The electromagnetic �EM� scattering by a multilay-ered sphere has many significant applications.1Light absorption by aerosols has a heating effect inthe atmosphere, which contrasts with the cooling ef-fect of nonabsorptive particles. The balance be-tween the cooling and the heating effects depends onthe absorption and scattering properties of the par-ticles. Also, the same amount of absorbing aerosolsfor different types of mixture may result in fairlydifferent climate effects.2,3 For the water-solubleaerosol, virtually most of the calculations have as-signed a volume-average refractive index to the aero-sol �e.g., the growth of the tropospheric water-solubleaerosol with humidity�. However, when the aerosolis water insoluble, we cannot assign it a volume-average refractive index; if this is done, the fact thatthe liquid is adsorbed on the surface will be neglect-ed.4 In fact, particles found in nature are frequentlynot homogeneous in composition but sometimes ex-hibit a layered or radially stratified structure or other
The author �[email protected]� is with the Cold and Arid Re-gions Environmental and Engineering Research Institute, ChineseAcademy of Sciences, 260 Dong Gang West Road, Lanzhou, 730000Gansu Province, China.
Received 8 June 2002; revised manuscript received 8 November2002.
inhomogeneous structures.3 In light of this, it is im-portant to be able to calculate the scattering field fora multilayered sphere.4
The theory of EM scattering from a coated spherewas first worked out by Aden and Kerker many yearsago.5 Many subsequent computations were basedon their method. Fenn and Oser6 applied this modeland considered the scattering properties of a concen-tric soot–water sphere. Bohren and Huffman7 alsodescribed the procedure of scattering by a coatedsphere and, furthermore, attached a calculation codeto this question. However, this code, as expressedby them, should not be used for a large and stronglyabsorbing sphere. Although the formulation is con-ceptually simple and straightforward, the computa-tion still suffers from several defects. Numericalinaccuracies can develop because of round-off errors,and some of the calculated quantities can become solarge as to cause computer overflow. To circumventthese various difficulties, Toon and Ackerman8 recastthe conventional expressions for a single-layeredsphere in a form amenable to accurate calculations.Unfortunately, their expressions can be applied onlyto a single-layered sphere. Bhandari9 proposed acomplete set of scattering coefficients for a multilay-ered sphere. The calculation procedure is based on aprescription that relates the scattering coefficientsfor a l-layered sphere to those for a �l � 1�-layeredsphere. The deducing procedure, frankly speaking,is long-winded, and the expressions are also a littlebit complicated. Theoretically, a computer program
for the scattering of a multilayered sphere is not in-conceivable.9 Practically, it could be difficult formore than four or five layers. At the same time, theabove two methods also encountered the round-offerrors when one deals with the large nonabsorbingsphere.
Recently Wu and Wang10 first developed an elegantrecursive method. After that, Wu et al.11 presentedan improved algorithm for a plane wave or a Gauss-ian beam. This algorithm is concise in form andmore stable for computing the scattering coefficients.Johnson12 also developed a recurrence algorithmwith a somewhat different form. However, the com-putational difficulties were still encountered in prac-tice, more specifically, for the case of a thin, stronglyabsorbing shell. Mackowski et al.13 implementeddifferent recursive formulas to estimate the thermalbehavior of particles. But Kai and Massoli14 foundthat Mackowski et al.’s method cannot perform thecalculation for a finely stratified sphere because ofthe round-off errors introduced by the ratios of theRiccati–Bessel functions. To overcome the round-offerrors for a finely stratified sphere, Kai and Massoli14
presented a new approach for the calculation of ratiosof Riccati–Bessel functions by using the Taylor ex-pansion. Actually, as pointed out by Wu et al.,11 wedo not require a Taylor expansion of the ratios ofRiccati–Bessel functions if a modified algorithm isused. Even though a number of authors have con-tributed to extend the computable region, the earliermethods, as mentioned above, are still subject to var-ious computational difficulties. In this paper theprimary objective is to present a more concise andnumerically stable expression for a multilayeredsphere with an arbitrary number of layers on top ofthe core.
In Section 2 the EM fields inside the sphere arefirst considered a superposition of two kinds of wave:inward and outward waves. A new set of scatteringcoefficients is derived for a multilayered sphere. InSection 3 a numerically stable and concise recursivealgorithm, which is not subject to round-off errorsand the numerical overflow associated with the com-putation of Riccati–Bessel functions of large complexarguments, is described for calculation of the scatter-ing fields. We also present the computationalschemes for calculating the logarithmic derivativesand the ratio of the Riccati–Bessel function and illus-trate the numerical stability of these schemes. Sec-tion 4 is devoted to various calculations and tests,which show that the expressions are simple and suit-able for computer coding, and the results are numer-ically stable and accurate.
