ENGN UMR1340 030601-2-T Scattering by a groove in an impedance plane Sunil Bindiganavale and John L. Volakis National Aeronautics and Space Administration NASA Langley Research Center Hampton VA 23681-0001 September 1993 THE UNIVERSITY OF MICHIGAN Radiation Laboratory Department of Electrical Engineering and Computer Science Ann Arbor, Michigan 48109-2122 USA

V-%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~pIN.,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~T~. *, -,- -.- - F. —,~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~> I-~~~~~~~~~ I: t&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:a 1:0 ll T I I-e 14'~!"u,"~~l~r.~ 451e 7LQ~a~ ~L`Q tS: OA WI jA a~~ I -1 V91~~~

030601-2-T TECHNICAL REPORT for NASA Grant NAG-1-1478 NASA Technical Monitor: Fred Beck GCD, Mail Stop 490 NASA Langley Research Center Hampton, VA 23681-0001 Telephone: (804) 864-1829 Grant Title: Analysis of Conformal Antennas in Composite Surfaces Report Title: Scattering by a Groove in an Impedance Plane Institution: Radiation Laborato ry, Department of Electrical Engineering and Computer Science The University of Michigan Ann.Arbor MI 48109-2122 Date: September 1993 Report Authors: S. Bindiganavale and J. L. Volakis Principal Investigator: John L. Volakis Telephone: (313) 764-0500 Volakis@ um.cc.umich.edu

Scattering by a Groove in an Impedance Plane Sunil S.Bindiganavale and John L.Volakis Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109-2122. September 24, 1993 Abstract An analysis of two-dimensional scattering from a narrow groove in an impedance plane is presented. The groove is represented by a impedance surface and hence the problem reduces to that of scattering from an impedance strip in an otherwise uniform impedance plane. On the basis of this model, appropriate integral equations are constructed using a form of the impedance plane Green's functions involving rapidly convergent integrals. The integral equations are solved by introducing a single basis representation of the equivalent current on the narrow impedance insert. Both transverse electric (TE) and transverse magnetic (TM) polarizations are treated. The resulting solution is validated by comparison with results from the standard boundary integral method (BIM) and a high frequency solution. It is found that the presented solution for narrow impedance inserts can be used in conjunction with the high frequency solution for the characterization of impedance inserts of any given width.

1 Introduction A topic of some concern in radar cross section studies is the scattering from gaps or cracks that may exist where two component parts of a target come together. Even if the crack is entirely or partially filled with some material, it can still provide a significant contribution to the overall scattering pattern of the target, and it is therefore necessary to develop methods for predicting its scattering. The scattering by a groove or an impedance insert in a ground plane has already been considered by a variety of techniques. Integral equation [1]-[3] and finite element-boundary integral solutions [4]-[7] have been effectively used in this respect. Also, in the case of narrow grooves or impedance inserts in a ground plane, closed form solutions have been obtained which were found quite accurate for widths 0.15A or less [8]-[10]. In this paper we present a similar solution to the scattering from narrow grooves in an otherwise uniform impedance plane. The solution is useful for computing the scattering by grooves in~ coated ground planes and is intended to supplement high frequency solutions which are suitable for large width impedance inserts [11],[12]. As in the solutions given in [13], the groove is simulated by an impedance insert, thus, forming a three-part impedance plane (see Figure 1). The solution of the scattering by the illustrated three-part impedance surface is obtained by introducing appropriate equivalent currents over the extent of the middle impedance strip and then constructing an integral equation for the solution of these currents. This integral equation is obtained by making use of the impedance plane Green's functions which are conveniently expressed in terms of rapidly converging semi-infinite integrals. An important aspect of our solution is the introduction of a one-term basis expansion for the equivalent current, appropriate for small width impedance inserts. The coefficient of the single basis function is found in closed form involving integrals which are functions of the insert's width but independent of the impedance characterizing the insert. Unfortunately, these integrals are also functions of the background impedance, thus, precluding their tabulation for different insert widths. In the following sections we first proceed with the construction of the integral equations for the three-part impedance plane for both TE and TM incidences. The solution of the integral equations are then considered on the assumption of a small width impedance insert. The validity of the results

