Perturbation theory for scattering from dielectricspheroids and short cylinders
Richard D. Haracz, Leonard D. Cohen, and Ariel- Cohen
The perturbation theory suggested by Shifrin is applied through the second order to the scattering of lightfrom dielectric spheroids and finite cylinders. In the case of short dielectric cylinders, this technique pro-vides an accurate prediction of the scattering pattern in its range of applicability, and this prediction is espe-cially useful as no exact scattering solution exists. The validity of the perturbation theory is established bycomparison with exact results for the spheroid, and excellent agreement is shown for ka(m - 1) 1, wherek = 2r/X, a is a representative target dimension, and m is the index of refraction. The results for the finitecylinder are refined from our previous work by a careful construction of the internal electrostatic solution.This allows the calculation of intensities for short cylinders. Comparisons are made between the spheroidsand cylinders of equal volumes for aspect ratios ranging from 1/2 to 5, and significant differences are notedin some cases.
I. IntroductionThe scattering of radiation by infinite cylinders has
been studied for perpendicular incidence from the earlyhistory of the application of Maxwell's theory by Ray-leigh' and Mie2 and more recently for oblique incidenceby Mentzner 3 for perfect conductors and Wait4 for ar-bitrary materials. The exact solution is presented indetail by Kerker.5
We recently presented the theory for finite dielectriccylinders6 using a perturbative approach due to Shifrin 7
and applied to spherical targets by Acquista.8 Acquistahas also given the necessary modifications for applica-tion to other target shapes including cylinders andspheroids.
In Ref. 6, the treatment of the internal field for thefinite cylinder was based on comparison to an inscribedspheroid. Here we make an accurate calculation of theinternal field in the electrostatic case. This allows usto present results for short cylinders. (Reference 6restricted application to cylinders of an aspect ratio of20 or greater.)
We also apply the perturbative approach to the di-electric spheroid and demonstrate the range of con-vergence by comparison with the exact results of Asanoand Yamamoto.9 The results for the finite cylinder andspheroid of equal aspect ratio and volume are then,compared to indicate the effect of the sharp edges of thecylinder.
The authors are with Drexel University, Department of Physics& Atmospheric Science, Philadelphia, Pennsylvania 19104.
11. Generalized Perturbation Theory for Scatteringfrom a Dielectric Target of Arbitrary Shape
We start with the solution of Maxwell's equations foran isotropic and nonmagnetic medium of index of re-fraction m:
E(r) = Eo exp(ikor) + V X V X fd 3 r' ( - 1)
exp(iko|r - r E(r') - (2
- 1)E(r),jr -r'l
(1)
where the integration is over the target. The electricfield inside the target can be expressed in terms of theambient electric field as
Ei(r') = AijEoj exp(ikor'), (2)
where A is the polarization matrix. The coordinatesimplied are shown in Fig. 1 as x,y 1,z with z the sym-metry axis of the target. However, A is known for onlya few special shapes in the electrostatic limit, and con-sequently Eq. (2) needs to be modified to be useful.We, therefore, represent the internal field as
Ei(r') = AijEeffj(r%),
where A is the polarization matrix for the electrostaticcase. It is known for spheres, spheroids, and infinitecylinders, and it can be constructed, as will be shown,for finite cylinders. For axially symmetrical targets,
aTE 0 0
= 0 aTE 0
0 0 aTM
(4)
Here TE refers to the electric field being perpendicularto the axis, and TM refers to the field being parallel to
In these equations, the principal part of the integral isto be taken, and
u(x) = Sd 3r'U(r') exp(ix r'). (12)
The unit vectors it1g, i1 are target coordinates as shownin Fig. 1, and
XI PX.* -x, (13)
xI
Fig. 1. Geometry of the scattering process. The axes xiyl,zl arethe target axes with z1 the axis of symmetry. The axes xy,z are thedetector axes with z the direction toward the detector. The incidentwave vector k has the spherical coordinates (k,3/2,01) in the target
frame.
the axis. Of course, Eeff in Eq. (3) is still unknown, andit is found term by term using an expansion in a (M2
- 1)/47r:
Eeff(r) = E0 exp(ik -r) + a a-Ee,'?(r). (5)n1l
This effective electric field is related to the internalfield in Eq. (3), which we rewrite as
Eeffi(r) = A-'Ej (r) = E (r) + BijEeffj(r), (6)
with
B = 1-A. (7)
For r outside the target, B = 0, and Eeff = E. Using Eq.(6), the integral equation (1) is transformed into anequation for the effective electric field:
Eeff,i(r) = Eoi exp(iko -r) + (ViVj + bijko
d t ,exp(ik. Ir- r'j) AE ( *X Sd3r'U(r') epi Ir-r')a AfjkEeffk(r')Ir -r'l
+ BijEeffj (r) U(r), (8)
where
UWr) =1, for r within the target, (9)
tO, for r outside the target.
