Scattering of linearly polarized microwave radiation froma dielectric target including the interactionbetween target elements
Richard D. Haracz, Leonard D. Cohen, Alice R. W. Presley, and Ariel Cohen
The theory for finding the internal field within a dielectric helix when the radiation has a wavelength largerthan the diameter of the helical wire is presented. Intensities are calculated and compared to an experimentand to the theoretical results of an earlier paper that does not include the self-interaction effect. The internalfield is defined in terms of a polarization matrix that is assumed to be constant across any cross section of thehelix. It is found that target self-iteractions have a significant effect on the internal field. It is also noted thatthis effect for the far field intensities, although significant and generally a better fit to the data, is notprofoundly different. That is, the effects of a more appropriately constructed internal field are lessimportant than the geometry effect in the far field.
1. Introduction
We recently described the scattering of microwaveradiation from a helical target and compared theory toexperiment.' The results of this comparison were en-couraging in spite of the seemingly naive nature of themodel assumption. This work was an extension of ourearlier efforts in extending the Shifrin approximation2
to nonspherical dielectric targets.3-6 The essentialfeature and the accomplishment of these efforts wereto show that the assumption that the polarization ma-trix is constant over the target gives very good results ifthe target is smaller than the wavelength of the inci-dent radiation. We showed this for such target shapesas short cylinders, spheroids, toroids, and helices.
In the treatment of the helix, the helical wire isdivided into disks perpendicular to the helical axis,and it is assumed that the polarization matrix is con-stant in the symmetry frame of each disk. The results,compared to a microwave experiment on a Plexiglashelix of known index of refraction, suggested a correct-ness of the model assumption especially for the largervalues of the scattering intensities where the experi-ment had its greatest reliability.
We take the next step in this paper-the self-inter-action effects on the polarization matrix are included.Including such effects is not new. Zeit et al.7 consid-ered a model of a helical target composed of interactingdipoles. Bustamente et al. 8 included the second-orderBorn approximation in their treatment of helical tar-gets. Singham and Bohren9 describe the effects of
The authors are with Drexel University, Department of Physics &Atmospheric Science, Philadelphia, Pennsylvania 19104.
multiple scattering and include second-order effectsfor a helical target. A recently published work byKattawar et al.10 solved the scattering integral equa-tion by the "resolvant-kernal" method, and their re-sults, which include self-interaction, are in good agree-ment with a scattering experiment on a cubical target.
Our approach is novel, however, as we strive to in-clude target self-interaction through the polarizationmatrix which we assume, and hope to verify, is a slowlyvarying spatial tensor function. The advantage is thatthe numerical machinery for the solution searches forrelatively small corrections. Hence a straightforwarditeration scheme can be employed. The price to bepaid is that we are restricting applications to targetsformed from a dielectric cylinder whose diameter(which can vary) must satisfy the Shifrin condition:2ka(m - 1) < 2, where k = 27r/X, with X the wavelength,a the diameter, and m the complex index of refraction.The length of the target axis is not restricted, as varia-tions in this dimension are calculated.
The next section of this paper presents an integralequation for the polarization matrix that includes theself-interaction integral. The target is again dividedinto disks, and the polarization assumed to be constantin each disk of infinitesimal thickness. The values ineach disk and their variation over the target axis will beestablished by solving the integral equation by itera-tion. We show that reasonable convergence isachieved after five iterations in the worse case consid-ered, and the resulting polarization matrix will be usedto calculate the far field intensities. The followingsections of the paper display and discuss the results.It is shown that the polarization matrix varies by afactor of 2 over the target. Moreover, a comparison ofthe results to the experimental points presented inRef. 1 shows that the inclusion of target self-interac-tion improves the agreement with experiment. How-ever, the essential features of the far field intensitiesare mainly controlled by the helical geometry.
Fig. 1. Reference or laboratory frame of reference (xoyo,zo), wherethe zo axis is in the direction of the incident radiation. k is the wavevector for the incident radiation. The axis of the helix, Zt, is locatedrelative to (xoyozo) by the polar and azimuthal angles ot and yt,respectively. The vector h(o) traces the axis of the helix, with k
ranging from zero to 2ir times the number of turns.