are examined by comparison with a closely related BIM solution [14] and a third order high frequency solution. We conclude by establishing the bounds of the proposed small-width approximations. 2 Formulation Consider a narrow filled groove of width w and depth d situated in an otherwise uniform impedance plane as shown in Figure 1. The groove is illuminated by the plane wave Ei(orHi) = zejikO(xcoso-+ysin0O) (1) for E- (or H) polarization, where ko = 2ir/Ao is the free space wavenumber and 0o is the angle of incidence. We shall consider the E- and H-polarizations separately. 2.1 E-Polarization For TM-incidence, the rectangular groove can be approximated by a strip of impedance [8] 77= jZo, tanh(pkod) (2) in which Zo is the free space intrinsic impedance, A)2 P=( ) - (3) d denotes the groove's depth, w is the groove's width, whereas C' and,' are the relative permeability and permittivity of the material filling the groove. The scattering by the subject impedance insert can be represented by introducing the equivalent electric and magnetic currents J=- x H M =Ex (4) over the extent of the groove. These currents are assumed to radiate in the presence of a uniform impedance plane satisfying the boundary condition x y x H- 71Zo0 x E (5)

Impedance surface (Normalized impedancel) )L d.:...;.., X W~~A (a) 7 ~1 r) 1~, (b) Satisfy the IBC Y X Y XE = -1 ZoYXH J M (C) 1: (a) Geometry of the groove in an impedance plane (b) Impedance (c) Equivalent scattering problem where r71 is the associated normalized impedance (see Figure 1) and (E, H) denote total fields. One can think of (5) as being imposed at y = 0-, but at y = 0+ E and H must satisfy the boundary condition x y x H= 7Zoy x E -w/2 < x < w/2, y 0+ (6) over the extent of the groove as dictated by the original problem. Using (4) in (6) we obtain that (assuming E = zEz) Jz= -rZoM., Jz = rZoEz (7) i.e, J and M are linearly related over the impedance insert, and thus we need only consider the solution of one of these, say JT.

To solve for J,, we construct an integral equation by enforcing (6). First we decompose EZ(EZ = Z E) as E = -E + E (8) where Ez denotes the scattered field and is equal to that radiated by J and M in the presence of the uniform impedance plane (see Figure l(c)). On using (7), EZ can be expressed as Ez = -jkoZo -/2 Jz(x')GIE(X, y/X', y')dx' Jw/2 Yo 9 w,/2 - -jkoZo]- I Jz(x')GIE(x, y/x', y')dx' where GIE is the electric Green's function of the uniform impedance plane (of normalized impedance 71). It is given by [15] GIE(xy/x',y') = - {H(2) (ko -/( x')2 + (y- y)2) + QE} (10) where QE - Ho(2) (ko(x-x')2 + (y + y)2) -2 a' H(2) ko/(x )2+(y + y' - j)2) dv (11) and a' = k. On substituting (9) into (8) and then into the second of (7) we obtain the integral equation qejkoxco'O =- YoJz(x) + jkoZorl I JZ(x')GIE((x, Y -- 0+/x', 0)dx 0' w12 cw/2 Y (w/2 +Yo-y I2 Jz(x')GIE(x,y -* 0+/x', O)dx (12) to be enforced at y = 0+ for the solution of Jz(x). 2.2 H-Polarization For TE-incidence, the rectangular gap can be approximated by a strip of normalized impedance [8] j Zop 7 =' tan(pkod) (13)

