Canonical problems and a generalized matrix technique

RADIO SCIENCE, VOL. 38, NO. 2, 8025, doi:10.1029/2001RS002581, 2003

Time domain theory of open waveguide resonators: Canonical problems and a generalized matrix technique Yuriy K. Sirenko Institute of Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkov, Ukraine

Nataliya P. Yashina Institute of Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkov, Ukraine

Received 29 November 2001; revised 12 July 2001; accepted 17 September 2002; published 25 April 2003.

[1] In this paper, we consider new solutions and algorithms for initial boundary-value

problems in the electromagnetic theory of open waveguide resonators. The approaches are based on a description of the scattering properties of such resonators in terms of the transform operators for an evolutionary basis of the non-stationary signal, that have the same significance in the course of analyses in time domain as generalized scattering matrixes in frequency domain. All suggested approaches imply using mathematically correct computational procedures at the key stages of forming the solution, in particular, INDEX TERMS: 0619 the analytical regularization method—the semi-inversion method. Electromagnetics: Electromagnetic theory; 0684 Electromagnetics: Transient and time domain; 0689 Electromagnetics: Wave propagation (4275); 0624 Electromagnetics: Guided waves; 0644 Electromagnetics: Numerical methods Citation: Sirenko, Y. K., and N. P. Yashina, Time domain theory of open waveguide resonators: Canonical problems and a generalized matrix technique, Radio Sci., 38(2), 8025, doi:10.1029/2001RS002581, 2003.

1. Introduction [2] In this paper we describe the complicated and canonic problems and accurate solutions to them that are to create the ‘‘basis’’ for Time Domain (TD) system of mathematical modeling of transmission lines (resonant phenomena based units like converters, filters, hermetic matching windows, multiplexes, changers of waveguide dimensions etc) that is complementary to the developed by authors [Shestopalov et al.,1986a, 1986b; Sirenko et al., 2000; Yashina, 1999; Pochanina et al., 1991] and references to them) in Frequency Domain (FD). The corresponding two parts of algorithm sets (algorithms for analysis in Time and Frequency Domain) being based on rigorous mathematical models, are to create together the unique system, providing simulation of rather complicated, requiring specific accuracy, resonant electromagnetic phenomena that is only possible when carried out in parallel and interacting both in TD and FD. [3] The partial analytical inversion of the operator by regularization of the problem is a powerful classical mathematical tool. Its gradual and well-validated appliCopyright 2003 by the American Geophysical Union. 0048-6604/03/2001RS002581

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cation in course of developing efficient methods of the solution to homogeneous (spectral) and inhomogeneous problems of the wave diffraction theory has allowed to almost solve the problem of an adequate electromagnetic modeling for a number of canonic open resonance structures (gratings, waveguide discontinuities, openended screens; see, e.g. books [Shestopalov et al., 1986a, 1986b] and references to them). The mathematical models of the analytical regularization methods and other rigorous analytical-numerical methods of FD have some incontrovertible advantages had been shown being worth by extensive and intensive studies of the physic nature of the resonance effects and phenomena. Unfortunately they are suitable only for a limited number of idealized scatterers having sufficiently simple geometry. The scope of problems that can be solved with these methods can be considerably enhanced by using the approaches that are based on the algorithms of the method of generalized scattering matrices. Due to these approaches, the complicated structures, whose elements are situated within a regular interaction domain of a finite length, can be efficiently analyzed, just as their separate parts. The interaction is accounted for by solving an additional, elementary, from the point of view computational complexity, problem—the canonical Fredholm set

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SIRENKO AND YASHINA: TIME DOMAIN THEORY

of equations with entries of the matrix operator exponentially decreasing along both the lines and columns. [4] Method of generalized scattering matrix is also well-known in the diffraction theory, its advantages are pronounced most well only if they are thoroughly matched with the algorithms of rigorous analyticalnumerical frequency domain methods, particularly, with the algorithms of the analytical regularization methods that are employed to create libraries of ‘‘elementary’’ units in the advanced modeling systems. [5] Suggesting direct analogues of these powerful methods for solving problems in time domain, we hope that here in TD they will acquire same significance as their prototypes in FD. For being specific we consider as examples initial boundary-value problems for circular and coaxial waveguides and restrain the study to the simplest case of excitation of such structures by symmetrical TE0p electromagnetic waves. Thus, the study of numerical implementation of the solution and several tests of the algorithm that are of principal importance for generic structures, are discussed in the framework of the most popular canonic problem that is a waveguide bifurcation.

