In the present work the diffraction of plane electromagnetic waves by an impedance loaded parallel plate waveguide formed by a twopart

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1 Progress In Electromagnetics Research, PIER 6, 93 31, 6 A HYBRID METHOD FOR THE SOLUTION OF PLANE WAVE DIFFRACTION BY AN IMPEDANCE LOADED PARALLEL PLATE WAVEGUIDE G. Çınar Gebze Institute of Technology Department of Electronics Engineering PO. Box 141, 414 Gebze, Kocaeli, Turkey A. Büyükaksoy Gebze Institute of Technology Department of Mathematics PO.Box 141, 414 Gebze, Kocaeli, Turkey Abstract The diffraction of E-polarized plane waves by an impedance loaded parallel plate waveguide formed by a two-part impedance plane and a parallel half plane with different face impedances is investigated rigorously by using the Fourier transform technique in conjunction with the Mode Matching Method. This mixed method of formulation gives rise to a scalar Modified Wiener- Hopf equation, the solution of which contains infinitely many constants satisfying an infinite system of linear algebraic equations. A numerical solution of this system is obtained for various values of the surface impedances and waveguide height. 1. INTRODUCTION In the present work the diffraction of plane electromagnetic waves by an impedance loaded parallel plate waveguide formed by a twopart impedance plane and a parallel half plane with different face impedances is analyzed. This problem is a generalization of a previous work by the authors [1] who considered the same geometry in the case where the half plane is perfectly conducting. In [1] the related boundary value problem is formulated as a matrix Wiener- Hopf equation which is uncoupled by the introduction of infinite

2 94 Çınar and Büyükaksoy sum of poles. The exact solution is then obtained in terms of the coefficients of the poles, where these coefficients are shown to satisfy infinite system of linear algebraic equations. When the half plane has non vanishing surface impedances, the resulting matrix Wiener- Hopf equation becomes intractable. To overcome this difficulty one recourses of a hybrid formulation consisting of employing the Fourier transform technique in conjunction with the mode matching method see for example [ 5]). By expanding the total field into a series of normal modes in the waveguide region and using the Fourier transform elsewhere, we get a scalar modified Wiener-Hopf equation of the second kind. The solution involves a set of infinitely many expansion coefficients satisfying infinite system of linear algebraic equations. Numerical solution to this system is obtained for different values of the surface impedances and height of the waveguide. In the case where the impedance of the semi-infinite plane vanishes, the results are compared numerically with those obtained in [1] by solving a matrix Wiener-Hopf equation, and it is shown than both results coincide exactly. FORMULATION OF THE PROBLEM Let a time harmonic plane wave E i x =, E i y =, E i z = u i = e ikx cos φ +y sin φ ) 1a) with time factor e iωt and wave number k, illuminates the parallel plate waveguide formed by a two-part impedance plane defined by {x, y, z),x, ),y =,z, )}, whose left and right parts are characterized by the surface impedances Z 1 = η 1 Z and Z = η Z, respectively, with Z being the characteristic impedance of the surrounding medium and a parallel half-plane with different face impedances located at {x <,y = b, z, )}. The surface impedances of the upper and lower faces of the half-plane are assumed to be Z 3 = η 3 Z and Z 4 = η 4 Z respectively see Figure 1). This problem is a generalization of the one considered in [1] where the half plane is assumed to be perfectly conducting. The method of formulation adopted in this work involves an appropriate definition of the total field such that in the waveguide region the field component may be expressed in terms of normal modes and the Fourier transform technique can be applied elsewhere. Notice that the problem considered in [1] can also be reduced to the solution of a scalar Modified Wiener-Hopf equation by using this hybrid method of Modal expansion and the Fourier transform technique.

3 Progress In Electromagnetics Research, PIER 6, 6 95 Figure 1. Impedance loaded parallel plate waveguide..1. Reduction to a Modified Wiener-Hopf Equation The total field can be expressed as { u T u i x, y)+u r x, y)+u 1 x, y), y > b x, y) = u 1) x, y) H x)+u ) x, y) H x), <y<b 1b) where u i x, y) is given by 1a), while u r x, y) stands for the field that would be reflected if the whole plane y = b were an impedance plane with relative surface impedance η 3 u r x, y) = 1 η 3 sin φ e ik[x cos φ y b) sin φ ] ) 1+η 3 sin φ In 1b), H x) is the Heaviside unit step function and φ is the angle of incidence, respectively. For the sake of analytical convenience we shall assume that k has a small imaginary part. The lossless case can then be obtained by making Imk) at the end of the analysis. u 1 and u j) j = 1, ), which satisfy the Helmholtz equation x + y + k )[ u1 x, y) u j) x, y) ] =, j =1, 3) are to be determined with the aid of the following boundary and continuity relations: 1+ η ) 3 u 1 x, b) =, x < 4a) ik y 1 η ) 4 u 1) x, b) =, x < 4 b) ik y 1+ η 1 ik y ) u 1) x, ) =, x < 4c)

