FRESNEL FORMULAE FOR SCATTERING OPERATORS

Transcription

1 elecounications and Radio Engineering, 70(9): (011) MAHEMAICAL MEHODS IN ELECROMAGNEIC HEORY FRESNEL FORMULAE FOR SCAERING OPERAORS I.V. Petrusenko & Yu.K. Sirenko A. Usikov Institute of Radio Physics and Electronics, National Acadey of Sciences of Ukraine 1, Acadeician Proskura St., Kharkiv 61085, Ukraine Address all corresondence to Yu.K. Sirenko E ail: yks@ire.kharkov.ua For the scalar roble of ode diffraction on the abrut waveguide discontinuity the Fresnel forulae for the reflection and transission atrix oerators are derived using the odearching techniue. his generalized for of the atrix odel is an iediate corollary of the roosed new stateent of the roble. Making use of the energy conservation law in oerator for, the correctness of the obtained Fresnel forulae for the scattering oerators is roved analytically. hus, the develoed aroach akes it ossible to substantiate coletely the widely used ode atching techniue for the class of diffraction robles under consideration. KEY WORDS: ode atching techniue, ode diffraction, Cayley transfor 1. INRODUCION he reflection of the tie-haronic lane wave fro the infinitely thin lane boundary S of two different aterial edia (i.e., fro a ste-like shift or an abrut discontinuity of electroagnetic roerties of sace) is discussed thoroughly in the ajority of anuals (here we restrict all ossible references to the fundaental work [1]). An aroriate atheatical law found by A. Fresnel for transverse waves of an elastic ether follows straightforwardly fro the boundary conditions E1 E ; on the boundary S, (1) H1 H also known as the atching conditions for tangential coonents of the electrical and agnetic hasors of the freuency. he Fresnel forulae take a articularly sile for under the noral incidence of the wave on the boundary S ; for instance, they can be written as /11/$ by Begell House, Inc. 749

2 750 Petrusenko & Sirenko d 1 d E R ; ; d 1 d 1 d 1 H () for two ossible E and H olarizations with resect to the observation lane. Here R is the reflection coefficient in the first ediu and denotes the factor of wave 1 transission fro the first ediu to the second one, whereas 1 is the 1 relative wave iedance/adittance. (Note the roerty of Es. (), in the substitution, which rovides for the forula for the reflection coefficient in the second ediu). It would aear reasonable that the law of the for as given in () will occur for all henoena of wave diffraction when the atching conditions (1) are et and the Poynting vector of the incident wave is noral to the lane boundary of the ste-wise variation in the waveguiding structure roerties. he goal of this aer is to substantiate rigorously this guess for the roble on N -furcation of a generalized waveguide, which siulates the class of scalar robles on ode diffraction by the abrut waveguide discontinuity. Relations (1) can also be regarded as the initial eualities of the ode-atching techniue widely used in coutational electroagnetics. herefore in this aer we assert that for the considered class of wave diffraction robles the atheatical odel of the ode-atching techniue can be written in the for of the Fresnel forulae for the reflection and transission atrix oerators. However, note that the coonly known version of the ode-atching techniue (resented, for exale, in the book []) results in the infinite systes of linear algebraic euations in which identifying the Fresnel forulae is uite a challenge. herefore, at first we need to carry out the rocedure of generalizing the odeatching techniue. In this ualitative odification of the ethod, the key oint is that the unknown vector of the Fourier coefficients is relaced by the atrix scattering oerator sought for. In the alied electrodynaics this constructive idea had aeared to be consistently realized for the first tie in the work [3]. his aroach will be subseuently referred to as the atrix oerator techniue [4]. he found Fresnel forulae in oerator for lead to the far-reaching conseuences. In this aer, the otentialities of the roosed aroach will be deonstrated with the solution to the actual roble of rigorous substantiation of the atrix odel for the ode diffraction on the abrut discontinuity in the waveguide.. GEOMERY OF HE PROBLEM AND HE MACHING CONDIIONS FOR HE PHASORS Consider the abrut N -furcation of a generalized rectangular cross-section waveguide by the syste of sei-infinite erfectly conducting finite-thickness screens. We will elecounications and Radio Engineering

