On electrostatics of finite planar system of strips applied in surface

acoustic wave interdigital transducers

Yuriy Tasinkevych, Eugene J. Danicki,

Polish Academy of Science, IPPT, Warsaw, Poland

Abstract

Planar systems of strips placed on a piezoelectric body are exploited in interdigital transducers forgeneration, detection and reflection of surface waves. The spatial field distribution in the plane ofstrips exhibits square-root singularities at the strips edges. As the result evaluation of charge spatialspectrum by direct application of the Fourier transformation to the spatial charge distribution neveryields satisfactory accuracy. This is due to the strong dependence of the spectrum on the distributiondetails over the entire system. A new method is proposed here for direct evaluation of the spatialspectrum; the spatial distribution is obtained from it by the inverse Fourier transformation for certainauxiliary purposes and final verification of numerical results. The solution is constructed as a linearcombination of the template functions, evaluated in spectral domain and satisfying the electric boundaryconditions on the strips. The same functions are applied in the solution of the complementary problemof strips embedded in a preexisting electric field, assumed to be spatially variable (harmonic). In thiscase the strip total charge is evaluated dependent on the preexisting (‘incident’) field, as well as theBloch harmonics of the ‘scattered’ field in a wide spectral domain. Particular attention is dedicated tothe numerical problems arising in evaluation of the template functions and the final charge spectrum.The computed examples show typical computational errors and how they were corrected by improvingthe numerical procedures.

1 Introduction

In classical electrostatics, the boundary value problem is formulated for electric field or its potential,governed by the Laplace equation and the boundary conditions on the surface of the conducting body. Thesolution provides the electric field in the space around the body and the electric induction (the electriccharge density) distribution on the body surface [?]. In this paper, the conducting body is actually asystem of in-plane perfectly conducting and infinitesimally thin strips of arbitrary width and spacing (seeexample in Fig. 1). Such systems found important applications in microelectronics, particularly in thesurface acoustic wave (SAW) devices, where they help to redistribute the time-harmonic electric field onthe surface of a piezoelectric substrate. Due to the piezoelectric effect, the field excites the harmonicRayleigh wave on the substrate. The conversion of electric to acoustic signal is most effective if thewave-number of Rayleigh wave falls within the spatial spectrum of the field distribution on the substrate.The temporal spectrum of the wave generated by the electric impulse applied to the strips resembles thespatial spectrum of the electric charge distribution on the strip system [?] (the involved coefficient betweenboth the spectra is the SAW velocity of Rayleigh wave vSAW ). In the analysis of such structures (calledinterdigital transducers, IDTs), the most important is the solution to the charge spatial spectrum on theplane of strips rather than the spatial charge distribution on the strips (this is why we call the problem‘nonstandard’).

Although they are simply interrelated by the Fourier spatial transform, the direct transformation poses aserious numerical problem due to the square-root singularity of the charge distribution on strips. Note that

1

a) b)

Figure 1: a) A broad view, including the 2nd and 3rd overtones, of the spatial spectrum (r - spectralvariable) of electric charge on the system of five strips alone (solid line) and placed beside a conductinghalf-plane (dashed); the inset presents the potential distribution on the strip plane in the latter case. b)The evaluated charge spectrum of dispersive interdigital transducer having 38 strips of different widthsand spacings (inset - the potential distribution). Its complicated form is meaningful.

there can be dozens or even hundreds of strips in the analyzed structures, and any interesting spectral linescorresponding to the SAW wave-number depends strongly on all these singularities. Moreover, wide domainof the spectrum is interesting for applications, even spanned over several transducer overtones (fundamentalharmonics). Naturally, the numerical results can be only obtained, and attempting to perform the Fouriertransformation (practically - using the fast Fourier transform, FFT), one has to sample the singular fieldat discrete values of the spatial variable, what inevitably leads to significant numerical errors (even if thefield is perfectly evaluated [?, ?]).

To overcome this difficulty, a new method is proposed in this paper that evaluates the charge spectrumdirectly, formally without any earlier evaluation of the spatial charge distribution with subsequent appli-cation of the FFT algorithm. The integrals of the field and induction are only necessary for evaluationof the strip potentials and charges in order to formulate the equations resulting from the circuit theory(the Kirchhoff laws), taking into account that some strips have given potentials, others can be isolatedor interconnected. In fact, several template solutions for spatial spectrum are constructed for given stripsystems, and the superposition of them is searched to satisfy the circuit equations.

The paper is organized as follows. The next section contains the preliminary matter introducing thespatial spectrum of electric field on the plane of strips, and the planar harmonic Green function that,establishing certain relation between the field components, replaces the Laplace equation in the consideredboundary-value problem. Section 3 presents details of efficient numerical evaluation of certain ‘generat-ing’ [?] or ‘template functions,’ presenting a series of independent solutions to the charge distributions onthe plane of strips expressed in the spectral domain. The system of equations from the Kirchhoff lawsfor strips is formulated and solved in the subsequent section. Unfortunately, the system is usually badlyconditioned due to several reasons and special attention is necessary to obtain the sought combination ofthe generating functions what makes the final solution of the discussed electrostatic problem for strips.

Section 5 presents a possible approach to the complementary problem of strips embedded in a preexistingelectric field. It is called an ‘electrostatic scattering’ here, because this preexisting field is assumed to be aspatial harmonic function like that in an electromagnetic scattering problem. The solution of this problem

2

is constructed by exploiting the above introduced generating functions, and again it is obtained directly inthe spectral domain, yielding a complete constructive theory of both the cases important in the analysisof SAW devices: for the field generated by the given strip voltages (the case of generating IDTs), and forthe field generated on the strip plane by the surface acoustic wave propagating on a piezoelectric substratesupporting the strips. The induced strip charge describes the SAW detection by strips, and the spectrumof the induced field is responsible for the Bragg scattering of SAW by strips [?].

