nonstationary model problems for waveguide open resonator theory

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Nonstationary Model Problems for Waveguide OpenResonator TheoryYuriy K. Sirenko a & Nataliya P. Yashina aa Institute of Radiophysics and Electronics NAS of the Ukraine , 12 Ac. Proscura St., Kharkov,310085, UkrainePublished online: 03 Feb 2007.

To cite this article: Yuriy K. Sirenko & Nataliya P. Yashina (1999) Nonstationary Model Problems for Waveguide Open ResonatorTheory, Electromagnetics, 19:5, 419-442, DOI: 10.1080/02726349908908661

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NONSTATIONARY MODEL PROBLEMS FOR WAVEGUIDE OPEN RESONATOR THEORY

Yuriy K. Sirenko and Nataliya P. Yashina Institute of Radiophysics and Electronics NAS of the Ukraine 12 Ac. Proscura St. Kharkov, 31 0085 Ukraine

ABSTRACT

The paper takes a fie& look at rigorous approaches and algorithms for model problems of electromagnetic theory of waveguide discontinuities. Numerical experiment is still the main tool for investigation the resonant wave scattering phenomena, which are characteristic for such structures. The mathematical approaches employed for such purposes have to fit certain requirements for accuracy, efficiency and versatility, for ability to be focused on particular details of various practically interesting regimes. The approaches for analysis of transient processes in waveguide open resonators considered herein satisfy all these requirements. They rely on the description of scattering properties of discontinuities in regular waveguides in terms of transform operators related to "evolutionary basis" of nonstationary signals, which are qualitatively the same for all guiding structures. The prototypes of such approaches in frequency domain (FD) have been verified by large body of boundary value problems (BVP) and their efficiency has been demonstrated. Their modification for time domain (TD) has been carried out for the &st time.

INTRODUCTION.

The regularization of a BVP by means of its partial inversion is by far not a new idea in functional analysis and theory of integral equation. Its devel- opment in wave =action and spectral theory provided considerable progress of the theory of electromagnetic modeling for basic canonical

Rmeived 4 January 1997; accepted 1 June 1998

Eledromagnetics. 19:419-442. 1999 Copyright @ 1999 Taylor 8 Francis 0272-6343199 912.W + .oO

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420 Y. K. SIRENKO AND N. P. YASHINA

open structures (gratings, waveguide discontinuities, unclosed scatterers) in parameter regions, where various (including non classical) dispersion laws are displaying. The considerable theoretical achievements in the de- velopment of methods and algorithms (Shestopalov et aL, 1971, 1983, 1984), in the investigation of physical processes (Shestopalov et aL, 1973, 1986, 1989), enabled the creation (within the framework of rigorous modeling) of CAD systems (Kirilenko, 1991) for advanced research and for modem applied problems. The corresponding numerical algorithms imple- ment the conventional procedure of generalized scattering matrix tecb- nique. This approach is well known in wave =action theory. The qualitatively new results are achieved because of the successll matching of this approach with analytical regularization algorithms that are basic underlying algorithms for building blocks (generic elementary structures). In ideological aspects TD cokideration of resonant discontinuities in guiding structures presented herein is close to those ones that have already found its persistent and rather successll implementation in the FD. In section 1 we dwell shortly at one of the simplest of them (Sirenko, 1987) just in order to demonstrate clearly the close relation between approaches and technique in FD and TD. The basic ideas of methods are, first, rigorous solution of electromagnetic problem for generic structures and, second, operator description of electromagnetic interaction between the generic elements, comprising complicated junction under investigation (with fiuther accurate numerical valida- tion and treatment of physical results). We then have modified them in such a way that they can be applied to the solving of wave scattering problems in TD without considerable loses in efficiency and versatility. In section 2 the basic notions required for the description of transient processes in terms of the transform operators related to "evolutionary basis" of nonstationary signal (Tretyakov, 1993, Sirenko, 1997, Bessonov, 1997), are introduced. The general scheme of algorithmization for problems of increasing complexity is outlined. Section 3 gives more detailed consideration of initial BVP for the circular and coaxial waveguides in the case of the excitation by TE,,waves. The choice of model problem has been done in concord with former scien- tific interests of the authors.