3 Fresnel Forulae for Scattering Oerators 751 describe the geoetry of the roble in the orthogonal coordinate syste,,, which is a generalization of the -D Cartesian xz,, and olar,, fraes in the lane 0. In this coordinate syste, the considered regular waveguide is a hoogeneous one along the Cartesian axis O, whereas its erfectly conducting walls are fored by two coordinate surfaces const1, searated by the interval. Assue that the waves roagate along axis O, which is directed towards furcation. An s -th screening layer that slits the regular waveguide is created by the coordinate surfaces of const tye; it has a thickness of s, s1, N 1. An interval between the screening layers,, N 1 is the width of -th regular waveguide which is geoetrically siilar to the generalized waveguide under consideration. In the general case the second or/and N 1 -th waveguide ay be non existent; then we have an additional ste discontinuity. A single reference lane is laced to the discontinuity lane N1 N1 S, 0 ; s1. (3) s1 he tie deendence exi t is oitted throughout. Let us consider the diffraction of LM -odes, 1,,..., with -coonents of the field H 0, E 0, or of LE -odes, n 0, 1,..., with E 0, H 0, which corresonds to the H -lane and, accordingly, to the E -lane roble. In the -th waveguide, 1, N 1, the travel of the -th ode in the ositive direction of axis O 0 is secified by the exonent ex n1 1, where is the roagation constant such that Re 0, I 0 or Re 0, I 0. (0)1 Next, let be the colun-vector of the real-valued transverse eigenfunctions of the -th waveguide. heir basic roerties are briefly described as ;, δ η- η I. (4) Here the Dirac delta has been used, the suerscrit denotes the transosition; the corresonding arentheses sybolize the integration over the interval and I is the identity atrix oerator (the ide-factor). Next, let the scalar function U denote the sought-for hasor, which deterines in the -th waveguide all the coonents of the electroagnetic field Volue 70, Nuber 9, 011

4 75 Petrusenko & Sirenko whose source is in the -th waveguide,, 1, N1. It would aear natural to assue that this source generates the field which is a colete set of odes LM or LE with any rescribed distribution of colex alitudes that 0 1 n 1 n0 we will collect into the row-vector hasor in the for b b (0)1. Let us reresent the south-for U b u, (5) ( ) where each eleent of the colun-vector u u (0)1 is the solution to the resective boundary-value roble in the -th regular waveguide. Naely, the ( ) function u has to satisfy (a) the -D Helholtz euation; (b) hoogeneous boundary conditions on the surface of erfect conductors; (c) the condition at infinity for waveguides and (d) the condition of energy boundedness in any closed volue inside the doain of field deterination. he validity of exression (5) stes obviously fro the linearity of the boundary-value roble under consideration. he field continuity conditions on the coon boundary of the first and -th,, N 1, waveguides have the for of eualities: U1 U; U1 U ; ; 0, (6) whereas on the reaining arts of reference lane (3) the hoogeneous boundary conditions are satisfied: LM od es U1 0; s, 0; s1, N 1. LE od es U1 0; (7) In eualities (6) and (7) the sybol / stands for the differentiation oerator. Substituting the forula (5) into Es. (6) we obtain the relations that hold true for all b. We then iediately arrive at the following eualities: u1 u; 1, N 1 ;, N 1, 1 ;, 0 (8) u u which reresent the sought-after atching conditions for hasors written as (5). Along siilar lines, we deduce fro (7) that the vector function or its derivative u u1 1 elecounications and Radio Engineering

5 Fresnel Forulae for Scattering Oerators 753 satisfies the hoogeneous boundary conditions on the face ends of erfectly conducting screens. 3. FRESNEL FORMULAE IN OPERAOR FORM In corresonding regular waveguides, the unknown function u can exlicitly be exanded into a series according to orthonoral waveguide odes [4,5]. In this case, the vector nature of this function dictates the eergence of atrix scattering oerators in these exansions. hus, we arrive at a new following forulation of ode-diffraction roble. he finite-ower wave is scattered by a given discontinuity in the waveguide; the field of this wave is an infinite set of odes with any known distribution of alitudes. It is necessary to find the atrix oerators of ode reflection and transission. For the roble in uestion we write at once the reuired relations on the reference lane (3): u 1/ I RI ; ; ; (9) 0, s ; s1, N 1; ( LM),0 1/ I ; ; 1; 1 1/ I RI ; ; 1; 1; 0, s ; s 1, N 1; ( LE). u,0 1; 1 1/ I ; ; ; (10) Here we have introduced a reflection atrix oerator in the -th waveguide R : and a atrix oerator of ode transission fro the -th waveguide into the -th waveguide :. In forulae (9) and (10) the diagonal oerator 1/ I n 1/ n 01 is defined under the condition that the cut-off oints (for which 0 at certain ) are absent. Here n is the Kronecker delta. Substituting (9) and (10) into the atching conditions (8), using the Galerkin rocedure and taking into account the roerties of eigenfunctions (4), we coe to a foral solution of the roble in ters of scattering oerators: Volue 70, Nuber 9, 011

