The scattering of a plane electromagnetic wave by asingle multilayered spherical particle has been exten-sively discussed within a theoretical framework sim-ilar to the Lorenz–Mie Theory ~LMT! in many areasof theoretical and applied research, such as combus-tion, chemical engineering, remote sensing, commu-nication, biology, and medicine. Furthermore, theclassical LMT was recently extended to the General-ized Lorenz–Mie Theory ~GLMT! describing the scat-tering of an arbitrary beam by a homogeneoussphere,1,2 with a more recent version devoted to thecase of a multilayered sphere.3–5 These theoreticaladvances also were applied extensively in variousfields.6–8 In the case of multilayered spheres, rele-vant examples of applied fields can include combus-tion, when fuel droplets in spray flames exhibit aradial temperature dependence associated with ra-dial gradients of the refractive index, or the study oflinear interactions between laser light or microwavesand blood cells or other biological materials for illnessdiagnosis.
From an algorithmic point of view, many analyt-ical and numerical methods were presented for the
Z. S. Wu and L. X. Guo are with the Department of Physics,Xidian University, Xi’an, 710071 Shaanxi Province, China. K. F.Ren, G. Gouesbet, and G. Grehan are with the Laboratoired’Energetique des Systemes et Procedes, VMR 6614, Complex deRecherche Interprofessionnel en Aerothermochimie, INSA deRouen B. P. 08, 76130 Mont-Saint-Aignan, France.
Received 25 January 1996; revised manuscript received 9 De-cember 1996.
5188 APPLIED OPTICS y Vol. 36, No. 21 y 20 July 1997
prediction of electromagnetic scattering by a coated,multilayered sphere with particular emphasis re-quired for the evaluation of the so-called scatteringcoefficients an and bn. In textbooks by Bohren andHuffman9 and by Kerker10 an analytical solution ispresented for the evaluation of the scattering coef-ficients an and bn of a multilayered sphere in termsof determinants. Toon and Ackerman11 rewrotethe Aden and Kerker equations12 to deal betterwith associated numerical problems. However,computational difficulties were still encountered inpractice. More specifically, the ill-conditionedcharacter of the problem and numerical round-offerrors can induce large accumulative inaccuracieswhen one computes the Ricatti–Bessel functions ~byan upward recursive procedure! involved in thescattering coefficients an and bn and in the internalfield coefficients. To overcome these difficulties,several researchers succeeded in recasting theproblem into more suitable forms and presentedvarious computational algorithms.
Bhandari13 proposed an iterative procedure inwhich the scattering coefficients for the ith layer inthe sphere are related to those for the ~i 2 1!th layer.However, the computational scheme requires the useof complex algorithms. More recently Mackowski etal.,14 Wu and Wang,15 and Kai and Massoli16 pro-posed more stable and accurate recursive proceduresfor numerically computing the scattering coefficientsan and bn. These procedures involve logarithmic de-rivatives and ratios of the Ricatti–Bessel functions.Wu and Wang15 discussed in detail the asymptoticform of these functions and their behavior in comput-ing the scattering coefficients.
Also, several authors3–5 independently extendedthe description of the scattering of a plane wave by acoated or multilayered sphere to the case of arbitrarybeam illumination with particular attention paid tothe case of Gaussian beams and laser sheets. Notethat Ref. 3 limits itself to coated spheres ~two layers!and that the formulation in Ref. 4 is similar to theGLMT stricto sensu, i.e., the GLMT for spheres. Animportant result is that scattering coefficients in thecase of shaped beams are still expressed in terms ofthe scattering coefficients an and bn for plane-waveillumination. Therefore any progress in algorithmsfor computing the plane-wave scattering coefficientscan be used immediately for the case of shaped beamillumination.
In particular, in all previous papers the number ofaffordable layers is always a steeply decreasing func-tion of the outer particle diameter. As a matter offact, according to recent results by Kai and Massoli,16
the maximal number of affordable layers for an inci-dent wavelength in the visible range ~l ' 0.5 mm! isapproximately 40 for a particle with a 10-mm diam-eter, whereas it decreases to approximately 12 for a60-mm-diameter particle and to only 2 for a 100-mm-diameter particle. However, for one to achieve anaccurate enough description of the light scattered bya multilayered sphere, the number of layers must beat least equal to approximately the outer size param-eter a 5 pdyl, where d is the outer particle diameterand l is the wavelength of the incident beam. Thenthis number of layers should be approximately equalto 600 for a 100-mm-diameter particle and for anincident wavelength equal to 0.5 mm, although onlytwo layers are numerically affordable. This examplepoints out the need for improved algorithms to whichthis paper is devoted.
