. INTRODUCTIONhe T-matrix has become an accepted tool for treating aariety of scattering problems involving objects of moreeneral shape than spheres or circular cylinders, with ap-lications in acoustics, electromagnetic theory, and elas-icity [1]. For surface scattering where separation of vari-bles can be employed, e.g., hard and soft cylinders oflliptical cross section or spheroids in three dimensionsworking in elliptic cylinder or spheroidal coordinates),he T-matrix method (using circular cylinder or sphericalolar coordinates) provides a roughly equivalent compu-ational alternative. The trade-off in those cases involveshe necessity of generating rather complex wave func-ions, on the one hand, versus numerical integration fol-owed by matrix inversion on the other. The simplest ex-mples of this are the perfectly conducting strip in twoimensions [2,3] and the circular [4] and elliptic disk [5]n three. For penetrable bodies, however, the orthogonal-ty of the wave functions is lost in nearly all coordinateystems and matrix inversion is unavoidable, making the-matrix method more attractive.Ström looked at the role of the T-matrix in multiple
cattering [6]; Ström and Zheng considered compositecatterers [7]. Schneider and Peden computed the differ-ntial cross section of a dielectric ellipsoid, probably therst T-matrix computation for an object without rota-ional symmetry [8]. Good recent review articles, includ-ng discussion of publicly available FORTRAN codes, haveeen given by Mishchenko and co-workers at NASA9,10]. Lakhtakia and the Varadans gave a database ofeneral applications in 1988 [11], and Mishchenko et al.ave a more extensive database for the electromagneticase in 2004 [12].
A major difficulty with the method has been the ten-ency for the auxiliary Q matrix to become ill-conditionedor objects very large compared with wavelength or thoseaving high aspect ratios. Regarding these problemsishchenko and Travis [13], and more recently Have-
ann and Baran [14], have shown that the use of quadrecision (31 decimal digits) gives greatly improved re-ults.
The present work is concerned primarily with electro-agnetic scattering by penetrable cylindrical obstacles,
llowing for absorption. After summarizing the basicquations, we look first at the low-frequency case, findingsimple explicit formula for the T-matrix for nonmag-
etic materials in the Rayleigh limit. This is then ex-ended to obtain the Rayleigh expansion, in which subse-uent terms give contributions from successively higherowers of the size/wavelength ratio. For the cases consid-red here, the expansion converges throughout the Ray-eigh region and somewhat beyond.
For high aspect ratios, one of the main problems in-olves significance loss during numerical integration for
[14]. A valuable paper bearing on this question wasublished in 1993, when Sarkissian and coworkers at theaval Research Laboratory (NRL) looked at acoustic scat-
ering from long but finite, thin rigid rods of circular crossection [15]. Effectively they were able to isolate theroublesome terms in the integrands of matrix elementsnd remove them. Here we describe a class of surfaces“complete quadrics”) for which a similar approach can bearried out for penetrable bodies. As a result, for the typi-al case of a 10:1 elliptic cylinder, one can prevent the lossf up to 12 significant figures. Using the technique, wean treat elliptic and rectangular cylinders, for example,aving aspect ratios up to 1000:1.
. BASIC EQUATIONSe begin by briefly summarizing the equations governing
he null-field approach to the T-matrix for penetrableodies, supplemented by more recent results relating tohe absorption matrix. A slightly different normalizationill be introduced for the basis functions, for reasons that
007 Optical Society of America
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2258 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 P. C. Waterman
ill become evident, and a new symmetric matrix isound, which is useful in the numerical analysis.
Consider a penetrable cylinder, with boundary definedn polar coordinates by �=����, as shown in Fig. 1, whenlluminated by an incident electromagnetic wave with Eeld parallel to the cylinder axis (the H parallel case fol-
ows by duality). Outgoing scattered waves are describedn the free-space region outside the smallest circle circum-cribing the cylinder (shown dashed in the figure) by theave functions
�n�k�� = �n1/2Hn�k��cos n�, n = 0,1, . . . , �1�
here the Hankel functions are expressed as Hn=JniNn in terms of the Bessel and Neumann functions. Forimplicity we suppose that both the cylinder and the inci-ent wave have a plane of mirror symmetry, so the oddunctions sin n� will not be needed (the more general caseas been described earlier [2]). The Neumann factor �01, �n=2 otherwise, and a time factor exp�−i�t� is sup-ressed. The field throughout the interior of the cylinders represented in terms of the regular wave functions
here k�=qk, with q= ���+ i�� /�0��1/2 the complex indexf refraction in terms of the relative dielectric constant,elative permeability, and conductivity of the cylinder ma-erial.
The incident, scattered, and interior fields are nowritten as
Ezinc�k�� = �
n=0an�n�k��, �3a�
Ezsca�k�� = �
n=0fn�n�k��, �3b�
Ezint�k��� = �
n=0n�n�k���, �3c�
nd given the incident wave through the an coefficients,e want to determine the scattered wave fn and possiblylso the interior coefficients n. Proceed by defining a ma-rix Q having elements [16]
Qmn =� d� · ���m�k��� � �n�k�� − ��m�k����n�k���,
m,n = 0,1, . . . , �4a�
hich is further resolved into regular and singular parts, S by replacing the Hankel functions by Bessel or Neu-
Fig. 1. Geometry of the penetrable cylinder.