It should be noted that Eq. (8) differs from its coun-terpart in Ref. 6 by the addition of the term containingthe matrix B, and this term does affect terms of theorder of 2 or greater.
The substitution of Eq. (5) into Eq. (8) leads to thefirst 2 orders of the perturbation theory (r - ):
with r a unit vector in the detector direction.Note that the geometry of the scattering is entirely
general and detailed in Fig. 1. The axes of the detectorare x ,y,z. The direction of incidence is along ko whosespherical coordinates in the target frame (x,yl,zi) aref/2 (tilt angle) and 01 (azimuthal angle). The scatter-ing angle is
0 = cos-l[cosO cos 3/2 - coskp sinO sin3/2], (14)
where 0 is the angle between the symmetry and detectoraxes. The incident electric field in terms of the detectoraxes is given by
Here 4 is the angle between E0 and the Y,Yi axis when01 = 0.
Ill. Application 1: Dielectric SpheroidIn this section, we compare the perturbation theory
with the exact results of Ref. 9 to determine the rangeof convergence of the expansion.
For the spheroid, the pupil function u is well known,10
and it has the formsu(p) = Vf[a(pI - e2pI)I,prolate,
u(p) = Vf[a(p 2 - e2 pII)],oblate,(16)
where p 1 and p 11 are the components of p perpendicularand parallel to the symmetry axis of the target, a is thesemimajor axis of the ellipsoidal cross section, E is theeccentricity, V is the volume, and
f(x) = 3[sin(x) - x cos(x)]/x3. (17)
The electrostatic polarization matrix is also well(10) known,11 and its elements for a prolate spheroid are
where 6 = 1/e.The scattering amplitudes calculated are
11 = E2 I Ec V-0, I2E2 IEsc L=9O0°
where
Ejr) = aEiWr) + a 2E(2fj(r). (21)
The results are compared with the exact calculations ofRef. 9 in Figs. 2-4 for the repsective aspect ratios 1/2, 1(sphere), and 5 taking the index of refraction m = 1.33.The incident radiation is along the symmetry axis (3/2= 0 and 0 = 0). The wave number ko = 1 m-1.
-510
-610
In these three figures, the second order is an -20%correction to the first order and improves agreementwith the exact results. The agreement in all these casesis excellent. For the largest spheroid chosen (Fig. 3) thequantity S = kol(m - 1) = 1, where is the spheroidalsize parameter, 1 2_ - b2. Convergence will befaster for S < 1, and it will be slower for S greater. Arough limit on the useful convergence of the expansionis S = 2. It should be noted that this criterion for con-vergence applies only to the Shifrin technique ofstarting from the electrostatic limit. Beginning theiteration with a polarization matrix tailored to shorterwavelengths would improve the convergence and allowapplication to larger indices and size parameters.
IV. Application 2: Finite Dielectric CylinderThe perturbation theory described here relies on an
accurate determination of the electrostatic polarizationmatrix. Though known for the infinite cylinder, thismatrix is not available for the finite cylinder. We nextpresent a method for finding A by a technique outlinedby Van Bladel.12
The electrostatic potential (ro) inside the cylinderis related to the surface potential as
-210
0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~-10
10'
-510
0 60 120 180
Fig. 2. Scattering amplitudes I, and I2 for light of wavelength X =2r tm incident along the axis of symmetry. The target has an indexof refraction m = 1.33 and is an oblate spheroid of size parameter C= 1 (see text) and aspect ratio /2. The volume is 3.22 m3. The solidlines represent the perturbative calculations through the second order.The open circles are the exact results of Ref. 9. The results for acylinder of the same aspect ratio and volume are virtually the same
as for the spheroid and, therefore, not shown.
I 0I I , 1 1 2 0 I I , I 180
o 60 120 180
Fig. 3. Amplitudes I, and I2 for light incident along the axis ofsymmetry. The target is a sphere of 1-,um radius. The solid linesrepresent the perturbative calculations through the second order, andthe open circles represent the exact (Mie) theory. The dashed linerepresents perturbative results through the second order for a cylinder
and aspect ratio 1. The volume of each target is 4.19 m3.