where p= V/c'' (14) As before, equivalent currents are introduced and the application of the impedance boundary condition to be satisfied over the groove yields the relations (assuming H = ZH,) Mz = TZo J, E. = qZoJ. (15) where E = -Ex + Ex` (16) The corresponding tangential scattered E field is given by E' =jo ko2 + J,(x')G (x, /' y') dx - ZO + )/2 J(x)G)GH(X, y/', y' ) d (17) Iy J-w/2 where GIH is the Green's function of the uniform impedance surface (of normalized surface impedance rl). It is given by [15] GIH(Xy/x',y') = -4{Ho (ko(x -x')2 + (y - y,)2) -QH} (18) where QH - H(2) (ko(x - x')2 + (y + y')2) -2 j p'e-'vH2) (ko /(x-x')2+ (y + y'-j)2) dv (19) and /' - k0o71. Substituting (17) into the second of (15) yields the integral equation sin qeikoxcosko = i-(o + aj 2) J/ Jx(x')GIH(x, y -- O/x, )dx' 9 [//2 +7- / 2Jx(x')GIH(X, Y 0+/X, O)dX +q7J (Jx) (20) for the solution of JT(x).

3 Integral Equation Solution Typically, the solution of (12) and (20) can be accomplished numerically using standard techniques. Such a numerical- solution is usually the only alternative for moderately sized grooves, but if the groove is small (kow << 1) then certain simplifications are possible. Once again, we shall consider the E- and H-polarizations separately. 3.1 E-Polarization It can be shown [16] that H,, and consequently J~, is of 0(1) near each of the impedance junctions (see Figure l(b)) provided neither /1 nor ar are zero. Thus, for kow < 1, we can assume that H, is nearly constant at y = 0+ over the extent of the groove. Based on this, we set Jz() = Xe (21) where Xe is a constant to be found. Substituting (21) into (12) and point matching at x = 0 yields -, -'~e7 (22) Xe Y + 7(11 + 2) + 3(22) where the integrals I1, 12 and 13 are given by I, kOZO 1/2 H 2) (kolxll) dx' 2 -u/2r o {- [ln ( 4 ) ]} (23) I2 = 2 j/2 GE(, O/x', O)dxz' (24) I3 = oo' (25) 2 aF w2 GIE(0, y/X', 0) l0d' (25) where y = 1.781 and GIE(X,y/x', y) = j (ae-,' H2) (ko /(X -x')2 + (y + y' - jv)2) dv (26) 110~~

In carrying out the integrals in (24) and (25), G0E is computed numerically. Because of the exponential factor e-"'V, the infinite integral defining G'E converges very rapidly. For our implementation G'E, was evaluated by subdividing the range of integration in 6 intervals and employing 4 point Gaussian quadrature integration in each interval. When the argument of the Hankel function is small, to avoid numerical difficulties, the integration from 0 to 0.1 is carried out analytically by introducing the small argument expansion of the Hankel function [17]. 3.2 H-Polarization From diffraction theory, it can again be shown that Hz is of 0(1) and thus we can approximate J, by Jx(() X= h (27) where Xh is a constant to be found. Substituting (27) into (20) and point matching at x = 0 yields sin qo Xh = I1 + 71 (I2 + 1) (28) The integrals I1 and I2 are given by 1 2 2\ J w/2 I, ( 2ko + GIH(X O/x' O)0 dx' (29) 2k0 - Ox2j w-/2 I2 = wy J-w/2GHI(O' yl/', O)Io dx'(30) aI2 y /(30) where G H(x, y/x', y') = j O'e-p v H(2) (ko(x -')2 + (y + y'- - j)2) dv (31) These can be evaluated numerically without difficulty since GIH converges rapidly due to the presence of the exponential factor e-'". Again, it is necessary to introduce the small argument expansion of the Hankel function and carry out the integration from 0 to 0.1 analytically as was done in [17].

