The T-matrix revisited

doi:10.1364/JOSAA.24.002257

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The T-matrix revisited

P. C. Waterman »View Author Affiliations

JOSA A, Vol. 24, Issue 8, pp. 2257-2267 (2007)
http://dx.doi.org/10.1364/JOSAA.24.002257

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Abstract

We consider electromagnetic scattering from penetrable cylinders of general cross section. After summarizing the basic T-matrix equations the low-frequency case is examined, which leads for nonmagnetic materials to the exact result $T = i R - R^{2}$ in the Rayleigh limit, satisfying both reciprocity and energy constraints. Here elements of R are given by integrals of regular wave functions over the cylinder surface. A “Rayleigh expansion” is then found that is convergent throughout the Rayleigh region and the lower end of the resonance region and requires no matrix inversion. For bodies of high aspect ratio, there is a problem with significance loss during numerical integration, due to large oscillatory terms. A class of surfaces has now been found for which these terms can be removed, however, enabling us to treat aspect ratios up to 1000:1. These methods are expected to apply also in three dimensions.

1. INTRODUCTION

The T-matrix has become an accepted tool for treating a variety of scattering problems involving objects of more general shape than spheres or circular cylinders, with applications in acoustics, electromagnetic theory, and elasticity [1

V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).

]. For surface scattering where separation of variables can be employed, e.g., hard and soft cylinders of elliptical cross section or spheroids in three dimensions (working in elliptic cylinder or spheroidal coordinates), the T-matrix method (using circular cylinder or spherical polar coordinates) provides a roughly equivalent computational alternative. The trade-off in those cases involves the necessity of generating rather complex wave functions, on the one hand, versus numerical integration followed by matrix inversion on the other. The simplest examples of this are the perfectly conducting strip in two dimensions [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

, 3

R. L. Weaver and Y.-H. Pao, “Application of the transition matrix to a ribbon-shaped scatterer,” J. Acoust. Soc. Am. 66, 1199–1206 (1979). [CrossRef]

] and the circular [4

G. Kristensson and P. C. Waterman, “The T-matrix for acoustic and electromagnetic scattering by circular disks,” J. Acoust. Soc. Am. 72, 1612–1625 (1982). [CrossRef]

] and elliptic disk [5

J. Björkberg and G. Kristensson, “Electromagnetic scattering by a perfectly conducting elliptic disk,” Can. J. Phys. 65, 723–734 (1987). [CrossRef]

] in three. For penetrable bodies, however, the orthogonality of the wave functions is lost in nearly all coordinate systems and matrix inversion is unavoidable, making the T-matrix method more attractive.

Ström looked at the role of the T-matrix in multiple scattering [6

S. Ström, “ T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties,” Phys. Rev. D 10, 2685–2690 (1974). [CrossRef]

]; Ström and Zheng considered composite scatterers [7

S. Ström and W. Zheng, “The null field approach to electromagnetic scattering from composite objects,” IEEE Trans. Antennas Propag. 36, 376–382 (1988). [CrossRef]

]. Schneider and Peden computed the differential cross section of a dielectric ellipsoid, probably the first T-matrix computation for an object without rotational symmetry [8

J. B. Schneider and I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36, 1317–1321 (1988). [CrossRef]

]. Good recent review articles, including discussion of publicly available FORTRAN codes, have been given by Mishchenko and co-workers at NASA [9

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “ T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996). [CrossRef]

, 10

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.

]. Lakhtakia and the Varadans gave a database of general applications in 1988 [11

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988). [CrossRef]

], and Mishchenko et al. gave a more extensive database for the electromagnetic case in 2004 [12

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “ T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transf. 88, 357–406 (2004).

A major difficulty with the method has been the tendency for the auxiliary Q matrix to become ill-conditioned for objects very large compared with wavelength or those having high aspect ratios. Regarding these problems Mishchenko and Travis [13

M. I. Mishchenko and L. D. Travis, “ T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994). [CrossRef]

], and more recently Havemann and Baran [14

S. Havemann and A. J. Baran, “Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 89, 87–96 (2004). [CrossRef]

], have shown that the use of quad precision (31 decimal digits) gives greatly improved results.

The present work is concerned primarily with electromagnetic scattering by penetrable cylindrical obstacles, allowing for absorption. After summarizing the basic equations, we look first at the low-frequency case, finding a simple explicit formula for the T-matrix for nonmagnetic materials in the Rayleigh limit. This is then extended to obtain the Rayleigh expansion, in which subsequent terms give contributions from successively higher powers of the size/wavelength ratio. For the cases considered here, the expansion converges throughout the Rayleigh region and somewhat beyond.

For high aspect ratios, one of the main problems involves significance loss during numerical integration for Q [14

]. A valuable paper bearing on this question was published in 1993, when Sarkissian and coworkers at the Naval Research Laboratory (NRL) looked at acoustic scattering from long but finite, thin rigid rods of circular cross section [15

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, “ T-matrix implementation of forward scattering from rigid structures,” J. Acoust. Soc. Am. 94, 3448–3453 (1993). [CrossRef]

]. Effectively they were able to isolate the troublesome terms in the integrands of matrix elements and remove them. Here we describe a class of surfaces (“complete quadrics”) for which a similar approach can be carried out for penetrable bodies. As a result, for the typical case of a 10:1 elliptic cylinder, one can prevent the loss of up to 12 significant figures. Using the technique, we can treat elliptic and rectangular cylinders, for example, having aspect ratios up to 1000:1.

2. BASIC EQUATIONS

We begin by briefly summarizing the equations governing the null-field approach to the T-matrix for penetrable bodies, supplemented by more recent results relating to the absorption matrix. A slightly different normalization will be introduced for the basis functions, for reasons that will become evident, and a new symmetric matrix is found, which is useful in the numerical analysis.

Consider a penetrable cylinder, with boundary defined in polar coordinates by

ρ = ρ (ϑ)

, as shown in Fig. 1 , when illuminated by an incident electromagnetic wave with E field parallel to the cylinder axis (the H parallel case follows by duality). Outgoing scattered waves are described in the free-space region outside the smallest circle circumscribing the cylinder (shown dashed in the figure) by the wave functions

ψ_{n} (k ρ) = ϵ_{n}^{1 ∕ 2} H_{n} (k ρ) \cos n ϑ, n = 0, 1, \dots,

(1)

where the Hankel functions are expressed as

H_{n} = J_{n} + i N_{n}

in terms of the Bessel and Neumann functions. For simplicity we suppose that both the cylinder and the incident wave have a plane of mirror symmetry, so the odd functions

\sin n ϑ

will not be needed (the more general case has been described earlier [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

]). The Neumann factor

ϵ_{0} = 1

ϵ_{n} = 2

otherwise, and a time factor

\exp (- i ω t)

is suppressed. The field throughout the interior of the cylinder is represented in terms of the regular wave functions

χ_{n} (k^{'} ρ) = ϵ_{n}^{1 ∕ 2} 2 {(1 + μ)}^{- 1} q^{- n} J_{n} (k^{'} ρ) \cos n ϑ, n = 0, 1, \dots,

(2)

where

k^{'} = q k

, with

q = {(μ ϵ + i μ σ ∕ ϵ_{0} ω)}^{1 ∕ 2}

the complex index of refraction in terms of the relative dielectric constant, relative permeability, and conductivity of the cylinder material.