ann functions, respectively, to get
Q = R + iS, �4b�
ith R and S both in general complex. The integral of Eq.4a) runs over the cylinder surface, with � set equal to��� after the gradients are taken. Equation (4a), inciden-ally, is precisely the form used also for acoustic scatter-ng in both two and three dimensions, with the wave func-ions representing velocity potentials and �, � replaced byhe compressibility and density ratios of the object, re-pectively [2]. For the three-dimensional electromagneticase, vector basis functions must be used. Rather than therdinary products of Eq. (4a) involving the gradient, onehen uses vector cross products and the curl operator [17].lthough we will focus on penetrable bodies, perfectlyonducting objects will also be considered, in which eventurface fields are represented by free-space wave func-ions. For E parallel to the cylinder axis (Dirichlet), dis-ard the terms multiplied by � in the integrand of Eq. (4a)o get QD, or alternately for H parallel (Neumann), dis-ard the terms not multiplied by � to get QN. In bothases the bulk parameters should be set to their free-pace values. These matrices are then used in place of Qnd R in the equations below to obtain the transition ma-rix. One warning here: if the surface fields themselvesre required, then the trial functions should incorporatedge conditions where necessary [16].
The expansion coefficients of Eqs. (3) can be shown toe connected through the equations
a = iQ�, �5a�
f = − iR�, �5b�
here the prime denotes transpose. Further defining T ashe operator that computes the scattered wave directlyrom the incident wave, i.e.,
f = Ta, �6a�
nd eliminating the n between Eqs. (5a) and (5b), the-matrix is seen to be given by
QT = − R. �6b�
n this last equation we have replaced T� by T, which isllowable because the two must be equal due to reciproc-ty.
In order to obtain the absorbed power, first define theatrix having elements [16]
ntegrating the inward normal component of the complexoynting vector over the surface of the cylinder nowields
12 Re��*W� = 1
4�*�W + W�*�
= 14a�*�Q*�−1�W + W�*��Q��−1a, �7b�
nd energy balance consequently requires that
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P. C. Waterman Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2259
− ReT = T�*T + 14A, �7c�
here
A = �Q*�−1�W + W�*��Q��−1 �7d�
s the (Hermitian) absorption matrix. Equation (7c),hich follows from the arbitrariness of the incident wave,
s a generalized optical theorem, ensuring for plane-wavencidence that (minus) the real part of the forward ampli-ude equals the sum of the scattering and absorptionidths. In terms of the scattering matrix S=I+2T relat-
ng incoming and outgoing waves, Eq. (7c) becomes sim-ly [17] (I is the identity matrix)
S�*S = I − A. �7e�
thus gives a measure of the departure of S from unitar-ty. Of course, for perfect conductors or lossless dielectrics,=0.The usual computational procedure with these equa-
ions is to truncate Eq. (6b) at NN and invert the re-ulting Q-matrix. The elements of T so obtained can thene checked for constancy versus truncation size andested against reciprocity and energy balance. There isnother constraint, however, that has gone unnoticed.rom Eq. (6b) T=−Q−1R, and reciprocity requires that−1R=R��Q��−1. Pre- and postmultiplying by Q, Q�, re-
pectively, one gets QR�=RQ�; i.e., the product
Z = QR� �8�
ust also be symmetric. This requirement is useful inroviding at least an indirect check on the accuracy ofoth Q and R, in addition giving a good indication of theppropriate truncation size to employ (at too small trun-ation, Z may not be symmetric), all before any matrix in-ersion is performed.
Once T is obtained, the scattered wave is given by
Ezsca�k�� = �
m,n=0
N
amTmn�n�k��
� f����2/i�k��1/2 exp�ik��, k� � 1, �9a�
here the far-field amplitude is
f��� = �m,n=0
N
amTmn�n1/2i−n cos n�, �9b�
nd the scattering width � is given by
� =�0
2�
d��f����2. �9c�
. LOW-FREQUENCY CASEe start with the low-frequency regime, where several
implifications occur. Cylinder dimensions are assumedmall compared with either the inside or outside wave-ength. Defining = �qk�max�, where �max is the maximumalue of ����, one requires that �1. It will also be con-enient, for this section only, to assume there are no mag-etic materials present so that �=1.
Consider the matrices R and S. Keeping only leadingowers of k� in the radial functions in the integrand for R,q. (4a), the result is proportional to
�m�k�� � �n�k�� − ��m�k���n�k��,
here the �m are potential functions regular at the ori-in. From this expression it appears that R is skew sym-etric, but this is illusory. Upon taking the divergence,
wo terms involving the Laplacian both vanish, and thewo remaining terms ±��m�k�� ·��n�k�� precisely cancel,iving a vanishing result. Consequently, R is gotten usinghe next higher-order terms in k� and turns out to havehe form
Rmn = �q2 − 1�rmn m+n+2, m,n = 0,1, . . . . �10a�
Here the rmn coefficients are a real, symmetric array,nd the complex nature of R is carried by the factor �q2
1� so that, for example, for q=1+ i one has Im�R�=2Re�R�.The integrand for S involves
�m�k�� � �n�k�� − ��m�k���n�k��,
here now the �m and �n are regular and singular poten-ial functions, respectively. The divergence theorem cantill be applied as above, but this time one must exclude aircular volume about the origin. The net result of all thiss that
Smn = Imn + smn m−n+2, m,n = 0,1, . . . �10b�
here the array smn may be complex. The terms Sm0, m1,2, . . . also contain ln terms regarded as constantsnd incorporated in the sm0. It should be noted that thisquation takes a slightly different form for elliptic cylin-ers. In that case all the integrals that would result in in-erse powers of in Eq. (10b) vanish identically [2], andne has instead
Smn = Imn + smn m−n+2, n � m
1� � for m + n even �odd�, otherwise,
m,n = 0,1, . . . . �10c�
In order to obtain the Rayleigh limit for the transitionatrix, first split off the identity by writing
Q0 = Q − iI. �11a�
ow T=−�Q0+ iI�−1R= �I− iQ0�−1iR→ �I+ iQ0�iR, using theinomial approximation with Q0 assumed small in an ap-ropriate sense (even though many elements of Q0 be-ome arbitrarily large). But Q0=R+ i�S−I�, so one canrite
T → i�R − �S − I�R� − R2. �11b�
f there are no losses, all the matrices on the right-handide of Eq. (11b) are real. With a little effort one sees fromqs. (10) that the product �S−I�R is smaller than R, ele-ent by element, by a factor at least of order 2 and can
hus be neglected to give
T = iR − R2 �12�
n the limit �1, where we have kept the leading termsn both real and imaginary parts separately. With losses
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2260 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 P. C. Waterman
resent, on the other hand, the term iR dominates botheal and imaginary parts of Eq. (11b), and Eq. (12) contin-es to be valid, the R2 term becoming negligible.We have computed R with the Bessel functions in place.
