Optical scattering by metallic and carbon aerosols ofhigh aspect ratio
Magdy F. Iskander, Steven C. Olson, Robert E. Benner, and Dawn Yoshida
The iterative extended boundary condition method (IEBCM) is utilized to calculate scattering and absorp-tion by metallic colloids and carbon aerosols in the 0.4-pm < X < 10-pm optical wavelength range. Thecolloids and aerosols were modeled by dielectric spheroids of high aspect ratio. The new IEBCM method isfound to be suitable for making calculations for particles with aspect ratios as high as 12. Results arepresented for silver and aluminum metallic aerosols as well as for atmospheric aerosols such as soot and ironoxides (magnetite). The various parameters used to examine the convergence of the IEBCM solution, such asthe number of subdomain expansions and the size of the incremental change in intermediate object sizes usedin the iterative process, are discussed. Using the internal field distribution to test the convergence of theresults is also found to be more accurate than the traditional procedure which utilizes extinction andscattering cross-section data.
1. Introduction
For many optical scattering applications such asthose involving surface-enhanced Raman scattering(SERS),1 obscuration and spacecraft survivability,2and examination of atmospheric absorption of visibleradiation,3 it is necessary to calculate the scatteringand absorption characteristics of elongated dielectricparticles. In SERS, for example, there is considerableinterest in identifying the shapes and properties ofenhancing structures for optimizing the Raman signalswhich are used for the chemical characterization ofsurfaces and interfaces.' Calculation of the scatteringand absorption by mixtures of absorbing and nonab-sorbing aerosols, on the other hand, is important incharacterizing atmospheric aerosols.3
In these calculations, aerosols have been modeled byspherical3 and spheroidal4 dielectric particles. TheMie solution is used for calculating scattering fromspherical particles, while a variety of methods havebeen utilized for spheroidal objects.4 In particular,the extended boundary condition method (EBCM)formulated by Waterman has been extensively used incalculations for spheroidal objects.4 5 As indicated inearlier publications, the EBCM has some convergence
When this work was done all the authors were with University ofUtah, Electrical Engineering Department, Salt Lake City, Utah84112. Steven Olson is now with Hughes Aircraft Company, RadarSystems Group, El Segundo, California.
problems, particularly when applied to high permittiv-ity dielectric materials and to objects of high aspectratio.6-8 This numerical instability arises from the ill-conditioned system of equations that results from tak-ing a large number of terms in the spherical field ex-pansion to force a single vector spherical expansion tofit a nonspherical object.67 Recently, a techniquecalled the iterative extended boundary conditionmethod (IEBCM) was developed to calculate the scat-tering and absorption by elongated dielectric objects.7-9The method utilizes several spherical expansions todescribe the internal fields in the dielectric object.Calculating the internal fields using these subdomainexpansions and enforcing the field continuity through-out the internal volume of the object by point match-ing the various expansions in the overlapping volumeswere found to provide adequate results for elongatedand composite dielectric structures.7 -10 An even bet-ter fit to the geometry of the dielectric object wasobtained by using mixed basis functions in the subdo-main expansions.9 To help carry out the calculationprocedure, an iterative method was utilized. The ini-tial estimate of the tangential fields on the surface ofthe object was obtained either by replacing the lossydielectric object by a perfectly conducting one7 or byapproximating the geometry by a spherical one of thesame dielectric properties.8 It can be shown that whilein the first case the iterative procedure effectivelybuilds in the original dielectric properties of the object,the second method effectively helps recover graduallythe original shape of the object. It should be empha-sized, however, that in both cases the key to success ofthe IEBCM is related to utilization of multiple basis
function expansions. Other procedures being usedpresently to obtain the initial estimate include theutilization of high-frequency (geometrical optics)and low-frequency (perturbation techniques) approxi-mations to start the iterations.
In this paper we present the results of our recentefforts to utilize the IEBCM at optical frequencies.Specifically, results for the scattering and absorptionby metallic and lossy dielectric elongated objects willbe described. Also, the various criteria used to checkthe convergence of the solutions will be discussed.
