DIFFRACTION OF ELECTROMAGNETIC PLANE WAVE BY AN IMPEDANCE STRIP

Transcription

1 Progress In Electromagnetics Research, PIER 75, , 7 DIFFRACTION OF ELECTROMAGNETIC PLANE WAVE BY AN IMPEDANCE STRIP A. Imran and Q. A. Naqvi Department of Electronics Quaid-i-Azam University Islamabad, Pakistan. Hongo , Nakashizu, Sakura city, Chiba, Japan Abstract This paper investigates the scattering of electromagnetic plane wave from an impedance strip. Both E- and H-polarizations are considered. The method of analysis is obayashi potential, which uses the discontinuous properties of Weber-Schafheitlin s integrals. Imposition of boundary conditions result in dual integral equations. Using the projection, equations reduces to matrix equations. The elements are given in terms of infinite integrals that contains the poles for particular values of surface impedance and these integrals are computed numerically. Far diffracted fields in the upper half space for different angles of incident are computed. To check the validity of the results, we have derived the physical optics (PO) approximate solutions. Numerical results for both the methods are compared. The agreement is good. Current distribution on the strip is also presented. 1. INTRODUCTION Scattering of electromagnetic waves from a strip is a classic problem in electromagnetics. This has been the subject of many investigations 1 1. A variety of methods may be used to analyze the problem When the width of the strip is very large as compared to the operating wavelength, high frequency approximate solution may be obtained by using the concept of geometrical theory of diffraction (GTD) 19. But when the width is not large compared to the wavelength, numerical approaches like the method of moment 8, 9, are more reliable.

2 34 Imran, Naqvi, and Hongo Other methods like Wiener-Hopf 18, 1, Maliuzhinet s techniques may be used to solve the problem. In this paper, we have formulated the problem by applying the obayashi potential method 3, 4. This method has been applied to various kinds of problems, such as the potential problems of electrified circu-lar disks 5, 6, the diffraction of acoustic waves by a circular disk and rectangular plate 7, 8, diffraction of electromagnetic plane wave by rectangular plate and hole, parallel slits, disk and circular hole In obayashi potential method, imposition of boundary conditions give us the dual integral equations. These equations are solved using the discontinuous properties of Weber-Schafheitlins integrals 3. Incorporating the edge conditions, we transform the resulting expressions into the matrix equations. The elements of the matrix are the infinite integrals which are difficult to solve analytically. Numerical computations are con-ducted and results for impedance strip obtained using obayashi method are compared with those based on physical optics method.. FORMULATION Consider an impedance strip of width a as shown in Figure 1. The strip is excited by a uniform electromagnetic plane wave. φ is the angle of incidence with the x-axis. Impedances of the upper and lower surfaces of the strip are Z + and Z respectively. If Ez i and Hz i be the incident fields for E- and H-polarization respectively, then ( ) E i z Hz i = exp jk(x cos φ + y sin φ ) (1) For simplicity we assumed the amplitude of the incident field is unity. The corresponding diffracted fields may be expressed as ( ) E d z = g 1 (ξ) cos(x a ξ)+g (ξ) sin(x a ξ) exp ξ κ y a dξ, H d z ( ) E d z = H d z y a > (a) h 1 (ξ) cos(x a ξ)+h (ξ) sin(x a ξ) exp ξ κ y a dξ, y a < (b) Since in D problems, the wave in each polarization (E- and H-) does not couple, we use the same symbols g(ξ) and h(ξ) for the unknown functions. In the above equations x a = x a and y a = y a are the normalized variables and κ = ka is the normalized wave number. We consider each polarization separately.