f desired, however, the limit formula can be gotten byeeping the first two terms in the low-frequency expan-ion of the Bessels [see Eq. (A4) of Appendix A], discard-ng any leading-order terms that cancel after integration,nd neglecting any higher-order terms in what remains.his alternate form for R should also prevent any minor
oss of significance due to the aforementioned cancella-ions.
Note that (except for circular cylinders for which R isiagonal) �R2�mn→Rm0R0n, so only scalar multiplications required in Eq. (12). R has just been seen to be sym-
etric in this limit, so reciprocity is built in. With noosses present R is real, the imaginary part of T is domi-ant, and from the energy constraint Eq. (7c) one hasReT= �ImT�2, which is satisfied by inspection. Extensiveumerical computations show that energy balance is alsoatisfied for absorbing cylinders.
This result is remarkable in its simplicity. Notice thatven for circular cylinders, standard separation of vari-bles leads to the limiting values of the quotientRmm /Qmm (with both matrices diagonal), involving bothessel and Hankel functions, while Eq. (12) gives iRmm�Rmm�2, involving only Bessel functions. In the low-
requency limit, the T-matrix also applies to boundary-alue problems of potential theory [18], and Eq. (12) (ac-ually just the term iR) should describe the correspondingielectric cylinder in an applied E field.Substituting Eq. (12) in the outgoing wave expansion
q. (9a), the term iR gives a series involving products ofegular and outgoing wave functions. Assuming the sum-ation can be taken inside the integral sign and differen-
iations, the sum can be collapsed to give an integral in-olving the Hankel function H0�k��−������, provided theeld point � is further from the origin than the largest ra-ial distance to the object’s surface [19]. Similar com-ents apply for the R2 term. This would seem to consti-
ute a proof that, at least in the Rayleigh limit, theutgoing wave expansion converges everywhere outsidehe smallest circular cylinder circumscribing the objectbut not necessarily in the annular region inside, see Fig.) as one would expect on physical grounds. Further studys called for here.
The derivation of Eq. (12) can be carried a step furthero obtain a numerical prescription equivalent to the fullayleigh expansion in powers of . Returning to the ex-ression T= �I− iQ0�−1iR, binomial expansion gives
T = �I + �p=1
�
ipQ0p�iR. �13�
eeping only one term in the summation, this equationeduces to Eq. (12) above, including the term neglectedherein, in the process roughly doubling the number ofignificant figures. As is increased, gradually moreerms must be included. In our experience, at most 2Nerms will suffice, where N is the matrix truncation sizeeing employed, if one is not too close to the radius of con-ergence.
The series will converge as long as the spectral radiusf Q0, given by the magnitude of the largest eigenvalue, isess than unity [20]. Results to date suggest that thisange covers the entire Rayleigh region and somewhat be-ond, extending to perhaps =1.5. The number of termsequired for a specified accuracy of course increases asne approaches the upper end of the range. An exampleill illustrate this. For a 2:1 rectangular cylinder with q1+ i (relatively high loss), one of the shapes discussed inome detail in the next section, Table 1 shows that the se-ies of Eq. (13) barely converges at =1.42. . . and divergesor slightly larger [exact sizes used were k��0�=0.45 and.46]. The spectral radius crosses through unity withinhis narrow window exactly as expected.
The above results for general values of dielectric con-tant and conductivity hinge on the normalizing con-tants q−n of Eq. (2), giving Im Q proportional to the iden-ity plus terms that can be neglected while at the sameime making R symmetric in the limit. So far we have noteen able to meet these conditions for magnetic materials,owever, and further study is required in that case. Inci-entally, the factor 2�1+��−1 included in Eq. (2) serves toake Im Qmm→1 for m�1, probably a worthwhile nor-alization in any case. This is in fact enough in some
ases to enable satisfaction of Eq. (13) if many terms arencluded, but this only indicates that the binomial seriess mathematically valid for those cases, not that it carrieshe physical significance of the Rayleigh expansion. Theethods of this section may not work for perfectly con-
ucting scatterers, either. Although R is again symmetricn the limit for those cases [2], the requirement that diag-nal elements of Im Q go to unity fails to be met.