6 754 Petrusenko & Sirenko D1 I ; 1; LM D1 I R R R; LE ;, I D (11) where 1 D1 I D0; 1; LM ;, D (1) ;, ; LE 1/, 1/ ; LM D0 I1 1 I LE N ; 0 1 D 0 D D D D D D I. An exression for the reflection oerator in the -th waveguide, recast in a ore convenient for:, can also be R D D ; I LM. (13) I LE Here the new oerator D,, is connected to the oerator D by the following relations ˆ 1 1 D D I D D D D I. In the secial case, N 1, 1, which is corresond to the canonical roble of ode diffraction on a ste discontinuity in the waveguide, the solution (11) and (1) takes a rather sile for: 1 DD 0 0 I R ; 1/ 1/ LM DD 0 0 I D0 I1 1, I1 ;, (14) 1 1 LE 0 0 D D I D0; which is siilar in for with the Fresnel forulae (). In addition, note that in order to find the scattering oerators in the -nd waveguide it would suffice to ake substitutions D0 D, 0 in forulae for scattering oerators (14) (with the above-entioned analogous roerty of Es. ()) ˆ elecounications and Radio Engineering

7 Fresnel Forulae for Scattering Oerators 755 hus, a new stateent of the considered ode-diffraction roble leads to the solution in the for of the Fresnel forulae for scattering oerators (11)-(14). As will be aarent fro the next Section, the existence and boundedness of all aforeentioned inverse oerators is guaranteed by the ower conservation law. 4. CORRECNESS OF HE OPERAOR FRESNEL FORMULAE We validate the correctness of the develoed atheatical odel by ilying that there is soe roof of existence and uniueness of solution (11)-(14) as well as of its stability on a set of bounded atrix oerators defined in the Hilbert sace. Note that for the finite value of 1 0 a air of linear-fractional transforation forally follows fro the first Fresnel forula (): R 1 1 R, (15) 1 1 R fro which in its turn a two-sided ilication follows: Re 0 R 1. (16) 1 In electrodynaics ters, these ineualities are in agreeent with the energy condition for ordinary assive edia 0; 0 and signify that the nuerical Fresnel forulae (), (15) are correct (i.e., 1 1, R 1). Let us show that the correctness of obtained oerator Fresnel forulae (11)-(14) also follow fro the fundaental energy law. Our roving is based on the reviously established (see [4]) a certain duality of roerties of the oerators, which for the first Fresnel forula (11) or (13). Naely, if for the given atrix oerator the localization of its sectru D D is unknown, then it is just the basic characteristics of the entire sectru R of the sought-for reflection oerator R are coletely defined by the generalized ower conservation law [6]. he interrelation of these two oerators in the for of the first Fresnel forula akes it ossible to find all of their reuired roerties. heore 1. he Fresnel forulae (11)-(14) are the colete and consistent solution of the considered roble of ode diffraction on the abrut discontinuity in the waveguide. Proof. It aounts to substantiating the condition 1 D, which is euivalent 1 :, 1, N 1. to the existence of the bounded oerator 1 D Volue 70, Nuber 9, 011