2 Basic solutions for planar systems

Let a harmonic potential on the plane y = 0, assumed independent of z, be

e−jrxejωt (1)

where the time-dependence is not much important for electrostatics, but will help us to formulate thestandard circuit equations for the strip currents rather than the strip charges; r is an arbitrary spatialspectral variable and ω is an angular frequency of the field (their time-dependence will be neglected infurther notations). On the basis of the Laplace equation for the potential of electric field ~E = −grad(ϕ)

∆ϕ = 0, (2)

it is seen that the y-dependence of the solution for ϕ vanishing with growing distance from the planey = 0 (in other words, satisfying the equivalent ‘radiation condition’ for electromagnetic waves) is e−|ry|.Assuming a vacuum space of dielectric constant ǫo, the following equations are obtained for the field, as afunction of spectral variable, on the plane y = 0+ (just above the plane)

E1(r) = jrϕ(r), E2(r) = rSrϕ(r), (3)

where Sr = 1 for r ≥ 0 and −1 otherwise is adopted (rSr = |r|; r is assumed to be real), and to shortenfurther notations, we applied E = E1 for tangential and D = E2 for normal components of the electricfield vector. Note that the surface charge on y = 0 plane is 2Dy because the induction at y = −0 is −Dy.It is convenient to introduce the planar harmonic Green’s function defined by

G(r) = E(r)/D(r) = jSr (4)

for the field components on the plane y = 0: the tangential and normal field components E and D satisfyingthe ”radiation conditions” (vanishing at |y| → ∞; the other, complementary class of the field satisfyingthe opposite condition, that is growing at infinity, will be considered in the scattering problem later). ThisGreen function will replace the Laplace equation in all the analysis that follows for the fields on the planey = 0 but, in order to simplify further notations, we assume here E = E1 and D = D2/ǫo (or, equivalently,the suitably modified units of normal induction).

It is known [?] that the charged half-plane x < 0 induces the following field on the plane y = 0+:

E(0)(x) =

{

1/√x, x > 0

0, x < 0

}

, E(0)(r) =1

2√

π|r|

{

ejπ/4, r > 0

e−jπ/4, r < 0

}

,

D(0)(x) =

{

0, x > 01/√−x, x < 0

}

, D(0)(r) =1

2√

π|r|

{

e−jπ/4, r > 0

ejπ/4, r < 0

} (5)

in the spatial, and spectral (at the right) representations. It is convenient to introduce the complex fielddefined by

Φ = D − jE, (6)

3

in both the spatial and spectral domains that yields for the half-plane

Φ(0)(x) =1√−x =

{

−j/√

|x| , x > 01/

√

|x| , x < 0

}

,Φ(0)(r) =

{

e−jπ/4/√πr, r > 0,

0, r < 0

}

. (7)

The square-root value is chosen to be positive for x ≥ 0, and√−1 = j otherwise. The above follows

the well-known electrostatic theorem that real and imaginary parts of any harmonic function represent asolution of a certain electrostatic problem. The boundary conditions for the considered problem here areE(x) = 0 on half-plane or strips, and D(x) = 0 outside the strips, appended by the ‘radiation condition’that the field vanishes at |y| → ∞.

This idea is exploited further below. Concerning the spectral representation, it is very important to notethat it has a semi-finite support, the feature of great importance for further numerical analysis. Namely, itresults from Eq. (??) that Φ(r) = D(r) − jE(r) = (1 + Sr)D(r) that is zero for r < 0, for fields satisfyingthe radiation condition, Eq. (??). The definition (??) has been chosen to obtain the convenient supportr ≥ 0. For given Φ(r), the representation of D and E in spectral domain can be inferred from the aboveresults as

D(r) =1

2

{

Φ(r), r ≥ 0Φ∗(−r), r < 0

}

, E(r) = jSrD(r) =j

2

{

Φ(r), r ≥ 0−Φ∗(−r), r < 0

}

, (8)

which can be easily checked by substitution to Eq. (??) and (??). Moreover, functions {D(r−p), E(r−p)}are constituents of another harmonic function [D(x)− jE(x)]e−jpx satisfying Eq. (??), but not necessarilyEq. (??) (this will be exploited later in Sec. 5).

Now consider a strip of width 2a > 0. The spatial field distribution [?] is described by

Φ(1)(x; a) = Φ(0)(x+ a)Φ(0)(a− x) =

{

|a2 − x2|−1/2, |x| < a,

jSx|a2 − x2|−1/2, |x| > a,

Φ(1)(r; a) = [Φ(0)(r)ejra] ∗[

Φ(0)(r)ejra]∗

(9)

(the superscript ∗ means the complex conjugation), respectively in the spatial and spectral domains, asresulting from the known convolution theorem for the Fourier transforms. Taking into account the semi-finite support of Φ(r), the convolution (??) is explicitly (hint: change the variable into y − r/2)

Φ(1)(r; a) =

∫ r

0

ej(r−y)ae−jπ/4

√

π(r − y)

e−jya

√πy

ejπ/4d y = J0(ra), (10)

where J0 is the Bessel function. It immediately results from the above relation that higher harmoniccomponents (over r) of the convolved functions do not contribute to the lower spectrum (at r and below)of the resulting function. This property is advantageous for the numerical analysis of Sec. 3, allowing oneto truncate the convolved functions. Repeating the above, one easily obtains similar expressions for thesystem of two strips:

