1023 J. Opt. Soc. Am., Vol. 70, No. 8, August 1980 0030-3941/80/081023-03$00.50 © 1980 Optical Society of America 1023

Asymptotic behavior of the efficiencies in Mie scattering Charles Acquista, Ariel Cohen,* and John A. Cooney

Department of Physics and Atmospheric Science, Drexel University, Philadelphia, Pennsylvania 19104

Jet Wimp Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104

(Received 24 December 1979; revised 1 March 1980) We prove that a previous derivation of the scattering and extinction efficiencies for infinitely large

absorbing spheres is incorrect, and leads to an erroneous limit for the scattering efficiency. We then derive an expression for the scattering efficiency limit from geometrical optics and that is valid when the refractive index has a small imaginary part.

Numerical computations of the scattering and extinction efficiencies, Qsca and Qext , for large spheres indicate that these efficiencies achieve finite, well-defined limits as the size pa­rameter x = πd/λ approaches infinity. The extinction effi­ciency approaches 2, a result that is in accordance with Ba-binet's principle; the scattering efficiency approaches a value dependent on the refractive index. Since the efficiencies can be expressed as infinite sums over the Mie partial wave coef­ficients an and bn,1

the mathematical problem is reduced to finding a method for evaluating these sums in closed form when x is large. Several years ago, Chýlek2 attempted to sum these series in a rigorous mathematical fashion. For the extinction efficiency, he ob­tained the accepted result: Qext → 2. For the scattering ef­ficiency, he obtained

a result based on an untested conjecture. Recently, Bohren and Herman3 showed that this limit for the scattering effi­ciency was not in agreement with accurate numerical calcu­lations.4 However, they were unable to reconcile this dis­agreement with Chýlek's proof of (3).

In this note we take issue with Chýlek's proof and demon­strate the nature of the error in his derivation. We likewise take issue with the asymptotic value of the scattering effi­

ciency as given in (3). Hence, the purpose of this work is threefold:

(i) To show that Chýlek's method of proving Qext → 2 is incorrect, regardless of the correctness of this result.

(ii) To show how this incorrect method yields an erroneous result for the limit of Qsca ∙

(iii) To derive an analytical expression for the limit of Qsca

from geometrical optics that is valid when |Im(m) | « 1.

The asymptotic limit Qext → 2 has, as noted elsewhere,1

universal acceptance based on physical concepts, experiments, and computations. The practical issue left concerning this limit turns largely on the question of mathematical rigor. Let us examine Chýlek's proof2 of this limit which was restricted to absorbing spheres with a refractive index given by m = μ — k , K > 0. In this work, the infinite sum in (1) was separated into three terms, p, Q, and R defined by

and an attempt was subsequently made to show that P → 2, Q→-0, and R → 0 as x → ∞. The proof of the first limit relies exclusively on earlier work in which it was shown that, for x/n » 1 and K > 0,

FIG. 1. Plot of Re(an+ bn) versus n/x for m= 1.33 — 0.01/and size pa­rameter x = 200.

In fact, the limit 'P → 2 was obtained by applying (7) in (4) which immediately yields

Following this procedure it is evident that P → 2 if and only if N1 → x. However, as x → ∞, the assumption x/n » 1 for all n in P is certainly not valid for n = N1. In what follows, we will show that as x → ∞ , the number of terms in P that do not satisfy x/n » 1 constitutes a nonvanishing proportion of the total number of terms. In particular, we will show that fully half the terms in P satisfy a contrary inequality, x/n < 2, in this asymptotic limit. For a fixed value of x, we define nr to be the number of terms in P which satisfy

where nr < N1. We will be interested in values of r that are not considered to be much greater than unity, e.g., r = 2. Chýlek2 defined the upper limit N1 by

where 0 < α < l , M is a large positive integer, and the angle brackets < ) indicate the greatest integer function. Since Nι is positive, we can rewrite (10) as

We obtain nr by subtracting the smallest value of n that satisfies (9), nm i n , from N1. Upon taking the reciprocal of each term in (9), we find

which leads to

(The last factor of 1 is necessary since both N1 and n m i n are included in nr.) The ratio of the number of terms in P that satisfy (9) to the total number is then

Dividing by x and treating l/x as a small parameter, we can rewrite this as

1024 J. Opt. Soc. Am., Vol. 70, No. 8, August 1980

Finally, taking the limit x → ∞, we obtain the desired re­sult:

For the particular case r = 2, we see that half the terms in P satisfy x/n ≤ 2. We have thus established that it is incorrect to use (7) in (4) since the assumption x/n » 1 is false, even though the ultimate result Qext → 2 is correct, physically speaking. (We will demonstrate later that applying this same procedure to Qsca leads to an incorrect result.)