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NONSTATIONARY MODEL PROBLEMS

1. COAXIAL CAVITY. DIFFRACTION PROBLEM

1.111 order to recall some basic facts and associated nc der the configuration m Fig. 1, where we assume that the coaxial cavity is exited by TE,, electromagnetic wave ( E, , E,, H, = 0 ) of unit amplitude 6om

waveguide A.

FIGURE 1. Coaxial cavity

T i e dependence is given by a factor exp(-iw t) and suppressed fiuther on. p,p are polar transverse coordinates. The scattered field in each region (A, B, C and D) is defined by the only non zero electric field mten- sity E, -

which may be found 6om reduced Maxwell's equation

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Here ~ , , p , are fiee space constants. Separating variables we obtain the

general solution

y , (p) -are transverse eigenhctions that satisfy the eigenvalue problem

2 2 % where: y , = (K' -A,, a ) , lm,Re y, t 0, K = ka is a dimensionless * 17-5

frequency parameter, and the signs + in e " define waves propagating towards increasing and decreasing z values. The total di5action field in each separate region may be presented as follows

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Formulae (1.4) introduce mto consideration so called scattering matrices that are bounded operators on the pair of "energy" spaces

2 a = @,I: zlanl n < and define all electromagnetic

" properties of the discontinuities for arbitrary monochromatic excitation. Upper subscripts (Like A 4 AD, DA etc) here show the direction of wave propagation m the different regions (for example propagation firom channel D to A corresponds to subscripts AD) lower subscripts shows space spec- tnun component (mode) of the field. In (1.4) the following notation have been introduced:

where J . . . , N . . . are Bessel and Neumann functions; v,, , p, , n = 1,2.. are strictly positive roots of equations

J, (v ) = 0 and G, (pJ) = 0. Eigenfimctions y,,(p),y,,,~)y,2 (p) com-

prise orthonormal basis m relevant plane domains (m two circular and m one annular waveguides, dependence on q, is not considered). After

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matching the total field tangential component in planes z = 0, z = -L, the problem reduces to the following dual system of linear algebraic equations of the first kind:

Here: 6,P -Kronecker's symbol; d: = (d, f c,) J,(v,B) exp (iy,, I ) 1 J,(v,); Ti = T 0 y$ 1 v, ; 1 = L I2a . The sequences of unknown amplitudes

d = {d,) and c = {c,,) have to be elements of the space < ; elements of the generalized scattering matrixes (Rny and others, not mentioned explicitly here) are defined in an obvious way. Thus we have for instance relations of the form:

2.System (1.6) represents an ordinary version of ill-posed operator equations of the fist kind arising in the mode matching technique. It is hardly possible to indicate in the general case the pair of spaces where bounded operator of the system would have inverse bounded operator (Suenko, 1983). But in certain particular cases the problem may be solved by cor- rectly chosen main diagonal in its reduced (finite - dimensional) equivalent. (The "Telative convergence" phenomenon is well known (Mittra et al., 1971)). Such regularization does not solve the problem on the whole: even if one manages to provide the convergence of approximate numerical solution to the explicit one. The convergence is only of coordinate-wise type; the convergence rate is rather different for phase computation (considerably lower) and energy characteristics; and deterioration of the results, as the order of the harmonic amplitude increases, is disastrous and unpre- dictable (Suenko, 1978, Shestopalov et al., 1984). Analytical regularization of mode matching operator equations of the first kind gives canonical Fredholm operator equations of the second kind that have unique solution in < almost everywhere wahin a parameter variation

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NONSTATIONARY MODEL PROBLEMS 425

region and may be approximately solved by means of truncation method. Numerical solutions of such equations converge rather quickly over < norm The extraction and analytical inversion (with respect to the right hand side part of particular equation) of a singular part of original ill posed BVP operator constitutes the key point of corresponding methods. In the case under consideration the singular component is defined by elements of

matrix operator (1.6) with the difference term [y,, - ymlj in denominator.

