθt.
(41)
r1' and r2' are similarly defined when the ray is refracted from the particle to the ambient medium. According to the amplitude of the scattered ray predicted from ray-optics [38

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35(3), 515–531 (1996). [CrossRef] [PubMed]

], the divergence correction factor is given by
f=2(m1)mlimη0[|Θ/ηΘ/ξΦ/ηΦ/ξ|sinΘ/η]1/2.
(42)
Note that Eq. (42) must be calculated numerically [38

38. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35(3), 515–531 (1996). [CrossRef] [PubMed]

]. The distance-scaling factor is given by
F={cp[p2cosϕ+ssinϕp2cos2ϕ+q2sin2ϕ+ssin(2ϕ)]}ac2sin2θ+a2cos2θ.
(43)
For the oblique incidence, the penumbra region is no longer a circle but an ellipse and the profile radius of curvature is dependent of the glancing point. Parameterized by the variable ξ, the radius of curvature is given by
c2a(sin2ξ+p2cos2ξ)1/2p3.
(44)
Thus, the edge-effect contribution has to be obtained by integrating the contribution along the boundary. With modifications, the edge effect term is written as
Qext,edge=q1c1βc2/3(ca)4/3+q2c2βc(ca)+q3c3βc4/3(ca)2/3+q4c4βc5/3(ca)1/3.
(45)
where the coefficients qi(i=14)are listed in Table 2.

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

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In the derivation of the coefficients, we use the following identity [42

42. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, 1954).

],
0π/2[sin2ξ+p2cos2ξ]ndξ={π2p2nF21(n,1/2;1;1p2),p>1π2F21(n,1/2;1;1p2),p<1.
(46)
The optimal number of terms in Eq. (45) is determined from comparing the results computed from the NW formula with the Lorenz-Mie solution for spheres with the same refractive index of spheroids.

4.3 Random orientation

The accuracy of Qext depends on the accuracy of Qext(μ). However, if Qext(μ) 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 (μ=1) and reaches a maximum for the side-on incidence (μ=0). Ref. [9

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

] 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, Qext(μ) 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 θ=0o,1o,2o90o 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.
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.
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, βa 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

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

], incorporating a Rayleigh formula, the ADT, and a bridge function, might be a better choice.

Figure 6 displays the results for spheroids with the refractive index of 1.1 + i0.5.
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.
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.

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.
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.
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

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

], 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).

Table 3 contains the extinction efficiencies for spheroids with excessively large size parameters.

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

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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

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

], the FE(b) includes only the leading edge-effect term, namely, 1.9924β2/3 (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, c2β1) have observable effects on the resultant accuracy because c2 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.

Figure 8 illustrates the accuracy of the new approximation in comparison with the FE approximation for spheroids with 91 fixed orientations.
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.
Assuming fto 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.

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

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The results computed from the three approximations have comparable accuracy.

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

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.
Fig. 10 The ratios of the extinction efficiencies computed from the present formula and the II-TM.
The particle is assumed to be randomly oriented. The difference is quantified through the ratio Qext,approximation/Qext,IITM. 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. 79 illustrate the differences.

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

9. G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

] 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

44. L. Bi, P. Yang, and G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010). [CrossRef] [PubMed]

]. 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

limη0ηΘ=m2(m1)1[mc/a(m1)c3/a3].
(63)

Acknowledgments

References and links

1.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, 2002).

2.

G. E. Elicable and L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. 129(1), 192–200 (1989). [CrossRef]

3.

G. Crawley, M. Cournil, and D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997). [CrossRef]

4.

M. Z. Li and D. Wilkinson, “Particle size distribution determination from spectral extinction using evolutionary programming,” Chem. Eng. Sci. 56(10), 3045–3052 (2001). [CrossRef]

5.

L. Wang, X. Sun, and J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012). [CrossRef]

6.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).

7.

A. A. Kokhanovsky, ed., Light Scattering Reviews 8 (Springer-Praxis Publishing, Chichester, UK, 2013).

8.

P. Chylek and J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A 8(2), 274–281 (1991). [CrossRef]

9.

G. R. Fournier and B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]

10.

B. T. N. Evans and G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33(24), 5796–5804 (1994). [CrossRef] [PubMed]

11.

A. J. Baran and S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transf. 63(2-6), 499–519 (1999). [CrossRef]

12.

P. Yang, K. N. Liou, and W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. 102(D18), 21825–21835 (1997). [CrossRef]

13.

J. Q. Zhao and Y. Q. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. 42(24), 4937–4945 (2003). [CrossRef] [PubMed]

14.

A. A. Kokhanovsky and E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. 28(1), 1–21 (1997). [CrossRef]

15.

H. M. Nussenzveig and W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]

16.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).

17.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).

18.

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [CrossRef] [PubMed]

19.

P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, and I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007). [CrossRef]

20.

L. Bi, P. Yang, G. W. Kattawar, and R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. 48(1), 114–126 (2009). [CrossRef] [PubMed]

21.

K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1980–1989 (2010). [CrossRef]

22.

K. N. Liou, Y. Takano, and P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011). [CrossRef]

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H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

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D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A 239, 338–348 (1957). [CrossRef]

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D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957). [CrossRef]

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P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971). [CrossRef]

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T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28(19), 4096–4102 (1989). [CrossRef] [PubMed]

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B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988). [CrossRef] [PubMed]

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L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013). [CrossRef]

32.

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013). [CrossRef]

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

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35.