The incident, scattered, and interior fields are now written as

E_{z}^{inc} (k ρ) = \sum_{n = 0} a_{n} χ_{n} (k ρ),

(3a)

E_{z}^{sca} (k ρ) = \sum_{n = 0} f_{n} ψ_{n} (k ρ),

(3b)

E_{z}^{int} (k^{'} ρ) = \sum_{n = 0} α_{n} χ_{n} (k^{'} ρ),

(3c)

(3)

and given the incident wave through the

a_{n}

coefficients, we want to determine the scattered wave

f_{n}

and possibly also the interior coefficients

α_{n}

. Proceed by defining a matrix Q having elements [16

P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999). [CrossRef]

]

Q_{m n} = \int d σ ∙ [μ χ_{m} (k^{'} ρ) \nabla ψ_{n} (k ρ) - \nabla χ_{m} (k^{'} ρ) ψ_{n} (k ρ)], m, n = 0, 1, \dots,

(4a)

which is further resolved into regular and singular parts R, S by replacing the Hankel functions by Bessel or Neumann functions, respectively, to get

Q = R + i S,

(4b)

with R and S both in general complex. The integral of Eq. (4a) runs over the cylinder surface, with ρ set equal to

ρ (ϑ)

after the gradients are taken. Equation (4a), incidentally, is precisely the form used also for acoustic scattering in both two and three dimensions, with the wave functions representing velocity potentials and ϵ, μ replaced by the compressibility and density ratios of the object, respectively [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

]. For the three-dimensional electromagnetic case, vector basis functions must be used. Rather than the ordinary products of Eq. (4a) involving the gradient, one then uses vector cross products and the curl operator [17

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 4, 825–839 (1971). [CrossRef] [CrossRef]

]. Although we will focus on penetrable bodies, perfectly conducting objects will also be considered, in which event surface fields are represented by free-space wave functions. For E parallel to the cylinder axis (Dirichlet), discard the terms multiplied by μ in the integrand of Eq. (4a) to get

Q_{D}

, or alternately for H parallel (Neumann), discard the terms not multiplied by μ to get

Q_{N}

. In both cases the bulk parameters should be set to their free-space values. These matrices are then used in place of Q and R in the equations below to obtain the transition matrix. One warning here: if the surface fields themselves are required, then the trial functions should incorporate edge conditions where necessary [16

P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999). [CrossRef]

The expansion coefficients of Eqs. (3) can be shown to be connected through the equations

a = i Q^{'} α,

(5a)

f = - i R^{'} α,

(5b)

(5)

where the prime denotes transpose. Further defining T as the operator that computes the scattered wave directly from the incident wave, i.e.,

f = T a,

(6a)

and eliminating the

α_{n}

between Eqs. (5a, 5b), the T-matrix is seen to be given by

Q T = - R .

(6b)

In this last equation we have replaced

T^{'}

by T, which is allowable because the two must be equal due to reciprocity.

In order to obtain the absorbed power, first define the matrix having elements [16

P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999). [CrossRef]

]

W_{m n} = \int d σ ∙ μ χ_{m} (k^{' *} ρ) \nabla χ_{n} (k^{'} ρ), m, n = 0, 1, \dots .

(7a)

Integrating the inward normal component of the complex Poynting vector over the surface of the cylinder now yields

\frac{1}{2} Re (α^{' *} W α) = \frac{1}{4} α^{' *} (W + W^{' *}) α = \frac{1}{4} a^{' *} {(Q^{*})}^{- 1} (W + W^{' *}) {(Q^{'})}^{- 1} a,

(7b)

and energy balance consequently requires that

- Re T = T^{' *} T + \frac{1}{4} A,

(7c)

where

A = {(Q^{*})}^{- 1} (W + W^{' *}) {(Q^{'})}^{- 1}

(7d)

is the (Hermitian) absorption matrix. Equation (7c), which follows from the arbitrariness of the incident wave, is a generalized optical theorem, ensuring for plane-wave incidence that (minus) the real part of the forward amplitude equals the sum of the scattering and absorption widths. In terms of the scattering matrix

S = I + 2 T

relating incoming and outgoing waves, Eq. (7c) becomes simply [17

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 4, 825–839 (1971). [CrossRef] [CrossRef]

] (I is the identity matrix)

S^{' *} S = I - A .

(7e)

A thus gives a measure of the departure of

S

from unitarity. Of course, for perfect conductors or lossless dielectrics,

A = 0

The usual computational procedure with these equations is to truncate Eq. (6b) at

N \times N

and invert the resulting Q-matrix. The elements of T so obtained can then be checked for constancy versus truncation size and tested against reciprocity and energy balance. There is another constraint, however, that has gone unnoticed. From Eq. (6b)

T = - Q^{- 1} R

, and reciprocity requires that

Q^{- 1} R = R^{'} {(Q^{'})}^{- 1}

. Pre- and postmultiplying by Q,

Q^{'}

, respectively, one gets

Q R^{'} = R Q^{'}

; i.e., the product

Z = Q R^{'}

(8)

must also be symmetric. This requirement is useful in providing at least an indirect check on the accuracy of both Q and R, in addition giving a good indication of the appropriate truncation size to employ (at too small truncation, Z may not be symmetric), all before any matrix inversion is performed.

Once T is obtained, the scattered wave is given by

E_{z}^{sca} (k ρ) = \sum_{m, n = 0}^{N} a_{m} T_{m n} ψ_{n} (k ρ) \sim f (ϑ) {(2 ∕ i π k ρ)}^{1 ∕ 2} \exp (i k ρ), k ρ ⪢ 1,

(9a)

where the far-field amplitude is

f (ϑ) = \sum_{m, n = 0}^{N} a_{m} T_{m n} ϵ_{n}^{1 ∕ 2} i^{- n} \cos n ϑ,

(9b)

and the scattering width σ is given by

σ = \int_{0}^{2 π} d ϑ {∣ f (ϑ) ∣}^{2} .

(9c)

3. LOW-FREQUENCY CASE

We start with the low-frequency regime, where several simplifications occur. Cylinder dimensions are assumed small compared with either the inside or outside wavelength. Defining

δ = ∣ q k ρ_{\max} ∣

, where

ρ_{\max}

is the maximum value of

ρ (ϑ)

, one requires that

δ ⪡ 1

. It will also be convenient, for this section only, to assume there are no magnetic materials present so that

μ = 1

Consider the matrices R and S. Keeping only leading powers of

k ρ

in the radial functions in the integrand for R, Eq. (4a), the result is proportional to

ϕ_{m} (k ρ) \nabla ϕ_{n} (k ρ) - \nabla ϕ_{m} (k ρ) ϕ_{n} (k ρ),

where the

ϕ_{m}

are potential functions regular at the origin. From this expression it appears that R is skew symmetric, but this is illusory. Upon taking the divergence, two terms involving the Laplacian both vanish, and the two remaining terms

\pm \nabla ϕ_{m} (k ρ) ∙ \nabla ϕ_{n} (k ρ)

precisely cancel, giving a vanishing result. Consequently, R is gotten using the next higher-order terms in

k ρ

and turns out to have the form

R_{m n} = (q^{2} - 1) r_{m n} δ^{m + n + 2}, m, n = 0, 1, \dots .