2. Evolutionary Basis of the Signal and Transform Operators 2.1. Essential Definitions [6] Let us first consider a model initial boundary-value problem for a certain type of discontinuity in waveguide of arbitrary cross section. (see Figure 1a). Formally this 3D scalar problem corresponds to the propagation of acoustical waves. The results derived in course of its solving can be easily extended for the case of electromagnetic waves propagation. We have chosen it as a model problem only because the 3D vector electromagnetic problems are quite cumbersome for the description of rather general principal questions. In the same time the description of 2D scalar problems would restrict the class of problems that are planned for consideration significantly. [7] Function U (g, t) describing wave scattering process is subject to the equations 8 @2 @ > > > þ L U ð g; tÞ ¼ F ð g; t Þ; t > 0; g 2 Q e ð g Þ s ð g Þ > > @t @t2 > < @ U ð g; 0Þ ¼ jðg Þ; U ðg; t Þjt¼0 ¼ yðgÞ; g 2 Q > > @t > > > > : M ½U ðg; t Þ ¼ 0; t 0 g2Sz

ð1Þ

We assume that the finite in the domain Q functions j(g), F(g, t), y(g), e(g) 1 and s(g) satisfy the conditions of the theorem about single-valued solvability of (1) in the

Figure 1. Modeled problems: (a) certain discontinuity in arbitrary waveguide and (b) compound waveguide discontinuity that is connection of generic structure via regular waveguide section. energy space (Sololev’s space) W12 (QT ), QT = Q (0, T ), T < 1 [see Ladyzhenskaya, 1985]. Here L is the differential elliptical operator of second order that takes in the local cylindrical coordinates xj, yj, zj the form @ 2/@zj2 + LxJ ;yJ ; M is the differential operator not higher than of the first order; Sz is the boundary of domain Q that is constituted by two semi-infinite regular (by zj > 0, j = 1, 2) waveguides that are bound by a compact discontinuity; g = {x, y, z} ; time t has the length dimension; the real functions e(g) 1 and s(g) 0 determine the influence of the discontinuity on the excitation propagation velocity and its dissipative characteristics. [8] In a particular case of an infinite regular waveguide (section int S of the domain Q by the planes zj = const is


constant along the axis z, and e(g) 1 = s(g) 0), let us write the solution to (1) like U ð g; t Þ ¼

X

vð z; t Þmn ð x; yÞ; g 2 Q; t > 0;

ð2Þ

n

where the sequence of functions v(z, t) = {vn (z, t)} satisfy the equations @2 @2 Dðln Þ½vn 2 þ 2 l2n vn ð z; t Þ ¼ an ð z; t Þ; ð3Þ @t @z jzj < 1; t > 0; n 2 fng

vn ð z; 0Þ ¼ bn ð zÞ;

@ nn ð z; t Þ ¼ cn ð zÞ; @t t¼0

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and time are determined exclusively by the system v (z, t) (or {G (ln) * fn}) whose properties allow it to serve as a universal evolutionary basis of any signal on any finite section of a regular guiding structure. [10] Let the wave of the type (2) serve as the excitation wave to the open waveguide resonator P shown in Figure 1a. The excitation field U i ð g; t Þ ¼ n vn1 ðz1 ; t Þmn1 ðx1 ; y1 Þ assumed to be non-zero only in the waveguide A that is regular for all z1 > 0. The field reaches the left boundary of the discontinuity lying in the plane z1 = 0 after the time t = 0. The secondary field that is induced in the regular semi-infinite waveguides A and B and propagates toward the increasing values of z1 and z2 we present like X