3 Progress In Electromagnetics Research, PIER 75, 7 35 Y X= - a X= a X Figure 1. Geometry of the problem. Z.1. E-polarization The required boundary conditions are given by Ez t = Z + Hx t, Ez t = Z H t x x a 1 (3a) y=+ y=+ y= y= Ez t = Ez t, Hx t = H t x x a 1 (3b) y= y=+ y= y=+ (4 (4 where t in superscript means total. From the condition (3b) we have { g 1 (ξ) h 1 (ξ) cos(x a ξ) } + g (ξ) h (ξ) sin(x a ξ) dξ = x a 1 a) { ξ κ g 1 (ξ)+h 1 (ξ) cos(x a ξ) } + g (ξ)+h (ξ) sin(x a ξ) dξ = x a 1 b) With the help of the discontinuous properties of the Weber- Schafheitlin s integrals, we can assume g 1 (ξ) h 1 (ξ) = A m J m+ 3 (ξ)ξ 3, g (ξ) h (ξ) = g 1 (ξ)+h 1 (ξ) = g (ξ)+h (ξ) = m= m= m= m= B m J m+ 5 (ξ)ξ 3 J m+ 1 (ξ) C m ξ κ ξ 1, J m+ 3 (ξ) D m ξ κ ξ 1 (5a) (5b)

4 36 Imran, Naqvi, and Hongo where we have taken into account the edge conditions of Ez t and Hx. t From the condition (3a) we have 1 j ζ + ξ κ κ g 1 (ξ) cos(x a ξ)+g (ξ) sin(x a ξ) dξ = 1 ζ + sin φ expjκx a cos φ (6a) 1 j ζ ξ κ κ h 1 (ξ) cos(x a ξ)+h (ξ) sin(x a ξ) dξ = 1+ζ sin φ expjκx a cos φ (6b) for x a 1 In the above expression ζ ± = Z ± /Z and Z be the impedance of free space. Now we substitute the weighting functions g 1 (ξ) h (ξ) determined from equations (5) into equations (4), comparing even and odd functions and then project the resulting equations into the functional space with elements p ± 1 n (x a). We have RE,E A + m + G + RE,E C m = 1 ζ + sin φ J E (7a) RE,E A m G RE,E C m =1+ζ sin φ J E (7b) B m D m J O RE,O + RE,O + B m G + RE,O G RE,O D m = j1 ζ + sin φ = j1 + ζ sin φ J O (7c) (7d) where π cos x = J 1 (x), π sin x = J 1 (x) and x m/ J m (ξ γ(n+m+1) J n+m+1 (ξ) x)= (n+m+1) p m n (x) Γ(n+1)Γ(m+1) ξ n= p m γ(n + 1)Γ(m +1) n (x) = x m/ J m ( xξ)j n+m+1 (ξ)dξ Γ(n + m +1) where the correspondence between the matrices and their elements are given by ( RE,E ± RE n + 1, m + 3 ) ; ζ ± ( RE,O ± RE n + 3, m + 5 ) ; ζ ±

5 Progress In Electromagnetics Research, PIER 75, 7 37 ( G ± RE,E G RE n + 1, m + 1 ) ; ζ ± (8a) ( G ± RE,O G RE n + 3, m + 3 ) ; ζ ± and RE (m,n;ζ) = G RE (m,n;ζ) = where RE,E + { J E J n+ 1 (κ cos φ ) (κ cos φ ) 1 J O J n+ 3 (κ cos φ ) (κ cos φ ) 1 1 j ζ κ J m (ξ)j n (ξ) Jm ξ κ (ξ)j n (ξ) dξ = ξ dξ j ζ (m, n) κ 1 = 4 sin (m n +1)π π(m+n+1)(m+n 1)(n m+1)(m n+1) j ζ κ (m,n) (8b) 1 j ζ Jm ξ κ κ (ξ)j n (ξ) ξ ξ κ dξ = = J m (ξ)j n (ξ) ξ ξ κ dξ j ζ κ J m (ξ)j n (ξ) ξ ξ κ dξ j ζ sin κ (m, n) = Equations (7) may be rewritten as { 1 G RE,E + + RE,E ξ J m (ξ)j n (ξ) dξ ξ 1 (m n)π m n ξ κ J m(ξ)j n (ξ) ξ dξ 1 G RE,E }C m = 1 ζ + sin φ RE,E ζ sin φ A m = RE,E + 1 G + RE,E C m 1 ζ + sin φ { RE,O + 1 G RE,O + RE,OE + 1 G RE,O }D m + RE,E + RE,E (8c) 1 }J E 1 J E