(10a)

Here the

r_{m n}

coefficients are a real, symmetric array, and the complex nature of R is carried by the factor

(q^{2} - 1)

so that, for example, for

q = 1 + i

one has

Im (R) = - 2 Re (R)

The integrand for S involves

ϕ_{m} (k ρ) \nabla ζ_{n} (k ρ) - \nabla ϕ_{m} (k ρ) ζ_{n} (k ρ),

where now the

ϕ_{m}

and

ζ_{n}

are regular and singular potential functions, respectively. The divergence theorem can still be applied as above, but this time one must exclude a circular volume about the origin. The net result of all this is that

S_{m n} = I_{m n} + s_{m n} δ^{m - n + 2}, m, n = 0, 1, \dots

(10b)

where the array

s_{m n}

may be complex. The terms

S_{m 0}

m = 1, 2, \dots

also contain

\ln δ

terms regarded as constants and incorporated in the

s_{m 0}

. It should be noted that this equation takes a slightly different form for elliptic cylinders. In that case all the integrals that would result in inverse powers of δ in Eq. (10b) vanish identically [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

], and one has instead

S_{m n} = I_{m n} + s_{m n} {\begin{matrix} δ^{m - n + 2}, & n ⩽ m \\ 1 (δ) for m + n even (odd), & otherwise \end{matrix}, m, n = 0, 1, \dots .

(10c)

In order to obtain the Rayleigh limit for the transition matrix, first split off the identity by writing

Q_{0} = Q - i I .

(11a)

Now

T = - {(Q_{0} + i I)}^{- 1} R = {(I - i Q_{0})}^{- 1} i R \to (I + i Q_{0}) i R

, using the binomial approximation with

Q_{0}

assumed small in an appropriate sense (even though many elements of

Q_{0}

become arbitrarily large). But

Q_{0} = R + i (S - I)

, so one can write

T \to i [R - (S - I) R] - R^{2} .

(11b)

If there are no losses, all the matrices on the right-hand side of Eq. (11b) are real. With a little effort one sees from Eqs. (10) that the product

(S - I) R

is smaller than R, element by element, by a factor at least of order

δ^{2}

and can thus be neglected to give

T = i R - R^{2}

(12)

in the limit

δ ⪡ 1

, where we have kept the leading terms in both real and imaginary parts separately. With losses present, on the other hand, the term

i R

dominates both real and imaginary parts of Eq. (11b), and Eq. (12) continues to be valid, the

R^{2}

term becoming negligible.

We have computed R with the Bessel functions in place. If desired, however, the limit formula can be gotten by keeping the first two terms in the low-frequency expansion of the Bessels [see Eq. (A4) of Appendix A], discarding any leading-order terms that cancel after integration, and neglecting any higher-order terms in what remains. This alternate form for R should also prevent any minor loss of significance due to the aforementioned cancellations.

Note that (except for circular cylinders for which R is diagonal)

{(R^{2})}_{m n} \to R_{m 0} R_{0 n}

, so only scalar multiplication is required in Eq. (12). R has just been seen to be symmetric in this limit, so reciprocity is built in. With no losses present R is real, the imaginary part of T is dominant, and from the energy constraint Eq. (7c) one has

- Re T = {(Im T)}^{2}

, which is satisfied by inspection. Extensive numerical computations show that energy balance is also satisfied for absorbing cylinders.

This result is remarkable in its simplicity. Notice that even for circular cylinders, standard separation of variables leads to the limiting values of the quotient

- R_{m m} ∕ Q_{m m}

(with both matrices diagonal), involving both Bessel and Hankel functions, while Eq. (12) gives

i R_{m m} - {(R_{m m})}^{2}

, involving only Bessel functions. In the low-frequency limit, the T-matrix also applies to boundary-value problems of potential theory [18

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979). [CrossRef]

], and Eq. (12) (actually just the term iR) should describe the corresponding dielectric cylinder in an applied E field.

Substituting Eq. (12) in the outgoing wave expansion Eq. (9a), the term

i R

gives a series involving products of regular and outgoing wave functions. Assuming the summation can be taken inside the integral sign and differentiations, the sum can be collapsed to give an integral involving the Hankel function

H_{0} (k ∣ ρ - ρ (ϑ) ∣)

, provided the field point ρ is further from the origin than the largest radial distance to the object’s surface [19

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.

]. Similar comments apply for the

R^{2}

term. This would seem to constitute a proof that, at least in the Rayleigh limit, the outgoing wave expansion converges everywhere outside the smallest circular cylinder circumscribing the object (but not necessarily in the annular region inside, see Fig. 1) as one would expect on physical grounds. Further study is called for here.

The derivation of Eq. (12) can be carried a step further to obtain a numerical prescription equivalent to the full Rayleigh expansion in powers of δ. Returning to the expression

T = {(I - i Q_{0})}^{- 1} i R

, binomial expansion gives

T = (I + \sum_{p = 1}^{\infty} i^{p} Q_{0}^{p}) i R .

(13)

Keeping only one term in the summation, this equation reduces to Eq. (12) above, including the term neglected therein, in the process roughly doubling the number of significant figures. As δ is increased, gradually more terms must be included. In our experience, at most

2 N

terms will suffice, where N is the matrix truncation size being employed, if one is not too close to the radius of convergence.

The series will converge as long as the spectral radius of

Q_{0}

, given by the magnitude of the largest eigenvalue, is less than unity [20

A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.

]. Results to date suggest that this range covers the entire Rayleigh region and somewhat beyond, extending to perhaps

δ = 1.5

. The number of terms required for a specified accuracy of course increases as one approaches the upper end of the range. An example will illustrate this. For a 2:1 rectangular cylinder with

q = 1 + i

(relatively high loss), one of the shapes discussed in some detail in the next section, Table 1 shows that the series of Eq. (13) barely converges at

δ = 1.42 \dots

and diverges for δ slightly larger [exact sizes used were

k ρ (0) = 0.45

and 0.46]. The spectral radius crosses through unity within this narrow window exactly as expected.

The above results for general values of dielectric constant and conductivity hinge on the normalizing constants

q^{- n}

of Eq. (2), giving

Im Q

proportional to the identity plus terms that can be neglected while at the same time making R symmetric in the limit. So far we have not been able to meet these conditions for magnetic materials, however, and further study is required in that case. Incidentally, the factor

2 {(1 + μ)}^{- 1}

included in Eq. (2) serves to make

Im Q_{mm} \to 1

for

m ⪢ 1

, probably a worthwhile normalization in any case. This is in fact enough in some cases to enable satisfaction of Eq. (13) if many terms are included, but this only indicates that the binomial series is mathematically valid for those cases, not that it carries the physical significance of the Rayleigh expansion. The methods of this section may not work for perfectly conducting scatterers, either. Although R is again symmetric in the limit for those cases [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

], the requirement that diagonal elements of

Im Q

go to unity fails to be met.