2. Scattering Coefficients
The particle illuminated by plane EM waves is as-sumed to be composed of a series of concentricspheres �Fig. 1�. Each is characterized by a size pa-rameter xl � 2�Nrl�� � krl and a relative refractiveindex ml � Nl�N � nl � ikl, l � 1, 2, . . . , L, where �is the wavelength of the incident wave, rl is the outerradius of the lth layer, N and Nl are the refractive
indices of the medium outside the particle and lthcomponent, and k is the propagation constant. Inthe region outside the particle, the relative refractiveindex is mL�1 � 1. The magnetic permeability isassumed to have the free-space value � � �0 every-where. Suppose the incident electric field is anx-polarized wave, Ei � E0 exp�ikr cos �ex. The totalspace is then divided into two regions: One is theregion inside the multilayered sphere, and the otheris the surrounding medium outside the particle. Letus first consider the case of an illuminating planewave with a time-harmonic factor exp��it�. In thetotal space �inside and outside the sphere�, the E, Mfields are considered a superposition of two kinds ofwave: inward and outward waves from a physicalpoint of view. The direction of propagation of thesewave fronts is inward when the radial dependence isgoverned by the first kind of spherical Bessel functionjn�kr� and outward when it is governed by the firstkind of spherical Hankel function hn
�1��kr�.15 For ex-ample, the electric field intensities of inward andoutward waves, Ein and Eout, are expressed in termsof complex spherical eigenvectors,
Ein � �n�1
�
En cn�l �Mo1n
�1� � idn�l �Ne1n
�1� �
and
Eout � �n�1
�
En ian�l �Ne1n
�3� � bn�l �Mo1n
�3� �,
respectively. This improvement not only makes theexpansions similar to those obtained from Mie theoryfor a single sphere but also avoids some of the nu-merical difficulties. It should be noted here that we
Fig. 1. Geometry of the scattering problem by a multilayeredsphere.
used only the first kind of Bessel function and thefirst kind of Hankel function. The choice is differentfrom the choice of the first and the second kinds ofBessel function in the traditional, earlier expressionsin some degree.7,10,13 As a consequence, the expan-sions of the fields in the lth region are given by
El � �n�1
�
En cn�l �Mo1n
�1� � idn�l �Ne1n
�1� � ian�l �Ne1n
�3� � bn�l �Mo1n
�3� �,
(1)
Hl � �kl
�� �n�1
�
En dn�l �Me1n
�1� � icn�l �No1n
�1� � ibn�l �No1n
�3�
� an�l �Me1n
�3� �, (2)
where En � inE0�2n � 1��n�n � 1�, � is the angularfrequency, and Mo1n
� j� , Me1n� j� , No1n
� j� , and Ne1n� j� � j � 1, 3�
are the vector harmonic functions with the radialdependence jn�klr� for j � 1 and hn
�1��klr� for j � 3.7Following the treatment of Bohren and Huffman7
for light scattering, we find that the internal EMfields �E1, H1� in the region 0 � r � r1 take theform7,10
E1 � �n�1
�
En cn�1�Mo1n
�1� � idn�1�Ne1n
�1� �, (3)
H1 � �k1
�� �n�1
�
En dn�1�Me1n
�1� � icn�1�No1n
�1� �. (4)
In the region outside the sphere, the total externalfields are taken to be the superposition of the incidentfield and the scattered field, E � Ei � Es. Accordingto Mie theory, the incident and scattered fields can beexpanded by7
Ei � �n�1
�
En Mo1n�1� � iNe1n
�1� �, (5)
Hi � �k
�� �n�1
�
En Me1n�1� � iNo1n
�1� �, (6)
Es � �n�1
�
En ianNe1n�3� � bnMo1n
�3� �, (7)
Hs � �k
�� �n�1
�
En �ibnNo1n�3� � anMe1n
�3� �. (8)
By comparing Eqs. �1� and �2�, �3� and �4�, and �5�–�8�,one can deduce an
�1� � bn�1� � 0, cn
�L�1� � dn�L�1� � 1,
an � an�L�1�, and bn � bn
�L�1�. The expansion coeffi-cients cn
�l �, dn�l �, an
�l �, and bn�l � and scattering coefficients
an, bn are obtained by matching the tangential com-ponents of EM fields at each interface. From theboundary conditions, there are four independent lin-ear equations in the aforementioned coefficients for agiven n. In a traditional way, however, we have tosolve the eight equations for the coefficients.7,10,13
The details are given in Appendix A.
We define the following:
Dn�1�� z� � ��n� z���n� z�, (9)
Dn�3�� z� � ��n� z���n� z�, (10)
Rn� z� � �n� z���n� z�. (11)
Equations �A2� in Appendix A can be easily solved forthe coefficients An
�l�1� and Bn�l�1�. By use of the no-
tations of Wu et al.,11 a recursion algorithm can bedeveloped in the following way:
An�1� � 0, Hn
a�m1 x1� � Dn�1��m1 x1�, (12a)
Hna�ml xl� �
Rn�ml xl� Dn�1��ml xl� � An
�l �Dn�3��ml xl�
Rn�ml xl� � An�l � ,
(12b)
An�l�1� � R�ml�1xl�
ml�1Hna�ml xl� � m1 Dn
�1��ml�1xl�
ml�1Hna�ml xl� � ml Dn
�3��ml�1xl�.