6 38Imran, Naqvi, and Hongo { 1+1 = j 1 ζ + sin φ RE,O + + ζ sin φ RE,O + 1 }J O B m = RE,O + 1 G + RO,E D m j1 ζ + sin φ RE,O + 1 J O (9) For the case of ζ + = ζ = ζ, equations (7) or (9) reduce to 1 1 A m = ζ sin φ RE,E J E, B m = jζ sin φ RE,O J O (1a) C m = G ± 1 RE,E J E, D m = j G ± 1 RE,O J O (1b) Far scattered field in the upper region can be evaluated by applying the saddle point method of integration. The result is given by Ez d = 1 { J m+ 3 (ξ) J m+ 1 (ξ) A m + C m cos(x a ξ) = + m= B m J m+ 5 ξ 3 ξ 3 ξ(ξ κ ) } (ξ) J m+ 3 (ξ) + D m sin(x ξ(ξ κ a ξ) ) exp ξ κ y a dξ π 1 exp jkρ + j π 8 kρ 4 +B m J m+ 5 (κ cos φ) j C m J m+ 1 m= tan φ { A m J m+ 3 (κ cos φ) (κ cos φ)+d m J m+ 3 (κ cos φ) } (κ cos φ) 1 (11a) where ξ = κ cos φ. And expansion coefficients A m,b m,c m,d m can be obtained from equations (1). The current density induced on the impedance strip is obtained as follows. J z = Hx t Hx t y=+ y= = jy C m J κ m+ 1 (ξ) cos(x a ξ)+d m J m+ 3 (ξ) sin(x a ξ) ξ 1 dξ m= = jy κ m= Γ(m + 1 ) { C m p 1 m (x Γ(m +1) a)+x a (m +1)D m p 1 } m (x a) (11b)

7 Progress In Electromagnetics Research, PIER 75, H-polarization The required boundary conditions of this problem are given by Ex t = Z + Hz t, Ex t = Z H t z x a 1 (1a) y=+ y=+ y= y= Ex t = Ex t, Hz t = H t z x a 1 (1b) y=+ y= y=+ y= The incident wave is given by (1) and the diffracted wave is given by (). From the condition (1b) we have we have the same equations as (4) and g 1 (ξ) h (ξ) are given by (5). From the condition (1a) we have ξ κ + jκζ + g 1 (ξ) cos(x a ξ)+g (ξ) sin(x a ξ) dξ = jκ(sin φ ζ + ) exp(jκx a cos φ ) (13a) ξ κ + jκζ h 1 (ξ) cos(x a ξ)+h (ξ) sin(x a ξ) dξ = jκ(sin φ + ζ ) exp(jκx a cos φ ) for x a 1 (13b) We substitute the weighting functions g 1 (ξ) h (ξ) of (13) determined by (5) and we project the resulting equations into the functional space with elements p ± 1 n (x a). Then we have RH,E A + m + G + RH,E C m = jsin φ ζ + J E, RH,E A m G RH,E C m = jsin φ + ζ J E (14a) B m D m J O RH,O + RH,O + B m G + RH,O G RH,O D m = sin φ ζ +, = sin φ + ζ J O (14b) where the correspondence between the matrices and their elements are given by RH,E ± RH (n + 1, m + 3 ) ; ζ ± RH,O ± RH (n + 3, m + 5 ) ; ζ ± G ± RH,E G RH (n + 1, m + 1 ) ; ζ ± G ± RH,O G RH (n + 3, m + 3 ) ; ζ ± (15)