Also, geometrical optics is often used to predictthe properties of large particles with a radial re-fractive index gradient. During the past years aspecial effort was devoted to the use of geometricaloptics in predicting the response of optical sizerswhen the particles under study were multilayeredspheres.17–22 Nevertheless, in some situations, asin the use of the rainbow angle approach, geomet-rical optics can predict only qualitative behavior,i.e., it fails to predict scattering patterns accuratelyenough, leading to uncertainties as discussed byAnders et al.23,24 As the rainbow refractometry isclaimed to measure droplet temperatures within anaccuracy of approximately 1 °C, the understandingof the effect of a radial temperature gradient in thesphere is relevant, particularly in combustion sys-tems in which the droplets are usually injected intoa high-temperature environment. This issue is ofparticular practical interest in discussing lightscattering by multilayered spheres.
This paper is therefore organized as follows. InSection 2 we provide a brief review of the theoreticalbackground. In Section 3 we describe the im-proved algorithms and their improved efficiency,including comparisons with the performance of Kai
and Massoli algorithms, in particular in terms ofcomputational times and extended ranges of appli-cability. Section 4 is devoted to some exemplify-ing results with particular attention paid to therainbow angle issue. Section 5 is the conclu-sion.
2. Theoretical Background
The geometry corresponding to the scattering of aplane wave or a Gaussian beam by a multilayereddielectric sphere is depicted in Fig. 1, in which mj isthe complex refractive index of the material in the jthlayer relative to the refractive index of the surround-ing medium. The jth layer is characterized by a sizeparameter of xj 5 2 prjyl, where rj is the outer radiusof the layer and l is the wavelength of the incidentwave in the surrounding free space.
Let us first consider the case of an illuminatingplane wave with an exp~iwt! time dependence, whichis omitted in all formulas as is the normal practice.The incident and scattered field components as wellas the field components in the different layers of thesphere can be expanded in an infinite series of spher-ical harmonic functions.
In accord with the conditions to be satisfied at in-finity, that is, the asymptotic behavior of the spheri-cal Bessel functions and the boundary conditions atthe interface in each different region, the coefficientsan and bn are derived ~see Ref. 15!. The scatteringcoefficients an and bn can then be evaluated by use ofa recursive computational scheme, which can be ex-pressed by the following set of formulas:
an 5Cn~xL!
jn~xL!zHn
a~mLxL! 2 mLDn~1!~xL!
Hna~mLxL! 2 mLDn
~3!~xL!, (1)
bn 5Cn~xL!
jn~xL!zmLHn
b~mLxL! 2 Dn~1!~xL!
mLHnb~mLxL! 2 Dn
~3!~xL!, (2)
Fig. 1. Geometry of the problem under study.
20 July 1997 y Vol. 36, No. 21 y APPLIED OPTICS 5189
with
Hna~mj, xj!
5@Cn~mjxj!yxn~mjxj!#Dn
~1!~mjxj! 2 AnjDn
~2!~mjxj!
Cn~mjxj!yxn~mjxj! 2 An~ j! , (3)
An~ j! 5
Cn~mjxj21!
xn~mjxj21!zmjHn
a~mj21xj21! 2 mj21Dn~1!~mjxj21!
mjHna~mj21xj21! 2 mj21Dn
~2!~mjxj21!,
(4)
An~1! 5 0, Hn
a~m1x1! 5 Dn~1!~m1x1!; (5)
Hnb~mj, xj!
5@Cn~mjxj!yxn~mjxj!#Dn
~1!~mjxj! 2 BnjDn
~2!~mjxj!