4 96Çınar and Büyükaksoy 1+ η ) u ) x, ) =, x > 4 d) ik y u 1 x, b)+u i x, b)+u r x, b) u ) x, b) =, x > 4e) y u 1 x, b)+ y ui x, b)+ y ur x, b) y u) x, b) =, x > 4f) u 1),y) u ),y)=, <y<b 4g) x u1),y) x u),y)=, <y<b. 4h) To ensure the uniqueness of the mixed boundary-value problem defined by the Helmholtz equation and the conditions 4a) 4n), one has to take into account the radiation and edge conditions as well which are [ ] u ρ ρ iku,ρ= x + y 5a) and u T x, y) =O x 1/) y u T x, y) =O x 1/), x 5b) respectively. Since u 1 x, y) satisfies the Helmholtz equation in the range x, ),y>b, its Fourier transform with respect to x gives [ d dy + k α )] F α, y) = 6a) with where F α, y) =F α, y)+f + α, y) F ± α, y) =± ± u 1 x, y) e iαx dx. 6b) 6c) By taking into account the following asymptotic behaviors of u 1 x, y) for x ± u 1 x, y) =O e ikx / ) x, x ) u 1 x, y) =O e ikx cos φ, x 7)

5 Progress In Electromagnetics Research, PIER 6, 6 97 one can show that F + α, y) and F α, y) are regular functions of α in the half-planes Im α) > Im k cos φ ) and Im α) < Im k), respectively. The general solution of 6a) satisfying the radiation condition for y reads F α, y)+f + α, y) =A α) e ikα)y b) with A α) being the unknown spectral coefficient and 8a) K α) = k α. 8b) The square-root function is defined in the complex α-plane, cut along α = k to α = k + i and α = k to α = k i, such that K ) = k. In the Fourier transform domain 4a) yields F α, b)+ η 3 F ik α, b) =. 9) where the dot represents first degree derivative with respect to y. By using 8a) and its derivative with respect to y, one can write and respectively where F + α, b)+ η 3 ik F + α, b) = F + α, b) =ik α) [ 1+ η 3 k K α) ] A α) P + α) [ 1+ η ] 3 k K α) F α, b), 1a) 1b) P + α) =F + α, b)+ η 3 F ik + α, b). 1c) On the other hand, u ) x, y) satisfies the Helmholtz equation in the range x, ),<y<bwhose half range Fourier transform with respect to x gives [ ] d dy + K α) G + α, y) =f y) iαg y) 11a) where G + α, y) is the Fourier transform of the field u ) x, y) in the specified region defined by G + α, y) = u ) x, y) e iαx dx 11b)

6 98 Çınar and Büyükaksoy and f y) and g y) stand for f y) = x u),y) 11c) g y) =u ),y). 11d) The general solution of the equation 11a) is G + α, y) = B α) sin Ky + C α) cos Ky + 1 K y [f t) iαg t)] sin [K y t)] dt. 1a) Here B α) and C α) are the unknown spectral coefficients. When the boundary condition 4d) is taken into account C α) = η K α) B α) ik 1b) relation is determined for these coefficients. substituted in 1a), G + α, y) becomes G + α, y) = [ sin Ky η ik K cos Ky ] B α) + 1 K y If this relation is [f t) iαg t)] sin [K y t)] dt. 1c) B α) can be expressed in terms of F + α, b) by using the continuity relation 4e) which yields with B α) = M α) = 1 K α) M α) {P + α) b [ft) iαgt)] ) η3 η cos Kb + ik [ sin Kb t) + η ] 3 K ik cos Kb t) dt 1d) 1 η η 3 k K By using 1d) in 1a) [ sin Ky η ] ik K cos Ky G + α, y) = {P + α) K α) M α) ) sin Kb K. 1e)