For partial inversion of (1.6) we apply conventional fbction-theoretic method, based on Mittag-Leffler's theorem for meromorphic function representation (Mittra et aL, 197 1, Suenko, 1978, Sirenko, 1983, Shes- topalov et al., 1984). We assume that there are meromorphic functions Q,(w) and Qp(w) in complex plane w that satisfy the following conditions: Q,(w) and QP(w) have simple poles at the points w = y,,p = 1,2 ... , zeros

at w = y W ;Q,&@)=o and ~ , & ~ ~ ) = 6 ~ 6 ~ , r n = 1 . 2 2 ..., j=1,2;

Res Q,(- y,)= 6;, Q,(w) and QP(w) decrease over any regular set of

contour in the plane w while lwl increases. f 17 1 The series z d , e , Q,(w) converges uniformly in the domain where

Q,(w) is analytic fbction. Then BVP (1.6) is equivalent to the following infinite system of linear algebraic equations of the second kind

and equation ( 1.7) is equivalent to

This follows fiom a comparison of Mittag-Lefner's expansion for meromorphic function Q* (w) = fzd:eWr'Q,(w) - T,'Qp(w) in the points

W = Ym, . rn=1,2 ,..., j=1,2 and w = - y m 2 with (1.6) and (1.7),

respectively.

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The regularization of the problem is completed by constructing functions Q,(w) and Q P ( w ) with all the properties mentioned above, acquiring the

form:

Above f ( 8 ) = 2[8 ln 0 + ( 1 - 8 ) ln(1- 8 )] and index over infinite

product means that the factor ( y ,, - w ) / ( y , , - y with n = p is omitted

in it. The asymptotic estimation of matrix operator elements and entries at the right hand side part of (1.8) yields

what defines the pair of spaces < + < as a class of correctness of the problem and provides the correctness of truncation method application in numerical implementation. Convergence rate over < norm is estimated

analytically and is as N-lIZ eq(-NI) , where N is a number of equations

accounted for in (1.8). 3. Several general comments may be relevant here. Semi analytical approaches are always taking into consideration geometrical particularities of scatterers. Their efficiency depends considerably on the level of completeness of extracting and analytical inversion of a singular part of original equation. This successful inversion then defines the right-hand side part of final operator equations of the second kind. In other words, efficiency of methods depends on the difference between the actual structure under consideration and that one for which analytical solution of mac t ion problem (maybe hypothetical one) may be obtained. Here, in reducing the original BVP to operator equations of the second kind the part of the problem related to the wave &action problem for the coaxial bifhation of circular waveguide has been solved analytically. From

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analytical estimation for convergence rate it is clear that employment of semi inversion technique for matrix operator of convolution type leads to efficient algorithms in the situation when the length of coaxial cavity L is rather large. When L decreases the error of numerical solution obtained fiom the systems of the same order increases. Waveguide discontinuities for which BVP can be solved analytically will be used as building blocs in the course of consideration of more complicated structures which can be decomposed into such simple "generic" cases. 4.Let us consider the coaxial bifurcation of circular waveguide to be such generic discontinuity for coaxial break of circular waveguide (Fig.2).

FIGURE 2. Circular waveguide bifurcation.

This simple example will help us to demonstrate application of the generalized scattering matrix scheme for the algorithmization of more complicated problems and it will show how one can construct the well posed equation (1.8) in this and many similar cases. We suppose that generalized scattering matrixes of such a structure (we

fouow former notation RM ={&:I, RUE = (x:) etc.) are known. They

may be exploited while solving next more complicated problem: to define the field, scattered by coaxial cavity while it is exited by set of eigen waves of circular waveguide A with vector of known amlitudes -

i = { i , , i ,,...... } = { i , } . ~ e t u s d e n o t e b y r = { r , } , s = { s , } andsoon(see

Fig. 1) the sets of unknown complex amplitudes of fraction field in the various regions. The changes that this step causes in representations like (1.4) are evident. Using conventional scattering matrix technique we get the following complete matrix system for vectors of unknown amplitudes:

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The transform conditions on the boundaries z = 0 and z = -L are ac-

counted for in (1.10). Diagonal matrix E = (E 1. kp( iyn~)~; 1 dehes nm

mode amplitude variation while they propagate along regular circular waveguide of radius a (region E). From (1.10) we get

AA . d f c = f T m [ i ] f RBBE [d fc ] ; r f t = R [ z ] + T A E & [ d k c ] that are the equations of the second kind equivalent to (1.8) and (1.9). From (1.10) we have another formally correct statement of the problem's algorithm, namely:

2.EVOLUTIONARY BASIS FOR NONSTATIONARY WAVE, AND TRANSFORM OPERATORS.

1.Let us consider the initial BVP for model structure (Fig. 3)

Assume that the hc t ions ~ ( g , t ) , p (g) , y (g) , E (g) - 1, a(&?), are finite in Q, satisfy the conditions of the theorem about unique solution to problem (2.1) in the energetic class (Sobolev's space) w;(QT), Q' = Qx(0, T), T< s (Ladizshenskaya, 1973).