Cn~mjxj!yxn~mjxj! 2 Bn~ j! , (6)
Bn~ j! 5
Cn~mjxj21!
xn~mjxj21!zmj2iHn
b~mj21xj21! 2 mjDn~1!~mjxj21!
mj21Hnb~mj21xj21! 2 mjDn
~2!~mjxj21!,
(7)
Bn~1! 5 0, Hn
b~m1x1! 5 Dn~1!~m1x1!, (8)
where Dn~1!~z! 5 Cn9~z!yCn~z!, Dn
~2!~z! 5 xn9~z!yxn~z!, and Dn
~3!~z! 5 jn9~z!yjn~z! are the logarithmicderivatives of the Ricatti–Bessel functions. Hn
a andHn
b are recursive functions introduced in the numer-ical procedure. In the above recurrence formulasthere are only three logarithmic derivatives and tworatios of the Ricatti–Bessel functions, Cn~z!yjn~z!and Cn~z!yxn~z!, which remain bounded when theouter size parameter becomes large and the core sizebecomes small.
Dn~1!~z! is evaluated by use of a downward recur-
rence formula. The other functions, such as Dn~2!~z!,
Dn~3!~z!, and Cn~z!yxn~z!, are all computed by use of
upward iterative formulas. The algorithm de-scribed above overcomes overflow problems otherwiseencountered for the high-order or large-argumentRicatti–Bessel functions involved in the formulation,avoiding erroneous erratic behavior as well as cumu-lative errors in matrix processing. An additional ad-vantage is that it can easily be simplified to the caseof the homogeneous sphere.15
Once the scattering coefficients an and bn for theplane wave are obtained, we can evaluate the scat-tering coefficients an and bn appearing in the GLMTfor multilayered sphere, i.e., in the theory of interac-tion between shaped beams and multilayeredspheres. These coefficients are given by relationsthat are structurally identical to those obtained inGLMT for homogeneous spheres4,5:
anm 5 gnm
TMan, bnm5gnm
TEbn, (9)
where gnm, TM and gn
m, TE are the beam-shape coef-ficients appearing in GLMT.1,2 It then becomes ap-parent that all expressions in the framework ofGLMT for multilayered spheres are structurally
5190 APPLIED OPTICS y Vol. 36, No. 21 y 20 July 1997
identical to those obtained in GLMT.4 In particular,the scattered wave in the far-field region is given by
Eu 5iE0 exp~2ikr!
krS2~u, w!, (10)
Ef 5 2E0 exp~2ikr!
krS1~u, w!, (11)
where the generalized amplitude functions S1~u, w!and S2~u, w! read as
S1~u, w! 5 (n51
`
(m52n
n 2n 1 1n~n 1 1!
@mgnm
TManpnumu ~cos u!
1 ignm
TEbntnumu cos u!# exp~imw!, (12)
S2~u, w! 5 (n51
`
(m52n
n 2n 1 1n~n 1 1!
@gnm
TMantnumu ~cos u!
1 imgnm
TEbnpnumu cos u!# exp~imw!. (13)
The intensity Is and the polarization degree P of thescattered field are given by
Is 5 uS1u2 1 uS2u2, P 5uS1u2 2 uS2u2
uS1u2 1 uS2u2. (14)
The extinction cross section Cext and the scatteringcross section Csca read as
Cext 5l2
p (n51
`
(m52n
n 2n 1 1n~n 1 1!
~n 1 1 1 umu!!~n 2 umu!!
3 Re$anugnm
TMu2 1 bnugnm
TEu2%, (15)
Csca 5l2
p (n51
`
(m52n
n 2n 1 1n~n 1 1!
~n 1 1 1 umu!!~n 2 umu!!
3 $uanu2ugnm
TMu2 1 ubnu2ugnm
TEu2%, (16)
The backscatter cross section s is defined by Kerker~Ref. 10, p. 128! as
s 5 l2yp U(n51
`
~n 1 1y2!~21!n~bn 2 an!U2
. (17)
Nevertheless, when the number of layers is large, theoriginal formulation of Wu and Wang15 fails becausethe order of the Bessel functions can become largewith respect to the argument. Then the ratiocn~mjxj!yxn~mjxj! and the coefficients An
~ j! and Bn~ j!
defined by Eqs. ~4! and ~7! tend to 0, especially forsmall j’s. As a result the denominators of Hn
a andHn
b @Eqs. ~3! and ~6!# can be very small, smaller in-deed than the digit accuracy permitted for a given
computer. To solve this problem, we introduce anew function Rn
~ j!, defined as
Rn~ j! 5
cn~mjxj21!
xn~mjxj21!
xn~mjxj!
cn~mjxj!. (18)
Eqs. ~3!, ~4!, ~6!, and ~7! now read as
Hna~mj, xj! 5
Dn~1!~mjxj! 2 An
jDn~2!~mjxj!