Φ(2)(r) = [Φ(1)(r; a1)ejrb1 ] ∗ [iΦ(1)(r; a2)ejrb2 ], (11)

where a1,2 and b1,2 are the corresponding widths and displacements of strips having the shifted spatial fieldrepresentation (??). The coefficient i is introduced in the second term to obtain D(x) = Re{Φ(x)} andE(x) = Im{Φ(x)}, conveniently analogous to Eq. (??). For three or more strips (N ≥ 3), we have

Φ(N)(r) = [Φ(N−1)(r)] ∗[

jΦ(1)(r; aN )ejrbN

]

,

Φ(N)(x) ∼ [|(x− x1)(x− x2)...(x− x2N )|]−1/2,(12)

for spectral and spatial representations, respectively, where xi are the strip edges. If there is a conductinghalf-plane, the convolution with Φ(0) in spectral domain is necessary, and the corresponding multiplication

4

by Φ(0)(x), perhaps shifted by x0 in spatial domain; this introduces another term (x − x0) in the aboveformula; x0 = 0 can be applied without any loss of generality by choosing properly the x-axis (Appendix A).

As it can be easily checked, the field discussed above vanishes fast with |x| → ∞, indicating themulti-pole character of the charge distribution on the strips. Thus, it cannot form a basis of completerepresentation of the arbitrary field generated by the system of strips. The system can, for instance,possess a net charge different from zero, inducing the electric field vanishing at infinity like 1/x. Theharmonic function representing such cases and exhibiting square-root singularity at the strip edges is theabove derived function Φ(x) multiplied by a polynomial function of a degree not exceeding N − 1. Hence,N independent template solutions result, all satisfying the boundary and radiation conditions (??)

Φ(N,0)(x) = Φ(N)(x), Φ(N,1)(x) = xΦ(N)(x), · · · ,Φ(N,n)(x) = xnΦ(N)(x), · · · , (13)

which yield the corresponding field components E(n) and D(n) on the plane y = 0. Applying the knownFourier transform theorem stating that multiplication by x in spatial domain corresponds to differentiationin spectral domain, one formally obtains the above functions in the spectral domain

Φ(N,0)(r) = Φ(N)(r), Φ(N,1)(r) = −j dd r

Φ(N)(r), · · · ,

Φ(N,n)(r) = (−j)n dn

d rnΦ(N)(r), n < N,

(14)

yielding the spectral field representations E(n)(r) and D(n)(r), according to Eq. (??) (in Sec. 3, thesefunctions will be redefined for numerical advantage). The superposition of these N independent functionssuffices for representation of an arbitrary field that can be generated by the system of N strips subjectedto N circuit constraints: given strip voltages or charges, or interconnections. A convenient method forevaluation of the above spectral ‘generating’ or ‘template functions’ for practical use is presented in thenext section. Their spatial counterparts can be computed by means of FFT, if necessary.

In practical systems, both the source terminals are connected to other strips in the system, renderingthe system electric neutrality (the currents flowing into and out of the system are in perfect balance asflowing through the same source). This means that the distant field excited by the charge distribution onstrips behaves like the field generated by an electric dipole, at most, vanishing faster than 1/x. Hence, thefield spectral representation must vanish at r = 0. It is evident that the function Φ(N,N−1)(x), having theFourier transform Φ(N,N−1)(r = 0) 6= 0, must be excluded from the set of functions representing the chargedistributions on strips because this and only this function represents a field vanishing like 1/x at infinity.The above is evident from Eq. (12):

D(r) =1

2

{

Φ(N,N−1)(r), r ≥ 0,

Φ(N,N−1)∗(−r), r < 0,

D(x) = F−1{D(r)} ∼ 1/|x| at |x| → ∞(15)

(F−1 means the inverse Fourier transform). The other functions represent field vanishing like x−k, k = 2, ..., N ,and their spectral representations behave like rn, n = 1, ..., N − 1 at r → 0, according to the limit theoremfor the Fourier transforms.

The above shows that at our disposal is a set of N − 1 template functions the superposition of which,with certain coefficients, must satisfy the circuit equations. Let each strip be connected to either of thesource terminals. It is clear that the conditions can be set only on the strip voltage differences, not on theabsolute voltages of strips. This shows that there are N−1 circuit equations in this example, and similarlyfor another cases, proving that there is a complete system of equations for evaluation of all superpositioncoefficients mentioned earlier. To formulate the equations however, we need to evaluate voltages Vi and

5

charges Qi (or, conveniently for application, the currents Ji = jωQi flowing to strips), for fields representedby each ‘generating function’. Details concerning the formulation and solution of the circuit equations arepresented in Sec. 4. It is worth to note here that in applications, we need to know the spatial spectrumof electric charge on strips in finite domain only (0, ru), where ru = ωu/vSAW and ωu is the upper edgeof the interesting frequency band of the considered IDT. The value of ru however, must be considerablyenlarged for the purpose of evaluation of the strips’ charges and potentials by means of the inverse Fouriertransformation (FFT). This evaluation provides good verification of the entire result of quite extensivecomputations involved. But formally, evaluation of Φ(r) can be constrained to rather small domain (0, ru),provided that the strips’ charges and potentials are evaluated by integration of the square-root singularspatial representations of surface electric fields, D(x) and E(x).