. HIGH-ASPECT-RATIO CYLINDERSroblems of significance loss might be anticipated in Q for
he elliptic cylinder because of the cancellations noted inq. (10c). It turns out there is a class of surfaces, includ-
ng the elliptic case, for which the integrands in questionontain large, oscillatory terms that integrate to zero. Inrder to avoid massive loss of significance it is essential toodify or remove these terms, following the lead of theRL group [15]. We consider penetrable bodies, which areore difficult to treat, but the underlying ideas are the
ame.Let the cylinder shape be defined by one or more
ounding surfaces. For the method to be applicable, ateast one of the surfaces must be describable by the quad-ic equation
Table 1. Convergence Limit of the RayleighExpansion [Eq. (12)] for a 2:1 Rectangular
Cylindera
ize Parameter
Spectral Radiusof Q0
TermsSummed Convergence
1.42 0.997 4000 Yes1.45 1.03 — No
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P. C. Waterman Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2261
ax2 + by2 + cxy + dx + ey + f = 0 �14a�
ith real coefficients. This is the general equation for aonic section, allowing for coordinate rotations and trans-ations. Changing over to polar coordinates by writing x����cos �, y=����sin �, the preceding formula becomes auadratic equation in ����:
�a cos2 � + b sin2 � + c cos � sin ���2���
+ �d cos � + e sin ������ + f = 0. �14b�
f the solutions to this equation, one open surface is al-owed, the infinite plane (angular range of integration �).wo parallel planes and elliptic and parabolic cylindersre also included (angular range 2�) but not hyperbolicylinders (which are open). In all cases 1/���� must be fi-ite and continuous everywhere, including the endpoints.e call these complete quadric (cq) surfaces.Now for the problem at hand consider Eq. (4a), rewrit-
en to give S. Derivatives of the radial functions will entern when the gradients are taken, but these may be reex-ressed in terms of standard relations involving contigu-us radial functions so that every term in the integrandill involve a product of the form Ju�k���Nv�k��. When-
ver v�u, both functions in the product are expanded andhe results consolidated into an ascending power series in����. The matrix S is now split into the sum S++S−,here S+ comes from the positive powers and constant
erms in this series and S− is due to the inverse powersany ln k���� terms are regarded as constants for this pur-ose]. The only nonzero elements of S− will fall above theain diagonal, and by the same token elements of S and
+ are identical on and below the main diagonal. Detailsre given in the Appendix.Whenever the surface passes close to the origin (and
his is guaranteed to happen for high aspect ratios withhe origin restricted to the interior) k���� will be small,he integrand of S− large and rapidly oscillating, and can-ellations may occur in the region during integration. Inact, based on extensive numerical computation, we makehe following ansatz:
Upon integration over a cq surface,
S− = 0. �15�
or these cases (plane, ellipse, parabola), total cancella-ion of the inverse powers occurs upon integration. Suchancellations have been known for some time for the el-ipse, both for the penetrable case, as noted in connectionith Eq. (10c), and the perfect conductor, and arise due to
rthogonalities of the angular functions [2]. Presumablyhis is the mechanism for the other surfaces, too, al-hough in some cases this may not be obvious. For ex-mple, for a circular cylinder
���� = x0 cos � + �1 − x02 sin2 ��1/2, �16a�
ith the coordinate origin translated across a diameteror −1�x �1, one of the simpler integrals is
0
I�x0� =�0
�
d� cos 3�/���� �Dirichlet case�. �16b�
sing Mathcad 11 in double precision ��15 sf� gives re-ults of order 10−15, from which one concludes that (i) thentegral does indeed vanish, and (ii) Mathcad can handleumerical integration very accurately. [An analyticalroof of the result might run as follows: inversion of aranslated circle in the unit circle gives another trans-ated circle. Thus it suffices to substitute the right-handide of Eq. (16a) in the numerator rather than the de-ominator of the integrand in Eq. (16b). At that point, theinomial expansion and standard orthogonality relationsan be employed.]
We will return to the translated circular cylinderhortly.
Although there is significance loss associated with S−,he problem is resolved if it, or at least that portion of itor which the surface of integration is near the origin, isimply discarded. This is best illustrated by example. Theimplest is the elliptic cylinder, defined within a scale fac-or by ( is aspect ratio)
���� = �cos2 � + −2 sin2 ��−1/2, �17a�
s shown in Fig. 1. Observe that the second symmetrylane enables one to compute R (and S) more efficiently,aking Rmn to be four times the integral over (0, � /2) form+n� even, zero otherwise. Because of this pattern of ze-os it follows that there is no coupling between matrix el-ments, with m and n both even, and the odd–odd terms,hich may be treated separately. The convergence limit
or the Rayleigh expansion given in Table 1, for example,eferred to the even–even case, while the odd–odd caseas still convergent up as far as =2.5, requiring only
our or five terms in the sum to get several significant fig-res.Because the integration involving Eq. (17a) is over a cq
urface, one can set
Q = R + iS+ �17b�
nd proceed normally thereafter. At this juncture, aimple check of both the ansatz and the computer pro-ram is available for aspect ratios that are not too largee.g., =2) by numerically comparing S+ and S, whichhould be identical except for possible significance lossbove the diagonal in the latter. With regard to actual sig-ificance loss, for a 10:1 nonmagnetic cylinder with q=1i, =21/2 (low-resonance region), plotting the log of theagnitudes of the integrands of S08
+ and S08− , for example,
ives the results shown in Fig. 2. In both cases one candentify a dominant term containing the factor cos 9�,nd visible weak minima are in fact situated at the zerosf that function (real and imaginary parts must be plottedndividually in order to display zero crossings and ampli-ude oscillations in full). Because the integrand of S08
− isarger by up to 12 orders of magnitude and yet makes noontribution, it follows that we can prevent the loss ofbout 12 significant figures in S08 (and losses in other el-ments as well) by simply omitting S−, as done in Eq.17b). At higher aspect ratios or larger truncation size N,he effect is even more pronounced.