P. Yang, M. Wendisch, L. Bi, G. W. Kattawar, M. I. Mishchenko, and Y. Hu, “Dependence of extinction cross-section on incident polarization state and particle orientation,” J. Quant. Spectrosc. Radiat. Transf. 112(12), 2035–2039 (2011). [CrossRef]

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K. F. Ren, F. Onofri, C. Rozé, and T. Girasole, “Vectorial complex ray model and application to two-dimensional scattering of plane wave by a spheroidal particle,” Opt. Lett. 36(3), 370–372 (2011). [CrossRef] [PubMed]

40.

F. Xu, K. F. Ren, X. Cai, and J. Shen, “Extension of geometrical-optics approximation to on-axis Gaussian beam scattering. II. By a spheroidal particle with end-on incidence,” Appl. Opt. 45(20), 5000–5009 (2006). [CrossRef] [PubMed]

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43.

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

44.

L. Bi, P. Yang, and G. W. Kattawar, “Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles,” Appl. Opt. 49(24), 4641–4646 (2010). [CrossRef] [PubMed]

45.

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OCIS Codes
(290.0290) Scattering : Scattering
(290.2200) Scattering : Extinction

ToC Category:
Scattering

History
Original Manuscript: March 10, 2014
Revised Manuscript: April 8, 2014
Manuscript Accepted: April 14, 2014
Published: April 21, 2014

Citation
Lei Bi and Ping Yang, "High-frequency extinction efficiencies of spheroids: rigorous T-matrix solutions and semi-empirical approximations," Opt. Express 22, 10270-10293 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10270


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References

  1. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, 2002).
  2. G. E. Elicable, L. H. Garcia-Rubio, “Latex particle size distribution from turbidimetry using inversion techniques,” J. Colloid Interface Sci. 129(1), 192–200 (1989). [CrossRef]
  3. G. Crawley, M. Cournil, D. D. Benedetto, “Size analysis of fine particle suspensions by spectral turbidimetry: potential and limits,” Powder Technol. 91(3), 197–208 (1997). [CrossRef]
  4. M. Z. Li, D. Wilkinson, “Particle size distribution determination from spectral extinction using evolutionary programming,” Chem. Eng. Sci. 56(10), 3045–3052 (2001). [CrossRef]
  5. L. Wang, X. Sun, J. Xing, “Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique,” Opt. Commun. 285(7), 1646–1653 (2012). [CrossRef]
  6. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 1999).
  7. A. A. Kokhanovsky, ed., Light Scattering Reviews 8 (Springer-Praxis Publishing, Chichester, UK, 2013).
  8. P. Chylek, J. D. Klett, “Extinction cross sections of nonspherical particles in the anomalous diffraction approximation,” J. Opt. Soc. Am. A 8(2), 274–281 (1991). [CrossRef]
  9. G. R. Fournier, B. T. N. Evans, “Approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30(15), 2042–2048 (1991). [CrossRef] [PubMed]
  10. B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33(24), 5796–5804 (1994). [CrossRef] [PubMed]
  11. A. J. Baran, S. Havemann, “Rapid computation of the optical properties of hexagonal columns using complex angular momentum theory,” J. Quant. Spectrosc. Radiat. Transf. 63(2-6), 499–519 (1999). [CrossRef]
  12. P. Yang, K. N. Liou, W. P. Arnott, “Extinction efficiency and single-scattering albedo of ice crystals in laboratory and natural cirrus clouds,” J. Geophys. Res. 102(D18), 21825–21835 (1997). [CrossRef]
  13. J. Q. Zhao, Y. Q. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. 42(24), 4937–4945 (2003). [CrossRef] [PubMed]
  14. A. A. Kokhanovsky, E. P. Zege, “Optical properties of aerosol particles: a review of approximate analytical solutions,” J. Aerosol Sci. 28(1), 1–21 (1997). [CrossRef]
  15. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45(18), 1490–1494 (1980). [CrossRef]
  16. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992).
  17. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).
  18. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975). [CrossRef] [PubMed]
  19. P. Yang, Q. Feng, G. Hong, G. W. Kattawar, W. J. Wiscombe, M. I. Mishchenko, O. Dubovik, I. Laszlo, I. N. Sokolik, “Modeling of the scattering and radiative properties of nonspherical dust-like aerosols,” J. Aerosol Sci. 38(10), 995–1014 (2007). [CrossRef]
  20. L. Bi, P. Yang, G. W. Kattawar, R. Kahn, “Single-scattering properties of triaxial ellipsoidal particles for a size parameter range from the Rayleigh to geometric-optics regimes,” Appl. Opt. 48(1), 114–126 (2009). [CrossRef] [PubMed]
  21. K. N. Liou, Y. Takano, P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1980–1989 (2010). [CrossRef]
  22. K. N. Liou, Y. Takano, P. Yang, “Light absorption and scattering by aggregates: application to black carbon and snow grains,” J. Quant. Spectrosc. Radiat. Transf. 112(10), 1581–1594 (2011). [CrossRef]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  24. D. S. Jones, “Approximate methods in high-frequency scattering,” Proc. R. Soc. A 239, 338–348 (1957). [CrossRef]
  25. D. S. Jones, “High-frequency scattering of electromagnetic waves,” Proc. R. Soc. Lond. A Math. Phys. Sci. 240(1221), 206–213 (1957). [CrossRef]
  26. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971). [CrossRef]
  27. S. C. Hill, A. C. Hill, P. W. Barber, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. 23(7), 1025–1031 (1984). [CrossRef] [PubMed]
  28. T. W. Chen, “High energy light scattering in the generalized eikonal approximation,” Appl. Opt. 28(19), 4096–4102 (1989). [CrossRef] [PubMed]
  29. N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993). [CrossRef]
  30. B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988). [CrossRef] [PubMed]
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  32. L. Bi, P. Yang, G. W. Kattawar, M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 123, 17–22 (2013). [CrossRef]
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