II. Basic Features of the IEBCM Technique
As indicated earlier, the IEBCM was developed toovercome the numerical instabilities of the regularEBCM.7 Table I summarizes a comparison betweenthe regular EBCM and the IEBCM solution proce-dures. One main feature of the IEBCM is its iterativenature. It requires an initial estimate of the tangentialfields on the surface of the scatterer. It also utilizesthe key process of representing the fields inside theobject by several overlapping subregional expansions.For objects which are characterized by large complexpermittivities, the initial estimate was obtained byreplacing the dielectric object by a perfectly conduct-ing one of the same shape and size and then solving forthe current densities of the substitute object.7 910 Inthis case the surface current density of the lth iterationis obtained by solving the following equation:
{P(r) + v X [(&') X (l-1)(r') *.0(kr)1kP]ds'
1 )- v x v x f--.-[h(r')
JWEo
X Hi' I; ) () G,(k TO/kr)ds'
= v XvX -. [nhr')VX JWe0
(1)
where h is the unit outward normal to the surface s,G(kr/ki') is the free-space transverse dyadic Green'sfunction, and k = 4y OEO is the wave number in freespace. The internal fields Ef&`) and Hfi(nt) are theknown fields from the internal problem in the previousiteration. The only unknown in Eq. (1) is the incre-mental surface current density n X AH) to be calculat-ed as the solution to the external problem. This incre-mental current, when determined, is added to h XH(1-1) from the previous iteration to give a new andrefined value of the total surface current h X H+.
To extend the application of the IEBCM techniqueto low-loss or lossless dielectric objects, the initial esti-mate was obtained by approximating the geometry ofthe object. A solution of the original geometry wasobtained by iteratively obtaining solutions for objectsof intermediate shapes between the initial approxi-mate (substitute) shape and the final geometry of in-terest. For example, for a highly elongated dielectric
Table 1. Comparison between the Basic Features of the Regular (EBCM)and the New Advantages of the New Iterative Technique
Comparison between EBCM and IEBCMEBCM IEBCM
Single spherical expansion
Analytical continuityimplicitly assumed
One-step solution
ApplicationLimited to small andmoderate aspect ratios
Multiple expansions; sphericaland mixed
Continuity explicitly enforced
Iterative and requires initialsurface fields
ApplicationsSmall and moderate aspectHighly elongated objectsObjects of large E* and KaComposite homogeneousobjects whose surface nat-urally divides into severalparts of different geometricshapes.
object, the initial estimate of the surface fields may beobtained from the Mie solution of a spherical objectwith the same dielectric properties. The iterative pro-cedure is then utilized to build in gradually the geome-try of the elongated object. In other words, the ob-ject's geometry was built in through the iterations inthe low-loss dielectric object's case. In the case ofscattering objects of high-complex permittivity, theinitial estimate was obtained by approximating theobject's properties (i.e., permittivity) by those of aperfectly conducting one, and the original dielectricproperties were built in gradually through the itera-tions. The iterative solution for the low-loss or loss-less dielectric case utilizes the following equation:
E(r) + V I [(r) X Ei(l)(j')] * G(kr/kr')ds
-v X v X ffl [f(jI) X Hnti)(T )]
= v X X . [h(') X AH(+)(r')]
* (kr/k;')ds'
-V X [h(r') X v' X Aa(+)(r')]
*0(kr/kr')ds', (2)
where AIH$) and v' X AR( °) are the incremental mag-netic and electric fields to be determined. As indicat-ed earlier, k- 1yn) and Htl) are known from the solutionof the approximate geometry. It should be noted thatin the high-loss object's case, we iterated only over thesurface electric current density [first term on the right-hand side of Eq. (2)]. In the low-loss case, we main-tained both the electric and magnetic current densityterms in Eq. (2) (i.e., complete integral equation of theEBCM approach). This is because in the low-loss case
Fig. 1. Magnitude of the axial electric field distribution in magne-tite, e = 2.4 -j0.83, semimajor axis = 100 nm, and aspect ratio a/b =4 and 9. The direction of propagation of the incident plane electro-magnetic wave is along the major axis of the spheroid. The incident
wavelength is = 0.4 pm
we iterate effectively between various geometries andsolve the complete equation (with both electric andmagnetic currents terms present) for each geometry.It should be emphasized, however, that the key advan-tage of the IEBCM,. which arises from the division ofthe internal volume of the dielectric object into severaloverlapping subregions and from utilizing separate ex-pansions in each, is utilized in both iterative methodsof obtaining the final solutions [i.e., Eqs. (1) and (2)].