1 2 An~ j! , (19)
An~ j! 5 Rn
~ j! zmjHn
a~mj21xj21! 2 mj21Dn~1!~mjxj21!
mjHna~mj21xj21! 2 mj21Dn
~2!~mjxj21!,
(20)
Hnb~mj, xj! 5
Dn~1!~mj 2 xj! 2 Bn
jDn~2!~mjxj!
1 2 Bn~ j! ,
(21)
Bn~ j! 5 Rn
~ j! zmj2iHn
b~mj21xj21! 2 mjDn~1!~mjxj21!
mj21Hnb~mj21xj21! 2 mjDn
~2!~mjxj21!,
(22)
where Rn~ j! is computed by upward recurrence.
With this improvement the algorithm introduced byWu and Wang15 becomes much more stable.
3. Algorithm Efficiency
A. Computational Domain
In optical particle sizing, particles with a large sizeand a continuous radial variation of the refractiveindex can be encountered. Such particles can be in-vestigated in the framework of the GLMT for multi-layered spheres if accurate enough recursivealgorithms are available. In this subsection wetherefore compare the ranges of applicability of thepreviously described algorithm with those used byother authors.14–16
In the improved algorithm the computation of in-dividual ratios of the Ricatti–Bessel functions is car-ried out in the same way as in the original paper byWu and Wang, in which the behavior of the numer-ical predictions was compared with various asymp-totic expansions. Furthermore, the code based onthe improved algorithm for computing the scatteringcoefficients was used extensively to compute the scat-tering properties of multilayered spheres with a largenumber of layers, in particular in the case in whichthe refractive index is the same anywhere in theparticle. These predictions were compared with suc-cess to the predictions of a classic LMT code for ho-mogeneous particles, including the case in which therefractive index of the particle is a complex number.
Figure 2 displays the number of affordable layersversus the size parameter for different algorithms forevaluation of the scattering coefficients an and bn.As in Kai and Massoli,16 the radial profile of therefractive index, m 5 n 1 ik, is described at the ithlayer level by ni 5 n1 1 0.5 ~nL 2 n1!~1 2 cos tp!, n15 1.01 nL, nL 5 1.33, ki 5 0, and t 5 ~i 2 1!y~L 2 1!,and the layer size parameters are set as xi 5 x1 1 t~xL
2 x1!, x1 5 0.001 xL, and ~i 5 1, 2, 3 . . . L!, where Lis the total number of layers.
In Fig. 2~a! domain A is the working domain ob-tained by Kai and Massoli16 with a FORTRAN 77 coderun on a UNIX reduced instruction set computingworkstation ~Apollo HP700!. The restricted charac-ter of domain A depends greatly on the computerused but also on the computational method and re-cursive schemes. Domain B defines the capabilitiesof our algorithm compiled with LF9025 on a PC 486DX. The program runs on the computer under theMS–DOS 6.2 operating system. The number of af-fordable layers is now compatible with the size pa-rameter criterion given at the end of the introductionfor an extended range of size parameters.
Figure 2~b! is the same as Fig. 2~a! but uses loga-rithmic instead of linear scales. However, we alsodisplay a third domain C, which was obtained withthe same FORTRAN code as that for domain B but com-piled with FORTRAN 77 from Sun OS on a UNIX work-station that possesses the same capacity as the oneused by Kai and Massoli.16,26 The domain C frontierat the top right-most part of the figure is caused by
Fig. 2. Computational domain on ~a! linear scale and ~b! logarith-mic scale.
20 July 1997 y Vol. 36, No. 21 y APPLIED OPTICS 5191
available memory limitations ~64 Mbytes!. How-ever, electromagnetic scattering by a multilayeredsphere is numerically available here in cases in whichthe number of layers used is as great as 15,000 and inwhich the outer size parameter is equal to approxi-mately 150,000. The critical size parameter forwhich its value is equal to the number of affordablelayers is approximately 103. Such improvements inefficiency are the results of algorithm modificationsas discussed below.