II. Integral Equation for the Polarization Matrix
The integral solution to Maxwell's equations for thescattering of radiation from a dielectric target is
E(r) = Eo exp(ikr) + V X V X dV'(m 2- 1)/(4ir)
X exp(ikir - r'I)/Ir - r'IE(r') + (1 - m 2)E(r). (1)
The effective field is defined by
Eeff(r) = (m2 + 2)/3E(r), (2)
and substitution of this into Eq. (1), noting that m = 1outside the target gives the equation
Eeff(r) = E. exp(ikr) + aV X V X f dV'
X exp(iklr - rI)/Ir - r'lEeff(r')
- 87r/3aU(r)Eeff(r), (3a)
where a = 3/(4r)(m 2 - 1)/(m2 + 2), and U(r) is one if ris inside the target and zero otherwise. This equationis the starting point for the Shifrin approximation,which is an expansion in powers of a (this has the valueof 0.06 for an index m = 1.5).
We are interested in the field point r being inside thetarget. For this situation, care must be exercised inbringing the operator inside the integral, as describedin Appendix V of Born and Wolf.11 The resultingequation is
Eeff(r)= E0 exp(ikr) + a lim J dV(V . V + k2)
X exp(iklr - rl)/Ir - r'lEeff(r'), (3b)
where a is a small sphere surrounding the point r' = r,and the integral excludes the volume of this sphere. Itshould be noted that this is the equation used in Ref. 10in their "resolvant-kernal" method.
k0
zt
-HELIX
Xt
Fig. 2. Target frame (xt,yt,zt) is shown along with two disks (crosssections of the helix). The internal field point is located at thecenter of the disk located by h(o). This disk is located relative to
any other in the helix by the vector d(o,p').
We now define the polarization matrix A as
[Eeff(r)]i = Aij[Ei..(r)]j (4)
using tensor indices and the summation convention (arepeated index denotes summation over the Cartesiancoordinates). Inserting Eq. (4) into Eq. (3) gives anintegral equation for the polarization matrix:
Aij(r) = bij + a dV' exp(ikR)/R
X [ri5ik + (hi + X;)(hk + x)r 2d
X Aj(r') exp[ik(r' - h)]. (5)
Here, R is the distance from the source point r' to thefield point r, and
r = [k2- (1/R)2 + ikiR],
r2 = -(k/R) 2 + 3/R4 - 3ik/R3.(6)
The vector h is directed from the origin of the targetframe of reference to the helix, as shown in Fig. 1. Thisfigure also shows the orientation of the spiral targetaxis located by the angles q5t and yt relative to thereference frame (xo,yozo) with z in the direction ofincident radiation.
Ill. Model Assumption and Evaluation of the PolarizationMatrix for a Helical Target
We assume that the polarization matrix A is con-stant throughout each disk that composes the target.This means that we are assuming that the wavelengthof the incident radiation is larger than the diameter ofthe largest disk in the target so that there are noresonances across the disk. Therefore, for the tensorfunction A(r), where r locates the internal field point,r can be taken at the center of the disk, which is on theaxis of the target.
The target (helix) frame (xt,yt,zt) is shown in Fig. 2where the vector
h(p) = a cos(O)it + a sin(o)jt + P/(27r)bkt (7)
locates the center of a disk, with Zt along the target axisand xt in the x0 - zo plane. The angle 0 locatespositions on the axis of the helix and varies between 0and 27rn, with n the number of turns in the helix. Thevector d(0,4') locates the 0 disk relative to the O' disk.The radius of the helix is a, P is the pitch of the right-handed helix, and itjtkt are the unit vectors for thehelix frame. Also shown in Fig. 2 is the direction ofincidence of the incident light (the zo axis).
Figure 3 shows the two disks in the helix and the diskframe of reference (x0,yp,z0) with zo perpendicular tothe disk (parallel to the helical axis) and x, in the z, -z0 plane. Thus, this frame changes with location onthe helix. As the integral Eq. (5) involves an integra-tion over the target, Fig. 3 also shows the vector p whichlocates positions on the O' disk, p and 0 are the polarcoordinates of p in the disk frame.
Using this geometry, the integral equation for thepolarization matrix, now denoted A(0) as the matrix isassumed to be constant in a disk, is
2. turnsA ij(0) = 6ij + a f V/(a2 + (P/(4r)) 2d0'
z+,
'Y,
xt
Fig. 3. Expanded view of the internal field disk () relative to anyother disk ('). The inclusion of self-interaction involves adding theeffects of all source points (located by r') at the center of the fielddisk. The integration discussed in the text in connection with Eq.