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2262 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 P. C. Waterman
Equation (17b) also applies to a translated or rotatedlliptical cylinder, allowing the origin to be set anywhereithin the cylinder (or on the surface, approached from
he inside). The effect is roughly unchanged for magneticaterials. Taking �=100 and reducing the cylinder size
y a factor of 10 to preserve the value of , the maximumatio of the two curves of Fig. 2 remains about the same.
Next, consider the rectangular cylinder defined (in therst quadrant) by
���� = �1���, 0 � � � tan−1
�2���, tan−1 � � � �/2 , �18a�
ith bounding planes �1���=1/cos � and �2���= /sin �s shown in Fig. 3, and for simplicity assume the aspectatio �1. It’s convenient to do the integrations sepa-ately over �1 and �2, writing R=R1+R2 and S=S1+S2 tovoid having discontinuities in the integrands.The problem in this case lies with S1 as it passes near
he origin. Write the integral as S1�pp� to indicate inte-ration over the physical part of �1 [solid line in Fig. 3 asistinguished from the nonphysical part (npp) given byhe dashed line]. Now S1�pp�=S1
+�pp�+S1−�pp�. But by an-
atz S1−�pp�+S1
−�npp�=0, and subtracting gives S1�pp�S1
+�pp�−S1−�npp�. Note that these last integrals select
nly positive powers of k�1 from the radial functions nearhe origin and only negative powers as k�1→�. Trouble-ome terms are removed, and the Q matrix itself becomesfor simplicity the common factor of 4 is not shown)
ig. 2. Disparity in magnitude of the integrands of S08+ and S08
−
s shown versus � for a 10:1 elliptic cylinder (see text). Real andmaginary parts of these quantities undergo sign changes nearhe minima seen here.
Fig. 3. Geometry of the rectangular cylinder.
Q = R1�pp� + R2�pp� + i�S1+�pp� − S1
−�npp� + S2�pp��.
�18b�
Notice that we did not bother to split up S2, which doesot pass near the origin in the example shown. If thehape were elongated horizontally instead, the roles of S1nd S2 would be interchanged in the above equations.quation (18a) can also be used for the square cylinder,
ncidentally, setting =1. In that event Eq. (18b) is noteeded, and Q can be used in its original form. In addi-ion, because of four-fold rotational symmetry one shouldompute only those even–even elements for which theum or difference of indices is a multiple of 4, the restanishing (the odd–odd elements are unaffected).
The translated circular cylinder example of Eq. (16a)equires further comment: for this one shape specifically,nverse powers of k� are unaffected, so S− is unchanged,ut additional cancellations occur in S above the diago-al, giving (correct to leading order)
Smn = Imn + smn m−n+2, n � m
n−m, otherwise, m,n = 0,1, . . . .
�19�
or S computed taking this into account we write S++ sohat S=S+++S− (see Appendix A). The new cancellationsppear to stem from the observation that inversion of thehifted circle in the unit circle gives another shifted circle,s noted earlier, and such an inversion interchanges theoles of the �u−v� powers preceding the constant term at=� (the terms omitted in forming S+) with those follow-
ng. Regardless of why they come about, however, just asith the ansatz given earlier they are easily confirmedumerically by noting that the computed S++, S+, and Sust all be identical to within successively increasing sig-ificance loss. In exact analogy with Eq. (17b), the Q ma-rix for the shifted circular cylinder is taken to be
Q = R + iS++. �20�
Now consider two such intersecting cylinders, their in-ersection forming a biconvex lens as shown in Fig. 4.aking x0 negative, S is gotten by integrating over thehysical part of the left-most cylinder from 0 to � /2again the solid curve) to obtain S++, then subtracting off− over the remainder of the cylinder to get
Q = R�pp� + i�S++�pp� − S−�npp��. �21�
ens thickness (or aspect ratio) is variable over a wideange, and the radii of curvature need not be equal; alano-convex lens could be considered, for example. Forhat matter, the curvature need not be constant, but one
ig. 4. Double-convex lens, formed by the intersection of twoircular cylinders.
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ittrtdl
autttSsrmls
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tt(evohwems
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Fo(i
Fif
P. C. Waterman Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2263
ould employ two intersecting elliptic or parabolic cylin-ers (in that event reverting back to S+, S−). The para-olic lens is sketched in Fig. 5. If there were no verticalymmetry plane Eq. (21) would become slightly more in-olved, as it would then be necessary to split the S inte-rals on both cylindrical surfaces. In several cases, inci-entally, we have found that S+ and S++ appear to workqually well.
The ansatz of Eq. (15) applies equally well to perfectlyonducting cylinders. Because only free-space wave func-ions are required, the resolution into S± can be drasti-ally simplified as described in the Appendix.
These examples illustrate the procedure for modifyingusing S±. To our knowledge there are only three cq sur-
aces, but clearly a great many shapes can be constructedsing them. In some of these cases at least, results haveeen obtained without difficulty for very high aspect ra-ios. For example, for both elliptical and rectangular cyl-nders, even though the condition number of Q increasesith , there is apparently no problem in obtaining the-matrix for aspect ratios up to 1000:1. Keep in mind also
hat the method may in fact prove useful in conservingignificance down to relatively small aspect ratios in someases, e.g., =2 or 3.
There’s a downside to all this, incidentally. The bulk ofhe computation time lies in the numerical integrationsor Q, and resolving S into S± can increase that time sig-ificantly. It may be possible to improve on the presentomputation, however. One technique for doing this isoted in Appendix A.