4 Far Zone Scattered Fields and Echowidth The far zone scattered fields are computed from (12) and (20) on using the large argument approximation of the Hankel function. By evaluating the integrals via the stationary phase method, we obtain the simple formulae Ez =-WZOX 2r 1 + l 2sinp (32) Hz= wZoXh ei(kop_) e 4 i sin2 (33) 2irp 1+q/sin (3 The corresponding echowidth is given by ZOjIw i (X, sin 5)2 E-pol = 1 + 7 sin (xh sin2 q)2 H-pol (34) (35) 5 Validation of Results The derived echowidth expressions are based on a low frequency solution of the exact integral equation. They are, thus, expected to be valid for small groove widths and it is, therefore, of interest to examine their accuracy limitations as the width of the groove increases. For this validation, we used the solution based on the BIM and a high frequency solution. The BIM reference solution was that presented by Moore and Ling [14] and applies to an isolated conductor-backed dielectric gap, such as that shown in Figure 2. To compare with this solution the dielectric coated conductor was approximated by the normalized impedance 771 = j tan (kox/e,#.d) (36) where Er and lr, are the permittivity and permeability of the dielectric layer and d is the thickness of the dielectric. Also the gap impedances are computed from (2) and (13). In Figures 3 and 4 we compare our small width approximation with the BIM data given in [14] for E and H polarizations, respectively. Each figure 9\

Pec backed Dielectric (represented by equivalent impedance ill) Dielectric pec Gap ( represented by equivalent impedance Ti ) Figure 2: Geometry of the gap at the junction of two semi-infinite dielectric coatings displays two curves, one corresponding to a gap of width O.l1o and the other to a gap of width w = 0.2o0. The other gap parameters are given in the figures. As can be expected, the agreement between the two solutions is good for those gaps in the smaller thickness coatings. However, for the 0.2A0 gap in a d = 0.4A0 thick coating, our small width and impedance approximations are no longer expected to be valid. Not surprisingly, the H-polarization echowidth for this geometry as computed via the small width approximation is 2-3 dB off from the BIM solution. The greater disagreement near grazing is due to the inaccurate simulation of the dielectric coating by an impedance surface. Having validated our solution we now proceed with an assessment of its accuracy range and limitation. Figures 5-8 show the backscatter echowidth as a function of the insert's width for different values of r71 and yr. The small width and high frequency solutions' are compared in these figures. Of course, by its derivation, the high frequency solution becomes more accurate as w increases, whereas the presented small width approximation does the same as w decreases. Consequently, the two solutions will have a certain intermediate 1The reference high frequency solution is a modification of that given in [11] using the impedance junction diffraction coefficients [16]

range of widths where agreement is likely to be expected. Figures 5 and 6 show E-polarization curves at some oblique (450) incidence. Similarly, Figures 7 and 8 display some H-polarization curves at oblique and normal incidences. From these figures, we observe that the presented approximate solutions are indeed in good agreement for 0.15A) < w < 0.3A0, and in many cases even outside this range. The conclusion that can be reached from Figures 5-8 is that the combination of the simple closed form high frequency and small width approximations provide an accurate evaluation of the scattering by the three part impedance plane for all insert widths. Specifically, the high frequency solution can be used for w > 0.25Ao, whereas the small width approximation is suitable for smaller w. 6 Summary Integral equations for the analysis of the scattering by a groove in an impedance plane were constructed by representing the groove as an impedance surface. The integral equations for the three-part impedance plane were solved on the assumption of a small width impedance insert. The coefficient of the single basis function for the equivalent current across the insert was explicitly given in terms of integrals which were functions of the insert's width and the background impedance. The accuracy and limitations of the derived small width approximations were evaluated by comparison with the boundary integral method and a high frequency solution. From these comparisons, we inferred that the presented solution was accurate for insert widths up to 0.25Ao and could be extended up to 0.4A0 for near normal incidences. Thus, the presented approximate small-width approximation and the high frequency solution complement each other for the evaluation of the scattering by a three part impedance plane for all insert widths.