111. Numerical Results
Figures 1-3 show some results obtained with theIEBCM when applied to light scattering by magnetiteand carbon soot aerosols of high aspect ratio. Figures1 and 2 show the electric field distribution along theaxis of the spheroidal particle, while Fig. 3 shows thecalculated scattering and extinction cross sections.From these results, it is clear that there is a tendency offield enhancement at the end of the spheroid with theincrease in aspect ratio, i.e., for a/b = 9. These resultsalso indicate that for spheroidal particles of semimajoraxis, a = 100 nm, e* = 2.4-jO.83, and as high an aspectratio as 12, solutions were possible at X = 0.4 Am.Needless to say, results in the 0.4 < X < 10-,um wave-length range (which is the range of interest for manyaerosol applications) are numerically easier to obtainbecause of the reduced optical size of the scatterer atlonger wavelengths X > 0.4 Am. The results in Figs. 1-3 are, therefore, at wavelengths which are among theworst cases from the numerical convergence point ofview.
IV. Computational Analysis of the IEBCM
In this section we examine the numerical accuracyand convergence criteria of the IEBCM. Convergence
.417 .765
.469 .472
.510 .435
.537 .438
.561 -. 442
.566 .444x x
.571 .483
.572 .779
a/b=3 a/b=9
Fig. 2. Magnitude of the axial electric field distribution in sootspheroidal particles of e* = 1.88 -10.69, semimajor axis a = 100 nm,and aspect ratio a/b = 3 and 9. The direction of the incident planewave is the same as Fig. 1, and the free space wavelength X = 0.4 ,m.
1.0
z00w
C,,C,)00C,z
C.)
0C.zw
0.1
0.01
0.001
0.0001.1. . . .( N
I i I a i I R i I I 2 3 4 5 6 7 8 9 10 11 12 13
ASPECT RATIO A/B
Fig. 3. Extinction and scattering cross sections of magnetite andsoot spheroidal particles (a = 100 nm) as a function of the aspect
ratio a/b.2 The incident plane wave wavelength is X = 0.4 ,um.
parameters such as the number and locations of thesubregional field expansions, the number of terms re-quired in each expansion and the size of the incremen-tal changes in the sizes of the intermediate objectswhen iterations are carried out over the geometry willbe examined.
First, to emphasize the key role played by the utiliza-tion of multiple internal field expansions in theIEBCM compared with a single expansion in theEBCM, numerical results for the calculated electric
a The results for spheroidal silver colloids are calculated at X = 0.621 pm, where e' =-17.4 and e" = 2.25. The semimajor axis of the spheroid
a = 100 nm, and results were calculated for a/b up to 7. Large field enhancements are not observed from the silver spheroids because of theirlarge sizes which were chosen to test the IEBCM in the regime X a.
Table ill. Comparison Between the EBCM and IEBCM Results for a Spheroidal Soot Particle of Semimajor Axis a =as a
a The results were calculated at X = 0.4,um where the complex permittivity of soot is e* = 1.88 - jO.69.
z
x
kdirection ofpropagation
H
E
Fig. 4. Geometrical locations of the points a-h along the axis z ofthe spheroid and utilized in the results given in Tables II and III.
field intensities at various points within the object areshown in Tables II and III. The locations of points a,b,.. ., h are illustrated in Fig. 4. The number of terms(NRANK) included in the single expansion of theEBCM are also indicated in the tables. It is clear fromTables II and III that for small aspect ratios, theEBCM and IEBCM give identical results for the fielddistributions. However, as the aspect ratio increases,the EBCM solution breaks down, and the field distri-bution at the ends of the spheroid becomes divergent.Specifically, with the increase in NRANK the EBCMsolution did not converge to a final value. Instead, theobtained results continued to diverge without indica-tion of the true value of the desired result. In otherwords, with the continuous divergence of the obtainedfield values it was not possible to determine the opti-mum value of NRANK. The divergent nature of thesolution is illustrated in Tables II and III by includingthe results obtained for NRANK = 20. The IEBCMresults, on the other hand, continue to behave properlyat the end of the spheroid, even with the increase inaspect ratio. A more quantitative validation of theIEBCM results will now be described.
A commonly used criterion for evaluating the accu-racy of numerical results is based on comparing calcu-lated data to experimentally measured values of theextinction and scattering cross sections of the object.This convergence criterion, which is based on compar-ing theoretical values to experimentally measured
(a) (b)Fig. 5. Extinction and scattering efficiencies as well as axial fielddistribution (Vim) values are presented to show the importance ofusing the distribution as an indicator of the convergence of thesolution instead of the efficiencies. The dielectric used was e* =3.058 + j.0 with ka = 1.5707, a = 100 nm, and a/b = 8.0. Five and
nine internal expansions were used for (a) and (b), respectively.