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Here, L= d2/ d z2j + L ~ ( x ~ , y,); is an elliptic differential operator of the

second order expressing in the local Cartesian coordinate system x, , y, ,z , ; j is related to the region number. M is a differential operator of

order non higher then one; S is the boundary surface of domain Q, that is constituted by two infinite regular (m the parameter region zJ > 0, j = 1,2.. .) waveguides, that are connected via compact cavity @art of the boundary surfaces is depicted in Fig.3); g = {x,y,z). The real fbctions

~ ( g ) 11 and a ( g ) 2 0 define the intluence of the discontinuity on propa-

gation velocity of scattered field and dissipative properties of the cavity In the case of infinite regular waveguide (the cross section of waveguide is

FIGURE 3. General view of resonant cavity in waveguide.

constant along Z, and ~ ( g ) - 1 = e 0) the solution to the problem

(2.1) can be presented m the form of

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Sequence of the functions v (z, t) = {v,,(z, t)) satisfies the equation

and initial conditions

{.u,(~,~))and {A,) are sets of eigedimctions and eigenvalues the prob-

lem, generated by (2.1) after transverse coordinates x , y have been sepa- rated. We presume also that operator L and M have such properties that associated eigedimctions form orthonormal basis in the relevant plane domain. Here a,(z, t) , b, (z) , c, (2 ) are the Fourier coefficients of the

expansion of functions~(g, t), q~(g), ~ ( g ) , over basis fimction series; {n)

is the ordered set of numbers n. Coming over to the generalized formulation of Cauchy problem (2.3),(2.4) (Vladimirov, 1988), and introducing known fundamental sobtion for the

operator D(A) G(A;Z, t) = (- l/2)x(t - lz~)~O[.(t2 - z2)']

(accordingly to the definition D(/z)[G(A)] = 6(z) 6(t) ), we get

Convolution procedure is denoted here by (*); fn(z, t) = a. (z,t)- 6("(t) b,(z)- 6(t)c,(z); X(. . .)- is the Heaviside step

function; 6'"' (. . .) is generalized delta-function derivative of m-th order.

Representation (2.2) and (2.5) give general, but correspondent to source given, form of nonstatonary wave field, that propagates down a regular waveguide. The wave variation in space in the course of time is completely defined by the system of functions v(z,t) (or {G(A,,) * f,,)). Known

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properties of these hc t ions allow them to be considered as universal "evolutionary basis" for nonstationary wave w i t h any finite interval of regular guiding structure (dependence on boundary configuration S and on non degenerate boundary conditions is not essential). 2.Let now a wave of the type (2.2) excites the open waveguide resonator drafted in Fig. 3. We consider the field of excitation uf(g,t) = v,,(z, t) p,,,(x,y) to be nonzero in waveguide A only, that is

" regular for all z, > 0 (left bound is placed in the plane z, = 0). The scattered field in regular semi infinite waveguides A and B and propagating along the waveguide towards increasing values of z, and z, may be expressed in the form of

Boundary transform operators RM and TBA (related to the boundaries z, = 0 ) of evolutionary basis we shaU introduce in following way:

Or the same in operator form:

The relation

defines diagonal operatorsEA(z,) and EB(z1)

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that describe the field variation along finite regular waveguide. The operators RM and T" together with "transport" operators E*,E' describe completely scattering properties of the cavity when it is excited from channel A. Fonnula (2.9) represents properties, which are common for all the solutions to inhomogeneous equations of type (2.3) within the half space z , 2 0. These solutions sat@ nonzero initial conditions and do not con- tain field components propagating towards decreasing z, ("outcoming"

components are equal to zero in every finite time t = T for sufficiently large values of z,) . (2.9) may be derived by application of integral trans-

forms : Laplace transform over !, or, as it has been done in (Maykov,1986) by Fourier cosine transform over z, . In a similar way, we define transform evolutionary basis operators R~~ and TAB for waves coming from waveguide B onto the boundaly z, = 0. Further on, these operators are assumed to be known.

FIGURE 4. Junction of resonators.