Chýlek's proof can be seen to be invalid from a second in­dependent argument. Suppose we attempt to calculate the value of Qext for very large spheres with a real refractive index using Chýlek's method. (Of course, it is universally accepted that the limit Qext → 2 applies in this case as well.) Although Chýlek states that his method cannot be used to obtain the result Qext → 2 when the refractive index is real, such a spe­cialization ought properly to be allowed. Hence, it is in­structive to investigate the consequences of using his method for real m. The only part of his proof that requires that m possess an imaginary component involves the application of (7) to the sum for P. If m is real, we must replace (7) with an analogous result for a large dielectric sphere:

The constant c does not reach a limit as x increases; instead, it oscillates between 0 and 2. Now, for a large value of x, P → 2c and everything else in Chýlek's procedure remains un­changed (viz., O → 0 and R → 0) leading to the obviously in­correct result that, for a very large dielectric sphere, the ex­tinction efficiency lies somewhere between 0 and 4, not nec­essarily near 2. Since we have shown that a straightforward and legitimate application of Chýlek's procedure leads to an incorrect result, we conclude once again that Chýlek's proof is defective.

Both of the above arguments may be viewed in perspective if we examine the behavior of the summand in (1). In Figs. 1 and 2 we plotted the summand Re (an + bn) as a function of n/x for a large absorbing sphere and for a large dielectric

FIG. 2. Plot of Re(an + bn) versus n/x for m = 1.33 and size parameter x = 200.

JOSA Letters 1024

sphere. In Fig. 1 we see clearly that when m is complex the summand is not equal to one over the entire interval n/x < 1. (Further calculations show that this behavior persists for larger spheres.) In Fig. 2 we see that when m is real the do­main n/x « 1 (over which the summand is nearly constant) contributes an insignificant amount to the total sum in (1). In conclusion, Chýlek's method of replacing the summand with its value for n/x « 1 is unjustifiable on mathematical, computational, and physical grounds.

The asymptotic limit of the scattering efficiency is some­what more complicated. After reviewing the history of the conjecture for this limit given in (3), Bohren and Herman3

showed that (3) was not in agreement with Mie calculations. However, they were unable to uncover the error in Chýlek's proof of (3) since, at that time, there was no reason to suspect an error in Chýlek's proof for the accepted result of the Qext limit. Since Chýlek obtained (3) using the same (incorrect) method described above for Qext it is not surprising that (3) is defective. The problem with Chýlek's proof of (3) is simply that the approximations for the Mie coefficients an and bn which are valid for n/x « 1 are applied for n/x ≈ 1. However, we need not evaluate (2) to obtain Qsca for large spheres. Van de Hulst1 pointed out several years ago that an accurate value for QSca could be obtained by combining the results of dif­fraction theory with geometrical optics:

where R1 and R2 are the Fresnel coefficients5 for light polar­ized parallel and perpendicular to the plane of incidence, and θ is the angle of incidence. Numerical evaluations of (18) compare favorably with the Mie sum (2). Often, in problems of atmospheric and underwater optics, scatterers have a re­fractive index with a small imaginary part. For these particles we can approximate (18) by ignoring the imaginary part of m. Now (18) can be evaluated analytically with the result

where μ = Re(m). To demonstrate the range of (19), we evaluated (18) and (19) for a refractive index m = 1.3-O.li which corresponds to a very appreciable absorption. Qsca

= 1.0649 using (18) while Qsca = 1.0611 using (19). For smaller values of Im(m), the agreement is even better. For m = 1.3-0.01i, Qsca = 1.0612 using (18), while the value of Qsca

using (19) is unchanged since μ remains 1.3.

ACKNOWLEDGMENTS

This research was funded in part by the Geosciences Di­rectorate of the Army Research Office, Durham, the Meteo­rology Branch of the National Science Foundation, and by the National Aeronautics and Space Administration under Grants NSG-6019 and NSG 2357. The authors gratefully acknow­ledge the assistance of A.C. Holland.

* Permanent Address: Department of Atmospheric Sciences, The Hebrew University, Jerusalem, Israel.

1For a discussion of Mie theory and Babinet's principle, see H.C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

2P. Chýlek, "Asymptotic limits of Mie scattering characteristics," J. Opt. Soc. Am. 63, 1316-1320 (1975).

3C. F. Bohren and B. M. Herman, "Asymptotic scattering efficiency for a large sphere," J. Opt. Soc. Am. 69, 1615-1616 (1979).

4Actually, the lack of agreement between (3) and Mie theory could have been demonstrated on the basis of much earlier calculations by van de Hulst (Ref. 1, p. 293).

5For a discussion of the Fresnel coefficients, see M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), pp. 38-41.

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