(8) involves the parameters 0 and p appearing here.
X j pdp exp[-ikd(0,')] X dOJc(0f) JO"
X exp(ikR/R) exp[ipr(o') cos(O)]
X T(,' - t)ij[r,,mA(0')mn + [d(0,01), - Pj]
X [d(0,/')m - pmjA(,')mnr21, T(t - /'),%. (8)
This integral involves a double integral over the k' disk(whose coordinates are p and 0). The p integration hasthe lower limit e(o), which is nonzero only when the q'and 0 disks coincide. The upper limit rw is the radiusof the wire that forms the helix. The lower limit is
eC ') = [-2 -[a 2 + (P(27r)]21, - ¢02W/2
for [a2 + (P/(2r)]210 - 0&12) < e, and zero otherwise.This accounts for the removal of a sphere of radius ethat surrounds the point r' = r. The disk that con-tains the field point dominates the integral, havingapproximately the same contribution as all the otherdisks combined. The parameters r, and r2 appearingin Eq. (8) are defined in Eqs. (6), the matrix T(O' - t)is a transformation from the disk frame to the targetframe, and [... .],p' means that the quantity in the brack-ets is in the disk frame. This repeated transformationis necessary as we initiate the calculation using a unit-diagonal form for A in the disk frame of reference, butas the disk frame continuously varies over the targetintegration, the pieces must be added coherently in thecommon helix frame of reference. Thus, Eq. (8) ac-counts for the interaction of the internal field effects ofall the disks at a point on the axis of the helix.
It should be noted that the parameters R, r1, and r2appearing in Eq. (8) depend on the disk angular vari-able 0 as
R2= 2a2 [1 - cos(- k,)] + [P/(27) ]2(¢ _ ()2
+ p2 - 2d(4,,')p(O). (9)
The last term is zero if the disks are stacked one on topof the other. It is not zero for the helix, however, as thedisks wind about the helical axis. Its contribution isonly significant for disks near the internal field pointand for disks situated side by side at each turn of thehelix. When the term is neglected, the 0 integral canbe done analytically leaving a double integral to benumerically evaluated. In the applications presentedhere, the contributions from the last term of Eq. (9)have been numerically evaluated when necessary asdetermined by a size check that is part of the computercode.
The evaluation of the integral is done by iteration,starting with the value for the polarization matrix for a= 0-the unit matrix. This evaluation can be accom-plished by taking n internal field points on the axis ofthe helix and fitting these with an nth-order polynomi-al:
where z = V/a2 + [P/(47r)]2 k' traces distance along thetarget axis. The disk frame is assumed, and the initialvalues of the coefficients are
C011 = C022 = C033 = 10, (11)
with all other components of cO zero. Moreover, as theinitial polarization matrix is assumed to be constant ineach disk, the coefficients cl, c2,. . . are all zero, initial-ly.
Once the polarization matrix is determined, the elec-tric field for r far from the target can be calculatedfrom
2-, turnsE(r)i = Eji exp(ikr) + aJ dpla 2
+ [P/(2wr)]2j112
X J1(q pera)/(qper) X exp[qh(G/)]jT(P -¢ ref)jmn
X A 0() - xx. 0A 0p(&)}E 0 .. (12)
The polarization matrix is evaluated in the frame ofreference of each disk and is transformed to the refer-ence frame so that the integration over the target canbe performed. The transformation matrix T(o -> ref)and the details for this equation are given in Ref. 1.
It is interesting to note that this result has the sameform as the first-order Shifrin approximation in Ref. 1,but Eq. (12) now contains all orders of self-interactionconsistent with the model assumptions. The simpleform of Eq. (12) depends on the assumption that A isconstant in a disk which is the assumption employed inRef. 1. The difference is that instead of guessing theconstant value, we now obtain the value by includingthe effects of all parts of the helix. Also, the value of Avaries over the helical axis as demanded by Eq. (8).Therefore, Eq. (12) is as exact as is allowed by themodel assumption and the precision of the numericaldetermination of A.