Fig. 4. Amplitude I1 for light incident along the axis of symmetry.The solid line represents the second-order perturbation theory fora spheroid of size parameter C = 3 and aspect ratio 5. The volumeis 4.81 ,um3. The open circles represent the exact results of Ref. 9.The dashed line represents the perturbative results for a cylinder of
the same aspect ratio and volume.
-210
-310
-410
-510
-610
0 60 120 180
Fig. 5. Amplitude 2 for the same case as Fig. 4.
m2 0(ro) = oo(ro) - m A (r) | I dS, (22)47r fs Ia - o
where 00 is the ambient potential, the integral is overthe cylindrical surface, and a/an is the outward normalgradient to the surface. It follows that the electric fieldinside the cylinder is
m2E(ro) = Eo + in 1 (r)Vo a 1 - ) dS. (23)4ii- f-s an r-1ro 0 /S (3
Here E0 is the ambient electrostatic field. This internalfield is related to the electrostatic polarization matrixas in Eq. (2), and we are led to the following equation for.the matrix elements:
(M2a -l i2_ 1 AS 0(r) a [(r-ro)i 1i - )Eoi = 4i- .'s a I -o 3 (24)
where aTE = a, = a2 and aTM = a3.Therefore, the matrix A is known if we know the
surface potential , and it is determined from theequation
2 in2 -11q5(r 8) = in2+1 X(r,)- ___rd -S,i2+ 2Jsnr -r.m + 1 m + 1 ~~~27r S an rr
(25)
where r, is on the cylindrical surface. We choose tosolve Eq. (25) by an iteration process starting with =00 as the first guess. For the TE elements of A, E0 =E 0 , and 00 = -r cosOE 0. For the TM element, E0 =Echi and 00 = -z 1 Eo. Convergence occurs rapidly foran index of refraction of 1.33 where only two iterationsare necessary. The integrations over the surface areperformed using the cylindrical coordinates shown inFig. 6.
ZC
ETM
IETEh…
Fig. 6. Geometry of the cylinder used to calculate the internalelectrostatic field. Polar coordinates are used in evaluating the
The results for the elements aTE and aTM are givenin Fig. 7 for the aspect ratios h/2a = 1/2, 1, and 5. Thevalues are given along the symmetry axis z1, but theyare similar along a perpendicular axis.
It is observed that for an aspect ratio of 5, the ele-ments are quite constant in the central region of thecylinder with values near those for an infinite cylinder.The values of the elements decrease near the ends. Thesame general variation occurs for the other aspect ra-tios.
V. Comparison of the Finite Cylinder and SpheroidThe amplitudes I, and 2 are evaluated using the
electrostatic polarization matrix elements of the lastsection. These results for the finite dielectric cylinderare plotted in Figs. 2-4 on which the values for thespheroids appear. In each case, the aspect ratios andvolumes of the cylinder and spheroid are the same.
For these cases the direction of incidence is along thesymmetry axis, and we note that the cylinder andspheroid give remarkably similar scatter patterns forthe aspect ratios /2 and 1. On the other hand, the cyl-inder with aspect ratio 5 scatters more radiation at largescattering angles than its spheroidal counterpart. Theindex of refraction is 1.33 to compare with results of Ref.9 (as is the selection of scattering amplitudes shown inthese figures). The cylindrical results are shown asdashed lines when they are different from the spheroidalresults (solid lines). The open circles shown in these
1.0
.9
.8
.7
.6
.5
1.01
.91
.8
.7
.6
I I I I I I I I Iinfinite 1.0
0 0 0 xX X
-: ~ disk .565
_ aTMdisk 1.0
_o o O O O 0 0 0
0 0~~~~~x * ' * 0 * * x
infinite .722
aTEI I I I I I I I
0 .1 .2 .3 4 5 6 .7 .8 .9 10X h
Fig. 7. Polarization matrix elements for the finite dielectric cylin-ders: X, aspect ratio of 5; *, 1; 0, 1/2. The results are plotted alongthe axis of the cylinder. The top portion of the figure is for an am-bient electric field directed parallel to the axis (TM), and the bottom
portion is for an ambient field perpendicular to the axis (TE).
figures are exact results. The wavelength of incidentradiation is 27r gim.
In Fig. 2, an oblate spheroid and cylinder of aspectratio 1/2 are shown, each with 3.22-/,m 3 volume. In theperturbative calculation, the second-order correctionis generally -26% for both targets. We see that that thescattering pattern for the amplitudes I and 2 arenearly the same.