Below we consider the field components D(0) and E(0) evaluated from Eq. (??) using Φ(N,0) in place ofΦ: D(0)(r) = Φ(N,0)(r)/2, r ≥ 0 or Φ(N,0)∗(−r)/2, r < 0, yielding results marked by the upper index (0);the analysis for other functions, involving multiplication by xn in Eq. (??) and indexed by (n), is similar.According to the definition of the electric potential E = −dϕ/d x, the strip voltages can be obtained byintegration of E(0)(x), what in spectral domain, corresponds to the division by r of the E(0) (with accuracyto an unimportant constant due to the voltage difference being only involved in the circuit equations).As concerns the strip charge, the function Q(0) is first defined being the integral of ǫoD(0)(x). Using theGreen’s function (??), one obtains:

V (0)(x) = F−1{−jE(r)/r} =1

2π

∫ ∞

−∞|r|−1D(0)(r)e−jrxd r,

Q(0)(x) = F−1{jD(r)/r} =j

2π

∫ ∞

−∞r−1D(0)(r)e−jrxd r.

(16)

The charge distribution is Q = 2Q because it is the integral of the induction difference on both sides ofthe plane y = 0, and D2(y − 0) = −D2(y + 0) = −ǫ0D. Naturally, V (x) is constant on the strips becauseE(x) = 0 there, and Q(x) is constant between the strips because D(x) = 0 there. Thus we apply for theith strip voltage Vi and current Ji

V(0)i = V (0)(ξi), J

(0)i = 2jω[Q(0)(ζi) −Q(0)(ζi−1)], i = 1, . . . , N, (17)

where ξi is a point within the ith strip domain (best - at its center), and ζi can be at the center of spacingbetween two neighboring strips, the i + 1 and ith; they can be easily evaluated from the strip edges xi,except ζ0 and ζN the value of which can be applied at certain distances in front of the first strip and afterthe last one; both values are equal because Q(N)(x→ ±∞) are equal. Note that all the above evaluationsinvolve only the spectral representation of D(0)(r). The other advantage of this approach is that we maynot bother with the square-root sign of Eq. (??) to evaluate the correct values of Vi, Ji.

3 Evaluation of template functions

As seen from the above discussion, the ‘generating functions’ Φ(N,i)(r), i = 1...N − 1 are crucial for evalu-ation of the charge spatial spectrum D(r), requiring numerical evaluation of N − 1 derivatives in Eq. (??).The template functions Φ(N,i)(r) (??) can however, be redefined to avoid numerical differentiation [?]. The

6

new definitions considered from now on are:

Φ(N,0)(x) = jN−1N∏

i=1

Φ(1)(x− bi, ai) = Φ(N)(x)

Φ(N,1)(x) = (x− b1)Φ(N)(x)

Φ(N,2)(x) = (x− b1)(x− b2)Φ(N)(x)· · ·

Φ(N,N−1)(x) = Φ(N)(x)N−1∏

i=1

(x− bi)

(18)

(bi − ai < bi + ai < bi+1 − ai+1 assumed). The new functions are a combination of the the old ones,Eqs. (??), with multiplication by the polynomial of same order n. Again, the modified function Φ takesreal values on the strips (representing the charge distribution there), and imaginary values between thestrips (representing electric field there). They can be rewritten in the form of a product of similar terms

Φ(N,i)(x) = −ji∏

m=1

j(x− bm)√

a2m − (x− bm)2

N∏

m=i+1

j√

a2m − (x− bm)2

, i = 0, . . . , N − 1,

f = −jN∏

m=1

j√

a2m − (x− bm)2

= −jk(x)|f |, k(x) = 1, 2, . . . , 2N + 1;

(19)

k(x) counts the consecutive free and metallized domains: k(x) = 2m on strips, m = 1, . . . , N , and k(x) =1, 2N + 1 on the left (x < b1 − a1) and right (x > bN + aN ) semi-infinite free domains, respectively. Thefirst product in Eq. (??) disappears for i = 0. The spectral representation of Φ(N,i) is

Φ(N,i)(r) = Φ′1(r) ∗ · · · ∗ Φ′

i(r) ∗ Φi+1 ∗ · · · ∗ ΦN (r), (20)

where (Appendix A)

Φm(r) = F{

1√

a2m − (x− bm)2

}

=

{

J0(ram) ejrbm , r ≥ 0,0, r < 0,

Φ′m(r) = F

{

x− bm√

a2m − (x− bm)2

}

=

{

−j[δ(r) − amJ1(ram)] ejrbm , r ≥ 0,0, r < 0.

(21)

It is due to the Dirac δ-function in Φ′m that the evaluation order in Eq. (??) is important: the con-

volutions with Φ′ should be evaluated last (counting from the right to the left) to obtain an integrableproduct of regular and δ functions. The algorithm would run as follows: ΦN−1 ∗ ΦN ⇒ Φ,Φ′

N−2 ∗ Φ ⇒Φ, · · · ,Φ′

1 ∗ Φ ⇒ Φ(N,N−2)(r). In fact, the definition of Φ(N,i) can be made in various ways, with am, bmchosen in arbitrary order, not necessarily for subsequent strips. This is equivalent to choosing Φ′

m + cmΦm

instead of Φ′m in Eq. (??), with arbitrary cm, resulting in arbitrary polynomials Pn(x) of degree n in

Eqs. (??) defining Φ(N,n)(x). This possibility will be exploited later in order to improve the numericalaccuracy of the analysis. It is worth to note here that in the original definition, Eq.(??), Pn = xn only.

The modified template functions Φ(N,i)(r), i = 0, . . . , N−1 require only numerical evaluation of multipleconvolutions. Nevertheless, it is still a rather difficult numerical task, especially for the case of large N .Here, the convolutions are evaluated using the convolution theorem:

∫ ∞

−∞f1(r − r′)f2(r′)d r′ = F{f1(x)f2(x)}, fi(x) = F−1{fi(r)},

f(x) ≈ ∆rM−1∑

i=0

fi e−jxri , ∆r = rM/M, ri = i∆r; fi = f(ri), i = 0, . . .M − 1,

(22)

7

where the FFT algorithm is used for numerical Fourier transformations. Its input data is the set of samplesof the transformed function evaluated at ri = i∆r, i = 0 . . .M − 1, with M being a power of 2. Thus, thefunction f(r) is truncated at rM = ∆rM chosen large enough if we want to verify the analysis with thethe figures of the spatial field distributions like the ones presented in this paper. Neglecting this purpose,the function f(r) could be truncated at lower ru mentioned in Sec. 2.