IV. Application
In Ref. 1 we applied the theory without self-interac-tion effects in the A matrix to an experiment per-formed at Florida State University by Ru Wang andhis collaborators. The helix used has seven turns, awire radius of rw = 0.24 cm, a helix radius a = 1.83 cm,and a pitch P = 0.553 cm. The complex index ofrefraction is m = 1.626 + iO.012.
As this seven-turn helix is very tightly wound, self-interaction effects are expected to be significant. Toshow how self-interaction affects the polarization ma-trix, we progress toward the final geometry of theexperimental helix: starting with the plastic wirestretched out into a straight cylinder, then wound insix turns, and finally wound in seven turns. The pitchof the six-turn helix is 6.936 cm, and this helix is ratherloosely wound. In each of these cases, the volume ofthe helix is 14.58 cm 3 .
In Ref. 1, self-interaction was not included. Thepolarization matrix was assumed to be diagonal andconstant in each disk frame of reference throughoutthe helix, and these diagonal matrix elements weretaken from the electrostatic theory for an infinite cyl-inder:
We will next show how the self-interaction effectsevolve as the helix is formed.
A. Target Wire Formed into a Long Cylinder
In this case, the polarization matrix converges towithin 1.5% in three iterations, and it is constantthroughout the cylinder to within -5%. The matrixfor a point at the center of the cylinder for light inci-dent perpendicular to the cylinder axis is
-(0.988,iO.125)A = O0
(-0.005,-iO.002)
0(1.002,iO.128)
0
(-0.005,-iO.001)
(1.684,iO.253)
This is somewhat different from the electrostatic re-sult, but not markedly. It is noted that for a wireradius of rw = 0.1 cm, the calculated polarization ma-trix is quite close to the electrostatic case.
B. Target Material Turned into a Helix of Six Turns
In this case, four iterations are needed to achieveconvergence to within -5%, and the variation of thepolarization matrix along the helix is -20%. The po-larization matrix at the center of the helix for lightincident perpendicular to the helix axis is
(0.711,iO.232)A =O
0(0.745,iO.189)
(-0.270,-iO.001)
0 1(-0.095,iO.030) (1.605,iO.646)
This matrix looks similar to the one for a straightcylinder, but the differences are significant, the largeimaginary term in A33 , for example, is due to targetself-interactions.
C. Target Material Turned into a Helix of Seven Turns
The evaluation of the polarization was done for twoorientations of the axis of the helix: t = 00 and ott =900. A second-order polynomial fit was tried takingthree points for the fit: two 1800 inside each end pointof the helix, and one point at the helix center. Fiveiterations are needed to achieve convergence to within10%. The polarization matrix is expressed in terms ofcoefficients defined in Eq. (10): Aij(o') = cOij + cijz +c2ijz2 , where z is the distance along the helical axis andranges from 0 to 147r(1.83). These coefficients aregiven below for two directions of incident radiation forthe diagonal and larger off-diagonal elements:
These coefficients form a polarization matrix that issignificantly different from the electrostatic matrixused in Ref. 1. The values vary over the helix by afactor of -2:1, the imaginary parts are large, and someoff-diagonal elements are relatively large.
These values for A are next used to calculate theintensities I,, and I22, where Ill is the case when boththe receiving and transmitting antennas are orientedto accept radiation polarized in the vertical direction(the scattering plane is horizontal). The intensity 122is for both directions of linear polarization horizontal.The results, compared to experiment and to our previ-ous results, appear in Figs. 4-9.
V. Discussion of Results
A comparison is made with the experiment per-formed by the Florida State group that we discussedand presented in Ref. 1. In that paper we remarkedthat the experimental points could not be relied onwhen the values descend to 3 orders of magnitudebelow their maximum values in the forward scatteringdirection. The figures that we are about to discuss area log plot of intensity vs scattering angle. The forwardvalues are all about 2 for this scale, but drop to valuesbelow -1, indicating a variation of 3 orders of magni-tude in the intensity. Experimental values in thislatter region probably have little meaning.
A. ot = 0 , yt = 00
Figures 4 and 5 have the helix oriented so that itsaxis is in the direction of incidence. In these figures,the experimental points are shown as crosses, the self-interaction results as lines through solid diamonds,and the results of Ref. 1 are shown as a continuous line.Figure 4 shows I(ot = 0, -t = 0), where ¢:t and yt arethe polar and azimuthal angles which establish thedirection of the helical axis in the reference frame(xoyoz o), with z in the direction of incidence and xo in
C I111ex(0,0)- 111c(0,0)_C 111(00)
100 200
Scat. Ang.