(12c)
Similarly, if the refractive indices appearing as fac-tors in the two terms are interchanged and An
�l�1� isreplaced by Bn
�l�1�, expressions defining An�l�1� auto-
matically become expressions for Bn�l�1� �l � 1, 2, . . . ,
L�,9,10 such that
Bn�1� � 0, Hn
b�m1 x1� � Dn�1��m1 x1�, (13a)
Hnb�ml xl� �
Rn�ml xl� Dn�1��ml xl� � Bn
�l �Dn�3��ml xl�
Rn�ml xl� � Bn�l � ,
(13b)
Bn�l�1� � R�ml�1xl�
ml Hnb�ml xl� � ml�1Dn
�1��ml�1xl�
ml Hnb�ml xl� � ml�1Dn
�3��ml�1xl�.
(13c)
Thus, from the known values of An�l � and Bn
�l � in the lthlayer, the values of An
�l�1� and Bn�l�1� in the �l � 1�th
layer are calculated in succession for l � 1, 2, . . . , L.The calculation starts in the core with the values An
�1�
� Bn�1� � 0, which results from initial conditions in-
side the core. The final coefficients in this series canbe identified with the scattering coefficients an, bn see Eqs. �A4� in Appendix A�, which are
an � An�L�1�
� Hn
a�mL xL��mL � n�xL��n� xL� � �n�1� xL�
Hna�mL xL��mL � n�xL��n� xL� � �n�1� xL�
,
(14)
bn � Bn�L�1�
� mL Hn
b�mL xL� � n�xL��n� xL� � �n�1� xL�
mL Hnb�mL xL� � n�xL��n� xL� � �n�1� xL�
(15)
because mL�1 � 1, which corresponds to the regionoutside the sphere.
The Riccati–Bessel functions �n�z� and �n�z� satisfythe general recursive relations10,14
Bn� z� � �n�z� Bn�1� z� � B�n�1� z�, (16)
B�n� z� � �n�z� Bn� z� � Bn�1� z�, (17)
beginning with
��1� z� � cos z, �0� z� � sin z,
��1� z� � cos z � i sin z, �0� z� � sin z � i cos z.
The Riccati–Bessel function Bn�z� represents �n�z�and �n�z�. From the two recurrence relations, thelogarithmic derivatives of them satisfy an identicalrecurrence expression:
Dn�i�� z� � n�z � Dn�1
�i� � z���1 � n�z, i � 1, 3.
(18)
Of the many practical applications, the scatteredfields in the far zone are of interest. Just like thetreatment of Mie theory for a single homogeneoussphere,7,11,13 once the scattering coefficients are cal-culated, the overall radiative properties of the com-posite particle are completely determined by thescattering coefficients an and bn. Thus we can de-termine all the measurable quantities associatedwith scattering and absorption, i.e., the scattering,extinction, and backscatter efficiencies �Qsca, Qext,and Qback� and scattering amplitudes �S1 and S2�, andthe polarization degree P.
3. Computational Scheme: Recursive Algorithm
Although the computation appears to be straightfor-ward, there are some pitfalls to be avoided. First,perhaps the worst of these occurs when the shell is astrongly absorbing particle such as soot aerosol.The case may lead to failure when one directly usesAden and Kerker’s equations, Bohren and Huffman’sprogram, or Wu et al.’s method or expressions de-scribed above. The reason for this is that the nu-merator and denominator involve the subtraction ofnearly equal terms, and the small round-off error will
cause the results to be largely erroneous. In addi-tion, the magnitude of a spherical Bessel functionincreases exponentially with the large complex argu-ment and may easily exceed the limits of any com-puter if the particles are sufficiently large andabsorbing.
However, for the scattering by a multilayeredsphere, the recursive method, except for the numer-ical difficulties mentioned above, is a more conve-nient form for computation; in addition, the basicexpressions Eqs. �14� and �15�� have the same formsas those for a homogeneous sphere, which are thewell-known Mie scattering coefficients. Then westill rely on the recursive method and modify theequations to remove the computational difficulties.