8 31 Imran, Naqvi, and Hongo J E J O κ J n+ 1 (κ cos φ ) (κ cos φ ) 1 κ J n+ 3 (κ cos φ ) (κ cos φ ) 1 and RH (m, n; ζ) = = jζκ G RH (m, n; ζ) =jζκ Jm jζκ + ξ κ (ξ)j n (ξ) dξ ξ J m (ξ)j n (ξ) ξ dξ + (m, n) (16a) J m (ξ)j n (ξ) ξ ξ κ dξ + J m (ξ)j n (ξ) dξ ξ (16b) A m B m { RH,E + 1 G + RH,E + RH,E 1 G RH,E }C m { = j sin φ ζ + RH,E sin φ + ζ RH,E + 1 }J E = { = RH,E + 1 G + RH,E C m RH,O + 1 G + RH,O + { sin φ ζ + + jsin φ ζ + 1 RH,O + RH,O 1 + sin φ + ζ 1 J E RH,E + G RH,O }D m RH,O 1 }J O = RH,O + 1 G + RH,O D m sin φ ζ + RH,O + 1 J O (17) For the case of ζ + = ζ = ζ, Equations (17) reduce to A m = j sin φ RH,E ± 1 J E B m = sin φ RH,O ± 1 J O C m = jζ G ± 1 RH,E J E D m = ζ G ± 1 RH,O J O (18a) (18b) Far scattered field in the upper region can be evaluated by applying the saddle point method of integration and the result has the same form as (11a), but the expansion coefficients A m D m are given by (17) or (18), instead of (1). The current density induced on the impedance strip is obtained as follows.

9 Progress In Electromagnetics Research, PIER 75, J x = Hz t Hz t y=+ y= J m+ 3 (ξ) = A m m= ξ 3 ( Γ m + 1 Γ(m +1) ) J m+ 5 (ξ) cos(x a ξ)+b m ξ 3 sin(x a ξ) = 1 A m m m= 4m +3 p m (x 1 a)+ m +1 p m+1 (x a) m +1 + x a 4m +3 B m + 3 m p 1 m (x a)+ m +1 p 1 m+1 (x a) (19) dξ 3. PHYSICAL OPTICS APPROXIMATE SOLUTIONS We consider here physical optics solutions for comparison with the previous solutions E-polarization The total field on the strip is Ez t = ζ + sin φ exp(jkxcos φ ) 1+ζ + sin φ (a) Hx t = Y sin φ exp(jkxcos φ ) 1+ζ + sin φ (b) The equivalent currents are M x = E t z, J z = H t x (c) Far field expression of the vector potential is given by A z = µy j Q C(kρ) a a = µy j Q C(kρ)S (φ), F x = ɛζ + j Q C(kρ) a a exp jk(cos φ + cos φ )x dx exp jk(cos φ + cos φ )x dx = ɛζ + j Q C(kρ)S (φ) (1)

10 31 Imran, Naqvi, and Hongo A z and F x are the components of magnetic and electric vector potential respectively. Q = sin φ, C(kρ) = ( jkρ 1+ζ + sin φ πkρ exp + j π ). 4 Thus far electric field is derived as E z = jωa z + 1 F x ɛ y = k 4 (1 ζ + sin φ)q C(kρ)S (φ) = 1 ζ + sin φ ka sin φ 1+ζ + sin φ ( jkρ + j π 4 exp ) sinc πkρ ka(cos φ + cos φ ) () 3.. H-polarization The total field on the strip is Hz t = sin φ exp(jkxcos φ ) ζ + + sin φ (3a) Ex t ζ + sin φ = Z exp(jkxcos φ ) ζ + + sin φ (3b) The equivalent currents are J x = Hz, t M z = Ex t (3c) Far field expression of the vector potential is given by A x = µ j Q 1C(kρ)S (φ) (4a) F z = ɛζ + j Z Q 1 C(kρ)S (φ) (4b) where Q 1 = sin φ ζ + +sin φ, C(kρ) = πkρ exp ( jkρ + j π ) 4. Thus far magnetic field is derived as E z = jωf z 1 F x µ y = k 4 (sin φ ζ +)Q 1 C(kρ)S (φ) = ζ + sin φ ka sin φ ζ + + sin φ ( jkρ + j π 4 exp ) sinc πkρ ka(cos φ + cos φ ) (5)