7 Progress In Electromagnetics Research, PIER 6, 6 99 b [ sin K b t) [f t) iαg t)] + η ] 3 K ik cos K b t) dt + 1 y [f t) iαg t)] sin [K y t)] dt 13) K is found. Although the left-hand side of the above equation is regular in the upper half-plane Im α) > Im k cos φ ), the regularity of the right-hand side is violated by the presence of the simple poles occurring at the zeros of M α), namely at α = ±α m satisfying M ±α m )=, Im α m ) > Im k), m =1,,... 14) These poles can be eliminated by imposing that their residues are zero. This gives [ ] P + α m )=Q η3 m ik K m sin K m b cos K m b f m iα m g m ) 15a) where f m and g m are specified by with and [ fm gm ] = 1 b [ f y) Q g y) m K m = ][ sin Km y η ] K m ik cos K my dy. k α m cos Km b K m + η Q K m = m ) ik sin K mb 15b) 15c) M α m ), 15d) α m where the prime sign on the above equation denotes first degree derivative with respect to α. Hence, taking into account equation 1b) and the Fourier transform of the continuity relation given by 4f) together ikχ η 3,α) P + α) F α, b) e ikb sin φo = k sin φ o η 3 sin φ o +1) α k cos φ o ) + b P + α) [ cos Kb + η ik K sin Kb ] M α) [ sin K b t) [f t) iαg t)] + η ] 3 K ik cos K b t) dt

8 3 Çınar and Büyükaksoy b + [f t) iαg t)] cos [K b t)] dt 16) is obtained where K α) χ η 3,α)= η 3 K α)+k. 17) Equation 16) can be arranged as χ η 3,α) P + α) χ η,α) N α) + F α, b) = e ikb sin φo k sin φ o η 3 sin φ o +1) α k cos φ o ) 1 b [ sin Kt [f t) iαg t)] M α) K η ] ik cos Kt dt 18) with N α) =M α) e ikb. 19) Incorporating the series expansions for the functions f y) and g y) [ ] f y) [ ][ fm sin Km y = η ] g y) gm K m=1 m ik cos K my, ) equation 18) becomes P + α) N α) + F α, b) = χ η 3,α) χ η,α) + f m iαg m ) m=1 K m α α m) e ikb sin φo k sin φ o η 3 sin φ o +1) α k cos φ o ) cos Km b + η ) K m ik sin K mb. 1) This is nothing but the scalar modified Wiener-Hopf equation to be solved. The kernel functions in 1), i.e. the functions N α) and χ η j,α) can be factorized by using known expressions. Thus, the factors of N α) are [6] and N + α) = [ ) η3 η cos kb +1 η η 3 ) ik e [ iαb π 1 C+ln π kb +i π )] m=1 sin kb k 1+ α α m N α) = N + α). ] 1/ e [ Kb π ) e iαb mπ ) α+ik ln k ] a) b)

9 Progress In Electromagnetics Research, PIER 6, 6 31 In the equation a) C is the well-known Euler constant which is, As to the factors of χ η j,α) they can be written in terms of the Maliuzhinets functions [7]: χ η j,kcos φ) = 4 sin φ { } Mπ 3π/ φ θ) M π π/ φ + θ) ηj M π π/) {[ 1+ )] [ 3π/ φ θ cos 1+ )]} π/ φ + θ 1 cos 3a) and χ + η j,kcos φ) =χ η j, k cos φ) 3b) with M π z) and θ are defined by sin θ = 1 η j 3c) and M π z) = exp 1 8π gives z π sin u π sin u/) + u cos u du. 3d) Hence, multiplying the Wiener-Hopf equation term by term with χ η,α) χ η 3,α) N α) χ + η 3,α) P + α) χ + η,α) N + α) + χ η,α) χ η 3,α) N α) F α, b) k sin φ o e ikb sin φo χ η,α) = η 3 sin φ o +1) α k cos φ o ) χ η 3,α) N α) + χ η,α) χ η 3,α) N K m cos Km b α) f m iαg m ) α m=1 αm) + η ) K m ik sin K mb. 4) The terms at the right-hand side of the equation must be decomposed with the help of Cauchy formula which leads