Fig. 4. Intensity I,1(0,0) refers to both incident and scattered radi-ation linearly polarized perpendicular to the scattering plane. Theexperimental points of Ref. 1 are shown as crosses, the theoreticalpoints using a constant A matrix is shown as a solid line, and thetheoretical points using A determined from Eq. (8) are shown as asolid line with diamonds. The orientation of the helical axis is
ft= and t = 0.
the scattering plane. We see that self-interactionsmooths out the deep minima formed when A is con-stant, and self-interaction generally improves theagreement with experiment.
In the case of I22(0,0), self-interaction improvesagreement at scattering angles up to 900, but it isapparent that the intensity is not very sensitive to self-interaction for the backscattering range of angles.Agreement with experiment is only qualitative wherethe experimental values fall below 2 orders of magni-tude of their forward value. But, as mentioned, suchexperimental points are not to be taken seriously.
Fig. 5. Intensity I22(0,0) refers to both incident and scattered radi-ation linearly polarized in the scattering plane. The experimentalpoints of Ref. 1 are shown as crosses, the theoretical points of Ref. 1are shown as crosses, the theoretical points using a constant A matrixis shown as a solid line, and the theoretical points using A deter-mined from Eq. (8) are shown as a solid line with diamonds. The
orientation of the helical axis is 'Pt = 00 and yt = 00.
.6eo
a06
an X I22ex(90.0)- 22.(90,0)- 122(90.0)
100 200
Scat. Ang.
Fig. 7. Same as Fig. 5, but the helical axis is oriented by 'pt = 900
and yt = 0.
01
-0
X 111ex(90,0)- 111c(900)- 111 (90.0)
I I11Cx(9090)
- I11c(90,90)-I 11 (90,90)
100 200
Scat. Ang.
Fig. 8. Same as Fig. 4, but the helical axis is oriented by t = 90°and yt = 900; in this case the two curves coincide.
100 200
Scat. Ang.
but the helical axis is oriented by t = 900and yt = 00.
B. (kt = 9 0 ° 'yt = 00
Figures 6 and 7 show I1(9O,O) and I22(90,0). Self-interaction improves the agreement with the experi-ment forIll as seen in Fig. 6. In the case of 122, the newtheory improves the comparison in the forward rangeof scattering angles, but large differences remain forbackscattering.
C. Ot = 90° ',Y = 900Figures 8 and 9 have the helical axis oriented per-
pendicular to the direction of scattering and in thescattering plane. It is remarkable that, even thoughthe polarization matrix that includes self-interactionis significantly different from the constant matrix usedin Ref. 1, the results for I,, are nearly identical to thosefor a constant A. The geometry of the helix, not theinternal field, controls the intensity for this orienta-
6
(00
X (22..(90,90)- (22,(90,90)- 122(90,90)
200
Scat. Ang.
Fig. 9. Same as Fig. 5, but the helical axis is oriented by 't = 90°and yt = 900.
tion of the helix relative to the incident light. In thecase of 122, our new approach significantly improvesagreement with the experimental points for the for-ward scattering angles.
We have presented a theory that makes use of thephilosophy of the Shifrin method. It is an approachthat does not rely on a perturbation technique, butwhich uses as a central feature the observation that adielectric target smooths out the internal field if thewavelength of incident radiation is larger than a strate-gic dimension of the target. As in our previous work,we restrict our calculation to targets that can be con-structed from an assembly of disks of infinitesimalthickness, but we now let Maxwell's equations in inte-gral form determine these constants and the variationof the internal field along the axis of the target. Oncethe polarization matrix is obtained in this way, we areable to construct a relatively simple form for the scat-tered field using the Fourier transform method insti-tuted by C. Acquista. 3
Judgment of how well we have taken account of self-interaction lies in a comparison with an experiment, anexperiment whose limitations have already been not-ed, but which remains rare by its very existences Inall the cases where differences are noted, the inclusionof self-interaction improves the comparison with ex-periment for scattering angles to -90°.
On the other hand, self-interaction does not signifi-cantly change the far field results for scattering anglesgreater than -150°. In this latter range, some of theexperimental points are significantly different fromthose predicted by theory even though the theoryseems to be relatively insensitive to the electric fieldwithin the helix. These cases, especially I(90,90),would be interesting candidates for further experi-mental study.