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3.Let us consider a complicated junction (Fig. 4), that is made up of two cavities connected via regular waveguide B of length L . Using notations introduced above (that are only modified because of the existence of two units) and definition (2.6>(2. lo), we arrive to (see Fig. 4):

By eliminating of certain unknowns, we can easily obtain an operator equation of the second kind with respect to unknown vector-function

I

wZf(1) = rm(l)[v] + R"(I)EB(L)R"(II)EB(L) [wz (I)] (2.12)

Thus, the original "compound" problem is reduced to such its form, which allows direct inversion (since the operator in the right hand side of (2.12)

acts on unknown vector-functions wZ1(1) with its values determined ear- lier). "Compound " junction after computing boundary operator's elements according (2.7), (2.8) may be transferred into the class of building blocks. 4.The model problems considered above demonstrate all key steps of the approach suggested. The technique is rather convenient and flexible in code implementation and it covers with small modification a considerable range of waveguide configurations.

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Important problem for the approach is the computation of concrete values of elements of operators R, T for various waveguide junctions like shown in Fig. 1 and Fig.4 in (Bessonov, 1997). There, the choice of structures to be considered has been made with respect to the specific research goals and was rather different for various modeling systems. Further on, we shall consider concrete problem, solution of that is necessary for the investigation of resonant electromagnetic processes in circular and coaxial waveguides.

3.COAXIAL WAVEGUJDE BIFURCATION.TD TREATMENT.

1,Investigation of TE symmetric electromagnetic waves in cylindrical structures, depicted in Fig. 1, has been reduced to the solving of initial BVP of the type (2.1)

Here: E ( Z ) is the relative dielectric permittivity of the media, filling the

waveguides, a ( z ) = (D,/E,)'~ a.(z); a, (z) is a conductivity; u(z,p,t)= E,, E, = E, = H, = 0. Nonzero components of density

vectors of electromagnetic field are given by relations:

Separation of variables in (3.1) allows us to use representation of general solution to the problem, which is the same in all geometrically similar waveguides, namely:

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where basis system of o r t h o n o d functions (v,,(P))(in relevant plane

region) is derived fiom (1.3), and elements v,(z,t) of evolutionary basis for nonstationary wave satisfy the equation

and initial conditions (2.4). In the following considerations, the field of excitation ui(z, ~ , t ) is sup- posed to be the wave of the kind (3.2), coming fiom the left onto the dis-

continuity (see Fig.2) with v,(o, t ) = 6 ~ 6 ( t - r l ) . The integer numbers p 2 1 and 7 > 0 have tixed values. Using of such an abstract non- physically existiig signal is motivated by methodological reasons. Namely, by means of such a signal using the procedure of "elementary" excitation of the structure is realized, that enables the extraction of "pure" ~ , , , ( t - r,~)

and ~ , , ( t - r l ) components (see definition (2.7)) in scattered field analysis.

The functions vPf(0,t) are present in expressions for these components. That is why it is necessary for confirming the correctness of (2.7), (2.8) to

prove that vP(0,t) = 6 ( t - r l ) defines vP1(0,t) in the unique way. The required result is derived fiom the Volterra equation of the first kind

In (2.9) the direction of incident wave propagation has to be accounted for. Inverting (3.4) by means of operator transform techniques we anive at:

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2.Following (2.6), (2.7) and (2.9) the total field ~ ( z , ~ , t ) , in regions z > 0

and transformed via bifiucation (see Fig.2, regions A and B) can be represented as

The functions v,,(P), Y n j ( P ) and eigenvalues A,,A ,, are defined by

(1.5). The continuity conditions for functions U and when z=O

(i.e. continuity conditions for the total field tangential components), pro- viding the unique extension of the solution of (3.1) 6om one partial region E into two others (A and B), may lead to functional equations of various type. In terms of Fourier coefficients for matching functions, one of them bas the following form:

Here:

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Equation (3.7) is the dual operator equation with respect to the set of

unknown functions { ~ : f ( t - v)}. Formulae (3.8) are for determining the

elements of the operator-functions TEE and T m , that are connected with the first (B, j = I ) and second (A, j = 2 ) waveguides in region z < 0. This mode matching technique can be represented in terms of equations different fiom those given by equation (3.7) and (3.8). However, herein we shall dwell on the form (3.7), which contains operators suitable for fiuther analytical processing while constructing rigorous solution to the problem. Regularization approach here is like in a FD and is relying on the extracting and inversion of the singular part in matrix equations of convolution type. This technique reduces the problem to straightfoxward time marching schema or (for more complicated problems) to canonical Fredholm operator equations, that can be solved numerically and the truncation error for the solutions may be estimated over the norm in Hilbert space. 3.Differentiating (3.7) with respect to t we obtain