. NUMERICAL RESULTSather than looking at scattering and absorption widths,e can better confirm the workings of the conservation
aws and low-frequency formulas discussed above byhowing results for matrix elements directly. Throughouthis section we consider for simplicity only nonmagneticaterials, �=1.As a first example consider a 10:1 absorbing elliptic cyl-
nder in the low-resonance region, q=1+ i, =21/2, withruncation at N=8. Figure 6 shows a bar plot of magni-udes of the T-matrix elements on a logarithmic scaleanging over 10 decades from 0 to −100 dB, amplitudeshus running from unity (at the uppermost tick mark)own to 10−10 at the base plane. Columns and rows areaid out parallel to the x and y axes, respectively, with T00
ig. 5. Parabolic lens, formed by the intersection of two para-olic cylinders.
t the origin. Some elements in the last few rows and col-mns do not appear because they fall below the arbi-rarily chosen cutoff. As already discussed, due to symme-ry entries vanish when the sum of the indices is odd sohat the even–even and odd–odd arrays are uncoupled.ymmetry of T itself is also evident. Rather than filling inmoothly, the odd–odd elements are seen to be smaller,elatively speaking, showing that the field itself becomesore nearly symmetric across the vertical plane as the el-
ipse becomes thinner. An extreme case of this will behown below.
The rapid falloff in magnitude going away from the ori-in ensures that this truncation is sufficient to obtain thear-field amplitudes and scattering widths to six or moreignificant figures using Eqs. (9). This falloff must alsoore than offset the growth of the Hankel functions as
ne moves into the near field if Eq. (9a) is to converge inhe vicinity of the cylinder as discussed earlier in the Ray-eigh case.
Figure 7 looks at truncation error by plotting the log ofhe absolute value of the inverse fractional difference ofhe preceding results with those obtained at N=10thereby increasing from a 44 to 55 system of even–ven terms and similarly for the odd–odd). The resultingertical scale ranging from 0 to 8 is basically the numberf significant figures of agreement of the two truncations,ere showing a largest entry of 6.4 (indicating agreementithin one part in 106.4). It’s interesting that columns are
ffectively identical, probably reflecting the fact that theatrix equation for T can alternately be thought of as N
ets of equations and unknowns for each of the columns
ig. 6. Bar plot showing the magnitudes of T-matrix elementsn a logarithmic scale covering ten decades from unit amplitudeuppermost tick mark) down to the base plane (10:1 elliptic cyl-nder, q=1+ i, =21/2, N=8).
ig. 7. Plot of estimated number of significant figures remain-ng after truncation error (term-by-term comparison of elementsor N=8, 10).
s
hpfit
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t6tspfists
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Fcan
2264 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 P. C. Waterman
eparately, but with the same coefficient array.In Fig. 8 the same significance scale is used to give a
igher resolution look at the symmetry and hence reci-rocity associated with elements of Fig. 6. One sees aboutve-figure agreement for the 0,2 and 2,0 elements, lesshereafter.
Figure 9 looks at the significance with which energy re-uirements are met, comparing the left- and right-handides of Eq. (7c). This plot also serves to confirm the rolef the absorption matrix, incidentally. Summarizing athis point, the results of these four plots appear to be selfonsistent and support the earlier statement regardingar-field accuracy.
As an exercise, the same plots were also made usinghe (proposed symmetric) Z matrix of Eq. (8) in place of T.nterestingly enough, with the exception of Fig. 9 dealingith energy balance, although actual numerical valuesere of course different, in the highly compressed scaleeing used, all three remaining plots were almost identi-al with those shown here. If this agreement holds upenerally, then Z may well serve as a useful guide to ap-ropriate values of input parameters in a computation of.In Fig. 10 T-matrix results are given for a 1000:1 ellip-
ic cylinder, with all other parameters the same as in Fig.(the uppermost tick mark again indicates unit ampli-
ude). Note that no increase in truncation size was neces-ary. The odd–odd coefficients have now vanished com-letely, indicating that both the scattered and internalelds are highly symmetrical across the vertical mirrorymmetry plane of the ellipse. (All missing elements inhe figure are still present in the computation and can beeen if the scale is extended down another ten orders of
ig. 8. Significant figures of agreement for corresponding ele-ents above and below the main diagonal in Fig. 6, testing
eciprocity.
ig. 9. Significant figures of agreement between the left- andight-hand sides of Eq. (7c), checking energy balance for the ex-mple of Fig. 6.
agnitude.) In comparison with Fig. 6 for the 10:1 case,ach of the elements here is smaller by a factor of about00, exactly the reduction in the amount of materialresent. Discarding S− prevented the loss of up to 28 sig-ificant figures in this case, incidentally.Figure 11 checks the Rayleigh limit formula by compar-
ng results of Eq. (12) with T as computed by matrix in-ersion. The scatterer here is a 10:1 rectangular cylinderith q=1+ i, =0.014, and again N=8, and agreement is
een to range from four to six significant figures. The odd–dd elements appear to agree somewhat better, probablyecause the associated spectral radius is smaller. Just as
ig. 10. Magnitudes of the T-matrix elements are shown for a000:1 elliptic cylinder. Scale is identical with that of Fig. 6 (q1+ i, =21/2, N=8).
ig. 11. Comparison of Eq. (12) with T using matrix inversionor a 10:1 rectangular cylinder shows four to six significant figuregreement throughout, checking the Rayleigh limit formula (q1+ i, =0.014, N=8).
ig. 12. Equation (13), using four terms in the summation, isompared with T (matrix inversion) for the cylinder of Fig. 11 athigher frequency � =2−1/2�, showing five to more than ten sig-
ificant figure agreement.
