< Ш<?а х. сообщения объединенного института ядерных исследований 1MB. дубна Е P.Exner, P.Seba

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1 < Ш<?а х сообщения объединенного института ядерных исследований дубна Е P.Exner, P.Seba TRAPPING MODES IN A CURVED ELECTROMAGNETIC WAVEGUIDE WITH PERFECTLY CONDUCTING WALLS 1MB

2 ф Обмарииииый HMcmiyi мирнит ассамдоиммё Ду те, 1Мв

3 1. Introuction The solution of the wave equation in an infinitely I on я curve planar strip was often iscusse in the e leotromagneti о an/or acoustic waveguie literature f 1-6 I. In я J I these works, however, people restricte their attention only i,o the "scattering" soiutione of the corresponing equations, calculating" the transmission an reflection coefficientp. for particular waveguie configurations. It was silently suppose that in a str i p with a constant non-zero with solutions of another type o гю ex ifit.in particular, it was suppose that there are no square. integrable solutions. Actually, this aer.umpt.ion can he mathematically confirme in the case of p,o?in propagation (Neumann bounary conitions on the strip hoima ry). ft seems to be Atkinson f 71 who first showe that the яооия Mo wave equation in a tube of a constant with has a continuous spectrum only. On the other han, from the theory of r.urfaoe waves fa wave equation with mixe bounary con i t i onr f + o--r = 0; a # 0 t <*n we know that trapping moes (square integrable solutions) appear even on the surface of a straight infinite canal - see[8/an also Г9] for the experimental verification ; the name trapping moe" was invente by F.Ursell in Г81- [n this context it seems to be interesting to investigate the existence of the trapping moes insie a bent electromagnetic waveguie with perfect iy conuct ing wa1 Is (Dirichlet bounary conitions). It in the aim of the present paper to show that such solutions really exist in this case, provie the waveguie is bent enough. The wave equat ion Ц + э Ц + k 2 f = о < u &к y Z with Dirichlet bounary conitions on bounaries of an infinite extent has been investigate in the mathematics 1 \iterature for a long time. For instance, for a semiinfinite tube it was shown that the spectrum of к infinitely narrow ut is purely iscrete uhen the tube becomes infinity [10] an purely continuous when the 1

4 tube is conical [11]. i.e., if the outwar normal at every point of the eeniinfinite tube makes an angle not less than ^ with a fixe irection. Ho results are known, however, for the case of a bent tube of a constant with. Recently the possible existence of square integrable solutions of the wave equation was iscusse by Popov [12], who investigate TM waves propagating in a straight waveguie which has a protrusion over a finite length but being otherwise of uniform with. In istinction to this paper we focus our attention to waveguies which are everywhere of a constant with but possibly bent. Our paper is organize as follows: The main result ie formulate in Section 2. Section 3 contains sone concluing remarks, while the proofs of the theorems are sketche in the Appenix. 2.The main results Let us investigate the electromagnetic wave propagating insie an infinite rectangular waveguie with a constant cross-section, the lower an upper walls of which lie in two parallel planes (Fig.l). We suppose the walls of the waveguie being perfectly conucting an we restrict ourselves to the ТИ-type waves only. Because of the constant height of the waveguie the problem can be reuce to investigation of the two-imensional wave equation (1) in a curve planar strip П (being the projection of the waveguie to the lower plane) with Dirichlet bounary conitions [13] f(x,y) = 0 ; (x,y) * 60 (2) on the strip bounary. Here f enotes the z component of the electric fiel, f = E z. He are intereste primarily in the spectral properties of the equation (1) with the bounary conitions (2). It is therefore reasonable to rewrite it in the operator form Др f = k 2 f (3) an to investigate its spectrum applying the methos of functional analysis (borrowe from the Schroeinger operator theory). Here A_ 2

5 ie- the Dirichlet l.aplacian in the ppa<--* L'fO^. i. = it IF Frierichs extension f. 14] of the map f > i 7? + Ы< efine on C f.(0). The strip О is suppose т.-. bt- smooth *n r.f a constant with. Using one of its bounaries as a reference Г, we can introuce natural curvilinear coorinates is,u) as x = a(s) - ub'(s) curw- у = Ь(в) + ua'(a). '''' where a,b are smooth functions characteri?ine the curve Г r={(a(r).b<s));ser*. (g ) We assume, moreover, a'(s) 2 + b'(s) 2 = 1, I7I so s is the arc length of Г an u means the istance of the point <x,y) from Г (Fig.2). The coorinates (s.u) are locally orthogonal. Therefore the metrics in О expresses with respect t, :, then-, through a iag'-.r.ui metric with tensor, 2 x- + y 2 = g e ss^ 4 g u oa,/ 1 1+u/Is)) where y(e) is the signe curvature of the reference curve Г r(e) - b (e)a'(s) - a (s)b is). ilm We suppose that the with of the strip is restricte by the inequality y(s) > -1 for all s <s IP an that the waveguie is bent in a finite region only, i.e. that у <= С (R). 3