11 Progress In Electromagnetics Research, PIER 75, DIFFRACTED FIELD PO E-Polarization P PO P κ =4. ζ =.1+.3ι φ = OBSRVATION ANGLE Figure. Comparison of diffracted field patterns for normal incidence E-Polarization PO DIFFRACTED FIELD PO P κ =4. ζ =.1+.3ι φ =6 P OBSERVATION ANGLE Figure 3. Far diffracted fields in the upper half plane for φ = 6.

12 314 Imran, Naqvi, and Hongo 4. DIFFRACTED FIELD H-Polarization P PO φ =9 κ =4. ζ =.1+.3ι OBSERVATION ANGLE Figure 4. Comparison between the two methods for φ = 9. FAR DIFFRACTED FIELD H-Polarization P PO κ =4. ζ =.1+.3ι φ = OBSEVATION ANGLE Figure 5. Comparison between PO and P for H-polarization.

13 Progress In Electromagnetics Research, PIER 75, CURRENT MAGNITUDE φ =9 φ =6 κ =4. ζ =.1+.3ι x/a Figure 6. Current distribution on the strip (E-polarization) H-polarization CURRENT MAGNITUDE φ =45 φ =9 κ =3. ζ =.1+.3ι x/a Figure 7. Current distribution H-polarization.

14 316 Imran, Naqvi, and Hongo 4. NUMERICAL RESULTS AND DISCUSSION To study the scattering properties of impedance strip, expansion coefficients A m D m are computed, for large number of m values, using equations (1) for E-polarization and equations (18) for H- polarization. Far diffracted fields are computed using these expansion coefficients in equation (11a) for both the cases. Fig. and Fig. 3 gives the far field patterns for different values of angle of incidence and κ = 4., ζ =.1+.3i. Line plots with circled symbols corresponds to kobayashi potential while Line plots with blocked symbols corresponds to physical optics. To check the validity of these results, we compared them with those of obtained using physical optics equation (). Fig. 4 and Fig. 5 give the diffracted patterns and their comparisons for h- polarization case corresponding to φ = π/ and φ = π/3, κ = 4., ζ =.1 +.3i with those obtained using PO equation (5). Fig. 6 and Fig. 7 show the numerical results for current distribution on the strip obtained from equation (11b) for E-polarization and equation (19) for H-polarization. The results are as expected. REFERENCES 1. Grinberg, G. A., Diffraction of electromagnetic waves by a strip of finite width, Soviet Phys. Doklady, Vol. 4, 1 15, Hansen, E. B., Scalar diffraction by infinite strip and circular disk, J. Math. Phys., Vol. 41, 9 45, Jones, D. S. and B. Nobel, The low frequency scattering by a perfectly conducting strip, Proc. Cambridge Phil. Soc., Vol. 57, , Fialkovskiy, A. T., Diffraction of planar electromagnetic waves by a slot and a strip, Radio Eng. Electron., Vol., , Herman, M. I. and J. L. Volakis, High frequency scattering by a resistive strip and extensions to conductive and impedance strips, Radio Sci., Vol., , May June Buyukaksoy, A., O. Bicakci, and A. H. Serbest, Diffraction of an E-polarized plane wave by a resistive strip located on a dielectric interface, Journal of Electromagnetic Waves and Applications, Vol. 8, Issue 5, , May Buyukaksoy, A. and G. Uzgoren, Secondary diffraction of a plane wave by a metal-lic wide strip residing on the plane interface of two dielectric media, Radio Science, Vol., Issue, , March Barkeshli,. and J. L. Volakis, Electromagnetic scattering by

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