4. HIGH-ASPECT-RATIO CYLINDERS

Problems of significance loss might be anticipated in Q for the elliptic cylinder because of the cancellations noted in Eq. (10c). It turns out there is a class of surfaces, including the elliptic case, for which the integrands in question contain large, oscillatory terms that integrate to zero. In order to avoid massive loss of significance it is essential to modify or remove these terms, following the lead of the NRL group [15

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, “ T-matrix implementation of forward scattering from rigid structures,” J. Acoust. Soc. Am. 94, 3448–3453 (1993). [CrossRef]

]. We consider penetrable bodies, which are more difficult to treat, but the underlying ideas are the same.

Let the cylinder shape be defined by one or more bounding surfaces. For the method to be applicable, at least one of the surfaces must be describable by the quadric equation

a x^{2} + b y^{2} + c x y + d x + e y + f = 0

(14a)

with real coefficients. This is the general equation for a conic section, allowing for coordinate rotations and translations. Changing over to polar coordinates by writing

x = ρ (ϑ) \cos ϑ

y = ρ (ϑ) \sin ϑ

, the preceding formula becomes a quadratic equation in

ρ (ϑ)

(a \cos^{2} ϑ + b \sin^{2} ϑ + c \cos ϑ \sin ϑ) ρ^{2} (ϑ) + (d \cos ϑ + e \sin ϑ) ρ (ϑ) + f = 0 .

(14b)

Of the solutions to this equation, one open surface is allowed, the infinite plane (angular range of integration π). Two parallel planes and elliptic and parabolic cylinders are also included (angular range

2 π

) but not hyperbolic cylinders (which are open). In all cases

1 ∕ ρ (ϑ)

must be finite and continuous everywhere, including the endpoints. We call these complete quadric (cq) surfaces.

Now for the problem at hand consider Eq. (4a), rewritten to give S. Derivatives of the radial functions will enter in when the gradients are taken, but these may be reexpressed in terms of standard relations involving contiguous radial functions so that every term in the integrand will involve a product of the form

J_{u} (k^{'} ρ) N_{v} (k ρ)

. Whenever

v > u

, both functions in the product are expanded and the results consolidated into an ascending power series in

k ρ (ϑ)

. The matrix S is now split into the sum

S^{+} + S^{-}

, where

S^{+}

comes from the positive powers and constant terms in this series and

S^{-}

is due to the inverse powers [any

\ln k ρ (ϑ)

terms are regarded as constants for this purpose]. The only nonzero elements of

S^{-}

will fall above the main diagonal, and by the same token elements of S and

S^{+}

are identical on and below the main diagonal. Details are given in the Appendix.

Whenever the surface passes close to the origin (and this is guaranteed to happen for high aspect ratios with the origin restricted to the interior)

k ρ (ϑ)

will be small, the integrand of

S^{-}

large and rapidly oscillating, and cancellations may occur in the region during integration. In fact, based on extensive numerical computation, we make the following ansatz:

Upon integration over a cq surface,

S^{-} = 0 .

(15)

For these cases (plane, ellipse, parabola), total cancellation of the inverse powers occurs upon integration. Such cancellations have been known for some time for the ellipse, both for the penetrable case, as noted in connection with Eq. (10c), and the perfect conductor, and arise due to orthogonalities of the angular functions [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

]. Presumably this is the mechanism for the other surfaces, too, although in some cases this may not be obvious. For example, for a circular cylinder

ρ (ϑ) = x_{0} \cos ϑ + {(1 - x_{0}^{2} \sin^{2} ϑ)}^{1 ∕ 2},

(16a)

with the coordinate origin translated across a diameter for

- 1 < x_{0} < 1

, one of the simpler integrals is

I (x_{0}) = \int_{0}^{π} d ϑ \cos 3 ϑ ∕ ρ (ϑ) (Dirichlet case) .

(16b)

Using Mathcad 11 in double precision

(\sim 15 sf)

gives results of order

10^{- 15}

, from which one concludes that (i) the integral does indeed vanish, and (ii) Mathcad can handle numerical integration very accurately. [An analytical proof of the result might run as follows: inversion of a translated circle in the unit circle gives another translated circle. Thus it suffices to substitute the right-hand side of Eq. (16a) in the numerator rather than the denominator of the integrand in Eq. (16b). At that point, the binomial expansion and standard orthogonality relations can be employed.]

We will return to the translated circular cylinder shortly.

Although there is significance loss associated with

S^{-}

, the problem is resolved if it, or at least that portion of it for which the surface of integration is near the origin, is simply discarded. This is best illustrated by example. The simplest is the elliptic cylinder, defined within a scale factor by (α is aspect ratio)

ρ (ϑ) = {(\cos^{2} ϑ + α^{- 2} \sin^{2} ϑ)}^{- 1 ∕ 2},

(17a)

as shown in Fig. 1. Observe that the second symmetry plane enables one to compute R (and S) more efficiently, taking

R_{m n}

to be four times the integral over (0,

π ∕ 2

) for

(m + n)

even, zero otherwise. Because of this pattern of zeros it follows that there is no coupling between matrix elements, with m and n both even, and the odd–odd terms, which may be treated separately. The convergence limit for the Rayleigh expansion given in Table 1, for example, referred to the even–even case, while the odd–odd case was still convergent up as far as

δ = 2.5

, requiring only four or five terms in the sum to get several significant figures.

Because the integration involving Eq. (17a) is over a cq surface, one can set

Q = R + i S^{+}

(17b)

and proceed normally thereafter. At this juncture, a simple check of both the ansatz and the computer program is available for aspect ratios that are not too large (e.g.,

α = 2

) by numerically comparing

S^{+}

and S, which should be identical except for possible significance loss above the diagonal in the latter. With regard to actual significance loss, for a 10:1 nonmagnetic cylinder with

q = 1 + i

δ = 2^{1 ∕ 2}

(low-resonance region), plotting the log of the magnitudes of the integrands of

S_{08}^{+}

and

S_{08}^{-}

, for example, gives the results shown in Fig. 2 . In both cases one can identify a dominant term containing the factor

\cos 9 ϑ

, and visible weak minima are in fact situated at the zeros of that function (real and imaginary parts must be plotted individually in order to display zero crossings and amplitude oscillations in full). Because the integrand of

S_{08}^{-}

is larger by up to 12 orders of magnitude and yet makes no contribution, it follows that we can prevent the loss of about 12 significant figures in

S_{08}

(and losses in other elements as well) by simply omitting

S^{-}

, as done in Eq. (17b). At higher aspect ratios or larger truncation size N, the effect is even more pronounced.

Equation (17b) also applies to a translated or rotated elliptical cylinder, allowing the origin to be set anywhere within the cylinder (or on the surface, approached from the inside). The effect is roughly unchanged for magnetic materials. Taking

μ = 100

and reducing the cylinder size by a factor of 10 to preserve the value of δ, the maximum ratio of the two curves of Fig. 2 remains about the same.

Next, consider the rectangular cylinder defined (in the first quadrant) by

ρ (ϑ) = {\begin{cases} ρ_{1} (ϑ), & 0 ⩽ ϑ ⩽ \tan^{- 1} α \\ ρ_{2} (ϑ), & \tan^{- 1} α ⩽ ϑ ⩽ π ∕ 2 \end{cases},

(18a)

with bounding planes

ρ_{1} (ϑ) = 1 ∕ \cos ϑ

and

ρ_{2} (ϑ) = α ∕ \sin ϑ

as shown in Fig. 3 , and for simplicity assume the aspect ratio

α > 1

. It’s convenient to do the integrations separately over

ρ_{1}

and

ρ_{2}

, writing

R = R_{1} + R_{2}

and

S = S_{1} + S_{2}

to avoid having discontinuities in the integrands.