cross sections, is inadequate, however. To illustratethis point, consider a spheroidal object of 8:1 aspectratio. When five internal field expansions were usedin the IEBCM to characterize this object, the extinc-tion and scattering cross sections, which should beequal for a lossless object, are different by only 0.34%as shown in Fig. 5(a). This agreement could lead tothe conclusion that the calculated results are conver-gent. However, when nine internal field expansionswere used, a significantly different field distributionalong the spheroidal object was obtained as shown inFig. 5(b), while the scattering and extinction efficien-cies differed by only 0.07%. The field distribution forthe nine spherical field expansions is the correct onebased on comparisons with results obtained using oth-er numbers of the expansions. Specifically, we notethat the fields at the end points when five sphericalexpansions were used are incorrect (Fig. 5). For fieldpoints in the central region, on the other hand, resultsobtained with five and nine expansions are identical.Hence, we conclude that the field distribution withinthe dielectric object is a better indicator of the conver-gence of the solution than is the extinction or scatter-ing cross section. In the following, the impact ofchanging the convergence parameters on the accuracyof the solution will be examined in terms of the fielddistribution.
Regarding the best number and locations of thesubregional spherical expansions of the internal fields,it is clear that they depend on the aspect ratio andoptical size of the object. In our calculations we usedup to eleven expansions to characterize objects with ashigh an aspect ratio as 15. The manner in which thecenters of the spherical field expansions were arranged
(
x
(a) (b)
BY
(c)Fig. 6. Origin locations and the number of internal expansions usedfor three of the many possible numerical models used for theIEBCM: (a) three internal expansions close together; (b) threeinternal expansions spread far apart; (c) five internal expansions.
0.400
0.428
0.441
0 -. 449
0.456
0.456
0.453
0.436
(a)
0.487 0.487
0.440 0.441
0.442 0.442
- 0.449 - 0.449
0.456 0.456
0.456 0.456
0.463 0.463
0.521 0.521
(b) (c)Fig. 7. Numerical results obtained from the IEBCM for the threemodels shown in Fig. 6. The dielectric used is soot evaluated atka =1.5707, X = 0.4 pm, a = 100 nm, a/b = 7, and e = 1.88-jO.69. It is tobe emphasized that the modeling for (b) and (c) resulted in the samesolution, whereas the model for (a) did not converge to the proper
solution.
is illustrated in Fig. 6. The objective was, of course, toobtain a convergent solution using the minimum num-ber of spherical expansions. The results of Fig. 7 showthat convergent values can be obtained [Figs. 7(b) and(c)] if the spherical expansions are distributedthroughout the object. The results of Fig. 7(a) areinaccurate (erroneous at the end points) because thespherical expansions are localized near the central re-gion of the spheroid. As the electrical size of the objectincreases along with an increase in the aspect ratio,more of these expansions are required, and their loca-tions should be distributed throughout the internalvolume of the spheroid.
Another convergence criterion for the IEBCM is thenumber of terms required in each internal expansion.
Fig. 8. Results demonstrating the impact on the convergence of thesolution by varying the total number of coefficients in each expan-sion used in the IEBCM solution procedure. Models (a)-(d) usedseven internal expansions. The models used the following numberof internal expansion coefficients: (a) 52; (b) 59; (c) 66; and (d) 73.The dielectric evaluated was magnetite with ha = 1.5707, X = 0.4 Am,
a = 100.0 nm, and a/b = 7.0, and e* = 3.058 + jO.0.
This number varies from one geometry to another andis influenced by the number of spherical expansionsused to describe the internal fields. If too many termsare included in a single expansion, the resulting systemof equations may become ill-conditioned. If, on theother hand, not enough terms are included, the fieldvariations will obviously be improperly modeled. Theresults in Fig. 8 illustrate the impact on the conver-gence of the solution of the number of terms used in theexpansion. For the lossless spheroidal object shown inFig. 8 with a = 100 nm, a/b = 7, and ka = 1.57 at X = 0.4,gm, it is clear that an insufficient number of expansioncoefficients may result in erroneous results as can beseen at the end points of the spheroid in Fig. 8(a).Comparison among the results of Figs. 8(b)-(d) showsthat every effort should be made to assure the conver-gence of the solution by varying the number of terms ineach expansion and noting the ultimate convergence.As expected, convergence problems in the IEBCM of-ten impact the field distribution at the end points moreseverely than at the object center.