A. Behavior of Riccati–Bessel Functions
Wu and Wang10 have discussed in detail the asymp-totic form of logarithmic derivatives and ratios ofRiccati–Bessel functions. However, as originallypresented, these behaviors are suitable only for thecase of a nonabsorbing or weakly absorbing sphereand do not provide enough information for the case ofthe large and strongly absorbing sphere. Theyfound that real and imaginary parts of �n�z���n�z�and �n�z���n�z� rapidly decrease to very small valuesclose to zero. In addition, Dn
2�z� � Dn3�z� � �Dn
1�z�when n � Nstop, where Nstop will be discussed inSubsection 3.C. However, for the case of a large andhighly absorbing sphere i.e., large Im�z��, these func-tions behave different. Tables 1 and 2 present, as anexample, the asymptotic behaviors of Dn
�i��z� �i � 1, 2,3�, �n�z���n�z�, and �n�z���n�z�. The size parameterand refractive index in Tables 1 and 2 are x � 80, m ��1.05, 1.0�, respectively. When n is small �e.g.,smaller than 60 in this case�, we may find that Dn
2�z�� Dn
1�z� � �Dn3�z�, the ratio �n�z���n�z� � i, and the
ratio �n�z���n�z� is always big. It is clear that allthese functions exhibit obviously different behaviorsfrom the ones shown in Ref. 10. Only when n is verybig, such as n �� Nstop, are their behaviors the sameas those described by Wu and Wang.10 One can de-rive the equations for ratios �n�z���n�z� and �n�z��
Table 1. Asymptotic Behavior of the Logarithmic Derivatives of Riccati–Bessel Functions Dn�i��z� �i � 1, 2, 3� for a Large, Absorbing Particlea
�n�z� by using the general recursive relations Eqs.�16�–�18� above, as follows,
�n� z�
�n� z��
�n�1� z�
�n�1� z�
Dn�2�� z� � n�z�
Dn�1�� z� � n�z�
, (19)
�n� z�
�n� z��
�n�1� z�
�n�1� z�
Dn�3�� z� � n�z�
Dn�1�� z� � n�z�
, (20)
with the starting ratios
�0� z�
�0� z�� i
1 � exp�2ai�exp��2b�
1 � exp�2ai�exp��2b�, (21)
�0� z�
�0� z��
1 � exp��2ai�exp�2b�
2, (22)
where z � mlxl � a � ib. When the radius, rl, islarger than the wavelength of light and the imagi-nary part of the refractive index of the shell is large i.e., large Im�mx��, the value of �0�z���0�z� is nearlyequal to a complex constant i. Both the real and theimaginary parts of �0�z���0�z� become big. At thismoment, the other algorithms, i.e., Aden and Kerk-er’s and Bohren and Huffman’s methods, will encoun-ter the numerical difficulties. Likewise, for a largeand strongly absorbing thin sphere, because of Dn
1�z�� Dn
2�z� and �n�z���n�z� � i �discussed above�, Eqs.�19� and �21� of Wu et al.11 are nearly equal to 0�0.The reason for this is that the numerator and denom-inator involve differences of terms, which are thesubtraction of nearly equal quantities to obtain asmall one, and result in numerically meaningless re-mainders. In addition to these problems, the mag-nitude of �0�z���0�z� increases exponentially with thecomplex argument; although the zero-order functionmight not exceed the limits of the computer, succes-sive higher-order functions computed by upward re-currence might.
B. Recursive Algorithm
To remove these difficulties, Eqs. �12� and �13� shouldbe rewritten in a form amenable to accurate calcula-tion. We introduce a ratio Qn
�l �, defined as
Qn�l � �
Rn� z1�
Rn� z2��
�n�ml xl�1�
�n�ml xl�1���n�ml xl�
�n�ml xl�. (23)
The presence of the ratio Qn�l � is a characteristic of our
recurrence expressions for scattering coefficients.Then Eqs. �12b� and �12c� are combined into one nu-merically stable form, namely,
Hna�ml xl� �
G2 Dn�1��ml xl� � Qn
�l �G1 Dn�3��ml xl�
G2 � Qn�l �G1
. (24)
As before, the determinant Hnb�mlxl� is obtained
from Hna�mlxl� by interchanging the positions of ml
and ml�1 and by the replacement of Hna with Hn
b.Thus
Hnb�ml xl� �
G2 Dn�1��ml xl� � Qn
�l �G1 Dn�3��ml xl�
G2 � Qn�l �G1
. (25)
The other notations in Eqs. �24� and �25� for l � 2,3, . . . , L are
G1 � ml Hna�ml�1xl�1� � ml�1Dn
�1��ml xl�1�, (26)
G2 � ml Hna�ml�1xl�1� � ml�1Dn
�3��ml xl�1�, (27)
G1 � ml�1Hnb�ml�1xl�1� � ml Dn
�1��ml xl�1�, (28)
G2 � ml�1Hnb�ml�1xl�1� � ml Dn
�3��ml xl�1�. (29)
Finally, the scattering coefficients, an and bn, can besuccessfully computed by Eqs. �14� and �15� and �24�and �25�. These equations constitute the basic ex-pressions for calculating the scattering fields.