The FFT algorithm is used mainly to take advantage of its speed, but due to the exploited trapezoidalrule of integration, the resulting accuracy of computations is not satisfactory for the case of larger number ofstrips, above 20 [?]. Better interpolation of the integrated function is necessary to overcome this difficulty.Typically, changing the integration rule would require integration section by section [?], but it is shownbelow that FFT can still be applied, provided that its output data are appropriately modified. This subjectis discussed below, after [?].

The transformed function can be approximated between its samples by means of higher order interpo-lation as follows

f(r) =

(M−1)−m∑

i=m

fiψ(r − ri

∆r) +

m∑

i=0

[fiϕi(r − ri

∆r) + fM−1−iϕM−1−i(

r − ri∆r

)], (23)

where ψ(0) = 1 and ψ(m) = 0, m± 1,±2, . . .; since the interpolation must yield the value of the sampledfunction at ri. The so-called ‘kernel function’ ψ is assumed taking non-zero value only for r in domainbetween the first and the last samples used for its evaluation. For the case of cubic interpolation, m = 3,there are four samples fi the kernel function ψ depends on. The explicit formula for ψ is presented inAppendix B after [?], which can be applied in the interior domains of r where s ∈ (−2, 2).

A different, non-centered interpolation formula must be applied for intervals close to the integral limits,0 and rM , resulting in the kernel functions ϕi(s), ϕM−1−i(s), respectively (Appendix B). Finally, theapproximation of the function f(r), Eq.(??) is:

f(r) =M−1∑

i=0

fiψ(s) +m∑

i=0

{fiϕ′i(s) + fM−1−iϕ

′M−1−i(s)}, s =

r − ri∆r

, (24)

where the first sum includes only the kernel ψ(s), and the last one depends on ϕ′(s) being the differencebetween the boundary kernels ϕ and the interior kernel ψ(s), presented again in the Appendix B. This isa formula analogous to Eq. (??) with the exception of the kernel functions ψ and ϕ′ which are convolvedwith fi (note that s ∼ r − ri). This allows one to write:

f(x) ≈ ∆[W (Θ)M−1∑

i=0

fie−jiΘ +

M1∑

i=0

{fiαi(Θ) + fM−1−iαM−1−i(Θ)}], Θ = x∆r,

W (Θ) =

∫

We−jsΘψ(s) d s, αi(Θ) =

∫

ϕ′

i

e−jsΘϕ′i(s− i) d s,

and αM−1−i(Θ)e−j(M−1)Θα∗i (Θ) since ϕ′

M−1−i(s) = ϕ′i(−s),

(25)

where the integration limits results directly from the support of the integrated functions presented inAppendix B. Applying the FFT algorithm to evaluate the first sum in (??) by analogy to Eq.(??), wefinally obtain the algorithm of Fourier transform evaluation exploiting the cubic interpolation:

[f(xn)] = ∆r{W (Θ)[FFT (f0 . . . fM−1, 0, . . . , 0︸ ︷︷ ︸

M

)] +3∑

i=0

(αi(Θ)fi + α∗i (Θ)fM−1−i)} (26)

(note the so-called ‘zero-padding’ the data set with an M -long array of zeros to avoid the so-called ‘circularconvolution’ that may spoil the most important lower spectrum of f ; this doubles at least the FFT data set).

8

a)

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

r/K b)0 5 10 15 20

−1

−0.5

0

0.5

1

1.5

x/ Λ

Figure 2: a) Comparison of the results obtained with the template functions Φ(N,i)(r) evaluated by meansof the standard FFT (dash lines) and using cubic interpolation (solid line), for 15 Λ-periodic strips andM = 217 samples in the domain (0, 105K). b)(right) Their inverse Fourier transformations show thecharacteristic distortion of the potential distribution resulting from the inaccuracy of the standard FFT isremoved when using the cubic interpolation.

Here [f(xn)] is the data vector of the sampled f(x) at x = nπ/M,n = 0, . . . , 2M − 1. The independenceof W (Θ) and αi(Θ) (presented explicitly in Appendix B) of the transformed functions (fi) means thatthey can be evaluated only once. Examples of the template functions evaluated by means of the cubicinterpolation method are shown in Fig. 2, and compared with the ones computed by standard FFT (solidcurve).

4 The spectrum of charge distribution

The linear combination of template functions

Φ(x) =N−2∑

i=0

αiΦ(N,i)(x),

D(x) =N−2∑

i=0

αiRe{Φ(N,i)(x)}, E(x) = −N−2∑

i=0

αiIm{Φ(N,i)(x)},(27)

according to Eq. (??), is discussed representing the spatial field distribution. The unknown coefficients αi

are to be found from the equations resulting from the circuit theory. To formulate these equations, oneneeds to know the voltage differences between the neighboring strips Uk = Vk+1 − Vk, k = 1, . . . , N − 1,and the strip charges Qk, k = 1, · · · , N , to be evaluated first:

Uk = −∫ bk+1−ak+1

bk+ak

E(x) d x, Qk = 2

∫ bk+ak

bk−ak

D(x)d x. (28)

In the most frequent case of given voltage differences Ui, the circuit theory yields the matrix equation:

Aα = U , U = [Ui], A = [Aki],

Aki =

∫ bk+1−ak+1

bk+ak

Im{Φ(N,i−1)(x)}d x, i, k = 1, . . . , N − 1.(29)

9

The above integrands are square-root singular at both integration limits. This poses certain difficulty inevaluation of the matrix elements Aij with sufficient accuracy in order to solve the system of equations. Thedifficulty can be overcome using the Gaussian quadrature formula [?] or other appropriate methods [?];the iterative integration scheme based on the so-called ‘extended midpoint rule’ has been found to bepromising [?].