The problem in this case lies with

S_{1}

as it passes near the origin. Write the integral as

S_{1} (pp)

to indicate integration over the physical part of

ρ_{1}

[solid line in Fig. 3 as distinguished from the nonphysical part (npp) given by the dashed line]. Now

S_{1} (pp) = S_{1}^{+} (pp) + S_{1}^{-} (pp)

. But by ansatz

S_{1}^{-} (pp) + S_{1}^{-} (npp) = 0

, and subtracting gives

S_{1} (pp) = S_{1}^{+} (pp) - S_{1}^{-} (npp) .

Note that these last integrals select only positive powers of

k ρ_{1}

from the radial functions near the origin and only negative powers as

k ρ_{1} \to \infty

. Troublesome terms are removed, and the Q matrix itself becomes (for simplicity the common factor of 4 is not shown)

Q = R_{1} (pp) + R_{2} (pp) + i [S_{1}^{+} (pp) - S_{1}^{-} (npp) + S_{2} (pp)] .

(18b)

Notice that we did not bother to split up

S_{2}

, which does not pass near the origin in the example shown. If the shape were elongated horizontally instead, the roles of

S_{1}

and

S_{2}

would be interchanged in the above equations. Equation (18a) can also be used for the square cylinder, incidentally, setting

α = 1

. In that event Eq. (18b) is not needed, and Q can be used in its original form. In addition, because of four-fold rotational symmetry one should compute only those even–even elements for which the sum or difference of indices is a multiple of 4, the rest vanishing (the odd–odd elements are unaffected).

The translated circular cylinder example of Eq. (16a) requires further comment: for this one shape specifically, inverse powers of

k ρ

are unaffected, so

S^{-}

is unchanged, but additional cancellations occur in S above the diagonal, giving (correct to leading order)

S_{m n} = I_{m n} + s_{m n} {\begin{matrix} δ^{m - n + 2}, & n ⩽ m \\ δ^{n - m}, & otherwise \end{matrix}, m, n = 0, 1, \dots .

(19)

For S computed taking this into account we write

S^{+ +}

so that

S = S^{+ +} + S^{-}

(see Appendix A). The new cancellations appear to stem from the observation that inversion of the shifted circle in the unit circle gives another shifted circle, as noted earlier, and such an inversion interchanges the roles of the

∣ u - v ∣

powers preceding the constant term at

u = ν

(the terms omitted in forming

S^{+}

) with those following. Regardless of why they come about, however, just as with the ansatz given earlier they are easily confirmed numerically by noting that the computed

S^{+ +}

S^{+}

, and S must all be identical to within successively increasing significance loss. In exact analogy with Eq. (17b), the Q matrix for the shifted circular cylinder is taken to be

Q = R + i S^{+ +} .

(20)

Now consider two such intersecting cylinders, their intersection forming a biconvex lens as shown in Fig. 4 . Taking

x_{0}

negative, S is gotten by integrating over the physical part of the left-most cylinder from 0 to

π ∕ 2

(again the solid curve) to obtain

S^{+ +}

, then subtracting off

S^{-}

over the remainder of the cylinder to get

Q = R (pp) + i [S^{+ +} (pp) - S^{-} (npp)] .

(21)

Lens thickness (or aspect ratio) is variable over a wide range, and the radii of curvature need not be equal; a plano-convex lens could be considered, for example. For that matter, the curvature need not be constant, but one could employ two intersecting elliptic or parabolic cylinders (in that event reverting back to

S^{+}

S^{-}

). The parabolic lens is sketched in Fig. 5 . If there were no vertical symmetry plane Eq. (21) would become slightly more involved, as it would then be necessary to split the S integrals on both cylindrical surfaces. In several cases, incidentally, we have found that

S^{+}

and

S^{+ +}

appear to work equally well.

The ansatz of Eq. (15) applies equally well to perfectly conducting cylinders. Because only free-space wave functions are required, the resolution into

S^{\pm}

can be drastically simplified as described in the Appendix.

These examples illustrate the procedure for modifying Q using

S^{\pm}

. To our knowledge there are only three cq surfaces, but clearly a great many shapes can be constructed using them. In some of these cases at least, results have been obtained without difficulty for very high aspect ratios. For example, for both elliptical and rectangular cylinders, even though the condition number of Q increases with α, there is apparently no problem in obtaining the T-matrix for aspect ratios up to 1000:1. Keep in mind also that the method may in fact prove useful in conserving significance down to relatively small aspect ratios in some cases, e.g.,

α = 2

or 3.

There’s a downside to all this, incidentally. The bulk of the computation time lies in the numerical integrations for Q, and resolving S into

S^{\pm}

can increase that time significantly. It may be possible to improve on the present computation, however. One technique for doing this is noted in Appendix A.

5. NUMERICAL RESULTS

Rather than looking at scattering and absorption widths, we can better confirm the workings of the conservation laws and low-frequency formulas discussed above by showing results for matrix elements directly. Throughout this section we consider for simplicity only nonmagnetic materials,

μ = 1

As a first example consider a 10:1 absorbing elliptic cylinder in the low-resonance region,

q = 1 + i

δ = 2^{1 ∕ 2}

, with truncation at

N = 8

. Figure 6 shows a bar plot of magnitudes of the T-matrix elements on a logarithmic scale ranging over 10 decades from

0 to - 100 dB

, amplitudes thus running from unity (at the uppermost tick mark) down to

10^{- 10}

at the base plane. Columns and rows are laid out parallel to the x and y axes, respectively, with

T_{00}

at the origin. Some elements in the last few rows and columns do not appear because they fall below the arbitrarily chosen cutoff. As already discussed, due to symmetry entries vanish when the sum of the indices is odd so that the even–even and odd–odd arrays are uncoupled. Symmetry of T itself is also evident. Rather than filling in smoothly, the odd–odd elements are seen to be smaller, relatively speaking, showing that the field itself becomes more nearly symmetric across the vertical plane as the ellipse becomes thinner. An extreme case of this will be shown below.

The rapid falloff in magnitude going away from the origin ensures that this truncation is sufficient to obtain the far-field amplitudes and scattering widths to six or more significant figures using Eqs. (9). This falloff must also more than offset the growth of the Hankel functions as one moves into the near field if Eq. (9a) is to converge in the vicinity of the cylinder as discussed earlier in the Rayleigh case.

Figure 7 looks at truncation error by plotting the log of the absolute value of the inverse fractional difference of the preceding results with those obtained at

N = 10

(thereby increasing from a

4 \times 4

5 \times 5

system of even–even terms and similarly for the odd–odd). The resulting vertical scale ranging from 0 to 8 is basically the number of significant figures of agreement of the two truncations, here showing a largest entry of 6.4 (indicating agreement within one part in

10^{6.4}

). It’s interesting that columns are effectively identical, probably reflecting the fact that the matrix equation for T can alternately be thought of as N sets of equations and unknowns for each of the columns separately, but with the same coefficient array.

In Fig. 8 the same significance scale is used to give a higher resolution look at the symmetry and hence reciprocity associated with elements of Fig. 6. One sees about five-figure agreement for the 0,2 and 2,0 elements, less thereafter.