C. Computational Scheme
According to the above analysis, the calculation ofscattering coefficients depends mainly on the loga-rithmic derivatives of Riccati–Bessel functions,Dn
�1��z� and Dn�3��z�. As discussed by Wiscombe,16 the
Dn�1��z� can be accurately calculated for a large do-
Table 2. Asymptotic Behavior of the Ratios of Riccati–Bessel Functions �n�z���n�z� and �n�z���n�z� for a Large, Absorbing Particlea
main of complex argument by use of the downwardrecurrence. The number of terms, Nmax, in thepartial-wave expansion is a function of the size pa-rameters. Theory and experiment show that a goodchoice for the number of terms is given by Nmax �max�Nstop, �mlxl�, �mlxl�1�� � 15, l � 1, 2, . . . , L.The value of Nstop is the integer closest to
Nstop � � xL � 4xL1�3 � 1 0.02 � xL � 8
xL � 4.05xL1�3 � 2 8 � xL � 4200
xL � 4xL1�3 � 2 4200 � xL � 20,000
� ;
(30)
these criteria were discussed by Wiscombe.16 Thelogarithmic derivative Dn
�3��z� is calculated by a newupward recurrence discussed in Ref. 13, and recur-rence is begun with
D0�3�� z� � i, (31)
�0� z��0� z� �12 1 � �cos 2a � i sin 2a�exp��2b��,
(32)
where z � a � ib. Mackowski et al.13 found that thismethod of computing Dn
�3��z� was stable for all valuesof z, as opposed to some methods of upwardrecurrence,8–10,12 which could become unstable forrelatively small Im�z�. As we will see, this is thereason why the Toon and Ackerman8 and the Bhan-dari9 methods cannot guarantee Qext � Qsca for alarge nonabsorbing sphere. The ratio Qn
�l � is calcu-lated by use of upward recursion. By using the gen-eral relations, we can write
Qn�l � � Qn�1
�l � Dn
�3�� z1� � n�z1 Dn
�1�� z1� � n�z1 � Dn�3�� z2� � n�z2
Dn�1�� z2� � n�z2
,
(33)
with the starting ratio
Q0�l � �
R0� z1�
R0� z2��
exp��i2a1� � exp��2b1�
exp��i2a2� � exp��2b2�
� exp �2�b2 � b1��, (34)
where z1 � mlxl�1 � a1 � ib1, z2 � mlxl � a2 � ib2.From Table 2 or Eq. �22�, we found the values ofRn�z1� and Rn�z2� will be ��1 for large Im�mx� andcan conceivably result in overflow, so they are all notbounded. The ratio, Qn
�l �, of these two functions,however, is always bounded because b2 � b1. Alter-natively, by use of the recursive relation �18�, theratio Qn
�l � gives
Qn�l � � Qn�1
�l � �xl�1
xl�2 z2 Dn
�1�� z2� � n�
z1 Dn�1�� z1� � n�
n � z2 Dn�1�3� � z2��
n � z1 Dn�1�3� � z1��
.
(35)
Note that the variable Qn�l � and its form that we ob-
tained by using Eq. �33� or �35� are different fromthose employed by other algorithms. When n is
large compared with the size parameters in question,the ratio, Qn
�l �, approaches zero rapidly. Particu-larly, the errors in evaluating Qn
�l � do not propagateand grow when n increases. For example, the ratioQn
�l � behaves as �x1�x2�2n for a two-layered spherewhen n �� x2. This has the consequence that, forsuch large values of n, the scattering coefficients re-duce to the Mie coefficients that correspond to a ho-mogeneous sphere with the parameters m2 and x2.For x1 �� x2 �a tiny core�, a less stringent condition,n �� x1, is enough.9 Furthermore, in the event thatfor �x2 � x1� �� 1 �a thin shell17 or a finely stratifiedsphere14�, the ratio Qn
�l � varies little and approximatesa constant with the n increasing. Then we can seethat this recursive formula is numerically stable, asthe fractional errors do not grow with n.8,9
The final terms to be computed are �n�xL� and�n�xL�. These are Riccati–Bessel functions of realargument and are therefore bounded. Thus the twofunctions are calculated by use of upward recursionin the same way as in Mie theory.8,16
4. Tests of the Algorithms and Results
To examine how this recursive algorithm works, wehave calculated the extinction efficiency, scatteringefficiency, and other scattering-related quantities as-sociated with the external field for various size andrefractive indices. Several comparisons are madewith calculations performed by other algorithms andwith the published results for quite different cases.All the calculations are performed with double-decision arithmetic.
A. Other Algorithms
We first performed various tests for the scattering,extinction, and backscattering efficiencies �Qsca, Qext,Qback� by comparing them with the correspondingvalues of other methods for a two-layered sphere sys-tem. First, the comparisons are performed whenthe imaginary parts of mlxl or mlxl�1 are relativelysmall �e.g., do not exceed 30, as suggested by Bohrenand Huffman7�. The results show good agreementin all instances with the values from the methods ofToon and Ackerman,8 Bhandari,9 Wu et al.,11 andBohren and Huffman.7 Second, the comparisons areperformed for the case of the large and strongly ab-sorbing coated sphere, such as when soot forms a thinshell on the outside of a nonabsorbing �a water drop-let� or weakly absorbing �sulfate aerosol� sphere.We find that the code of Bohren and Huffman7 cannotperform this calculation. The method of Wu et al.11
may be subject to some numerical difficulties.Whereas the first two algorithms and ours do notencounter the numerical problems, the results showagreement even for this difficult case that could causeother algorithms to fail. As an illustration, the com-parison of different algorithms for the scattering andextinction efficiencies and the single-scattering al-bedo is shown in Fig. 2. This example is a waterdroplet �m1 � 1.33 � i0.00� on which the soot �m2 �1.59 � i0.66� has formed a thin shell; it is opposed toFig. 5 in Wu and Wang10; the volume fraction of soot
here is equal to 0.01. We see in Fig. 2 that theresults of Wu et al.11 are obviously unreasonablewhen the size parameter is larger than 30. Finally,if the sphere is large and nonabsorbing, the algo-rithms of Toon and Ackerman8 and Bhandari9 alsocannot guarantee Qext � Qsca because of the use ofunstable upward recurrence for Dn
�3��z�. From ourpractice, this deficiency can be remedied by applica-tion of the new upward recurrence13 for Dn
�3��z�. Fur-thermore, to compare the algorithm’s efficiency, weshow the computational domain �number of layersversus the size parameter� in Fig. 3, where the re-fractive index and the size parameter are the same asthose in Ref. 11. Figure 3 demonstrates that ourmethod can also increase the affordable number oflayers and the size parameter by several orders ofmagnitude. This limit is due only to the memory ofthe computer.