6 The first step in proving the existence of the trapping moes is to transform the operator A» to the coorinates (e,u). Using the unitary map 0 : L 2 (0)» L 2 (Rx(0,)) given by <Uf)<e,u) = g 1/4 (s,u)f(e>u) (11) не get after straightforwar ifferentiation that Д~ is unitarily 2 equivalent to an operator A efine on L (Rx(0,)) by the ifferential expression Af = (fe^ 1 fc +^2 ) f + V(e ' u)f < 12 > an by the Dirichlet bounary conitions all в E R. Here f(e,0) = f(e,) = 0 for V(s.u )= I 8-3/^- g - 2 [^] 2 -l g -l^] 2. ( 13) Hence to prove that the wave equation (1) has square integrable solutions, one has to check that the operator A has a non-empty iscrete spectrum. Theorem_l Let us suppose that the strip Ci is bent nontrivially, i.e..that Г (в) is non-zero for some s. Then there is a positive number о such that for each <, A has at least one boun state, о The magnitue of can be estimate by Theore»_2 The operator A hae at least one boun state if Г f f-, + "^ ( 8 ) 2. 1 ein( P )u a < 0. (14) y ( 8 ) 2 J J I <1+и>-(в)Г (1+иг(в)Г J К 0 Proofs of these two Theorems are sketche in the Appenix. 3 DiscussIons Let us now iscuss sone physical consequences of Theorem 1. First of all we have to nention that as far as we know the existence of trapping noes insie a curve waveguie has been overlooke in both the theoretical an aathenatical literature. On the other han, the trapping noes can manifest theaselves in an experimentally verifiable way. In orer to illustrate this we 4

7 woul briefly iscuss the energy transmission through a bent waveguie of a finite length. Mathematically speaking, once we eal with a waveguie of a finite length L the trapping moes isappear from the spectrum turning into resonances the imaginary parts of which approach zero as L > e>. (This can be rigorously proven using the asymptotic perturbation theory an spectral concentration results [14].) Physically it means that the trapping moes woul contribute to the energy transfer through the truncate waveguie. This contribution woul be substantial especially for waves with frequency below the first transversal moe of the waveguie, leaing in euch a way to a resonant peak in the energy transfer plot. The peak is suppose to be sharp for waveguies long enough, when the imaginary part of the trapping moe resonance tens to zero. A similar resonant peak was observe in a soun transmission through bent pipes (see [15] for experimental an [16] for theoretical reeulte). Moreover the peak was localize below the frequency of the secon transversal moe. This ie in a close analogy with our results, since a irect computation shows that for a rectangularly bent waveguie the trapping moe appears 2 I n at к = 0.93 к,, к, = т being the first transversal moe. 1 l a Let us now comment briefly on Theorem 2. Using it one can obtain simple bouns on the with below which at least one trapping moe appears. For a simply bent strip ( i.e., г(в) i 0 for all s e R ) we fin that where o s 277 ( / i ^ V ^ - i ) < 15 > _. z = f J(r'(s)) 2 s ] J г 2 (в) e (16) К IP an у = шах y( s). This boun showp, i hat tht inverse of the + о vritical with ie of the sane orer of magnitue as the maximal curvature r + Appenix Before proving Theorems 1 an 2 we formulate a simple Lemma concerning the operator H^ = - JL + XV on L 2 {Rx[0,]), where Ap is the Dirichlet Laplacian. 5

8 Lemma Suppose that ViKxTO.] > such that-.1 К is a boune an measurable function J J Vlx.vl 11 ( x' I x y <». (17) m о Then H. has at least one. houn state E(M below the bottom of the 2 2 continuousfipeet.rum, E(X) < I n / ), if J J V(x.yl sin"(g y) x y < oo. (18) 0? CI Proof:We use the Birman-Schwinger principle Г14] accoring to which E(M Is a boun state of Нч if an only if the operator >^K 2 has an eigenvalue -1 for о = E(X). Here К is an integral operator with the kerne] K a(x.y.x'.y I - V(x.yl,/2 R 0(a,x.y,x.y - )V(x',y ) 1 / 2, (19) a 2-1 where R ( J a,... ) is the kernel of ^n^^ = ( " л г> ~ a * V ' = V sgn V. Separating the variables: we can ecompose a n Б ()(а,х,у,х,у' ) = ^ *п ( у ' r n n = l ( a ' x ' x < y ' * n ' ( 2 C ) ) where * (y) =(2/) sin (-3 у ) is the n-th transversal-moe wavefimction an г (а,х,х') is the kernel of x' Using (20) we can ivie the kernel (19) into two parts K a = M a L, (21) where,/o " exp(-k (a) x-x ),, M (x,y,x,y') = V(x,y) V"-} X <y> * (y )V(x,y y" 1 *. " 2k (c.) n n=2 n,. exp(-k.(a) x-x' ) - 1, + V(x,y) \ w x.(.y) *,<y')v(x',y' ) ' ; (22) i 2kj<<*) 1 l L a(x,y,x',y') = 2 k 1 ( a ) V(x,y) 1/2 * 1(y)a: 1(y-) V(x,y) 1 / 2 a n.2 2,,1/2 v«> = ( n -J--* 2 ) 6