The incremental change in geometry from one shapeto the next in the iterative procedure, which involvesapproximating the geometry and recovering the origi-nal one iteratively, is yet another convergence criterionthat should be examined carefully. To save computa-tional time, the incremental change should be made aslarge as possible. Large geometrical changes, howev-er, may result in inaccurate results because of theinadequacy of the initial estimate (from the previousgeometry) for the new geometry. In an attempt todevelop some guidelines for the best incrementalchanges in the object's shape, we found that they de-pend on the aspect ratio of the object, its permittivity,and its optical size. Figure 9 shows an example of theeffect of iterating between initial objects of variousaspect ratios to solve for an object with an aspect ratio
Fig. 9 IEBCM results demonstrating the effect on the convergenceof the solution due to the size of the incremental change in geometryfrom one solution to the next. The same dielectric, geometrical, andelectrical parameters used for Fig. 8 were used in this case also.Seven internal expansions were used in all cases. The change in theshape of the geometries in terms of aspect ratios is as follows: (a)
3:1-9:1; (b) 5:1-9:1; (c) 7:1-9:1; and (d) 8:1-9:1.
of 9:1. Figure 9 shows that for the 9:1 aspect ratioobject with a small dielectric constant, the size of theincremental change in shape can be quite large. Forexample, it is possible to obtain reasonable results forthe object of 9:1 aspect ratio if the initial surface fieldsare taken from an object with a 3:1 aspect ratio. Formore accurate results, however, objects of larger aspectratios such as 7:1 or 8:1 should be used to obtain theneeded initial estimate of the fields to obtain a finalsolution for the 9:1 object. Reducing the extent of theincremental change in geometry was found to be moreimportant for objects of larger permittivity and largeroptical sizes. Much smaller incremental steps in thesizes of the intermediate objects are essential in ob-taining a convergent solution for objects of large opti-cal size. It is, therefore, suggested that the size of theincremental change in the geometry of the intermedi-ate objects should be carefully evaluated before con-tinuing the iterative process to prevent erroneous re-sults.
V. Discussion
In this paper several field distributions within elon-gated spheroidal dielectric objects were determined atoptical wavelengths using the IEBCM. The resultswere compared with the EBCM solution which hasbeen used extensively in calculating the scattering andabsorption characteristics of small particles in the op-tical regime. It was shown that the IEBCM solutiondoes indeed agree with the EBCM solution for lowaspect ratios; however, as the aspect ratio increases,the EBCM solution no longer models the internalfields accurately. Therefore, the multiple sphericalexpansions employed in the IEBCM play a major rolein maintaining a convergent solution as the particle ofinterest becomes highly elongated. Of course, the iter-ative nature of the IEBCM is also required in obtaining
convergent solutions, but more emphasis was placedon the multiple internal expansions which maintainthe field continuity throughout the interior of the elon-gated speroidal object.
It was shown that obtaining convergent solutionsusing the IEBCM is not a trivial matter and usuallyrequires varying different IEBCM parameters to veri-fy the accuracy of the solution. Such parameters in-clude the number of terms used for each expansion andthe incremental change in geometry for the case of thelow-loss IEBCM solution procedure. We also exam-ined the effect of varying these solution parameters onthe convergence of the IEBCM. The number andlocations of the internal spherical expansions used tosubdivide the spheroid played a major role in obtainingconvergent solutions. The solution is also sensitive tothe number of terms taken in the various expansionsused in the subregions. Another parameter of interestis the size of the incremental change in geometry.Since the iterative nature of the low-loss IEBCM solu-tion procedure involves solving for the correct incre-mental electric and magnetic current densities at eachintermediate geometry by using a previous geometry(with smaller aspect ratio) as an initial assumption, theincremental step in geometry should be carefully eval-uated. The convergence parameters may vary greatlyfrom one geometry to another depending on the per-mittivity, optical size, and aspect ratio of the object.
It was also shown that using the extinction and scat-tering efficiencies as an indicator of the solution con-vergence in a lossless scatterer is not a good method foridentifying a convergent solution. The axial field dis-tribution inside the object was found to be a betterindicator for the convergence of the IEBCM solution.
Our present efforts in extending the applications ofthe IEBCM include its application to optical scatter-ing and absorption by elongated particles with chain-and cluster-type structures and also its hybridizationwith the geometrical optics approximations and thegeometrical theory of diffraction (GTD). In the twolatter cases the geometrical optics approximation andthe GTD will be used basically to obtain the initialassumption of the surface fields which are required tostart the iterative procedure of the IEBCM. The dif-fraction coefficient in the GTD may be obtained from
the rigorous solution of the canonical problems de-scribed in Ref. 12.
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