omt
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6TRcfOtriEfE
epdotcatdacswslBtt
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i
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Hs
P. C. Waterman Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2265
ccurred for the elliptic cylinder of Fig. 6, odd–odd ele-ents of the T-matrix (which is not shown) are smaller
han the even–even elements.Figure 12 makes a comparison for the same rectangu-
ar cylinder at a higher frequency so that =2−1/2. This iso longer the Rayleigh limit, and this time T is comparedith its value from the Rayleigh expansion Eq. (13), using
our terms of the series. Agreement ranges from five toetter than ten significant figures, and the distinction be-ween even–even and odd–odd terms is now more clear-ut. Using more terms in the series improves the agree-ent, by the way, although the significance distinction is
till present. One concludes that the differences seen hereetween T (from matrix inversion) and the Rayleigh ex-ansion are due solely to truncation of the infinite seriesf Eq. (13). The uniformity of the significance is perhapsn artifact of truncation of a matrix binomial series, al-hough to our knowledge there is nothing in the literaturen this point.
. DISCUSSIONhe main results of this work are Eqs. (12) and (13), theayleigh limit and expansion applicable at low frequen-ies, and the ansatz of Eq. (15), which becomes more use-ul the higher the aspect ratio of the scattering object.nly a few examples were given, but it should be noted
hat a number of other cases were considered, all givingesults fully consistent with those reported here. Thesencluded magnetic materials, except in connection withqs. (12) and (13), as well as several cases using the odd
unctions, obtained by replacing the cosines by sines inqs. (1) and (2).Although only cylindrical objects were considered, we
xpect most of these procedures to have their counter-arts for acoustic and electromagnetic scattering in threeimensions. Indeed, extensive cancellations are known toccur when evaluating the Q matrix for both surface scat-ering (Dirichlet or Neumann conditions in the acousticase or perfect conductivity in the electromagnetic case)nd volume scattering by spheroids or ellipsoids [2,17]. Athis writing, one might anticipate the cq surfaces in threeimensions to be planes, ellipsoids, and paraboloids,long with truncated versions of the elliptic and parabolicylinders considered above. Notice that the cylindricalurfaces are then being asked to play a new role, one inhich the Q matrix involves surface integrals of the
pherical wave functions as, for example, the finite circu-ar cylinder [15]. It should also be noted that splitting theessel function products should be somewhat simpler in
hree dimensions, because there are no logarithmic termso deal with.
PPENDIX A: EVALUATION OF S±
egin with Eq. (4a) using Bessel and Neumann functionsm�qk�� and Nn�k�� and note that [2]
d� · � = d��k��/�k� − �1/k��k������/���.
or derivatives of the Bessel functions we use the stan-ard relations [21]
Jm� �z� = − Jm+1�z� + �m/z�Jm�z�,
Nn��z� = Nn−1�z� − �n/z�Nn�z�. �A1�
ubstituting in Eq. (4a), elements of S take the form
Smn = dmn�0
2�
d� �k�Jm�qk��Nn−1�k��cos m�
cos n� + �nJm�qk��Nn�k��cos m���1/�������
sin n� − cos n�� + qk�Jm+1�qk��Nn�k��cos m�
cos n� − mJm�qk��Nn�k���cos m�
+ �1/�������sin m��cos n��, �A2�
ith normalizing constants given by
dmn = ��m�n�1/22�1 + ��−1q−m. �A3�
f the object has two planes of mirror symmetry, then aarity function par�m ,n�=1 for m+n even, zero other-ise, should also be included in the dmn. To obtain the
egular matrix elements Rmn, replace Nn�k�� by Jn�k�� inq. (A2).Now the ascending series for the radial functions are
21]
Jm�z� = �k=0
�
�− 1�k�z/2�2k+m/k!�m + k�!,
Nn�z� = − �1/���k=0
n−1
�n − k − 1�!�z/2�2k−n/k!
+ �2/��ln�z/2�Jn�z� − �1/���k=0
�
�− 1�k���k + 1�
+ ��n + k + 1���z/2�2k+n/k!�n + k�!, �A4�
n terms of the � function
��n� = − � for n = 1
��n − 1� + 1/�n − 1� otherwise, �A5�
here n�1 in the second of Eqs. (A4) and �=0.57721. . . isuler’s constant. Writing Jm�qz�Nn�z�=JNsum�m ,n ,q ,z�,ne now has
N sum�m,n,q,z�
= − �1/���k=0
K
�s=0
n−1
a�m,k�b�n,s�q2k+m�z/2�2k+2s+m−n
+ �1/��Jm�qz�2 ln�z/2�Jn�z� − �k=0
K
���k + 1�
+ ��n + k + 1��a�n,k��z/2�2k+n . �A6�
ere we have put the right-hand side in a form betteruited for computation by defining
a�m,k� = 1/m! for k = 0
− a�m,k − 1�/k�m + k� otherwise , �A7�
acc
oPssspess
Nmtan
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ds+wToff[
cto
h
HmtctpiEpTsseutAtfi
ti(Etrsc
ig=tncaowadp
Ncfpf
troooMhlcEt
2266 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 P. C. Waterman
b�n,s� = �n − 1�! for s = 0
b�n,s − 1�/s�n − s� otherwise , �A8�
nd the infinite series have been truncated at K, whichan conservatively be set at 2N, with N the matrix trun-ation size used in the text.