Another favorable method of evaluation of the integrals (??) exploits the correspondence of integrationin spatial domain, and division by spectral variable r in spectral domains. This applied to E(r) and Q(r)yields the formula (??) for the corresponding functions V (N,i)(x) and Q(N,i)(x) evaluated from Φ(N,i)(r)according to Eqs. (??) and (??). This yields the matrix components; explicitly:

Aki = V (N,i−1)(bk) − V (N,i−1)(bk+1), (30)

without an extra numerical integration in spatial domain, but taking advantage of an earlier effort ofevaluation of the spatial spectrum with high accuracy, and subsequent application of the FFT in orderto evaluate V (N,i)(x) in discrete representation. The value of V (N,i)(bk) is taken from the correspondingnearest point of the FFT output series. After solving the system of equations (??), the spatial spectrumD(r) is evaluated from Eq. (??) using

Φ(r) =N−2∑

m=0

αnΦ(N,m)(r). (31)

Example charge spatial spectra were already presented in Fig. 1, for strips placed beside a groundedconducting half-plane. This case can be treated in the same way as that presented above. The minorproblem arising in evaluation of the convolution with square root singular function Φ(0), Eq. (??), can beovercome as shown in Appendix C.

It must be remarked here that the template functions Eqs. (??)–(??) span large range of values forlarger number of strips N . In the case of 25 periodic strips, for example, the components in the sum(??) differ by 12 orders of magnitude. Thus, the sum may become severely inaccurate even in doublecomputation with double precision. The nature of this problem originates from the functional form ofΦ(N,i)(x), Eq. (??). To overcome this difficulty, we can take advantage of a certain degree of freedom,mentioned earlier, allowing us to choose arbitrary polynomials in numerators of these functions within thesame order i. It is evident that polynomials having zeros distributed uniformly over the domain occupiedby the strip system would temper the growth of Φ(N,i)(x) over subsequent strips or spacing where V and Qare evaluated, thus they would temper the values of the matrix coefficients Aij . For example, roots can beplaced on the selected strip centers [?]. The polynomials of higher template functions (larger index i) wouldhave roots placed on almost all strips. For lower i - only on some of them chosen to obtain quasi-uniformroot distribution. The order of magnitude of Φ(N,i)(x) with larger i is however, generally lower and theroots distribution is less significant. The method is effective enough to reduce the above mentioned spanof orders of magnitude from the value of 12 to lower and acceptable value of 7 (in the case of 25 periodicstrips). The corresponding examples are shown in Fig. 3.

5 The scattering problem

This section strongly relies on the earlier paper [?] presenting the ‘electrostatic scattering’ problem forperiodic strips (or groups of strips) with period Λ = 2π/K. There are also defined the ‘generating functions’which, due to the system periodicity, are actually the discrete spectral functions represented by the Fourierseries Fn. They are obtained by convolutions like these in Eq. (??) and (??) but involving Legendrepolynomials Pn(·) instead of Bessel function J0(ra). On the strength of the asymptotic expansion [?]

Pn(cos θ) = J0([2n+ 1] sin[θ/2]) +O(sin2[θ/2]), θ ∼ aK → 0, (32)

10

a) b)

Figure 3: a) The erroneous charge spectrum caused by loss of numerical precision in evaluation of thesum (??) which components span several orders of magnitude. b) The modification of the numeratorpolynomials corrects the problem . The insets represent the corresponding potential spatial distributions.This confirms the value of verification of numerical results by the inverse Fourier transformation of theevaluated spectrum, lacking one when the spatial distribution is computed first, and then transformedto obtain the spectrum (note that the problem with magnitude span concerns both methods because theevaluation of strip potentials is necessary in both).

we infer that the difference between the functions obtained in both cases vanish if r = nK,K → 0. Weare going to exploit this approximation for K small but finite, in developing the solution analogous to thescattering problem using the earlier evaluated function D(r), Eq. (??).

As presented in Sec. 3, this function is evaluated at discrete values of the spectral variable r = iK,K =∆r. Thus, it is the discrete series in the numerical analysis and it actually represents, on the basis of thetheory of FFT [?], the periodic function in spatial domain with a certain large period 2π/K. This enablesus to introduce the notations

Di = D(iK), Ei = iSiDi; D−i = D∗i . (33)

In the periodic case, only one series Fn is evaluated, and all other which correspond to Φ(N,n) for differentn ∈ (0, N − 1), are obtained by shifting the index of Fn to obtain the series Fn−m. Moreover, if m ≤ N/2,the ‘generating functions’ Fn−m satisfy the radiation conditions and thus are accounted for in the fieldrepresentation in order to satisfy the corresponding circuit equations, including electric neutrality of thesystem. This is because F|k|≤N/2 = 0. In the case of aperiodic systems, the functions Φ(N,n) have an

analogous property that, behaving like rn−N , they have small values over a broader domain of r ≈ 0.As it will be shown below, the series Di introduced in Eq. (??), with D0 6= 0, will play important role

in the solution of the considered scattering problem. Note that the pair of functions with shifted indicesDn−m and En−m places the values D0, E0 at the spectral line r = mK, what corresponds to multiplicationby e−jmKx in the spatial domain. Naturally, such multiplied functions still satisfy the boundary conditionson the plane of strips: E(x) = 0 on the strips and D(x) = 0 outside the strips. This property will beexploited below with this important remark that D and E evaluated from Eq. (??) using the shifted Di−m

