Scattering of TM waves by an impedance cylinder immersed halfway between two half spaces

Transcription

1 Scttering of TM wves by n impednce cylinder immersed hlfwy between two hlf spces Aldin H. Kmel(l@urech.com) PO Box 433 Heliopolis Center 757, Ciro, Egypt Scttering of TM wves by n impednce cylinder hlfwy immersed between two hlf spces of different electromgnetic properties hs been studied. Solutions re obtined from n ppliction of discrete index of Hnkel function trnsform. Expressions for the fields in both hlf spces re given. Introduction: To investigte the fetures of vrious medi by mens of electromgnetic rdition it is necessry to know the field scttered by inhomogeneties of these medi. This problem cn be tckled by using s bsis rigorous solution of bsic structure. One of the bsic structures is the one considered in this letter. The problem under considertion hs lso cquired prcticl relevnce in fields of opticl engineering such s the study of contminted surfces nd the detection of defects. Additionlly the solution of cnonicl problems such s the one under considertion is importnt in the sense of scttering nd diffrction theories. The im of this letter is to present solutions, in terms of discrete index of Hnkel function trnsform, to the problem of the scttering of TM wves by circulr impednce cylinder immersed hlfwy between two hlf spces of different electromgnetic properties. Other configurtions of cylinder nd two hlf spces hve been delt with before in the literture (references [,] using Fourier series expnsions on the ngulr vrible nd reference [3] using integrl equtions methods) but the one discussed in this letter, to the best of the uthor s knowledge, hs not been ddressed before. Formultion: We look into the problem of scttering by n impednce cylinder of surfce impednce Z s,rdius with hlf of which in medium one nd the other hlf in medium two. The interfce between thetwomediisthey =0plne. The xis of the cylinder is the xis of the coordinte system. The cylinder is infinite in the direction. k,², nd µ re respectively the wve number, permittivity, nd permebility in medium one. k,² nd µ re corresponding quntities in medium two. Continuity of tngentil field components pply t the interfce between the two medi nd impednce boundry conditions pply on the cylinder surfce.

2 Atimehrmonicinfinite electric line current source (in the direction locted t (ρ 0, φ 0 ) in medium one provides the illumintion. Time dependence e iωt is ssumed nd suppressed from the nlysis. It should be pointed out tht other forms of excittion re dmitted. To pss from result derived from unit strength line source locted t (ρ 0, φ 0 ) to the result for unit mplitude plne wve incident long the direction φ 0,onefirst lets ρ 0 in the fields expressions then sets q k ρ 0 [ e i(k ρ ) =]. The fields due to bem excittion re derived 4 from those of plne wve excittion by ttching profile to the incident plne wve nd mking use of the superposition principle. By using the symmetry of the problem structure with respect to the plne ϕ = ±/, we split the problem into two independent subproblems. The boundry conditions on the symmetry plne correspond to either perfect electric conductor (PEC) or perfect mgnetic conductor (PMC). So without loss of generlity, we confine our ttention to thecseofpecwllonly. Mthemticl model: We propose to solve the problem by mens of discrete index of Hnkel function trnsform. The trnsform pir is given by [4] f(ρ) = P p A p Φ p (kρ) A p = R f(ρ)φ ρ p(kρ)dρ Φ p (kρ) ={ i ν pb(ν p ) [ ν d(ν)]ν p b(ν) =Jν(k)+iCJ 0 ν (k) d(ν) =H 0() ν (k)+ich ν () } / H () ν p (kρ) (k) J ν () is Bessel function; Jν() 0 = d J d ν(); H ν () of type one; H 0() ν () = d d H() ν () is Hnkel function () ; C = Z/Z s ; d(ν p )=0for {ν p } locted in the first qudrnt of the complex ν plne; nd Z = q µ/² the medium impednce. Pssivity requirement is met if Re C º 0. Such n index trnsform hs been used before [4] to nlye diffrction by n impednce cylinder in free spce. We represent the electric fieldinthe direction in medium one, E (), s the sum over E (d) which ccounts for the source discontinuity (in the φ direction) nd n dditionl field E (r) which ccounts for the rest of the field. In medium two we represent the electric fieldinthe direction s E (). E (d) = X p [ sin ν p( φ >)cosν p φ < Φ p (k ρ 0 )]Φ p (k ρ) () ν p cos ν p E (r) = X p A (ν p )sinν p (φ )Φ p(k ρ) ()