Figure 3 compares the cylinder with a sphere eachwith 4.19-gm 3 volume. Here the second-order correc-tion is generally -22%, and the results for the cylinderdiffer noticeably with the sphere at the larger scatteringangles.
Figures 4 and 5 show the respective amplitudes I andI2 for a prolate spheroid and cylinder of aspect ratio 5.The volume of each target is 4.81 giM3. The second-order correction here is -25%. In these cases, the cyl-inder scatters light quite differently at the larger scat-tering angles. A resonance at -100° appears for thespheroid in the amplitude I2 but does not appear at allin the corresponding cylinder.
Figure 8 depicts the amplitude I, for incidence per-pendicular to the axis of symmetry (/2 = 90° and 01 =0). Two targets are shown, one with an aspect ratio of2 and 1.62-gm3 volume, and the other has an aspectratio of 5 and 4.81-gm 3 volume. The second-ordercorrection for the first is -20 and -26% for the longertarget. We note that the results for the cylinder differappreciably from the corresponding spheroid only forthe larger aspect ratio.
-210
-310
-210
-3100 60 120 180
Fig. 8. Amplitude I, for light incident perpendicular to the axis ofsymmetry. The solid lines represent the perturbative spheroidalresults and the dashed lines the cylindrical results. The top portionof the figure corresponds to targets of aspect ratio 2 and 1.61-,um,while the bottom portion corresponds to targets of aspect ratio 5 and
VI. Discussion and ConclusionsWe continue our study of the Shifrin perturbation
theory for the scattering of radiation of wavelength Xfrom particles of size a in the range where 27r(m - 1)/X_ 1. This theory has two obvious advantages. First,the analytic first-order contribution is a fair represen-tation of the scattering and provides an easy-to-calcu-late approximation. Second, this perturbation theoryprovides an accurate means for calculating the scat-tering pattern for the short dielectric cylinder where noexact theory exists.
This theory relies on an accurate determination of theelectrostatic field within the target. This internal fieldis used as the first-order kernel term upon which thesuccessive approximations are built. The results arethen applicable to the scattering of radiation of wave-length only slightly larger than the target.
In the case of spheroids of index m = 1.33, the elec-trostatic field is known, and exact solutions for sizeparameters ranging from 1 to 3 are available.9 Thisallows us to test the perturbation theory, and we findexcellent agreement for aspect ratios ranging from 1/2to 5. The second-order correction in these calculationsis generally -25%.
In the case of the short dielectric cylinder (aspectratios <10), the internal electrostatic field is not known.A technique is presented for constructing this field bya rapidly converging iterative scheme. For the samesize parameters as used in the spheroid calculations, itis found that the second-order correction is about thesame as found in the spheroid calculations. Thissuggests that the perturbation theory applied to theshort cylinder is providing equally accurate results.
A comparison is made between the results for thespheroids and the finite cylinders of equal aspect ratiosand volumes for light incident parallel and perpendic-ular to the symmetry axes. It is found that the resultsagree for small scattering angles in all the cases con-sidered, but there are significant differences in the in-tensity patterns at larger scattering angles in some ofthe cases.
This work was supported in part by the U.S. ArmyChemical System Laboratory at Edgewood Arsenalunder contract DAAK 11-80-0047.
References1. J. W. Strutt, Philos. Mag. 12, 81 (1881).2. G. Mie, Ann. Phys. Leipzig 25, 377 (1908).3. J. R. Mentzner, Scattering and Diffraction of Radio Waves
(Pergamon, Oxford, 1955).4. J. R. Wait, Electromagnetic Radiation by Small Particles (Wiley,
New York, 1957).5. M. Kerker, The Scattering of Light and Other Electromagnetic
Radiation (Academic, New York, 1969).6. L. D. Cohen, R. D. Haracz, A. Cohen, and C. Acquista, Appl. Opt.
22, 742 (1983).7. K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow,
1951; NASA TTF-477, Washington, D.C., 1968).8. C. Acquista, Appl. Opt. 15, 2932 (1976).9. S. Asano and G. Yamamoto, Appl. Opt. 14, 29 (1975).
10. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill,New York, 1968).
11. See Ref. 8 and C. Acquista, "Shifrin's Method Applied to Scat-tering by Tenuous Nonspherical Particles," in Light Scatteringby Irregularly Shaped Particles, D. Schuerman, Ed. (Plenum,New York, 1980).
12. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York,1964), pp. 68-77.
Meetings Calendar continued from page 435
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