9 The operator M a is boune for a e [0,n/]. Therefore for a «[0,n/] an x sufficiently small не have so Hence XK II XM II < 1 (23) 01 (l+xk e) _1 = t l+x(l+xm a) -1 L a }" 1 (1+ХИ а)" 1. (24) has the eigenvalue -1 iff the same is true for X(1+XM )~ L. This operator is, however, of rank one an has therefore just one non-гего eigenvalue (X) given by f ( X > = ЯГТЬО -f J" У < Х 'У> 1/2^1<У> [(l+^m a) _1 V 1/2 a: 1](x )y)xy. (25) 1 R 0 Using the formula (25) we fin for X ? ( M = 2k X (a) [ X i" V<x,y)Zj(y) 2 xy + 0<X) j (26) an Bolving the equation?(x) = -1 we get finally 4 1(X) = - J J V(x,y)* 1(y) 2 xy + 0(x 2 ). (27) Bt 0 Moreover k,(x) correspons to a boun state of H^ iff k-(x) > 1, which proves the Lenna Proof of ТЬеогею 1: We estinate first the operator A from above by an operator A where r_ = inf r(e). 1» 2» 2 ~, - Z-s + V(e.u) (28) (l+rj*»s ЛГ The function V(s,u) fulfils the conition (17) of the Lenna. We fin in such a way that A has at least one boun state if J J" V(s,u) ein 2 ( j"-) s u < 0 (29) R 0 which is obviously true for snail enough. Since A S A, an both A an A + have the sane continuous spectra, the existence of the boun state of the operator A state of the operator A. inplieb the existence of the boun Proof of Theore» 2: As alreay nentione in the proof of Theore» 1, we have only to check (29) an to apply the Lenna. The part of V(e,u) containing 7

10 secon cterivat. i ve of *( s ) may be integrate by parts, 1 els the con it ion of t-.he. Theorem. which erences : Campbell J.J., Jones W.R.:I EE Transactions on Microwave Theory an Technique, MTT 16 (1968) 517 Kirilenko A. A., Ru.i L.A., Tkatchenko V. Г. : Raiotechnika i Elektronika 30 (1985) 918 {in Russian) Kirilenko A.A., Pu.i L. A.. Shestopalov: Priklanaja Elektroti ika, Vyscha.ia Shkola, Moscow 1978 {in Russian) Kirilenko A.A., Ku.i L.A.,ShestopaJov: Raiotechnika i Klektronika 4 (1974) 687 fin Russian) Mehran R: leett Transactions on Microwave Theory an Technique MTT 26 ( 1978) 4ГЦ) Mareuritz M.: Waveguie Hanbook, Dover Publ,, New York 1964 Atkinson K.V.: Phil.Mag.40 (1949) 645 (free И F.: Pmc.Camb. Phil. Soc. 47 (1951) 347 Ursell p. : Hror>.Rny. Яос. 214 ( 1952) 79 He J1 ich К. : i n Stuies an Essays Presente to R. ','ourant; New York, 1946 Jones D.S.: Proc.Camb.Phii.Soc. 49 (1953) 668 Popov A.N.:.Journal of Technical Physics 56 (1986) 1916 fin Russian) Lew in L.: Theory of Waveguies; Butterworth Publ. Lonon 1975 Fee M.,Simon В.: Methos of Moern Mathematical Physics vol.4, Acaemic Press. New York 1978 Lippert W.K.k.: Acustica 5 (1955) 274 Kacenelenbaum IB. Z. : The Theory of Nonregular Waveguies with Slowly Varying Parameters; Moscow 1961 (in Russian) Petlenko V.A, Khizhniak N. A. : Izvesti.ia VIIZ 21 (1978) 1325 (in Ruasian ) Receive by Publishing Department on May 30,

11 Экснер П., Шеба П. Существование связанного состояния в изогнутом волноводе с идеально проводящими стенками Е Пользуясь методами, разработанными в теории операторов Шредингера, мы показываем, что волновое уравнение для изогнутого электромагнитного волновода с идеально проводящими стенками имеет квадратично интегрируемые решения /запертые моды/. Работа выполнена в Лаборатории теоретической физики ОИЯИ. Сообщаю* Объединенного института ядерных исследований. Дубна 1988 Exner P., Seba P. Trapping Moes in a Curve Electromagnetic Waveguie with Perfectly Conucting Walls E Using methos evelope in the Schroeinger operator theory we show the existence of square integrable solutions (trapping moes) of the wave equation insie a cur ve electromagnetic waveguie with perfectly conucting walls. The investigation has been performe at the Laboratory of Theoretical Physics, JINR. of ЙМ Mat fiihihif for Nuclear Reeearcfc. Dttbna 1988

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