By inspection, inverse powers of z are seen to occurnly in the double sum in Eq. (A6), and provided n�m.ositive and negative powers can be sorted out using thetep function h�x�=1 for x�0, zero otherwise. Thus, in-erting a factor h�2k+2s+m−n+1/2� within the doubleum, with Eq. (A6) otherwise unchanged, gives the sum ofositive powers (including a constant term if m−n isven); we call this quantity JNsum+�m ,n ,q ,z�. By theame token, the negative powers are gotten by making aign change in the argument of the step function:
JNsum−�m,n,q,z� = − �1/���k=0
n/2
�s=0
n/2
h�− �2k + 2s + m − n + 1/2��
a�m,k�b�n,s�q2k+m�z/2�2k+2s+m−n.
�A9�
otice that the upper limits have been reduced here to re-ove some vanishing terms and are understood to be in-
erpreted as the integer part of n /2. The factor 1/2 in thergument of the step function prevents inclusion of erro-eous terms due to h�−0�.Now define the function
JN±�m,n,q,z� = JNsum±�m,n,q,z� if n � m
Jm�qz�Nn�z� otherwise .
�A10�
he quantities S± are gotten by inserting JN± respec-ively in place of the product of Bessel functions whereverhey occur in the integrand of Eq. (A2). At this point onean note that both of Eqs. (A1) are valid for both theessel and Neumann functions. Using the equations ashown, however, will prevent any inverse powers from ap-earing in the products on the main diagonal of S.In order to obtain S++ for the translated circular cylin-
er, it is only necessary to modify the argument of thetep function in the equation for JNsum+ to read h�k+sm−n+5/2�. The first nonvanishing power of will occurhen the argument equals 1/2, i.e., when k+s=n−m−2.he power in question, k+s, thus increases linearly asne moves further above the diagonal, and the remainingactor of 2 is presumably due to the indices of the Besselunctions and the S matrix not being entirely in registersee Eq. (A2)].
The above equations can also be used for the perfectlyonducting case, but there is a simpler alternative. Usingechniques described elsewhere [2,22], for n�m one canbtain the identity
Jm�z�Nn�z� = Jn�z�Nm�z� − �1/�� �s=0
�n−m−1�/2
cmns�2/z�n−m−2s,
�A11a�
aving coefficients
cmns = �m − n + s + 1�s�n − s − 1�!/s!�m + s�!. �A11b�
ere �a�0=1, �a�s= �a+s−1��a�s−1 otherwise, is Pochham-er’s symbol. These equations can be verified by induc-
ion on n=m+1, m+2, . . . (notice that the case n=m+1orresponds to the Wronskian relation for the radial func-ions). The summation is seen to contain all the inverseowers, so we can replace the quantities JNsum± appear-ng in Eq. (A10) by the two terms on the right-hand side ofq. (A11a), respectively, giving the simpler formulas. Un-ublished numerical computations show that the-matrix for perfectly conducting cylinders with thehapes discussed in Section 4 also satisfies the three con-istency conditions, although with edges present it is nec-ssary to truncate at higher numbers of equations andnknowns in order to obtain accuracy comparable withhe penetrable cases, especially at higher aspect ratios.lternately, perfectly conducting bodies with edges can be
reated by incorporating edge conditions in the surfaceelds [16].Incidentally, it is fairly easy to see from the above equa-
ion that S will be symmetric for elliptic cylinders. Turn-ng this argument around, if it can be verified that Eq.A11a) holds also for spherical Bessel functions [althoughq. (A11b) may look somewhat different], then, because of
he well-known symmetry of S for surface scattering (Di-ichlet and Neumann bc) by ellipsoids in three dimen-ions, the ansatz S−=0 will presumably hold for thoseases also.
Equation (A11a) also provides some interesting insightnto the structure of S. Because the equation holds foreneral surface shapes, it follows that S+ (and hence Q+
R+ iS+) is always symmetric for those cases. Now noticehe gap between Smn
+ and Smn− , i.e., the successively larger
umber of (positive) powers of z missing as n−m in-reases. If magnitude can be associated with power of z,s occurred with the high-aspect-ratio cases, then againne would expect large magnitude disparities, whichould be lost if matrix elements of S+ and S− were everdded together. In short, rather than invert Q=Q++ iS−
irectly, it might be preferable to invert Q+ and then em-loy the formal identity
Q−1 = �I + i�Q+�−1S−�−1�Q+�−1. �A12�
ote also that S−=0 for the perfectly conducting ellipticylinder, suggesting that the binomial series obtainedrom Eq. (A12) might be appropriate for cylinder shapeserturbed from the elliptical. We leave this as a questionor further study.
As mentioned in the text, the splitting of S± can be ex-remely time consuming for the penetrable cases; manyepetitions of the above equations are required to carryut the numerical integrations to high accuracy. The sec-nd and fourth lines of Eq. (A2) involve the same productf Bessels and can be combined, giving a 25% saving.ore significant improvement can probably be realized,
owever, if for each matrix element one were to first col-ect and either analytically or numerically evaluate theoefficient of each power of z in the double summations ofqs. (A6) and (A9), and probably also the single summa-
ion of Eq. (A6), before starting the integration. This
wS
aemTo
ATpAda
R
1
1
1
1
1
1
1
1
1
1
2
2
2
P. C. Waterman Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2267
ould be done for m, n, q, and K all fixed, separately for+ and S−, and should eliminate a great deal of repetition.Larger objects of course require that more equations
nd unknowns be kept. It should be kept in mind, how-ver, that the Neumann functions do not begin to grow inagnitude until their index exceeds their argument.hus the splitting operation need not begin until the sec-nd index of S is greater than k�min.
CKNOWLEDGMENThe author is indebted to Dennis Roser and Science Ap-lications International Corp. and Jeff Hale and the U. S.rmy Edgewood Chemical and Biological Center, Aber-een Proving Ground, Maryland, for their encouragementnd sponsorship.
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