Instead of using ratios of the same Ricatti–Besselfunctions with different arguments, as in Ref. 16, wecompute ratios of two kinds of Ricatti–Bessel func-tions with the same argument, such as Cn~z!yxn~z! orCn~z!yjn~z!. For example, the ratio Cn~z!yxn~z! isthen computed by use of an upward recursive for-mula, reading as
Cn~z!
xn~z!5
Cn21~z!
xn21~z!
@Dn~2!~z! 1 nyz#
@Dn~1!~z! 1 nyz#
, (23)
which is different from those used by Toon and Ack-erman,11 Bhandari,13 and Kai and Massoli.16
When n .. uzu, the real and imaginary parts ofCn~z!yxn~z! rapidly decrease to very small valuesclose to zero. ~Of course, the convergence speed es-sentially depends on the complex argument.! In ad-dition, Dn
~2! ' Dn~3! ' Dn
~1! ' nyz when n is bigenough, say, as n . Nstop. By investigating Eq. ~23!,we then see that errors in evaluating Cn~z!yxn~z! donot propagate and grow when n increases. In par-ticular, the ratio Cn~z!yxn~z! is bounded and the re-cursion is stable whatever the complex argument.Furthermore, the asymptotic behavior of the ratioCn~z!yjn~z! is similar to the one of the ratio Cn~z!yxn~z!. A detailed discussion of such properties waspresented in an earlier article.15
Conversely, with the upward recursive relationsused by Kai and Massoli, namely,
jn~z1!
jn~z2!5
jn21~z1!
jn21~z2!
@Dn21~3!~z1! 2 nyz1#
@Dn21~3!~z2! 2 nyz2#
, (24)
Cn~z1!
Cn~z2!5
Cn21~z1!
Cn21~z2!
@Dn~1!~z2! 1 nyz2#
@Dn~1!~z1! 1 nyz1#
, (25)
significant round-off errors are introduced.Also, in our recursive scheme there is no special
difficulty in dealing with the limit cases uzi11 2 ziu30 and umj11 2 mju3 0 because of the absence of anysignificant round-off error. Therefore, in contrastwith other algorithms, dealing with such limit casesdoes not require us to Taylor expand the Ricatti–Bessel function ratios or factors. As a matter of fact,accurate and stable results for small or large spherescan be obtained: even the refractive indices of twosuccessive layers are close in the sense that mj 5mj11 or mj xj 5 mj11 xj11.
B. Computational Time
In Table 1 we compile the computational time ob-tained on a PC 486 DX at 33 MHz for various num-
5192 APPLIED OPTICS y Vol. 36, No. 21 y 20 July 1997
bers of layers L and the outer radius of the particle.The profile refractive index is taken to be linear from1.33 at the center to 1.31 at the surface. We alsomention that the version of the code used computesnot only the scattering coefficients an and bn but alsothe angular distribution of the scattered intensity~from 0 to 180° with a step D of 1°!. The computationof the scattering coefficients is actually the most time-consuming task.
Furthermore, if the number of layers and the outersize parameter are large, the two-dimensional arraycorresponding to the logarithmic derivatives, Dn
~1!~z!5 Cn9~z!yCn~z!, which is calculated by a downwardrecursive scheme, requires a large amount of RAMand can be larger than the memory of the computer.
C. Large Nonabsorbing Spheres
In the case of large nonabsorbing spheres with a contin-uous radial profile of the refractive index, we need toknow how many discrete layers are required to approxi-mate the original continuous profile accurately enough.On one hand, Kai et al.25 propose the criterion
uQext 2 QscauuQext 1 Qscau
, 1% (26)
to determine a correct number of layers. This crite-rion is, however, not satisfactory because both Qextand Qsca tend to 2 for a large nonabsorbing sphereilluminated by a plane wave. On the other hand, thebackscatter cross section s is found to be dependentsignificantly on the number of layers in the presentcase of a large nonabsorbing sphere. As an example,let us consider the profile of the refractive index dis-played in Fig. 3~a!, which is described by ni 5 n1 10.5~nL 2 n1!~1 2 cos tp! and ki 5 0, where t 5 ~i 21!y~L 2 1!, n1 5 1.43, and nL 5 1.30. Figure 3~b!displays the evolution of the corresponding extinctionand backscatter cross section versus the number oflayers used. In this figure the parameter is the
Table 1. Computation Time in minutes on a PC 486 DX at 33 MHz
wavelength of the incident beam ~l 5 5, 2.5, and 1mm! and the outer radius is 50 mm. It is seen that stends to become a constant when the number of lay-ers increases. Then the number of layers requiredto describe a given continuous profile can be definedas Lf, where s approximately reaches its constantbehavior. Of course, Lf depends on the size param-eter, and the larger the size parameter, the larger thenumber Lf of layers. Figure 3~b! also shows that theefficiency factor Qext converges too quickly to a con-stant value of approximately 2 in agreement with ourprevious discussion of inequality ~26!, therefore pro-viding too loose a criterion. Note that, relying on ourLf criterion, we find that the number of layers mustbe of the order of the outer size parameter. Also, thethickness Dr of each layer has to be as small as ly4.