Figure 9 looks at the significance with which energy requirements are met, comparing the left- and right-hand sides of Eq. (7c). This plot also serves to confirm the role of the absorption matrix, incidentally. Summarizing at this point, the results of these four plots appear to be self consistent and support the earlier statement regarding far-field accuracy.

As an exercise, the same plots were also made using the (proposed symmetric) Z matrix of Eq. (8) in place of T. Interestingly enough, with the exception of Fig. 9 dealing with energy balance, although actual numerical values were of course different, in the highly compressed scale being used, all three remaining plots were almost identical with those shown here. If this agreement holds up generally, then Z may well serve as a useful guide to appropriate values of input parameters in a computation of T.

In Fig. 10 T-matrix results are given for a 1000:1 elliptic cylinder, with all other parameters the same as in Fig. 6 (the uppermost tick mark again indicates unit amplitude). Note that no increase in truncation size was necessary. The odd–odd coefficients have now vanished completely, indicating that both the scattered and internal fields are highly symmetrical across the vertical mirror symmetry plane of the ellipse. (All missing elements in the figure are still present in the computation and can be seen if the scale is extended down another ten orders of magnitude.) In comparison with Fig. 6 for the 10:1 case, each of the elements here is smaller by a factor of about 100, exactly the reduction in the amount of material present. Discarding

S^{-}

prevented the loss of up to 28 significant figures in this case, incidentally.

Figure 11 checks the Rayleigh limit formula by comparing results of Eq. (12) with T as computed by matrix inversion. The scatterer here is a 10:1 rectangular cylinder with

q = 1 + i

δ = 0.014

, and again

N = 8

, and agreement is seen to range from four to six significant figures. The odd–odd elements appear to agree somewhat better, probably because the associated spectral radius is smaller. Just as occurred for the elliptic cylinder of Fig. 6, odd–odd elements of the T-matrix (which is not shown) are smaller than the even–even elements.

Figure 12 makes a comparison for the same rectangular cylinder at a higher frequency so that

δ = 2^{- 1 ∕ 2}

. This is no longer the Rayleigh limit, and this time T is compared with its value from the Rayleigh expansion Eq. (13), using four terms of the series. Agreement ranges from five to better than ten significant figures, and the distinction between even–even and odd–odd terms is now more clear-cut. Using more terms in the series improves the agreement, by the way, although the significance distinction is still present. One concludes that the differences seen here between T (from matrix inversion) and the Rayleigh expansion are due solely to truncation of the infinite series of Eq. (13). The uniformity of the significance is perhaps an artifact of truncation of a matrix binomial series, although to our knowledge there is nothing in the literature on this point.

6. DISCUSSION

The main results of this work are Eqs. (12, 13), the Rayleigh limit and expansion applicable at low frequencies, and the ansatz of Eq. (15), which becomes more useful the higher the aspect ratio of the scattering object. Only a few examples were given, but it should be noted that a number of other cases were considered, all giving results fully consistent with those reported here. These included magnetic materials, except in connection with Eqs. (12, 13), as well as several cases using the odd functions, obtained by replacing the cosines by sines in Eqs. (1, 2).

Although only cylindrical objects were considered, we expect most of these procedures to have their counterparts for acoustic and electromagnetic scattering in three dimensions. Indeed, extensive cancellations are known to occur when evaluating the Q matrix for both surface scattering (Dirichlet or Neumann conditions in the acoustic case or perfect conductivity in the electromagnetic case) and volume scattering by spheroids or ellipsoids [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

, 17

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 4, 825–839 (1971). [CrossRef] [CrossRef]

]. At this writing, one might anticipate the cq surfaces in three dimensions to be planes, ellipsoids, and paraboloids, along with truncated versions of the elliptic and parabolic cylinders considered above. Notice that the cylindrical surfaces are then being asked to play a new role, one in which the Q matrix involves surface integrals of the spherical wave functions as, for example, the finite circular cylinder [15

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, “ T-matrix implementation of forward scattering from rigid structures,” J. Acoust. Soc. Am. 94, 3448–3453 (1993). [CrossRef]

]. It should also be noted that splitting the Bessel function products should be somewhat simpler in three dimensions, because there are no logarithmic terms to deal with.

Appendices

Appendix A: Evaluation of $S^{\pm}$

Begin with Eq. (4a) using Bessel and Neumann functions

J_{m} (q k ρ)

and

N_{n} (k ρ)

and note that [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

]

d σ ∙ \nabla = d ϑ [k ρ \partial ∕ \partial k ρ - (1 ∕ k ρ) k ρ^{'} (ϑ) \partial ∕ \partial ϑ] .

For derivatives of the Bessel functions we use the standard relations [21

F. W. J. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions , M. Abramowitz and I. A. Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355–433.

]

J_{m}^{'} (z) = - J_{m + 1} (z) + (m ∕ z) J_{m} (z),

N_{n}^{'} (z) = N_{n - 1} (z) - (n ∕ z) N_{n} (z) .

(A1)

Substituting in Eq. (4a), elements of S take the form

S_{m n} = d_{m n} \int_{0}^{2 π} d ϑ {μ k ρ J_{m} (q k ρ) N_{n - 1} (k ρ) \cos m ϑ \cos n ϑ + μ n J_{m} (q k ρ) N_{n} (k ρ) \cos m ϑ [(1 ∕ ρ) ρ^{'} (ϑ) \sin n ϑ - \cos n ϑ] + q k ρ J_{m + 1} (q k ρ) N_{n} (k ρ) \cos m ϑ \cos n ϑ - m J_{m} (q k ρ) N_{n} (k ρ) [\cos m ϑ + (1 ∕ ρ) ρ^{'} (ϑ) \sin m ϑ] \cos n ϑ},

(A2)

with normalizing constants given by

d_{m n} = {(ϵ_{m} ϵ_{n})}^{1 ∕ 2} 2 {(1 + μ)}^{- 1} q^{- m} .

(A3)

If the object has two planes of mirror symmetry, then a parity function

par (m, n) = 1

for

m + n

even, zero otherwise, should also be included in the

d_{m n}

. To obtain the regular matrix elements

R_{m n}

, replace

N_{n} (k ρ)

J_{n} (k ρ)

in Eq. (A2).

Now the ascending series for the radial functions are [21

F. W. J. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions , M. Abramowitz and I. A. Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355–433.

]

J_{m} (z) = \sum_{k = 0}^{\infty} {(- 1)}^{k} {(z ∕ 2)}^{2 k + m} ∕ k! (m + k)!,

N_{n} (z) = - (1 ∕ π) \sum_{k = 0}^{n - 1} (n - k - 1)! {(z ∕ 2)}^{2 k - n} ∕ k! + (2 ∕ π) \ln (z ∕ 2) J_{n} (z) - (1 ∕ π) \sum_{k = 0}^{\infty} {(- 1)}^{k} [ψ (k + 1) + ψ (n + k + 1)] {(z ∕ 2)}^{2 k + n} ∕ k! (n + k)!,

(A4)

in terms of the ψ function

ψ (n) = {\begin{cases} - γ for n = 1 \\ ψ (n - 1) + 1 ∕ (n - 1) & otherwise, \end{cases}

(A5)

where

n ⩾ 1

in the second of Eqs. (A4) and

γ = 0.57721 \dots

is Euler’s constant. Writing

J_{m} (q z) N_{n} (z) = J N sum (m, n, q, z)

, one now has

J N sum (m, n, q, z) = - (1 ∕ π) \sum_{k = 0}^{K} \sum_{s = 0}^{n - 1} a (m, k) b (n, s) q^{2 k + m} {(z ∕ 2)}^{2 k + 2 s + m - n} + (1 ∕ π) J_{m} (q z) {2 \ln (z ∕ 2) J_{n} (z) - \sum_{k = 0}^{K} [ψ (k + 1) + ψ (n + k + 1)] a (n, k) {(z ∕ 2)}^{2 k + n}} .