0.000 -10.0 -20.0 A \ -30.0 | Small width approx.(w=O.lX,d=0.2X,r-=2) E0 BIM [9](w=0.lX,d=0.2X,F =2)...... —- Small width approx.(w —0.2X,d= —O.X,er-10) -4. A BIM [9](w=0.2X,d —=.1X,Er=1 0) -40.0 0.00 15.00 30.00 45.00 60.00 75.00 90.00 Observation Angle (q) [deg] Figure 3: E-polarization backscatter echowidth for a narrow gap in.a grounded dielectric layer as in Figure 2. (a) w=0.1A, d=0.2A and c, = 2 (b)w=0.2A, d=O0.1A and e, = 10 12

4 -'a. A -10.0 A o A -20.0 Fge:H orai bkatcSmall width approx.(w —0o.la,d —0.2,cr=2) -40.0 g IM [9](w=O.l.,d —O.2an.,df2) \ -------- Small width approx.(w=0.2A,d_-0.4F,er=2) \ A BIM [9](w —0.2X,d —0.4.,er=2) 1. -50.0, 0.00 15.00 30.00 45.00 60.00 75.00 90.00 Observation Angle ({) [deg] Figure 4: H-polarization backscatter echowidth for a narrow gap in a grounded dilectric layer as in Figure 2. (a) w=0.1A, d=0.2A and c = 2 and (b) w=0.2A, d=0.4A and r = 2 13

o 1 —=0.25 ll=-j0.25 --- r=0.25 ll=-j0.25 5.00 A rl=0.25 10=-j.5..-.... 1 —=0.25 rl=-jO.5 0.000 > =0.25'1=-j0.75 i -5.00 - - --- q r=0.25 T=.-jO.75 -10.0 I A 1-25.0 0 A-... o.. -30.0 A. -35.0 A 0.. -40.0 o... 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Width[R] Figure 5: E-polarization backscatter echowidth at oblique incidence (450) for an impedance insert of normalized impedance = 0.25 in various impedance planes as a function of the insert's width (symbols: small width approx, lines: high freq solution) 14

11=0.15i 1=-jO.25 - =O. 15 r1 =-jO.25 5.00 A rl=jO.15 1l=-jO.5 - n=jO.15 l l=-jO.5 0.000'0 li=jO.25 rl=-j0.75 m~ -5.00 ---- n=jO.25 nl=-jO.75 -10.0 --- " -15.0.- - 0 20.0 0 0 0.- 0 0 A -25.0 - A - 30.0 -35.0 - o -40.0 I I.. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Width[R] Figure 6: E-polarization backscatter echowidth at oblique incidence (450) for various three-part impedance planes as a function of the insert's width (symbols: small width approx, lines: high freq solution) 15

0 i1=0.25 1ll=0 - r=0.25 1ll=0 5.00 A n=0.15 n1=~....- - rl=O. 15,1m=0 0.000 0 il=0.5 ll=O m_ -5.00 ---—. =0.5 ri1=0 I 15 0 < G, -h -'i 0 -15.0 _..... —-A AA —------------ ~ -20.0 - - A o 0 o -30.0 -. O A. -35.0 0 A -40.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Width[R] Figure 7: H-polarization backscatter echowidth at oblique incidence (450) for various impedance inserts in a metallic plane as a function of the insert's width (symbols: small width approx, lines: high freq solution) 16

5.00 0.000 0 - - Ck -5.00 - A b0 -10.0. 0 O OA;, -15.0 - A -20.0 00 n=.25 n1=j0.75. - O J. 3 —,-25.0 1 - 1=.25 1r1=j0.75 -25.0 0~~~, AA q=j0.25 ~]=j0.75 -30.0 ------ =j0.25 11l=j0.75 - 1=-j0.25 hi=j0.75 -35.0 - -- -- -- rl=-j0.25 ill=j0.75 -40.0, - - 0.00 0.10 0.20 0.30 0.40 0.50 Width[X] Figure 8: H-polarization backscatter echowidth at normal incidence for various impedance inserts in an impedance plane of normalized impedance r1l = j0.75 as a function of the insert's width (symbols: small width approx, lines: high freq solution) References [1] R.F.Harrington and J.R.Mautz, "Generalized network formulation for aperture problems," IEEE Trans. Antennas Propagat., vol.24, pp.870-873, Nov. 1976. 17