We also checked results for extinction efficiencyfactors by comparing them with the tabulated valuesof Espenscheid et al.18 for real refractive indices, asize parameter of 10 for the total particle, and a largerange of core sizes. There is good agreement to fourdecimal digits that were reported by Ref. 18. Wealso performed several tests, all of which are correctin comparison with Mie theory �Mie code was pro-vided by Wiscombe� to at least six decimal digits, forvarious efficiency factors and an asymmetry param-eter. These tests include setting the refractive indi-ces of the core and shell equal to each other, settingthe refractive index of the shell equal to that of freespace �1 � 0i�, setting the radius core and shell equalto each other, and setting the core radius to zero.
B. Kai and Massoli Model
As in Kai and Massoli,14 the radial profile of therefractive index, ml � nl � ikl, is described at thelth-layer level by nl � n1 � 0.5�nL � n1��1 � cos t��and kl � 0, where t � �l � 1���L � 1�, n1 � 1.43, andnL � 1.33. The size parameter is xl � x1 � t�xL �x1�, l � 1, 2, . . . , L, and L is the total number oflayers. Figure 4 displays the scattering efficiency�Qsca� as a function of dimensionless core size param-eter x1�xL. We found that our results are in goodaccord with the graphic results of Kai and Massoli�see their Fig. 4�.14 Therefore, as pointed out by Wuet al.,11 it is not necessary to use the Taylor expan-sions for the ratios of the Riccati–Bessel function.The numerical results do not exhibit so-called second-kind or third-kind round-off errors, as mentioned byKai and Massoli.14 In Fig. 5 the vertically polarizedintensity i� is plotted against the scattering angle .As shown, the present formulations yield results inagreement with Fig. 6 of Kai and Massoli for twodifferent cases.14 In this stratified sphere model, forcase 1, ml � xL�xl, and, for case 2, ml � 1.01�xL�xl�
2,
Fig. 2. Extinction and scattering efficiencies �Qext, Qsca� andsingle-scattering albedo as a function of the size parameter of theoutmost layer for a soot-coated water sphere. A comparison of theresults obtained from Wu et al.,11 Bhandari,9 and Toon and Ack-erman8 is shown. The refractive indices of water and soot arem1 � 1.33 � i0.00, m2 � 1.59 � i0.66, respectively. The volumefraction of soot is 0.01.
Fig. 3. Computational domain for a multilayered sphere. Thesolid curve is the affordable number of layers, and region B is thecomputational domain obtained by Kai and Massoli.14 Region Ais the computational domain of our algorithm. The refractiveindex ml � nl � ikl is nl � n1 � 0.5�nL � n1��1 � cos t��, and kl �0, where t � �l � 1���L � 1�, n1 � 1.01nL, and nL � 1.33. The sizeparameter is xl � x1 � t�xL � x1�, where l � 1, 2, . . . , L, x1 �0.001xL, and L is the total number of layers.
where L � 1500 and xl � x1 � �xL � x1��l � 1���L �1�. As the total number of layers become largeenough, the estimated values by a multilayeredsphere are in good agreement with the analytic solu-tions. Therefore the multilayered sphere model canbe used to describe all types of radially inhomoge-neous sphere, and the converged solutions of the mul-
tilayered sphere model can be considered to be thetrue solutions of realistic radially inhomogeneousspheres, to a good approximation.14
C. Mackowski et al. Model
For the purposes of illustration, the particular type ofparticle considered here consists of a mixture of ab-sorbing, sub-micro-sized particles suspended in adroplet of water. Such a mixture has relevance todroplet combustion of solid fuel and water slurries.13
The mixture is assumed to be characterized by aradially dependent effective refractive index, meff,reading as see Eq. �72� of Mackowski et al.13�
meff2 �r� � mmed
2 �1 � 3fv�r��
1 � fv�r���� , (36)
where � � �mabs2 � mmed
2 ���mabs2 � 2mmed
2 �, mmed is therefractive index of the water medium, mabs is therefractive index of the absorbing particle, and fv�r� isthe volume fraction of the absorbing particle. Fol-lowing Mackowski et al.’s treatment,13 five cases ofdistributions of fv�r� are calculated. They are �1� auniform distribution, fv�r� � f�v � 0.1; �2� a two-layered sphere with the absorbing material existingas the core, x1 � f� v
1�3 xL; �3� a two-layered sphere withthe absorbing material forming the shell, x1 � �1 �f�v�
1�3xL; �4� the absorbing material distributed ac-cording to fv�x� � 4�3�x�xL� f�v; and �5� the absorbingmaterial distributed according to fv�x� � 1�4�1 �x�xL� f�v.
The effect of absorbing material distributions onthe overall radiative properties, i.e., Qsca, Qext, andQback, are given in Table 3. These results agree al-most exactly with the values calculated by Mac-kowski et al.13 The only minor difference is thevalue of Qback in the second line of Table 3. Theeffect of the distribution of absorbing material haslittle influence on the overall extinction efficiency ofthe particle, but the scattering efficiency is greatestfor core absorption and smallest for uniformly distrib-uted absorption. Also, the single-scattering albedo behaves quite differently. It should be particu-larly pointed out that other algorithms could encoun-ter numerical difficulties when they are used withcase 3.