8 756 Petrusenko & Sirenko he diagonal blocks of the generalized ower conservation law (forula (0) fro [6]) yield the following relation: I R U I R U, 1, N 1, (17) N 1 1, where the dagger is for Heritian conjugation. It follows fro the definition of (0)1 the craed unitary oerator ex arg n i U that its nuerical range lies coletely within the fourth uadrant of the colex lane. Hence, all the colex nubers of the tye N 1, 1, b U b b are also belong to the sae uadrant. It then follows fro (17) that the sectru R lies within the unit disc and each nonreal oint of this sectru is an eigenvalue of finite ultilicity [6,7] (see also [,4,8]). Hence, there exists the Cayley transfor, which in Weyl s notation takes the for [9]: I R W. (18) I R As corollary of the sectru aing theore (see, e.g., [10]) the sectru of the oerator W lies entirely within the right-hand half-lane, Re 0, W. he aarent relation D 1 W1, 1; LM 1 (19) W,, LE coletes the roof. heore. he oerator D is an accretive one, whereas the reflection oerator is a contraction, N R 1, 1. Proof. he two-sided ilication identical-in-for with E. (16) Re W 0 R 1 is the basic roerty of the Cayley transforation (18) (e.g. see [7]). herefore it would suffice to rove the accretiveness of oerator D, i.e., Re D 0. elecounications and Radio Engineering

9 Fresnel Forulae for Scattering Oerators 757 In ters of the Cayley transfor the energy law (17) takes the for: N 1 1 WU W I U W I. (0) 4 1, Fro this euality it follows that the nuerical range of oerator WU lies entirely within the fourth uadrant of the colex lane. he literally relicated roof of theore fro the aer [7] results in euality Re W 0. Identity (19) coletes the roof. he stability of the found solution (11)-(14) is established by the following heore 3. he oerator 1 Proof. hrough direct calculation we obtain A ID is an accretive contraction: Re A AA. Re A AA A ReD A 0 as a corollary to the accretiveness of oerator D. 5. CONCLUSIONS he H E lane N 1 -ort junction of the waveguides of rectangular crosssection, which are regular in the generalized frae above, has been analyzed via the ode-atching techniue. A new stateent of the roble of ode diffraction on the discontinuity in the waveguide has been forulated. his forulation is as follows. he finite-ower wave is incident uon the waveguide discontinuity. he field of this wave consists of an infinite set of odes with any known alitude distribution. here are just the scattering oerators to be found. As a direct result of this stateent of the ode-diffraction roble, the atrixoerator odel in a erfect for of Fresnel forulae for the reflection and transission oerators has been obtained. It has been found that the fundaental law of energy conservation ilies the correctness of the derived Fresnel forulae in oerator for. hus, the unsettled roble of justification of the atrix odel of the ode-atching techniue has been solved for the considered class of diffraction robles. Finally, owing to a new stateent of the roble of ode diffraction the atrixoerator nature of the ode-atching techniue has been clarified. REFERENCES 1. Born, M. and Wolf, E., (1999), Princiles of otics: Electroagnetic theory of roagation, interference and diffraction of light, Cabridge: University Press, Volue 70, Nuber 9, 011

10 758 Petrusenko & Sirenko. Mittra, R. and Lee, S.W., (1971), Analytical techniues in the theory of guided waves, New York, he Macillan Coany, Shestoalov, V.P. and Shcherbak, V.V., (1968), Matrix oerators in diffraction robles, Izvestiya VUZ. Radiofizika, 9():85-95 (in Russian). 4. Petrusenko, I.V., (004), Analytic nuerical analysis of waveguide bends, Electroagnetics, 4: Petrusenko, I.V., (006), Basic roerties of the generalized scattering atrix of waveguide transforers, Electroagnetics, 6: Petrusenko, I.V. and Sirenko, Yu.K., (009), Generalization of the ower conservation law for scalar ode-diffraction robles, elecounications and Radio Engineering, 68(16): Petrusenko, I.V. and Sirenko, Yu.K., (008), Abrut discontinuities: the reflection oerator is a contraction, elecounications and Radio Engineering, 67(19): Shestoalov, V.P., Kirilenko, A.A., and Masalov, S.A., (1984), Convolution-tye atrix euations in the theory of diffraction. Naukova Duka, Kyiv: 96. (in Russian). 9. Weyl, H., (1997), he classical grous: heir invariants and reresentations, Chichester, Princeton University Press, Richtyer, R.D., (1978), Princiles of advanced atheatical hysics. Vol. 1, New York Heidelberg - Berlin, Sringer-Verlag, - 4. elecounications and Radio Engineering