It should be stressed that the polarization matrixapproach seems to be particularly well suited to nu-merical techniques as it does not vary much within thetarget as long as the wavelength remains larger thanthe disk diameter. Moreover, Eq. (8) shows that thevalue of the polarization matrix approaches the identi-ty matrix as a tends to zero. Thus, the polarizationmatrix initiates from the simplest of forms.
The results presented here stem from a calculationof the polarization matrix at only three points in thetarget-two near the end points and one at the center-which establishes a second-order polynomial fit for therest of the target. Five iterations are needed toachieve results that converge to within 10% for theexperimental target-which is an especially difficulttarget as the coils of the helix are nearly touching.Convergence is much more rapid for a target of one lessturn where the coils are much further apart. We ob-serve that self-interaction makes a significant differ-ence in the structure of the internal field, as represent-ed by A. But the effect of this internal field is not theoverwhelming effect for the far field intensities, thetarget geometry is the controlling characteristic.
One would expect the internal effects of the self-interactions to be of greater importance as the diame-
ter of the target approaches the wavelength of theradiation used, and we intend to see how our approachworks in such applications as the dielectric spheroid.
Our approach is designed to be computationallysimple. It is a straightforward iterative calculationwhich converges quickly for the cases studied in thiswork. We have little desire to refine the model as-sumption to allow for more variation of A in each disk,as such a refinement would greatly increase the nu-merical complexity of our technique.
In short, we propose that this approach will giveaccurate scattering predictions with minimal calcula-tional apparatus for the kinds of target that can beformed from a dielectric wire that has a diameter lessthan the wavelength of the incident radiation. Thiseconomy is especially important if one intends to cal-culate the scattering patterns from many such targetsoriented in many directions.
This research was partially supported by the U.S.Army CRDEC. Ariel Cohen was on leave from theDepartment of Atmospheric Sciences of the HebrewUniversity, Jerusalem.
References
1. R. D. Haracz, L. D. Cohen, A. Cohen, and R. T. Wang, "Scatter-ing of Linearly Polarized Microwave Radiation from a DielectricHelix," Appl. Opt. 26, 4632 (1987).
2. K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow,1951; NASA TTF 447, Washington, DC, 1968).
3. C. Acquista, "Light Scattering by Tenuous Particles: A Gener-alization of the Rayleigh-Gans-Rocard Approach," Appl. Opt.15, 2932 (1976).
4. L. D. Cohen, R. D. Haracz, A. Cohen, and C. Acquista, "Scatter-ing of Light from Arbitrarily Oriented Finite Cylinders," Appl.Opt. 22, 742 (1983).
5. R. D. Haracz, L. D. Cohen, and A. Cohen, "Perturbation Theoryfor Scattering from Dielectric Spheroids and Short Cylinders,"Appl. Opt. 23, 436 (1984).
6. R. D. Haracz, L. D. Cohen, and C. Acquista "Light Scatteringfrom Dielectric Targets Composed of a Continuous Assembly ofCircular Disks," Appl. Opt. 25, 4386 (1986).
7. S. Zeitz, A. Belmont, and C. Nicolini, "Differential Scattering ofCircularly Polarized Light as a Unique Probe of Polynecleon-some Superstructures," Cell Biophys. 5, 163 (1983).
8. C. Bustamente, M. F. Maestre, D. Keller, and I. Tinoco, "Differ-ential Scattering (CIDS) of Circularly Polarized Light by DenseParticles," J. Chem. Phys. 80, 4817 (1984).
9. S. B. Singham and C. F. Bohren," Lights Scattering by anArbitrary Particle: A Physical Reformation of the CoupledDipole Method," Opt. Lett. 12, 10 (1987).
10. G. W. Kattawar, C-R Hu, M. E. Parkin, and P. Herb, "MuellerMatrix Calculations for Dielectric Cubes: Comparison withExperiments," Appl. Opt. 26, 4174 (1987).
11. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1965), see Appendix 5.
12. A paper has just appeared in Applied Optics that also makes acomparison of theory to the same experiment as used in Ref. 1for a right-handed helix; see P. Chiappetta and B. Torresani,"Electromagnetic Scattering from a Dielectric Helix," Appl.Opt. 27, 4856 (1988).