D. Bhandari Model
To have a comparison with the results of Bhandari,17
we considered the absorption of visible light by tinysoot particle forming the core of a water droplet.
Fig. 4. Scattering efficiency Qsca versus the dimensionless sizeparameter of the core x1�xL. The refractive index ml � nl � ikl isnl � n1 � 0.5�nL � n1��1 � cos t��, and kl � 0, where t � �l �1���L � 1�, n1 � 1.43, and nL � 1.33. The size parameter is xl �x1 � t�xL � x1�, l � 1, 2, . . . , L, and L is the total number of layers.
Fig. 5. Vertically polarized intensity i� as a function of the scat-tering angle for a radially multilayered sphere with a size pa-rameter xL � 5. The refractive indices in the lth layer, are, forcase 1, m1 � 1.0834 � i0.0, ml � xL�xl, and, for case 2, m1 �1.0779 � i0.0, ml � 1.01�xL�xl�
2. The size parameter is xl � x1 ��xL � x1��l � 1���L � 1�, where the total number of layers is L �1500.
Table 3. Extinction, Scattering, and Backscattering Efficiencies and Single-Scattering Albedo �Qext, Qsca, Qback and ��a
This is one of the cases of practical interest in thestudy of scattering of light by polluted clouds and fog.For the calculation, r2 � 5 �m, � � 0.5 �m, m2 �water�� 1.33 � i0.00, and m1 � 2.0 � ik. For the specialcase of soot, take k � 0.66. The other values of k areset to 0.22 and 0.44. The absorption cross sectionper unit volume of absorbing material, �abs�V, is plot-ted against the volume fraction fv of the absorbingmaterial for different values of k in Fig. 6. For thesake of illustration, we consider again the case of awater droplet �m1 � 1.33 � i0.00� with the soot �m2 �2.0 � i0.66� forming a thin shell on the outside. Theresults are shown in Fig. 7. As can be seen, thedifferent distributions of the soot in a water droplet�i.e., the soot existing as the core or shell� profoundlyaffect the absorption cross section per unit volume ofthe absorbing material. The graphical results agreewell with those computed by a perturbative approx-imation method.17 More important, other algo-
rithms could suffer from numerical difficulties in thecase when the soot exists as a thin shell. This agree-ment is also a good test of the numerical stability ofthe computational procedure outlined in Section 3.
E. Luneburg Lens
This test case is the scattering from a sphericalLuneburg lens, which is also approximated as a mul-tilayered sphere. The Luneburg lens is character-ized by a refractive index that varies according to therelation12
m�r� � 2 � �r�a�2�1�2, (37)
where a is the radius of the outer rim of the sphericallens and r is the radial distance measured from thecenter. The refractive index is a radially varyingcontinuous profile. The radially multilayeredsphere model may be used not only to calculatespheres with discontinuous profiles of refractive in-dex but also to approximate spheres with continuousprofiles of refractive index by use of a large number oflayers. In the present calculation, the Luneburglens is modeled as a multilayered sphere with 500layers. The refractive index of each layer is definedto be equal to m�r� at the midpoint of the layer, i.e.,ml � m�r��, r� � �rl�1 � rl��2 for l � 2, 3, . . . , L. Thesize parameter is obtained when the sphere is dividedinto 500 equally spaced layers.12 According to geo-metrical optics theory, the differential cross sectioncan be expressed as12
d��d� � a2 cos��. (38)
Results of our procedure are shown in Fig. 8 for sizeparameter xL � 60. The quantity plotted is the re-duced differential cross sections d��d�a2�� �i.e., thedifferential cross section divided by a2�. Compari-sons are made with the calculations performed pre-viously by Johnson12 for the situation of a
Fig. 6. Absorption cross section per unit volume of absorbingmaterial, �abs�V, as a function of its volume fraction fv for a two-layer sphere model with a radius r2 � 5 �m at the outer rim. Therefractive index of the water shell is m2 � 1.33 � i0.00, and thatof the absorbing core is m1 � 2.0 � ik, where k � 0.22, 0.44, 0.66.The case of k � 0.66 corresponds to a soot particle. The wave-length is � � 0.5 �m.
Fig. 7. As in Fig. 6, but the absorbing material in the two-layersphere model exists as a thin shell on the outside of a waterdroplet.
Fig. 8. Reduced differential cross section d��d�a2�� as a functionof scattering angle for a spherical Luneburg lens with a sizeparameter xL � 60. The size parameter in the lth layer is xl �lxL�500. The refractive index is ml � 2 � �x��xL�2�1�2, with x� ��x� l�1 � x� l��2, for l � 2, 3, . . . , L. For the geometrical opticstheory, d��d� � a2 cos��.
multilayered sphere model and geometrical opticstheory. As a further test, when we compare resultswith those plotted in Fig. 3 of Johnson,12 it can beseen that there was good agreement in all instances.