It is clear &om the following expression, that elements of the unknown

vector-hction RE(^ - v)) appearing in the right hand side part of

(3.9) iduence actually on its value in left part at the moment t only by means of their values in the moment T that is strictly less then t:

That allows us after fixing t (next step in moving along time layers, the starting point is t = 7 ) to consider (3.9) as a dual infinite system of linear algebraic equations of the first kind with respect to the set of unknown functions RE(^ - o))). Elements F,,, are time independent and that is

why the solution to the problem for various t is reducible to single inversion of the corresponding operator.

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Let us introduce new notation (see (1.5)) by the identities

In these t e r n oc equation (3.9) acquire the form

For inversion of this system we apply the residue calculus method (Mittra et al., 1971), relying on the Mittag-Leffler theorem about expansion of a meromophic function via its singularities. Assume that there are meromorphic functions Qrj(w) in plane of the complex variable w such that:

Q,,(w) has simple poles in the points w = v i , n = 1,2 ... ; Q , hi)= -6:, Q,,(v:/@~) = 0 3 Q r 2 ( ~ ; ) = 0 r Q a ( v ~ / o ~ ) = - ~ : , m r , n ~ = 1,2 ... ;

Q,,(w) decreases over certain regular contour system in the plane w when

I w I increases. Assume also that the elements w decrease with increasing

r rapidly enough to provide uniform convergence of the series CQ,J(w)o,l over the domain in the w plane where Q,,(w) is an ana- I

lytical function. Under above-mentioned assumptions, we get:

R,, = C ~ e s Q,](v,z) a,] ; n = 42.. r.1

The functions Q,,(w), with all required properties, have the form of

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d Here ~ , ' ( w , l ) = -G,(w,I). A straightfonvard check, relying on proper-

dw ties of cylindrical fimction (Abramowitz et al., 1972) provides following results: 0 the fimctions Q,(w) have simple poles and zeros in required points of

the plane and only in them; when w = 0, there is a removable singularity (branch point of the h c -

tion we and simple pole of N,(w));

the necessary normalization is provided by the factors introduced in (3.12), independent of W ; for large (wl, n and m the following estimates are valid:

Thus, the problem has been sotved and its solution is given in the form (3.1 1). It is clear that the problem concerning excitation fiom the other side (i.e. delinition of operators R" , RBB , Tm ) can be solved without any principal modifications. The same operator as in (3.10) has to be inverted but for another right hand side part now.

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Y. K. SIRENKO AND N. P. YASHINA

The extracting qualitatively the same for all guiding structures "evolutionary" basis of nonstationary wave defined the space within which the description of scattering characteristics of waveguide discontinuities in terms of relevant transform boundary and "transport" operators became possible. Such operators, that are formally decomposing (by means of their elements) total physical pictures into separate transformation acts of various partial components of signal may be an efficient tool for qualitative and quantitative analysis of the processes considered, especially in situa- tions when resonant (anomalous) wave scattering is possible. Such approaches considerably s impw the algorithmization of problems of increasing complexity. The compiling procedures (like several other details of the approach considered herein) have a well-known analogue in FD method of generalized scattering matrix. Numerical algorithms for computation of the scattering characteristics of a generic structure rely on well-posed problems, which have been analytically regularized. The implementation of the final algorithm employ conventional numerical methods and provide results with any required accuracy (the error is estimated analytically). Such property of algorithm is rather important for solving several basic and applied problems. Resonant discontinuities are able to perform efficient ~equency, polariza- tion and space signal selection. These qualities are caused, mainly, by anomalous spectral properties of the relevant open resonators (Shestopalov et al., 1989). This makes it mandatory to use only reliable and mathemati- cally proved methods and algorithms in the numerical experiments. The approach suggested herein and in (Bessonov, 1997) may be efficient in solving model problems of wave acoustics and electromagnetics, wave optics and in the theory of periodic gratings. Only the type of operators L and M in (2.1) restricts the range of solvable problems: the system of transverse eigenfimctions {,u,(~,~)) has to make up a complete basis.

ACKNOWLEDGMENT.