3 E () = X p A (ν p )sinν p (φ + )Φ p(k ρ) (3) From E the rest of the field components re derived H ρ = iω² E k ρ φ H φ = iω² E k ρ The continuity of the tngentil fields on the interfce between the two medi leds to E (d) + E (r) = E () t φ =0 ² ( E(d) + E(r) )=² φ φ E() t φ =0 φ Pp{ sin ν p( φ0 ) ν p cos ν p Φ p (k ρ 0 ) A (ν p )sinν p } Φ p(k ρ)= P p A (ν p )sinν p Φ p (k ρ) Pp ν p A (ν p )cosν p Φ p(k ρ)=r P p ν p A (ν p )cosν p Φ p(k ρ) with r = ² ² The impednce boundry condition E = Z s H φ on the surfce of the cylinder is built in the eigen functions Φ p (k, ρ). Thisisonedvntge of using the discrete index of Hnkel trnsform for problems with boundries long ρ = constnt. We utilie the orthogonlity reltion of the ρ eigen functions to rech sin ν q ( φ0 ) ν q cos ν q Φ q (k ρ 0 ) A (ν q )sinν q = P p A (ν p )sinν p C pq q ν q A (ν q )cosν q = r P p ν p A (ν p )cosν p C pq q C pq = R Φ ρ q(k ρ)φ p (k ρ)dρ We cst the liner system s S D[sin ν ]A = CD[sin ν ]A D[ν cos ν ]A = CD[rν cos ν ]A where S is the vector S = { sin ν q( φ0 ) ν q cos ν q Φ q (k ρ 0 )}; D[.] re digonl mtrices with digonl elements [.] nd A, re the vectors of spectrl mplitudes. From the bove system we derive A = M S, A = D [rν cos ν ]C D[ν cos ν ]A where M = D[sin ν ]+CD[sin ν ]D [rν cos ν ]C D[ν cos ν ] It is not difficult to show, from the symptotic expnsion of Bessel nd Hnkel functions s q [4], tht the series representtions in equtions (-3) converge exponentilly with order of the p th term O(e iν p φ φ 0 ) for eqution () nd O(e iν p( φ +φ 0) ) for equtions (,3). The only exceptions re when φ = φ 0 in equtions () nd when φ = φ 0 =0 in equtions (,3) where the corresponding series diverges. A remedy is given in the next section. 3

4 Hd we used Fourier expnsion on the ngulr vrible insted of the discrete Hnkel index trnsform, the lck of n orthogonlity reltion for the Hnkel functions of integer order would hve resulted in the ppernce of dditionl dense mtrices of the form R q (k ρ)h p () (k ρ)dρ, R ρ 0 ρ H() ρ H() q (k ρ)j p (k ρ)dρ nd R ρ 0 ρ H() q (k ρ)h p () (k ρ)dρ in the liner system; nd the convergence of the series is lgebric. These re other dvntges of using the discrete Hnkel index trnsform for problems with boundries long φ = constnt. The isovelocity cse: It is lwys instructive to look into the isovelocity cse (k = k )wherein,forz s =0or, ν p = ν p, C pq turns digonl nd the liner system simplifies to sin ν q ( φ0 ) ν q cos ν q A (ν q )=ra (ν q ) A (ν q )= +r Φ q (k ρ 0 ) A (ν q )sinν q = A (ν q )sinν q sin ν q ( φ0 ) ν q cos ν q Φ sin ν q q (k ρ 0 ) leding to the bove closed form nlytic expressions for the coefficients A, (ν q ). Therefore E () = +r E (r) = r +r For E (d) P q P q sin ν q ( φ0 ) ν q cos ν q sin ν q sin ν q ( φ0 ) ν q cos ν q sin ν q sin ν q (φ + )Φ q(k ρ 0 )Φ q (k ρ) sin ν q (φ )Φ q(k ρ 0 )Φ q (k ρ), if one must compute the fields long φ = φ 0 where the series representtion diverges nd is not, relying on field continuity, stisfied by interpolting from neighboring points then convergent representtion for φ = φ 0 is derived by ) mnipulting the summnd prt [ sin ν p( φ > )cosν pφ < ν p cos ν p ] into [ 4 {ieiν p φ φ 0 ie iν p cos ν +e iν p p(φ φ 0 )+ sin ν p( φ φ0 ) cos ν p }]. The second nd third terms result in convergent series for ll observtion ngles; b) pplying Wtson trnsform [5] on the prtil series of the first term reduces it to i 6 [R γ eiν φ φ0 {H ν () (k ρ < )H ν () (k ρ > ) H ν () (k ρ)h ν () (k ρ 0 ) H() ν (k ) }dν] H ν () (k ) with γ the contour round the eros of d(ν) =0in the first qudrnt of the complex ν plne. After symptotic evlution, the integrls re recognied s the geometricl opticl incident nd reflected fields respectively. The sme tretment is pplicble to E (r),() when φ = φ 0 =0. The presented solution method is extendble to other geometries: cylinder in the vicinity of two hlf spces nd cylinder less more thn hlfwy buried. These nd extension to source excittions leding to coupled TE TM polritions will be presented elsewhere. References 4

5 RAO, T.C. nd BARAKAT, R.: Plne wve scttering by conducting cylinder prtilly buried in ground plne.. TM cse, J. Opt. Soc. Am., 989, 6, (9), pp RAO T.C. nd BARAKAT, R.: Plne wve scttering by conducting cylinder prtilly buried in ground plne II. TE cse, J. Opt. Soc. Am., 99, 8, (), pp SAIZ, J.M., VALLE, P.J., GONZALEZ, F., ORTIZ, E.M., nd F. MORENO,: Scttering by metllic cylinder on substrte: burying effect, Optics Letters, 996,, (7), pp FELSEN, L.B. nd MARCUVITZ, N., Rdition nd scttering of wves (Prentice-Hll inc., New Jersey, USA 973). 5. WATSON, G.N., Diffrction of electric wves by the erth, Proc. Roy. Soc. (London), 99, A95, pp