Di = Di−m and Ei = Ei−m = jSi−mDi−m (34)

fail to satisfy Eq. (??) (Ei 6= jSiDi, this can be easily verified by inspection), and thus they fail to satisfythe radiation condition (the corresponding field does not vanish at |y| → ∞). Note however that this

11

a) b)

c) d)

Figure 4: Scattering by 4 strips: a) the surface electric field, and b) the induction D(x). The integrals ofc) the surface electric field −V (x) (note its constant values on strips) and d) the charge distribution Q(x);equal values of Q(x) on both sides of the structure indicates the structure electric neutrality. Solid and dotlines represent real and imaginary values, respectively. Vertical scales are arbitrary; thick lines in lowerfigures represent the strips (at the zero level of the corresponding integrals).

failure takes place only due to the spectral lines 0 ≤ i ≤ m; all other lines satisfy well the Eq. (??).To proceed further, another class of field is introduced here - the ‘incident’ harmonic field of wavenumber

rI = IK and complex amplitudes DI and EI . This field does not vanish at infinity; being the incidentwave, it rather grows with |y| → ∞ and thus is a function of type e|ry|. In contrast to the scattered field,Eq. (??), the incident field obeys:

EI = −G(IK)D = −jSIDI . (35)

To simplify the analysis, I > 0 is applied in this paper.The field on the strip plane y = 0 is the sum of the scattered (marked below by the upper index s)

and ‘incident’ spatial waves; it must satisfy the boundary conditions on this plane. Noticing that thefield {Di−m, Ei−m} does it, we attempt to express the surface field by the combination (the summationconvention applied over repeated indices)

Ei = Esi + EIδiI = jαmSi−mDi−m,

Di = Dsi +DIδiI = αmDi−m,

(36)

where δ is the Kronecker delta and αm are unknown coefficients. One needs only to add the requirementthat the scattered field obeys Eq. (??), that is Es

i = jSiDsi ; explicitly

αm(1 − SiSi−m)Di−m = 2DIδiI . (37)

It may be checked by inspection that the solution to this infinite system of equations (for i in infinitelimits) can be solved with αm, 1 ≤ m ≤ I + 1, for the assumed I > 0. Indeed, for any i > I, the termin bracket turns to zero satisfying the homogeneous equation (δI,i>I = 0), and similarly for any i ≤ 0,provided that m takes values in the above limits. For i = I we have αI+1 = 1/D∗

1. Other αm can beevaluated in recursive manner, starting with equation i = I down to i = 1.

The resulting field distribution may include a net charge, D0 6= 0. To assure the system electricneutrality, one needs to add α0Di to the evaluated Di with coefficient α0 chosen to obtain

D0 + α0D0 = 0. (38)

This completes the solution of the scattering problem; the resulting surface field in the spectral represen-tation

Di = αmDi−m, Ei = jαmSi−mDi−m; 0 ≤ m ≤ I + 1, (39)

12

satisfies the boundary conditions at the strip plane; Fig. 4 presents the computed example. Moreover,the induced strip voltages and charges can be evaluated in the already presented manner, Eq.(??), in thecorresponding discrete form (r = iK). These should be treated like originating from the negative externalvoltage source. If, for example the ith strip is assumed to be grounded but gets certain potential due tothe incident field, then a combination of template functions Φ(N,n) must be added with proper coefficientsevaluated in the earlier sections.

6 Conclusions

Two nonstandard complementary electrostatic problems are solved by a novel method exploiting the specialtemplate functions constructed in spectral domain and satisfying the electric boundary conditions on thestrips. Detailed discussion concerning the numerical evaluation of these functions with sufficient accuracyis presented. This is perhaps the most important result of the paper; the same functions and analogousapproach exploiting them can be applied in the analysis of the electrostatic problems for anisotropicmedia [?, ?], and also numerous problems of electromagnetic and elastic wave scattering by strips orplanar cracks [?, ?].

In contrast to standard electrostatics, here we construct solutions in the spectral domain. Note thatthe field spatial distribution is the least important in applications: measured are either the mutual stripcapacitances depending on the total strip charges (the integrals of the charge distribution), or the spatialspectrum (in the scattering cases, like Bragg scattering by strips or frequency characteristics of surfacewave transducers). The solution constructed in spectral domain with satisfactory accuracy is advantageousas compared with the standard evaluation of spatial charge distribution at least in the latter case, becauseof high sensitivity of the charge spectrum on the charge distribution details over the entire system. Havingthe spatial spectrum, one can easily evaluate the spatial distribution and its integral (in order to evaluatethe measured total charges of strips) with similar high accuracy, what is not true in the reverse way.Moreover, the inverse Fourier transformation yields the tool for verification of the spectral results, lackingin the standard analysis [?, ?, ?].

In practical problems, the spatial spectrum needs to be evaluated in a rather narrow domain (rumentioned in the Sec. 3). In this case however, the total charges of strips must be evaluated by integrationof the spatial representation of the template functions. This subject is not discussed in detail in this paper;refer to the works [?, ?], for instance. It is worth to note that the Bessel functions can be well representedby their power-series expansions in a narrow spectral domain, allowing one to evaluate the correspondingconvolutions, Eq. (??), exactly (within the power-series representation). This would yield the frequencycharacteristics of the interdigital transducer (like the one presented in Fig. 1 [?], or comb transducer [?]),or characterization of the Bragg scattering by the system of strips [?], in explicit analytical forms in thelower spectral domain.