4. Exemplifying Results
A. Scattering Diagrams and Cross Sections
Figures 4~a! and 4~b! display the scattering inten-
Fig. 3. Definition of a criterion for the number of layers requiredto describe a refractive index profile: ~a! continuous profile index,~b! extinction and backward efficiency factors versus the number oflayers for a plane wave.
sity and the polarization degree, respectively, of ablood cell illuminated off-axis by a Gaussian beam.The beam-shape coefficients of the Gaussian beamwere computed in the framework of the localizedapproximation.2,27 The radii of the nucleus andthe cytoplasm layer are r1 5 3 mm and r2 5 3.5 mm,with refractive indices relative to water equal to m15 1.05 1 0.05i and m2 5 1.03, respectively. Thewavelength of the incident beam is l 5 0.6328 mm.In the beam coordinates the blood cell is located atx0 5 2 mm, y0 5 2 mm, and z0 5 10 mm, the electricvector being polarized in the x direction at the focalwaist. When the beam-waist radius is w0 5 10 mmand w0 5 20 mm, i.e., more than three times largerthan the outer cell diameter, there is almost nodifference for both the angular distribution of thescattering intensity and for the polarization degreewith respect to the plane-wave illumination case.When the beam-waist radius becomes comparable
Fig. 4. Scattering properties of a blood cell with an off-axis loca-tion in a Gaussian beam: continuous curve, w0 5 20 mm, filledcircles, w0 5 10 mm, filled squares, w0 5 3 mm, filled triangles, w0
5 1 mm. ~a! Scattering intensity versus scattering angle, ~b! po-larization degree versus scattering angle.
20 July 1997 y Vol. 36, No. 21 y APPLIED OPTICS 5193
with and smaller than the outer cell diameter, scat-tered intensities decrease greatly when the beamwidth is reduced.
In Figs. 5 and 6 the extinction and scattering effi-ciency factors Qext and Qsca, here defined as the cor-responding cross sections Cext and Csca over pr2,where r is the outer radius of the particle, are plottedas a function of the beam-waist radius in Fig. 5 andthe distance between the nucleated blood cell and thebeam-waist center in Fig. 6. In Fig. 5 Qext and Qscaare almost constant when w0 5 10 mm and remainclose to the values for an incident plane-wave illumi-nation. In contrast, for narrower beams when w0 is2, 3, or 5 mm, Qext and Qsca significantly decreasewhen z0 increases. Figure 6 shows, for different val-ues of the coordinates x0 and y0 ~with z0 set to 100mm!, that Qext and Qsca depend greatly on the beam-waist radius w0. It is not worthwhile to present re-
Fig. 5. Qext and Qsca versus the distance to the beam waist. Theparticle is located at x0 5 2 mm, y0 5 2 mm. The parameter is thebeam waist that takes on the values of 2, 3, 5, and 10 mm. Theincident wavelength is 0.6328 mm.
Fig. 6. Qext and Qsca versus the beam-waist radius. The param-eter is the particle location that takes on the values, in mm, of ~x0
5 0, y0 5 0!, ~x0 5 2, y0 5 2!, ~x0 5 5, y0 5 5!, and ~x0 5 2, y0 510!. z0 is set to 100 mm. The wavelength is 0.6328 mm.
5194 APPLIED OPTICS y Vol. 36, No. 21 y 20 July 1997
sults for different values of z0 because they areobviously more dependent on w0 than z0.