5. Summary and Conclusions
An improved and more efficient algorithm for scat-tering coefficients of a multilayered sphere has beenpresented. The procedures include three opera-tions. First, the derivation is based on the physicalconsideration: The internal and external EM fieldsare considered a superposition of two kinds of wave:inward and outward waves. The radial dependenceof the inward wave is governed by the first kind ofspherical Bessel function jn�kr�. In contrast, the ra-dial dependence of the outward wave is governed bythe first kind of spherical Hankel function hn
�1��kr�.The final recursive expressions are similar in form tothose of Mie theory for a homogeneous sphere. Nospecial restriction on the number of layers is imposedby this method. More specially, this recursive algo-rithm is concise and convenient for program coding.Second, the Riccati–Bessel functions, �n and �n, arereplaced with their logarithmic derivatives. Thistechnique, long used in Mie theory and for coatedspheres, circumvents the exponential dependence ofthe functions on the size parameter. Finally, a suit-able ratio of Riccati–Bessel functions is bounded asthe core size becomes small or as the size parameterbecomes large. Presence of the ratio Qn
�l � is one of thecharacteristics of our recurrence expressions for scat-tering coefficients. The round-off errors that pre-vented previous algorithms from dealing withproblems of highly absorbing and finely stratifiedspheres or large nonabsorbing stratified spheres havebeen eliminated. With those operations, the finalrecursive algorithm avoided the numerical difficul-ties in earlier methods that were due to the subtrac-tion of nearly equal terms, leaving a small residual.The various tests show the recursive algorithm isnumerically stable and correct for a large range ofsize parameters and refractive indices. This behav-ior contrasts with that of some other recurrence al-gorithms for solving highly absorbing thin-shellsphere problems, which could be subject to relativelysevere round-off errors.
Appendix A: Boundary Conditions at the Interfaces
Following Bohren and Huffman’s treatment,7 we canexpand the problem of scattering by a coated sphereinto the problem of scattering by a multilayeredsphere. The boundary conditions for different inter-faces are
�El�1 � El� � er � 0, �Hl�1 � Hl� � er � 0, (A1)
where r � rl and l � 1, 2, . . . , L � 1, L.From the orthogonality, together with the field ex-
pansions, Eqs. �1�–�8�, and the expressions for vectorharmonics Bohren and Huffman,7 Eq. �4.50��, theboundary conditions above yield four independent
linear equations in expansion coefficients cn�l �, dn
�l �,an
�l �, bn�l �:
dn�l�1�m1��n�ml�1xl� � an
�l�1�ml��n�ml�1xl�
� dn�l �ml�1��n�ml xl� � an
�l �ml�1��n�ml xl� � 0,
cn�l�1�ml�n�ml�1xl� � bn
�l�1�ml�n�ml�1xl�
� cn�l �ml�1�n�ml xl� � bn
�l �ml�1�n�ml xl� � 0,
cn�l�1���n�ml�1xl� � bn
�l�1���n�ml�1xl�
� cn�l ���n�ml xl� � bn
�l ���n�ml xl� � 0,
dn�l�1��n�ml�1xl� � an
�l�1��n�ml�1xl�
� dn�l ��n�ml xl� � an
�l ��n�ml xl� � 0, (A2)
where the Riccati–Bessel functions are �n�z� � zjn�z�and �n�z� � zhn
�1�. Let An�l � � an
�l ��dn�l � and Bn
�l � �bn
�l ��cn�l �. In the core �l � 1�, an
�1� and bn�1� must be zero
because no outward fields exist in the region r � r1 see Eqs. �3� and �4��, which is different from thetraditional treatment7,10,13; then we get An
�1� � Bn�1� �
0. The auxiliary arguments An�l � and Bn
�l �, where l �2, 3, . . . , L � 1, L, are easily obtained when the setof linear equations �A2� is solved.
For the outermost layer rl � rL, the external fieldsare
EL�1 � Ei � Es,
HL�1 � Hi � Hs. (A3)
Thus, from Eqs. �1� and �2� and �5�–�8�, we have cn�L�1�
� dn�L�1� � 1, an � an
�L�1� � An�L�1�, and bn � bn
�L�1� �Bn
�L�1�. For the sake of clearness, an alternativeform of the set of Eqs. �A2� can be rewritten as
mL��n� xL� � an mL��n� xL� � dn�L���n�mL xL�
� an�L���n�mL xL� � 0,
mL�n� xL� � bn mL�n� xL� � cn�L��n�mL xL�
� bn�L��n�mL xL� � 0,
��n� xL� � bn��n� xL� � cn�L���n�mL xL� � bn
�L���n�mL xL� � 0,
�n� xL� � an�n� xL� � dn�L��n�mL xL� � an
�L��n�mL xL� � 0(A4)
because mL�1 � 1, which corresponds to the regionoutside the sphere.
The author thanks very specially the two anony-mous reviewers for their comments and suggestions,which improved the presentation of our method.The author also appreciates our computer center andWu-Zhou Wei for providing the use of the computer.This research was supported in part by ChineseAcademy of Sciences �KZCX2-201�, by the Coordi-nated Enhanced Observing Period Asia-AustraliaMonsoon Project on Tibetan Plateau �CAMP�Tibet�,and by the National Natural Science Foundation ofChina �Grant 40275003�.
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