Ujs ð g; t Þ ¼

and the initial conditions

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wnj zj ; t mnj xj ; yj ; zj 0;

n 2 fnð jÞg;

n

j ¼ 1; 2: ð4Þ

ð6Þ

Through the relations and {mn (r, f)} and {ln} are sets of the eigen functions and eigen values of the homogeneous boundary-value problem 8

> Lx;y þ l2n mn ð x; yÞ ¼ 0; f x; yg 2 int S > < ; > > M m ½ ð x; y Þ ¼ 0 : n S

that is derived from (1) by separating the cross coordinates x, y (S is the boundary of the domain intS and the supposed properties of operators Lx, y and M are so that the function system {mn (x, y)} constitute an orthonormal basis in the corresponding plane domain). Here an (z, t), bn (z), cn (z) are the Fourier coefficients of functions F (g, t), j (g), (g), by expanding the latter into series over the basis system {mn (x, y)}; and {n} is an ordered set of numbers n. [9] Going to the generalized definition of the Cauchy problems (3), (4), see [Vladimirov, 1971] and using the fundamental solution G(l; z, t) = (1/2) c (t jzj) J0 [l (t 2 z 2 ) 1/2 ] of operator D(l) (by definition D(l)[G(l)] = d(z)d(t)), we yield X

U ð g; t Þ ¼ Gðln Þ * fn mn ð x; yÞ: ð5Þ n

Here the asterisk stands for the convolution operation, fn (z, t) = an(z, t) d(1) (t)bn(z) d(t)cn(z), c is the Heaviside function, Jm—Bessel function, d(m) is the generalized derivative of the delta-function of the order m. Equations (2) and (5) demonstrate the general and specific, corresponding to the given sources, form of wave propagating in a regular waveguide. Its alteration in space

w0nj ð0; t Þ ¼

@ wnj zj ; t jzj ¼0 @zj

Zt X h 0

i 1 2 BA RAA ð t t Þd þ T ð t t Þd nm nm j j vm1 ð0; tÞdt;

m

j ¼ 1; 2

ð7Þ

we introduce the boundary (at the boundaries of the discontinuity zj = 0; dmn is the Kronecker delta) transform operators RAA and T BA of the evolutionary basis of the non harmonic wave arriving from the left, from the waveguide A: w0j ð0; t Þ ¼

n o h i w0nj ð0; t Þ ¼ RAA d1j þ T BA d2j ½v1 ð0; tÞ ;

t 0; j ¼ 1; 2: ð8Þ AA BA The entries Rnm (t t) and Tnm (t t) of these operators define the space-time distribution of the incident wave energy U i (g, t) in the reflected and the passed through the waveguide resonator field. The connection of the domains is defined in following way: the upper index from the domain whose identifier stands on the right to the domain with the identifier on the left. The connection of the modes: inferior index; on the right is the mode number of the incident wave, on the left is the mode number of the secondary field. [11] Obvious that RAA and T BA operating in the domain of the signal’s evolutionary basis are the

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characteristics of the waveguide unit itself and just sum up properly the results of a multiple elementary disturbances that can constitute any primary signal Ui (g, t). Thus, if vn1 (0, t) = dnp d (t h), where p is an 0 (0, t) = RAA integer and h > 0, we have wn1 np (t h) 0 BA (0, t) = Tnp (t h). The use of such an and wn2 abstract, physically impossible signal is justified by the methodological backgrounds, namely through this signal the structure gets elementary ‘‘excited’’ that enables one to extract from the generated field the ‘‘pure’’ compoAA BA (t t) and Tnp (t t) of the transform nents Rnp operators. The elements of the matrix-functions RAA and T BA can be certainly computed in a different way by searching the sequence of values p in the excitations waves with Ui (g, t), with vn1 (z1, t) = dnpvn1 (z1, t), n 2 {n(1)}, where vn1 (z1, t) are now sufficiently arbitrary functions. From (7) follows the relation between the amplitudes wnj0 (0, t) of the secondary field corresponding to the fixed value of p and the sought values: w0nj ð0; t Þ ¼