10 3 Çınar and Büyükaksoy e ikb sin φo k sin φ o χ η,α) η 3 sin φ o +1) α k cos φ o ) χ η 3,α) N α) k sin φ o e ikb sin φo = η 3 sin φ o +1)α k cos φ o ) [ χ η,α) χ η 3,α) N α) χ ] η,kcos φ o )N k cos φ o ) χ η 3,kcos φ o ) k sin φ o e ikb sin φo χ η,kcos φ o ) N k cos φ o ) + η 3 sin φ o +1)α kcos φ o ) χ η 3,kcos φ o ) 5a) and χ η,α) χ η 3,α) N K m cos Km b α) f m iαg m ) α m=1 αm) + η ) K m ik sin K mb K m cos Km b = + η ) α + α m=1 m ) K m ik sin K mb [ fm iαg m ) χ η,α) α α m ) χ η 3,α) N α)+ f ] m+iα m g m ) χ + η,α m )N + α m ) α m χ + η 3,α m ) f m +iα m g m ) cos α m=1 m α+α m ) K Km b m + η ) K m ik sin K χ+ η,α m )N + α m ) mb. χ + η 3,α m ) 5b) Collecting the terms which are regular in the upper half-plane at the left-hand side and those regular in the lower half-plane at the righthand side, the equation becomes χ + η 3,α) P + α) χ + η,α) N + α) k sinφ o e ikb sinφo χ η,kcosφ o )N k cosφ o ) η 3 sinφ o +1)α k cos φ o ) χ η 3,kcosφ o ) f m +iα m g m ) cos + α m=1 m α+α m ) K Km b m + η ) K m ik sin K χ+ η,α m ) N + α m ) mb χ + η 3,α m ) = χ η,α) χ η 3,α) N α) F k sin φ o e ikb sin φo α, b)+ η 3 sin φ o +1)α kcos φ o ) [ χ η,α) χ η 3,α) N α) χ ] η,kcos φ o ) N k cos φ o ) χ η 3,kcos φ o ) K m cos Km b + + η ) α + α m ) K m ik sin K mb m=1

11 Progress In Electromagnetics Research, PIER 6, 6 33 [ fm iαg m ) χ η,α) α α m ) χ η 3,α) N α)+ f ] m+iα m g m ) χ + η,α m )N + α m ). α m χ + η 3,α m ) 6) Taking into account the analytical continuation principle followed by Liouville s theorem, the solution of the Wiener-Hopf equation yields χ + η 3,α) P + α) χ + η,α) N + α) = k sin φ o e ikb sin φo η 3 sin φ o +1)α kcos φ o ) χ η,kcos φ o ) N k cos φ o ) f m + iα m g m ) χ η 3,kcos φ o ) α m=1 m α + α m ) cos Km b K m + η ) K m ik sin K χ+ η,α m ) N + α m ) mb χ + η 3,α m ) 7).. Determination of the Unknown Coefficients The important difference of this kind of formulation from the one applied in [1], was the simultaneous use of Mode-Matching technique with the Fourier transform. The Mode-Matching technique allows us to express the field component defined in the waveguide region in terms of normal modes as u 1) sin x, y) = a n e iβnx ξn y η ) 1 ξ n ik cos ξ ny. 8) n=1 Here β n s and ξ n s are solved through η 1 + η 4 ) cos ξ n b 1+ η ) 1η 4 sin ξn b ik k ξ n =, ξ n n =1,,... 9a) and β n = k ξn, Im β n ) > Im k), n =1,,... 9b) If the continuity relations 4g) and 4h) are used simultaneously f y) iαg y) = x u1),y) iαu 1),y) 3) relation is determined which is sin Km y f m iαg m ) η ) K m=1 m ik cos K my = sin ξn y i a n α + β n ) η ) 1 ξ n=1 n ik cos ξ ny. 31)

12 34 Çınar and Büyükaksoy Multiplying both sides by sin K j y η K j ik cos K jy integrating over <y<band taking j m results with f m iαg m )= i Q a n α + β n ) nm, 3) m n=1 where nm is ) ξn Km nm = η 1 η ) + η 3+η 4 ) cos ξn b K m ξ n + η ) 1 ik ik ξ n ik sin ξ nb cos Km b + η ) K m ik sin K mb. 33) This result can be substituted in the equations 15a) and 7) giving A n α j ) a n = B α j ), 34) n=1 with [ ] η3 A n α j ) = i ik K j sin K j b cos K j b α j + β n ) nj in + α j ) χ + η,α j ) K m β n α m ) nm χ + η 3,α j ) m=1 cos Km b + η K m ik sin K mb α m Q m α j + α m ) ) χ+ η,α m ) N + α m ), 35) χ + η 3,α m ) and B α j ) = k sin φ ikb sin φo oe χ η,kcos φ o ) N k cos φ o ) η 3 sin φ o +1) χ η 3,kcos φ o ) N + α j ) χ + η,α j ) α j k cos φ o ) χ + η 3,α j ). 36) The infinite system of algebraic equations in 34) will be solved numerically. To this end we truncate the infinite series and the infinite system of algebraic equations after the first N terms. Figure shows the variation of the modulus of the diffracted field against the truncation number N. It is seen that the amplitude of the diffracted field becomes insensitive to the increase of the truncation number after N =3.