[2] J.S.Asvestas, "New integral equations for scattering from an indented screen," in 1989 Radio Sci. Meet. Dig., Boulder, CO, p.106. [3] K.Barkeshli and J.L.Volakis, "TE scattering by a two-dimensional groove in a ground plane using higher order impedance boundary conditions," IEEE Trans. Antennas Propagat., vol.38, pp.12801286, Aug. 1990. [4] S.K.Jeng, "Scattering from a Cavity-Backed Slit in a Ground Plane - TE case," IEEE Trans. Antennas Propagat., vol.38, pp.1523-1529, Oct. 1990. [5] J.M.Jin and J.L.Volakis, "TE scattering by an inhomogeneously filled aperture in a thick conducting plane," IEEE Trans. Antennas Propagat., vol.38, pp.1280-1286, Aug. 1990. [6] J.M.Jin and J.L.Volakis, "TM scattering by an inhomogeneously filled aperture in a thick conducting plane," Proc. Inst. Elec. Eng., pt. H, vol.137, pp. 153-159, June 1990. [7] K.W.Whites, E.Michielssen, and R.Mittra, "Approximating the scattering by a material-filled 2-D trough in an infinite plane using the impedance boundary condition," IEEE Trans. Antennas Propagat., vol.41, pp.146-153, Feb. 1993. [8] T.B.A.Senior and J.L.Volakis, "Scattering by gaps and cracks," IEEE Trans. Antennas Propagat., vol.37, no.6, pp.744-750, June 1989. [9] T.B.A.Senior, K.Sarabandi and J.R.Natzke, "Scattering by a narrow gap," IEEE Trans. Antennas Propagat., vol.38, pp.1102-1110, 1990. [10] K.Barkeshli and J.L.Volakis, "Electromagnetic scattering from an aperture formed by a rectangular cavity recessed in a ground plane," J. Electromagn. Waves Appl., 1991.

[11] M.I.Herman and J.L.Volakis, "High frequency scattering by a double impedance wedge," IEEE Trans. Antennas Propagat., vol.36, May 1988, pp.664-678. [12] R.G.Rojas, H.C.Ly, P.H.Pathak, and R.Tiberio, "Electromagnetic plane wave diffraction by a three-part thin, planar dielectric/magnetic slab," Radio Sci., vol.26, no.5, pp.1267-1280. [13] K.Barkeshli and J.L.Volakis, "Scattering from Narrow Rectangular Filled Grooves," IEEE Trans. Antennas Propagat., vol.39, no.6, pp.804-810, June 1991. [14] J.Moore and H.Ling, "Scattering by gaps in coated structures," J. Electromagn. Waves Appl., vol.7, no.3, pp.325-344, 1993. [15] Kamal Sarabandi, "Scattering from Dielectric Structures Above Impedance Surfaces and Resistive Sheets," IEEE Trans. Antennas Propagat., vol.40, no.1, pp.68-78, Jan. 1992. [16] T.B.A.Senior, "Diffraction by half plane junctions," Dept. EECS, Univ. Michigan, Radiation Lab. Technical Rep. RL 892, March 1993. [17] M.A.Ricoy and J.L.Volakis, "Integral equations with reduced unknowns for the simulation of two-dimensional composite structures," IEEE Trans. Antennas Propagat., vol.37, no.3, pp.362372, March 1989. (see correction in IEEE Trans. Antennas Propagat., vol.41, no.3, pp.393, March 1993)

UNIVERSITY OF MICHIGAN IIIIII 111011IIlllII I1111111 fl 3 9015 02827 3038