(A6)

Here we have put the right-hand side in a form better suited for computation by defining

a (m, k) = {\begin{cases} 1 ∕ m! for k = 0 \\ - a (m, k - 1) ∕ k (m + k) otherwise \end{cases},

(A7)

b (n, s) = {\begin{cases} (n - 1)! for s = 0 \\ b (n, s - 1) ∕ s (n - s) otherwise \end{cases},

(A8)

and the infinite series have been truncated at K, which can conservatively be set at

2 N

, with N the matrix truncation size used in the text.

By inspection, inverse powers of z are seen to occur only in the double sum in Eq. (A6), and provided

n > m

. Positive and negative powers can be sorted out using the step function

h (x) = 1

for

x ⩾ 0

, zero otherwise. Thus, inserting a factor

h (2 k + 2 s + m - n + 1 ∕ 2)

within the double sum, with Eq. (A6) otherwise unchanged, gives the sum of positive powers (including a constant term if

m - n

is even); we call this quantity

J N {sum}^{+} (m, n, q, z)

. By the same token, the negative powers are gotten by making a sign change in the argument of the step function:

J N {sum}^{-} (m, n, q, z) = - (1 ∕ π) \sum_{k = 0}^{n ∕ 2} \sum_{s = 0}^{n ∕ 2} h [- (2 k + 2 s + m - n + 1 ∕ 2)] \times a (m, k) b (n, s) q^{2 k + m} {(z ∕ 2)}^{2 k + 2 s + m - n} .

(A9)

Notice that the upper limits have been reduced here to remove some vanishing terms and are understood to be interpreted as the integer part of

n ∕ 2

. The factor

1 ∕ 2

in the argument of the step function prevents inclusion of erroneous terms due to

h (- 0)

Now define the function

J N^{\pm} (m, n, q, z) = {\begin{matrix} J N {sum}^{\pm} (m, n, q, z) if n > m \\ J_{m} (q z) N_{n} (z) otherwise \end{matrix} .

(A10)

The quantities

S^{\pm}

are gotten by inserting

J N^{\pm}

respectively in place of the product of Bessel functions wherever they occur in the integrand of Eq. (A2). At this point one can note that both of Eqs. (A1) are valid for both the Bessel and Neumann functions. Using the equations as shown, however, will prevent any inverse powers from appearing in the products on the main diagonal of S.

In order to obtain

S^{+ +}

for the translated circular cylinder, it is only necessary to modify the argument of the step function in the equation for

J N {sum}^{+}

to read

h (k + s + m - n + 5 ∕ 2)

. The first nonvanishing power of δ will occur when the argument equals

1 ∕ 2

, i.e., when

k + s = n - m - 2

. The power in question,

k + s

, thus increases linearly as one moves further above the diagonal, and the remaining factor of 2 is presumably due to the indices of the Bessel functions and the S matrix not being entirely in register [see Eq. (A2)].

The above equations can also be used for the perfectly conducting case, but there is a simpler alternative. Using techniques described elsewhere [2

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969). [CrossRef]

, 22

G. N. Watson, Theory of Bessel Functions , 2nd ed. (Cambridge U. Press, 1962), pp. 145–150.

], for

n > m

one can obtain the identity

J_{m} (z) N_{n} (z) = J_{n} (z) N_{m} (z) - (1 ∕ π) \sum_{s = 0}^{(n - m - 1) ∕ 2} c_{m n s} {(2 ∕ z)}^{n - m - 2 s},

(A11a)

having coefficients

c_{m n s} = {(m - n + s + 1)}_{s} (n - s - 1)! ∕ s! (m + s)! .

(A11b)

Here

{(a)}_{0} = 1

{(a)}_{s} = (a + s - 1) {(a)}_{s - 1}

otherwise, is Pochhammer’s symbol. These equations can be verified by induction on

n = m + 1

m + 2, \dots

(notice that the case

n = m + 1

corresponds to the Wronskian relation for the radial functions). The summation is seen to contain all the inverse powers, so we can replace the quantities

J N {sum}^{\pm}

appearing in Eq. (A10) by the two terms on the right-hand side of Eq. (A11a), respectively, giving the simpler formulas. Unpublished numerical computations show that the T-matrix for perfectly conducting cylinders with the shapes discussed in Section 4 also satisfies the three consistency conditions, although with edges present it is necessary to truncate at higher numbers of equations and unknowns in order to obtain accuracy comparable with the penetrable cases, especially at higher aspect ratios. Alternately, perfectly conducting bodies with edges can be treated by incorporating edge conditions in the surface fields [16

P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999). [CrossRef]

Incidentally, it is fairly easy to see from the above equation that S will be symmetric for elliptic cylinders. Turning this argument around, if it can be verified that Eq. (A11a) holds also for spherical Bessel functions [although Eq. (A11b) may look somewhat different], then, because of the well-known symmetry of S for surface scattering (Dirichlet and Neumann bc) by ellipsoids in three dimensions, the ansatz

S^{-} = 0

will presumably hold for those cases also.

Equation (A11a) also provides some interesting insight into the structure of S. Because the equation holds for general surface shapes, it follows that

S^{+}

(and hence

Q^{+} = R + i S^{+}

) is always symmetric for those cases. Now notice the gap between

S_{m n}^{+}

and

S_{m n}^{-}

, i.e., the successively larger number of (positive) powers of z missing as

n - m

increases. If magnitude can be associated with power of z, as occurred with the high-aspect-ratio cases, then again one would expect large magnitude disparities, which would be lost if matrix elements of

S^{+}

and

S^{-}

were ever added together. In short, rather than invert

Q = Q^{+} + i S^{-}

directly, it might be preferable to invert

Q^{+}

and then employ the formal identity

Q^{- 1} = {[I + i {(Q^{+})}^{- 1} S^{-}]}^{- 1} {(Q^{+})}^{- 1} .

(A12)

Note also that

S^{-} = 0

for the perfectly conducting elliptic cylinder, suggesting that the binomial series obtained from Eq. (A12) might be appropriate for cylinder shapes perturbed from the elliptical. We leave this as a question for further study.

As mentioned in the text, the splitting of

S^{\pm}

can be extremely time consuming for the penetrable cases; many repetitions of the above equations are required to carry out the numerical integrations to high accuracy. The second and fourth lines of Eq. (A2) involve the same product of Bessels and can be combined, giving a 25% saving. More significant improvement can probably be realized, however, if for each matrix element one were to first collect and either analytically or numerically evaluate the coefficient of each power of z in the double summations of Eqs. (A6, A9), and probably also the single summation of Eq. (A6), before starting the integration. This would be done for m, n, q, and K all fixed, separately for

S^{+}

and

S^{-}

, and should eliminate a great deal of repetition.