Zt h

i 1 2 BA RAA np ðt tÞdj þ Tnp ðt tÞdj vp1 ð0; tÞdt;

0

j ¼ 1; 2;

After its inversion using the operation method we obtain i 3 2 h L w0nj ð0; t Þ ðsÞ 1 2 BA 1 4

5ðt Þ; RAA np ðt Þdj þ Tnp ðt Þdj ¼ L L vp1 ð0; t Þ ðsÞ

0

c ðt tÞ zj w0nj ð0; tÞdt; zj 0; t 0;

where L and L1 are the direct and inverse Laplace transforms. There are many ways to implement the corresponding presentation with the minimal error and acceptable computational efforts. Properly chosen function vp1 (0, t) is a crucial factor here. In an ideal case vp1 (0, t) has to have such a form that the necessary integral transformations could be done analytically. [12] The operators RAA and T BA are the boundary ones, they determine all the characteristics of the transient processes directly at the discontinuity boundary of a regular guiding structure. The secondary field, departing from this boundary, propagates ‘‘freely’’ in the semi-infinite regular channels and in course of it becomes deformed. The spacetime amplitudes of the modes {wnj (zj, t)} (signal’s evolutionary basis) varies in time and distance in different way for various values of n and j. These changes can be described by using the diagonal transport operators ZA (z1) and ZB (z2) that are subject to the rule

ð9Þ

ð10Þ

that represent the common feature of the solution to the homogeneous equations of the type (3) at the semi-axes zj 0. The solutions that satisfy the zero initial conditions and are free from any components propagating along the decreasing zj. This formula is derived through the integral Laplace transform over t or the Fourier cosine transform over zj [Maikov et al., 1986]. Actually, on the assumption that at the initial moment of time t = 0 the excitation wave U j (g, t) has not yet reached the discontinuity boundary in the plane z1 = 0, we obtain for the amplitudes {wnj (zj, t)} a sequence of homogeneous initial boundary value problems: # 8" > @2 @2 > 2 > < 2 þ 2 lnj wnj zj ; t ¼ 0; zj 0; t > 0; @t @zj >

@ > > : wnj zj ; 0 ¼ 0; wnj zj ; t ¼ 0; @t t¼0 zj 0;

j ¼ 1; 2;

i h wj zj ; t ¼ wnj zj ; t ¼ Z A ðz1 Þd1j þZ B ðz2 Þd2j h i w0j ð0; tÞ ; j ¼ 1; 2

The structure of these operators is detailed by the formula Z 1=2 2 2 wnj zj ; t ¼ J0 lnj ðt tÞ zj

j ¼ 1; 2;

n 2 fnð jÞg:

ð11Þ

By using in (11) the Fourier cosine transform over zj at the semi-axes zj 0 (image $ original) rffiffiffi Z1 2 ~ ðwÞ ¼ Fc ½w ðwÞ wð zÞ cosð wzÞdz $ wð zÞ w p 0 rffiffiffi Z1 2 1 ~ ð zÞ ~ ðwÞ cosð wzÞdw ¼ Fc ½w w p 0

we obtain the following Caushy problems for the images ~ nj(w, t): w 2 8 1=2

@ 2 2 2 > 2 ~ > ~ nj ðw; t Þ D lnj þ w þ lnj þ w w > > @t 2 > > > 1=2 < 2 ð w; t Þ ¼ w0nj ð0; t Þ ~ w nj > p > > > > > @ > :w ~ nj ðw; 0Þ ¼ 0; ~ nj ðw; t Þ ¼ 0; w 0 w @t t¼0 ð12Þ


Here we had taken into account that the waves Ujs (zj, t) are free of components propagating along the decreasing zj. The components traveling toward zj = 1 at any finite moment of time t = T are zero for sufficiently large zj. ~ (l; t) = c The convolution of the fundamental solution G 1 ~ (t)l sin lt of operator D(l) with the right-hand part of ~ nj (w, t) in the form (12) enables us to write w

