Bistatic Scattering in .NET

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For bistatic scattering, we take the average of (1/J~l)1/Jil)*) and have
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(1/J~1) (kx)1/J~l)' (k~))
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= Ik~12 (kxk ix - k 2)2 0(k x - k~)W(kx - kix ) = I(k x )8(kx - k~) (9.3.67)
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4 Kirchhoff Approach
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(9.3.68) z Similar to the Dirichlet case of Section 3.1, we use the first-order fields to calculate the bistatic scattering coefficient. Keeping first-order fields only, we have . (1)* ] z (5s . z) = Re 2k (1/J~1) a~sz ) [
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dk~ei(kx-k~)X( -ik~*)ei(k.-k:*)Z(1/J~l)(kx)1/J~l)* (k~))]
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1 - Re [ 2k
Joo dkxk*ze i(k.-k;)z'T(kx)] -1
1 -k 'T(k 2k jk d kx k z.J., x )
( 9.3.69 )
Transforming to that of the angles k x
= k sin Os
and k z = k cos 0s, we obtain
1 j~ 2 2 . (Ss . z) = 2k _.!!. dOsk cos OsI(kx = k sm Os)
"21.!!. dOs4k
(sin Os sin Oi - 1)2 W(k sin Os - k sin Oi)
The bistatic scattering coefficient (J"(()s) is the integrand of (9.3.70) divided by cosOi/2 = lSi' il. Thus,
LJ) . . (Us -_ 4k (sinOssinOi -1)2 W (k SIn () s _ k sIn ().) t COSOi
Kirchhoff Approach
In this section we derive the bistatic scattering coefficients using the Kirchhoff approach (KA). The KA approximates the surface fields using the tangent plane approximation. Using the tangent plane approximation, the fields at any point on the surface are approximated by the fields that would be present on the tangent plane at that point. For that to be valid, it is required that every point on the surface has a large radius of curvature relative to the wavelength of the incident field. The Kirchhoff approach casts the form of the solution in terms of diffraction integral. We first illustrate the approximation by considering the Dirichlet problem of a one-dimensional surface. Consider an incident wave impinging on a rough surface (Fig. 9.3.1). Let the incident wave vector be ki = kixx - kizz (9.4.1)
The surface has l-D height profile specified by z = f(x). We first derive Huygens' principle for 2-D scattering problem. The boundary is at z = f(x). The 2-D Green's function is (\72
+ k2 )g(r, 7") = -8(1' -
(9.4.2) (9.4.3)
g(r,r') = lH61)(klr - 7"1)
where l' = xx + zz is the two dimensional position vector. In (9.4.3) H6 Hankel function of the first kind. We utilize the 2-D Green's theorem.
dr [1/;1\721}J2-1}J2\721}Jl] =
dSn '[1}Jl\71/;2-1/;2\71/;d
where n is the outward normal to the two dimensional volume V2 and S is the one dimensional surface enclosing V2. Letting 1}Jl = 1/;(1') and 1/;2 = g(r,r') in (9.4.4) gives
1/;(1") = 1/;inc(r')
dS [1/;(r)n . \7 g(r, 7") - g(r, 7")n . \71/;(1')]
is the upward normal to the surface.
4.1 Dirichlet Problem for One-Dimensional Surface
The Dirichlet boundary condition specifies that
1/;(r) = 0
at z = f(x) on the surface. In the electromagnetic context,1/; corresponds to the electric field in the y-direction, and the Dirichlet boundary condition corresponds to TE wave impinging upon a perfectly conducting surface. Since ds = dxJl + (df jdx)2, we have 1/;(7")
= 1/;inc(r') -
dx g(r, 7") N . \71/;(1')
where r = (x, f(x)) and
N = --x+z
of ~
4.1 Dirichlet Problem for I-D Surface
tangent plane
Figure 9.4.1 Surface field tangent plane at r = (x, f(x)).
is a vector along the normal direction. In (9.4.8), N "V'ljJ(r) is the unknown surface field. An integral equation can be obtained from (9.4.8) by letting r' lie on the surface so that the left-hand side of (9.4.8) becomes zero. Kirchhoff approximation, on the other hand, consists of using an approximate expression for the surface field N . \7'ljJ(r). To obtain an approximate expression, we note that if a tangent plane is drawn at the point r = (x, f(x)), we will have a flat surface oriented as shown in Fig. 9.4.1 and the plane is perpendicular to the normal vector n. We note that for a hypothetical tangent plane with normal n that is a perfect electric conductor, the tangential magnetic field is