Larger objects of course require that more equations and unknowns be kept. It should be kept in mind, however, that the Neumann functions do not begin to grow in magnitude until their index exceeds their argument. Thus the splitting operation need not begin until the second index of S is greater than

k ρ_{\min}

ACKNOWLEDGMENT

The author is indebted to Dennis Roser and Science Applications International Corp. and Jeff Hale and the U. S. Army Edgewood Chemical and Biological Center, Aberdeen Proving Ground, Maryland, for their encouragement and sponsorship.

References and links

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9.	M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “ T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996). [CrossRef]
10.	M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.
11.	V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Comments on recent criticism of the T-matrix method,” J. Acoust. Soc. Am. 84, 2280–2284 (1988). [CrossRef]
12.	M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, “ T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database,” J. Quant. Spectrosc. Radiat. Transf. 88, 357–406 (2004).
13.	M. I. Mishchenko and L. D. Travis, “ T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994). [CrossRef]
14.	S. Havemann and A. J. Baran, “Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 89, 87–96 (2004). [CrossRef]
15.	A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, “ T-matrix implementation of forward scattering from rigid structures,” J. Acoust. Soc. Am. 94, 3448–3453 (1993). [CrossRef]
16.	P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999). [CrossRef]
17.	P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 4, 825–839 (1971). [CrossRef] [CrossRef]
18.	P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979). [CrossRef]
19.	P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.
20.	A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.
21.	F. W. J. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions , M. Abramowitz and I. A. Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355–433.
22.	G. N. Watson, Theory of Bessel Functions , 2nd ed. (Cambridge U. Press, 1962), pp. 145–150.

Table 1 Convergence Limit of the Rayleigh Expansion [Eq. (12)] for a 2:1 Rectangular Cylinder^{^a}

Size Parameter δ	Spectral Radius of $Q_{0}$	Terms Summed	Convergence
1.42	0.997	4000	Yes
1.45	1.03	—	No

μ = 1

q = 1 + i

, matrix truncation size

N = 10

Fig. 1 Geometry of the penetrable cylinder.

Fig. 2 Disparity in magnitude of the integrands of

S_{08}^{+}

and

S_{08}^{-}

is shown versus ϑ for a 10:1 elliptic cylinder (see text). Real and imaginary parts of these quantities undergo sign changes near the minima seen here.

Fig. 3 Geometry of the rectangular cylinder.

Fig. 4 Double-convex lens, formed by the intersection of two circular cylinders.

Fig. 5 Parabolic lens, formed by the intersection of two parabolic cylinders.

Fig. 6 Bar plot showing the magnitudes of T-matrix elements on a logarithmic scale covering ten decades from unit amplitude (uppermost tick mark) down to the base plane (10:1 elliptic cylinder,

q = 1 + i

δ = 2^{1 ∕ 2}

N = 8

Fig. 7 Plot of estimated number of significant figures remaining after truncation error (term-by-term comparison of elements for

N = 8

, 10).

Fig. 8 Significant figures of agreement for corresponding elements above and below the main diagonal in Fig. 6, testing reciprocity.

Fig. 9 Significant figures of agreement between the left- and right-hand sides of Eq. (7c), checking energy balance for the example of Fig. 6.

Fig. 10 Magnitudes of the T-matrix elements are shown for a 1000:1 elliptic cylinder. Scale is identical with that of Fig. 6 (

q = 1 + i

δ = 2^{1 ∕ 2}

N = 8

Fig. 11 Comparison of Eq. (12) with T using matrix inversion for a 10:1 rectangular cylinder shows four to six significant figure agreement throughout, checking the Rayleigh limit formula (

q = 1 + i

δ = 0.014

N = 8

Fig. 12 Equation (13), using four terms in the summation, is compared with T (matrix inversion) for the cylinder of Fig. 11 at a higher frequency

(δ = 2^{- 1 ∕ 2})

, showing five to more than ten significant figure agreement.

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(080.3630) Geometric optics : Lenses
(290.3770) Scattering : Long-wave scattering
(290.4020) Scattering : Mie theory
(290.5870) Scattering : Scattering, Rayleigh

ToC Category:
Scattering

History
Original Manuscript: March 22, 2007
Manuscript Accepted: April 5, 2007
Published: July 11, 2007

Citation
P. C. Waterman, "The T-matrix revisited," J. Opt. Soc. Am. A 24, 2257-2267 (2007)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-8-2257

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References

V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).
P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969). [CrossRef]
R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979). [CrossRef]
G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982). [CrossRef]
J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987). [CrossRef]
S. Ström, "T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties," Phys. Rev. D 10, 2685-2690 (1974). [CrossRef]
S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988). [CrossRef]
J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988). [CrossRef]
M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996). [CrossRef]
M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.
V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988). [CrossRef]
M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).
M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994). [CrossRef]
S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004). [CrossRef]
A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993). [CrossRef]
P. C. Waterman, "Surface fields and the T-matrix," J. Opt. Soc. Am. A 16, 2968-2977 (1999). [CrossRef]
P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971). [CrossRef]
P. C. Waterman, "Matrix methods in potential theory and electromagnetic scattering," J. Appl. Phys. 50, 4550-4566 (1979). [CrossRef]
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.
A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.
F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355-433.
G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1962), pp. 145-150.

Babenko, V. A.
Baran, A. J.
Björkberg, J.
Dragonette, L. R.
Feshbach, H.
Gaumond, C. F.
Havemann, S.
Householder, A. S.
Khlebtsov, N. G.
Kristensson, G.
Lacis, A. A.
Lakhtakia, A.
Mackowski, D. W.
Mishchenko, M. I.
Morse, P. M.
Olver, F. W. J.
Pao, Y.-H.
Peden, I. C.
Sarkissian, A.
Schneider, J. B.
Ström, S.
Travis, L. D.
Varadan, V. K.
Varadan, V. V.
Videen, G.
Waterman, P. C.
Watson, G. N.
Weaver, R. L.
Wriedt, T.
Zheng, W.

Can. J. Phys.

J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987). [CrossRef]

IEEE Trans. Antennas Propag.

S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988). [CrossRef]
J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988). [CrossRef]

J. Acoust. Soc. Am.

P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969). [CrossRef]
R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979). [CrossRef]
G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982). [CrossRef]
V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988). [CrossRef]
A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993). [CrossRef]

J. Appl. Phys.

P. C. Waterman, "Matrix methods in potential theory and electromagnetic scattering," J. Appl. Phys. 50, 4550-4566 (1979). [CrossRef]

J. Opt. Soc. Am. A

P. C. Waterman, "Surface fields and the T-matrix," J. Opt. Soc. Am. A 16, 2968-2977 (1999). [CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).
S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004). [CrossRef]
M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996). [CrossRef]

Opt. Commun.

M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994). [CrossRef]

Phys. Rev. D

S. Ström, "T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties," Phys. Rev. D 10, 2685-2690 (1974). [CrossRef]
P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971). [CrossRef]

Other

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.
V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.
A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.
F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355-433.
G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1962), pp. 145-150.

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