We finally consider an inhomogeneous sphere witha continuous radial variation of the refractive indexm defined by the Cauchy formula
m 5c
1 1 fr2yb2 ,
where c 5 1.2, f 5 0.253, and the outer size param-eter is equal to kb 5 5 ~to compare with Kerker,10 p.249!. The sphere is divided into 50 layers. In Fig.7 the average intensity ~i1 1 i2!y2 is plotted versusthe scattering angle for an incident plane wave andfor on-axis Gaussian beam illuminations. The pa-rameter is the beam-waist radius that takes on thevalues of 1, 3, and 5 l.
B. Rainbow Angle
There has been a long tradition in the study of therainbow phenomenon, as exemplified by recent pa-pers such as Refs. 28–30, among many others.Nowadays the rainbow phenomena is used by work-
Fig. 7. Scattering diagrams for a 50-layer particle. The param-eter is the beam-waist size, expressed in number of wavelengths.
Fig. 8. Scattering diagram in the rainbow angular range for awater droplet diameter of 250 mm.
Fig. 9. Evolution of the scattering diagram in the rainbow angular range for a 250-mm-diameter droplet with a linear variation of therefractive index ~1.33 at the core center and 1.36 at the outer surface!. The droplet is divided by use of n layers of equal thickness.
ers in spray combustion to extract droplet sizes andtemperatures by analysis of the scattered-light dis-tribution at the rainbow angle.17,18,20–22 As a mat-ter of fact, Fig. 8 displays the computed scattered-light distribution at the rainbow angle for ahomogeneous water droplet ~m 5 1.33 1 0i! with adiameter equal to 250 mm. One identifies the firstor main rainbow at an angle of approximately 138°,then the Alexander’s dark band in the range of
132–137°, and the secondary rainbow at approxi-mately 128°.
According to the approach used by van Beeck andRiethmuller,20 the refractive index, i.e., the temper-ature of the droplet, is extracted from the location ofthe first maximum M of the main rainbow, whereasthe particle size is extracted from the differencebetween the locations M and P of the first two peaksof the main rainbow. In this approach, based on
20 July 1997 y Vol. 36, No. 21 y APPLIED OPTICS 5195
Table 2. Study of the Influence of the Number of Layers on Rainbow Technique Measurementsa
a For water only the homogeneous sphere case is considered. For alcohol the number of layers ranges from 1 ~homogeneous case! to 128.
the Airy theory of the rainbow, the location of therainbow and the diameter d of the particle are givenby22,20
u 5 22* arcsinF13
~m2 2 1!G1y2
1 4* arcosF13
~12 2 3m2!1y2
m G , (27)
d 5l
4 Scos ~rrg!
sin3~rrg!D1y2 Sai 2 aj
ui 2 ujD3y2
, (28)
where ai’s are normalized angular deviations corre-sponding to i peak maxima31,20 and tabuled by vanBeeck and Riethmuller20 with respect to the geomet-ric rainbow, ui’s are the actual locations of the ithpeak expressed in radians, and rrg is given by
sin rrg 5 @~m2 2 1!y3#1y2. (29)
Here m is the real part of the refractive index of theparticle relative to the surrounding medium.
When these formulas are applied to the data in Fig.8, m is estimated to be 1.3360 and the particle diam-eter, to be 224 mm. These estimations still can berefined by use of iteration in the Airy formalism, thenafterward by use of the full LMT approach.
Fig. 10. Light scattered in the rainbow angle range for a nonlin-ear profile of the refractive index.
5196 APPLIED OPTICS y Vol. 36, No. 21 y 20 July 1997
We now focus our attention on the influence of arefractive index radial gradient on the measure-ments. Therefore, we are content in quantifyingmeasurement shifts produced by such a gradientwithout accounting for the aforementioned refine-ments, which are second-order for our purpose.
In addition to investigating the influence of refrac-tive index gradients, it is also worthwhile to addressthe influence of sphericity departures. Relying onMarston’s work,32 van Beeck and Riethmuller22 pro-posed to detect nonsphericity of droplets by compar-ing the size obtained from the Airy pattern ~caused bythe refracted rays! and the size obtained from theripple structure superimposed on the Airy rainbowpattern ~caused by the reflected rays!. The sensitiv-ity of the refractive index to radial gradients wasstudied by Anders et al.23,24 by use of the geometricaloptics, which, however, cannot accurately describethe rainbow ~see van de Hulst, pp. 240–24931!. Thepreviously introduced algorithm provides an oppor-tunity for an exact investigation of this issue.