1=2

2 ~ nj ðw; t Þ ¼ w p

Zt sin

l2nj

þw

2

1=2

l2nj þ w2

1=2

U ðBÞ ¼ w0nj ð0; tÞdt; w 0; t 0: ð13Þ

@ wnj @zj

zj ¼0

¼

Zt X h 0

AB Tnm ðt tÞd1j

m

ð14Þ

or in operator form

j ¼ 1; 2:

½vn1 ðz1 ðIÞ; t Þ þ wn1 ðz1 ðIÞ; t Þ

X

½wn2 ðz2 ðIÞ; t Þ þ wn1 ðz1 ðIIÞ; t Þ

n2fnðBÞg

mn2 ðx2 ðIÞ; y2 ðIÞÞ; U ðCÞ ¼

X

wn2 ðz2 ðIIÞ; t Þmn2 ðx2 ðIIÞ; y2 ðIIÞÞ:

n2fnðCÞg

The first group of components here corresponds to the waves propagating from the left to the right, the second groups - to the wave moving from the right to the left (see Figure 1b). Following the expressions (7) – (10), (14), and (15), and using the denotations w0j ðIÞ

@ wj zj ðIÞ; t

; @zj ðIÞ zj ðIÞ¼0

wj ðIÞ ¼ wnj zj ðIÞ; t zj ðIÞ¼0 ;

we obtain the following set of operator equations:

2 þ RBB nm ðt tÞdj vm2 ð0; tÞdt; j ¼ 1; 2;

w0j ð0; t Þ ¼

X

mn1 ðx1 ðIÞ; y1 ðIÞÞ;

[13] The above-introduced operators describe on the whole the scattering properties of discontinuities that are being excited from the channel A. Let us determine now, similarly to RAA and T BA, the transformation operators AB of the evolutionary basis of the wave RBB and TP i U ðg; t Þ ¼ n vn2 ðz2 ; t Þmn2 ðx2 ; y2 Þ arriving from the right, through the waveguide B, at the boundary z2 = 0: w0nj ð0; t Þ

U ðA Þ ¼

ðt tÞ

2.2. The Method of Transform Operators For Evolutionary Basis of Signal

zj ; t

frameworks of the above accepted terms, (the obvious changes are due to two different types of discontinuities I and II), we present the solution of the corresponding initial boundary value problem in the regular domains A, B and C in the form

Formula (10) that defines the diagonal transport operators describing the field variation by a ‘‘free’’ propagation of the wave Ujs (g, t) in the finite sections of the regular waveguides A and B we obtain after transforming (13) to the originals wnj(zj, t)).

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n2fnðAÞg

0

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n o h i w0nj ð0; t Þ ¼ T AB d1j þRBB d2j ½v2 ð0; tÞ ;

8 0 w1 ðIÞ ¼ RAA ðIÞ½v1 ðIÞ þ T AB ðIÞZB ðz1 ðIIÞ ¼ LÞ w01 ðIIÞ > > > > > >

> > < w02 ðIÞ ¼ T BA ðIÞ½v1 ðIÞ þ RBB ðIÞZB ðz1 ðIIÞ ¼ LÞ w01 ðIIÞ

> > w01 ðIIÞ ¼ RBB ðIIÞZB ðz2 ðIÞ ¼ LÞ w02 ðIÞ > > > > > >

: 0 w2 ðIIÞ ¼ T CB ðIIÞZB ðz2 ðIÞ ¼ LÞ w02 ðIÞ

ð16Þ ð15Þ

Assuming the sets of operators R and T to be known for separate simple discontinuities, we develop an algorithm for analyzing the scattering properties of a compound unit constituted by such discontinuities. In the situation modeled that is shown in Figure 1b, the unit contains two sequent discontinuities I and II connected by a section B of the regular waveguide of a finite length L, and is excited by a wave of the type (2) arriving at the boundary z1(I) = 0 from the waveguide A. Remaining in the