13 Progress In Electromagnetics Research, PIER 6, 6 35 Figure. Modulus of the diffracted field versus the truncation number N. 3. THE DIFFRACTED FIELD AND COMPUTATIONAL RESULTS The diffracted field u 1 x, y) can be determined through u 1 x, y) = 1 π L P + α) [1 + η 3 K α) /k] eikα)y b) e iαx dα. 37) Applying the change of variables α = k cos t, x = ρ cos φ and y = ρ sin φ, the above integral becomes u 1 x, y) = 1 π P + k cos t) η 3 sin t +1) e ikb sin t e ikρ cost φ) k sin tdt. 38) This integral can be solved asymptotically via the saddle point technique. Here the saddle point occurs at t = φ whose contribution is u 1 ρ, φ) = eikρ e iπ/4 k sin φ kρ π η 3 sin φ +1) e ikb sin φ P + k cos φ). 39)

14 36Çınar and Büyükaksoy Figure 3. The variation of the diffracted field with respect to η 1. Taking into account equation 7), the diffracted field can be cast into the following form: u 1 ρ, φ) = eikρ e iπ/4 k sinφ kρ π η 3 sinφ+1) e ikb sinφ χ η,kcosφ) χ η 3,kcosφ) N k cosφ) { sin φ o e ikb sin φo χ η,kcos φ o ) η 3 sin φ o + 1) cos φ + cos φ o ) χ η 3,kcos φ o ) N k cos φ o ) f m +iα m g m ) + α m=1 m α m k cosφ) K coskm b m + η ) } K m ik sink χ+ η,α m )N + α m ) mb. χ + η 3,α m ) 4) Now, some graphical results showing the effects of various geometrical and physical parameters on the diffraction phenomenon are presented. From the graphical results displayed so far one can conclude that the diffracted field is insensitive to the the variations of η 1 and η 4,as expected. The diffracted field amplitude is affected notably when η 3 is capacitive and a decrease in the amplitude of the diffracted field is observed when η 3 increases.

15 Progress In Electromagnetics Research, PIER 6, 6 37 Figure 4a. The variation of the diffracted field with respect to η. Figure 4b. The variation of the diffracted field with respect to η.

16 38 Çınar and Büyükaksoy Figure 5a. The variation of the diffracted field with respect to η 3. Figure 5b. The variation of the diffracted field with respect to η 3.

17 Progress In Electromagnetics Research, PIER 6, 6 39 Figure 6. The variation of the diffracted field with respect to η CONCLUDING REMARKS In this work, the diffraction of E-polarized plane waves by an impedance loaded parallel plate waveguide is investigated rigorously by using a mixed method of formulation consisting of employing the mode matching method in conjunction with the Fourier transform technique. For the special case when b tends to zero, the diffracted field given in 4) becomes u 1 e iπ/4 η 3 η ) sin φ sin φ π 1 + η 3 sin φ )1+η 3 sin φ) cos φ + cos φ) χ η, cos φ ) χ η, cos φ) e χ η 3, cos φ ) χ ikρ, η 3, cos φ) kρ which is the well-known solution for the two-part impedance plane diffraction problem whose parts x< and x> are characterized by the relative surface impedances η 3 and η, respectively. Figure 7 is a comparison between matrix Wiener-Hopf solution derived in [1] and the hybrid method used in this work, when the half plane is perfectly conducting Z 3 = Z 4 = ). It is shown that the two curves coincide exactly.

18 31 Çınar and Büyükaksoy Figure 7. The comparison of hybrid and direct Fourier transform formulations. REFERENCES 1. Büyükaksoy, A. and G. Cınar, Solution of a matrix Wiener- Hopf equation connected with the plane wave diffraction by an impedance loaded parallel plate waveguide, Math. Meth. Appl. Sci., Vol. 8, , 5.. Büyükaksoy, A. and F. Birbir, Analysis of impedance loaded parallel plate waveguide radiator, J. Electrom. Waves and Appl. JEMWA), Vol. 1, , Büyükaksoy, A. and B. Polat, Plane wave diffraction by a thick walled parallel plate impedance waveguide, IEEE Trans. Antennas and Propagat, Vol. 46, , Çınar, G. and A. Büyükaksoy, Diffraction by a set three impedance half-planes with the one amidst located in the opposite direction, Can. J. Phys. Vol. 8, ,. 5. Çınar, G. and A. Büyükaksoy, Diffraction of a normally incident plane wave by three parallel half planes with different face impedances, IEEE Trans. Antennas and Propagation, Vol. 5, , February Noble, B., Methods Based on the Wiener-Hopf Techniques, Pergamon Press, London, Senior, T. B. A. and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics, IEE Press, London, 1995.