The authors would like to thank Professors S.Strom and 0.A.Tretyakov for their interest to this work and fruitful discussions. The support of the Royal Swedish Academy of Sciences is gatelidly acknowledged.

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REFERENCES.

1. Abramowitz M. and LStegun, "Handbook of Mathematical Functions." New York: Dover, 1972.

2. Bessonov E. G., Yu. K Sirenko, N. P. Yashina," Non stationary model problems of the theory of waveguide open resonators", Zarub. Radioel. Uspehi Sovr. Radioel ( Journal of Advanced Radio electronics). Vol. 14, pp.24-35, December 1997. (In Russian).

3. Kirilenko A.A., V.LTkachenko, "Systematic approach to computer- aided electrodynamic simulation of discretespectrum microwave de- vices." MMET IV Int. Seminar, Alushta, 1991. Kharkov: Test-Radio, 1991, pp. 54-64.

4. Ladizshenskaya O.A., 'Boundary Problems of Mathematical Physics." Moscow: Nauka, 1973. (In Russian).

5. Maykov A.R, A.G.Sveshnikov, S.A.Yakunin, 'Werence schema for nonstationary Maxwell equations in wave guiding systems." Zshurnal Vichisl. Matemi Matem Fisiki". Vol26, pp. 85 1-863, December 1986. (In Russian).

6. Mittra R and S.W.Lee, "Analytical Techniques in the Theory of Guided Waves." The Macmillan Company 1971.

7. Pochanina I.E., V.P. Shestopalov, N.P.Yashina, Wybrid modes of open waveguide cavities. (Numerical and analytical investigation)". English trans. in 'Radiophysics and Quantum Electronics". Vo1.32, pp. 744-752, August 1989.

8. Shestopalov V.P., 'Method of Riemann-Gilbert Problem in W a c t i o n Theory and Electromagnetic Wave Propagation Theory. 'Kharkov: Kharkov State Univ., 1971. (In Russian).

9. Shestopalov V.P., A.A. Kirilenko, S.A.Masalov "Convolution Type Matrix Equations in Difliaction Theory." Kiev: Naukova Dumka, 1984. (In Russian).

10. Shestopalov V.P., 'Dual Series Equations of Modem DifEaction The- ory." Kiev: Naukova Dumka, 1983. (In Russian).

1 1. Shestopalov V.P., L.A. Litvinenko, S.A.Masalov, V.G.Sologub, 'Wave W a c t i o n on Gratings." Kharkov: Kharkov State Univ., 1973. (In Russian).

12. Shestopalov V.P., A.A. Kirilenko, S.A.Masalov, Y.K Sienko, 'Resonant Wave scattering. Vol. I". Kiev: Naukova Dumka, 1986. (In Russian).

13. Shestopalov V.P., A.A. Kirilenko, L.A.Rud' 'Resonant Wave Scatter- ing. VoL II". Kiev: Naukova Dumka, 1986. (In Russian).

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442 Y. K. SIRENKO AND N. P. YASHlNA

14. Shestopalov V.P., YKSuenko, 'Dynamical Theory of Gratings". Kiev: Naukova Dumka, 1989. (In Russian).

15. Sirenko Y.K, "Certain mathematical questions in wave &action on gratings." Kharkov, IRE of AS of Ukrainian SSR, Print No. 103.1978. (In Russian).

16. Sirenko Y.K, V.P. Shestopalov and N.P.Yashina, "Free oscillations in a coaxial-waveguide resonator". English trans. in "Soviet Journal of Communications Technology and Electronics." Vol. 32, pp. 60-67, January 1987.

17. Sirenko Y K , " Upon the prove of the semi inversion method for matrix operators in &action theory." Zshurnal Vichisl. Matem Matem Fiiki". Vol23, pp. 1381- 1391, December 1983. (In Russian).

18. Sirenko Y.K, V.P. Shestopalov, N.P. Yashina, 'New methods in the dynamic linear theory of open waveguide resonators". English trans. in '7. Comput. Maths Mathem Phys." Vol. 37, pp. 845-853, July 1997.

19. Tretyakov O.A., 'Essentials of nonstationary and nonlinear electromagnetic wave theory". In "Analytical and Numerical Methods in Electromagnetic Wave Theory". Ed. by M.Hashioto, M. Idemen and Tretyakov O.A.. Tokyo: Science House Co., Ltd., 1993.

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nonstationary model problems for waveguide open resonator theory

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