This set clearly demonstrates all the stages of forming a feedback of a complex structure to the excitation by the P signal U i ðg; t Þ ¼ n vn1 ðz1 ðIÞ; t Þmn1 ðx1 ðIÞ; y1 ðIÞÞ with the evolutionary basis v1(z1(I), t) = {vn1(z1(I), t)}, or simply the signal n1(I). Thus, e.g. the first equation of the set is a sum of signals first of which appears as a result of the reflection of the primary signal n1(I) from the discontinuity I, and the second one is produced by the signal w1(II) that has passed through this discontinuity and the regular space B. [14] By eliminating from (16) one unknown function, we arrive to an operator equation of the second kind with

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respect to the unknown vector-function w20(I) w02 ðIÞ ¼ T BA ðIÞ½v1 ðIÞ þ RBB ðIÞZ B ðz1 ðIIÞ ¼ LÞ

RBB ðIIÞZ B ðz2 ðIÞ ¼ LÞ w02 ðIÞ

ð17Þ

and to the ‘‘recalculating’’ formulas determining all the components of the field generated by the unit. The operator in the right-hand part of (17) influences the sought vector-functions w20(I) whose argument t is strictly less than the time t in the argument of the same function standing on the left (the lag effect that is conditioned by the finite velocity of the signal propagation). Hence, the numerical solution of the final equation can be obtained within the standard scheme of marching over time layers, thus, the initial ‘‘complicated’’ problem is equivalently re-formulated to the form allowing the direct inversion by using the conventional methods of computational mathematics. The complex unit is reduced to the category of elementary basic blocks after calculating the elements of boundary operators by the formulas (7) and (14). [15] Let us return once more to the expressions (7) – (10), (14), and (15) to prevent the eventual questions concerning the definition of boundary transformation operators. The point is that these operator act in a different way if compared with their analogues in the frequency domain. Theoretically, one can choose the traditional way in the time domain establishing relations by the boundary transformation operators, the pair ‘‘field ! field’’ and not the pair ‘‘field ! the field derivative toward the propagation’’, as it was done in (7) and (14). Naturally, in such case we would have obtained RAA pp (t t) = d (t t) instead of, e.g., quite a complicated expression [see Shestopalov et al., 1986a, 1986b] describing the characteristics of the simplest discontinuity ‘‘waveguide end cap’’. Let us consider now more detailed the structure of the transport operators ZA(z1) and ZB(z2) (formulas (9) and (10)). It was the optimal, in terms of the computational costs, scheme of including them into the algorithm of analyzing the complex unit (see (16) and (17)), that predetermined the quite not physical choice made in (7) and (14). The transformation ‘‘field ! field’’ is implemented by the product of the operators ZR and ZT directly, without any intermediate differentiating operator toward the signal propagation direction.

3. Canonical TD Problems [16] The separation of the evolutionary basis of the signal that is qualitatively the same for all guiding structures has determined the domain where the scattering properties of the waveguide discontinuities can be described in terms of corresponding boundary and transport transform operators. These operators allow studying

the details of the general physical situation on the level of partial signal components, and, hence, they prove to be an efficient tool of the qualitative and quantitative analysis of the resonance (irregular) wave scattering processes. The operators considerably simplify the algorithmization of the problems of increasing complexity, which may encase simpler and already solved ones. In addition, they enable one, as it has been done in FD, to construct, using the sets of standard basic units, the systems of electromagnetic unit modeling whatever complicated they might be. The approach can obviously be recognized as an advantageous one; the operation principle and the formal mathematical description can be easily implemented in numerical form and remain unchanged by essential wide-range variations of the waveguiding structures’ configuration and the conditions on their boundaries, as well as by going from the scalar problems to the vector ones. There is only one principal limitation on the applicability of these approaches, namely the type of operators L and M in (1): the basis character of the system {mn} of the eigen cross functions is required. One more serious problem of this approach is associated with the calculation of the values of operators R and T for quite a wide set of simple discontinuities. Obviously this set should be in line with the general objectives of the study and can be substantially different depending on the purpose and the level of complexity of this very model system. Here we consider in details only one of such problems, comprising almost all peculiarities of the approach. 3.1. Statements of the Problem [17] The study of the TE0p symmetric (@/@f 0) electromagnetic waves in cylindrical structures is reduced to solving the following initial boundary-value problems of the type (1) 8 @2 @ @2 @ 1 @ > > U þ ð rU Þ ¼ 0; e ð z Þ U s ð z Þ U þ > > @t @r r @r @t2 @z2 > > > > > > g ¼ fr; f; zg 2 Q; t > 0 > > > < @ i ¼ @ U i ð z; r; t Þ ; > U ð z; r; t Þ U ð z; r; 0 Þ ¼ U ð z; r; 0 Þ; > > > @t @t > t¼0 t¼0 > > > > g2Q > > > > : U ð z; r; t Þjsz ¼ 0; t 0