Acknowledgments

Present work was sponsored by State Committee of Scientific Research, Grant 3 T11B 046 26. Valuablecomments by Prof. Czes law Bajer (Polish Academy of Science) and Prof. Micha l Mrozowski (GdanskTechnical University) are greatly acknowledged.

13

Appendix A

It is known that[?]

F{

1√

a2m − (x− bm)2

}

=

∫ ∞

−∞

ejrxd x√

a2m − (x− bm)2

= J0(ram)ejrbm , r ≥ 0, (40)

and 0 for r < 0 (note the exponential term resulting from the displacement bm). The generalized transformof the function (x− bm)/

√

a2m − (x− bm)2 can be found using the rule F{xf(x)} = −j.F (r)/r., yielding

F{x/√

a2m − x2} = −j d

d rJ0(ram) = −j[δ(r) − amJ1(ram)]. (41)

Finally, for the shifted function:

F{(x− bm)/√

a2m − (x− bm)2} = −j[δ(r) − amJ1(ram)]ejrbm . (42)

Appendix B

The cubic interpolation functions are, after [?]:

ψ(s) =

(s+ 3)(s+ 2)(s+ 1)/6, s ∈ (−2,−1),−(s+ 2)(s+ 1)(s− 1)/2, s ∈ (−1, 0),(s+ 1)(s− 1)(s− 2)/2, s ∈ (0, 1),−(s− 1)(s− 2)(s− 3)/6, s ∈ (1, 2),

(43)

ϕ0(s) =

{

−(s− 1)(s− 2)(s− 3)/6, s ∈ (0, 1),ψ(s), s ∈ (1, 2),

ϕ1(s) =

{

(s+ 1)(s− 1)(s− 2)/2, s ∈ (−1, 0),ψ(s), s ∈ (0, 2),

ϕ2(s) =

{

−(s+ 2)(s+ 1)(s− 1)/2, s ∈ (−2,−1),ψ(s), s ∈ (−1, 2),

ϕ3(s) =

{

(s+ 3)(s+ 2)(s+ 1)/6, s ∈ (−3,−2),ψ(s), s ∈ (−2, 2),

ϕM−1−i(s) = ϕi(−s), i = 1, . . . ,m = 3,

(44)

ϕ′0(s) =

{

−ψ(s), s ∈ (−2, 0),−2s(s− 1)(s− 2)/3, s ∈ (0, 1),

ϕ′1(s) =

{

−ψ(s), s ∈ (−2,−1),s(s+ 1)(s− 1), s ∈ (−1, 0),

ϕ′2(s) = −2s(s+ 2)(s+ 1)/3, s ∈ (−2,−1),

ϕ′3(s) = (s+ 3)(s+ 2)(s+ 1)/6, s ∈ (−3,−2),

ϕ′M−1−i(s) = ϕ′

i(−s), i = 1, . . . ,m = 3.

14

The functions W (Θ) and αm are:

W (Θ) =6 + Θ2

3Θ4(3 − 4 cos Θ + cos 2Θ);

α0 =(−42 + 5Θ2) + (6 + Θ2)(8 cos Θ − cos 2Θ)

6Θ4+

+j(−12Θ + 6Θ3) + (6 + Θ2) sin 2Θ

6Θ4,

α1 =14(3 − Θ2) − 7(6 + Θ2) cos Θ

6Θ4+ j

30Θ − 5(6 + Θ2) sin Θ

6Θ4,

α2 =−4(3 − Θ2) + 2(6 + Θ2) cos Θ

3Θ4+ j

−12Θ + 2(6 + Θ2) sin Θ

3Θ4,

α3 =2(3 − Θ2) − (6 + Θ2) cos Θ

6Θ4+ j

6Θ − (6 + Θ2) sin Θ

6Θ4;

(45)

the funcions αi can also be easily expanded in the convenient power series.

Appendix C

Presence of semi-infinite conducting screen in the considered system requires adequate modification oftemplate functions, Eq. ??, as follows:

Φ(N,0)(x) ⇐ Φ(0)(x)Φ(N,0)(x), Φ(N,0)(r) ⇐ Φ(0)(r) ∗ Φ(N,0)(r),

Φ(N,1)(x) ⇐ Φ(0)(x)Φ(N,1)(x), Φ(N,1)(r) ⇐ Φ(0)(r) ∗ Φ(N,1)(r),. . .

Φ(N,N−1)(x)) ⇐ Φ(0)(x)Φ(N,N−1)(x), Φ(N,N−1)(r) ⇐ Φ(0)(r) ∗ Φ(N,N−1)(r),

(46)

where Φi(r) and Φ′i(r) are given by (??) and the template function of the screen Φ(0) - by Eq.(??).

Consider the following convolution accounting for the semi-finite support of both Φ(r) and Φ(0)(r):

I = Φ(0)(r) ∗ Φ(r) =

∫ r

0J0(ay)ejyb 1√

r − yd y. (47)

Integrating (??) by parts, one obtains

I = −2√r − yJ0(ay)ejyb|r0 + 2

∫ r

0

√r − y [δ(y) + J (y)] d y,

J (y) = [jbJ0(ay) − aJ1(ay)]ejyb, y ≥ 0(48)

(note the semi-finite support of J ), where δ(y) is the Dirac δ-function. Finally,

I = 4√r + 2

∫ r

0

√r − yJ (y)d y = 4

√r + 2

√r ∗ J (r), r ≥ 0. (49)

The resulting I(r), being evaluated avoiding the inconvenient integration of the square-root singular func-tion Φ(0)(x), has the semi-finite support like any template functions Φ(r).

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