Figure 9 displays the scattered-light distribution inthe rainbow angle range for a 250-mm-diameter drop-let with a linear variation of the refractive index thatis set to 1.33 at the center and 1.36 at the outersurface. The particle is divided by successive use of2, 4, 8, 16, 32, 64, and 128 layers of equal thickness.The incident wavelength is 0.5145 mm.
Some particularities of the light scattered by a mul-tilayered sphere in the rainbow angle range can beunderlined as follows. There is a significant amountof light that is scattered in the Alexander dark bandat angles smaller than approximately 140° @see Figs.9~a!–9~d!#. Subsidiary peaks in the main rainbow donot exhibit a regular decrease as in the case of homo-geneous particles @Figs. 8 and 9~f !#. The scatteringdiagrams are identical for 64- and 128-layer particles@Fig. 9~f !#. In practice, this means that the numberof layers is now sufficient to represent correctly acontinuous profile of refractive index. In otherwords, the other cases ~n 5 2, 4, 8, 16, 32! concernmultilayered spheres stricto sensu, i.e., spheres ex-hibiting refractive index jumps.
The data in Fig. 9 are then low-pass filtered toremove the high-frequency ripple structure by use ofa procedure described by van Beeck and Riethmuller,afterward allowing an easy angular-location determi-nation of peak maxima. Column ~b! in Table 2 in-dicates the biggest amplitude peak locations fordifferent kinds of particles. Column ~c! provides therefractive index value, which is measured from thesepeak locations by use of Eq. ~22!. The particle size iscomputed from the angular separation between thepeak of maxima amplitude and the next peak by useof Eq. ~23! @see column ~d!#. Column ~e! exhibits theAiry theory rainbow-shift reading as
u1 2 u0 5 a1ly4l, (30)
where l 5 ~3ld2y4h!~1y3! and h 5 3y4sin2 rrgtanrrg asdefined by van de Hulst.31 Column ~f ! provides therefractive index value, which is corrected when therainbow shift is taken into account. Finally, column~g! shows the particle diameter extracted from Eq.~23! by use of the refractive index given in column ~f !.In Table 2 the cases of homogeneous particles with arefractive index of 1.33 and 1.36 were added as limitcases. These two cases are labeled water and alco-hol, respectively.
For the two-layer case the measured refractive in-dex is found to be 1.3628, i.e., larger than the actualrefractive index at the outer layer ~1.36!. Thenwhen the number of layers increases, we observe anoverall trend of a decrease in the index to 1.3361 and1.3364 for the cases of 64 and 128 layers, respectively.Therefore in the case of a linear continuous profile ofthe refractive index, the measured refractive index ismuch closer to the one at the particle center ~1.33!than to the one at the surface ~1.36!. Such is also thecase for a nonlinear radial variation of the refractiveindex. For example, for the index profile of Fig. 3~a!within a 250-mm-diameter particle, we obtain thescattered-light distributions in the rainbow rangethat are displayed in Fig. 10 for both 128 and 256layers. For this case the measured refractive indi-ces are m 5 1.4127 for both.
5. Conclusion
An efficient algorithm allowing us to compute theproperties of the light scattered by a multilayeredsphere illuminated by a plane wave—or more gener-ally, by a shaped beam—was presented. This algo-rithm allows us to increase the affordable range ofcomputations, both in the size and in the number oflayers, by several orders of magnitude.
With a criterion relying on the computation ofbackscatter cross sections, requiring that the numberof layers be of the order of the size parameter, we canpredict the scattering properties of multilayered par-ticles with a size parameter as large as 1000 ~corre-sponding to a particle diameter of 150 mm for awavelength of 0.5 mm! with an arbitrary profile of therefractive index. Nevertheless, an accurate descrip-tion of the scattering properties often can be obtainedfor a number of layers smaller than the one given by
the above criterion, as exemplified by the rainbowangle for a 128-layer particle with a size parameterequal to 1500 ~particle diameter of 250 mm, incidentwavelength of 0.5 mm!.
Exemplifying results were provided for the compu-tation of extinction, scattering, absorption, and back-scatter cross sections. Also, several scatteringdiagrams, including a study of the sensitivity of themain rainbow description to a radial variation of therefractive index, were presented.
This work was supported by the National NaturalScience Foundation of China and the French Minis-try of Environment in the framework of a program ofcooperation between China and France ~PRA 94-E4!.
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