ð18Þ

Here e(z) is the relative dielectric permittivity of the filling material, s(z) = h0s0(z), h0 = (m0/e0)1/2 and s0(z) is the impedance of free space and the specific conductivity, U(z, r t) = Ef is the only non-zero component of the electrical field strength vector ~ E, Ez = Er = Hf = 0,

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Figure 2. Canonic problem—bifurcation in circular waveguide.

and the non-zero components of the magnetic field ~ are determined by the expressions strength vector H 1 @ @ 1 @ @ H ¼ E ; r f @t h0 @z @t Hz ¼ h r @r irEf . [18] The excitation wave U (z, r, t) is assumed to be non-zero only in the regular semi-infinite waveguide (in the domain A for the structure presented in Figure 2, and in the domain B for those in Figure 3), from which the wave at the time t > 0 reaches the discontinuity. In this waveguide U(z, r, t) = Ui(z, r, t) + Us(z, r, t). The geometry and the variable coefficients in the equations (18) allow presenting their common solution in all the regular partial domains like 0

U ð z; r; t Þ ¼

X

vn ð z; t Þmn ðrÞ;

ð19Þ

n

where the orthonormal in the corresponding plane domain int S (circular or ring-shaped) basisfmn ðrÞg is determined by the set of nontrivial solutions of homogeneous (spectral) problems 8 > < @ 1 @ ðrm Þ þ l2 m ¼ 0; n n @r r @r n > : mn j s ¼ 0

fr; fg 2 int S

; ð20Þ

and the space-time amplitudes vn (z, t) (elements of the signal’s evolutionary basis) satisfy the equation @2 @2 Dðln Þ½vn ¼ ðeð zÞ 1Þ 2 þ sð zÞ 2 ½vn ; t > 0; @t @z n ¼ 1; 2; . . . ð21Þ

The considered problems can be subdivided into three types of domains int S (see Figure 2); two circular ones (r < a and r < b, a > b) and one ring-shaped (b < r < a). The sets {mn}, {ln} that are solutions to the spectral problems (20) are known: i1 r h mn ðrÞ ¼ J1 xn p1=2 aJ0 ðxn Þ ; ln ¼ xn =a; r < a; a r

1=2 ; p a2 G20 ðgn ; 1Þ b2 G20 ðgn ; qÞ mn1 ðrÞ ¼ G1 gn ; a ln1 ¼ gn =a;

b < r < a;

i1 r h mn2 ðrÞ ¼ J1 xn p1=2 bJ0 ðxn Þ ; b ln2 ¼ xn =b ¼ xn =ðqaÞ;

q ¼ b=a;

r < b:

Figure 3. Waveguide open resonator, illustrating application of scattering matrix technique analog in TD.

ð22Þ

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Here Jq and Nq are the Bessel and Neumann functions; xn and gn, n = 1, 2. . . are the strict positive roots of equations J1(x) = 0 and G1(g, 1) = 0, and r r r Gq gn ; ¼ Jq gn N1 ðgn qÞ Nq gn J1 ðgn qÞ: a a a

alently written in the terms of Fourier coefficients of matchingZfunctions in the form: X

Fnmj J0 ½ln ðt tÞ þ J0 lmj ðt tÞ cðt tÞ n

0

8