#### 5.1. Normalization of the WH Equations

[34] The method to numerically factorize the WH kernel and to evaluate the far-field quantities is illustrated, for the sake of clarity, only for the simpler case of a PEC half plane. The extension of this method to deal with imperfect half planes is rather straightforward.

[35] In terms of the dimensionless normalized admittance

the PEC half plane problem yields (see (13) and (20))

where **R**_{o} = *jE*_{0}**y**_{t}(*η*_{0})**e**_{0t} is a known term whereas **A**_{−}^{s}(*η*) is the Fourier transform of the total scattered current induced on the half plane, that is the total minus the Physical Optic (PO) current.

[36] Factorization in closed form is possible whenever **y**_{t}(*η*) commute with a polynomial matrix; if this does not happen, one has to use the approximate technique described by *Daniele* [2004a], which is based on the numerical solution of a Fredholm integral equation of the second kind. This technique requires that the WH matrix kernel and its inverse exist and are finite for *η* ±∞ on the real axis. Therefore we need to modify the kernel **y**_{t}(*η*) because of the following asymptotic behaviors for *η* ∞

where *a*_{ij} and *b*_{ij} are not vanishing constants. By introducing the matrices

equation (84) yields

with

and where **G**_{e}(*η*) and **G**_{e}(*η*)^{−1} exist and are finite for *η* ±∞.

[37] By introducing the normalized quantities _{o} = *η*_{o}/*k*_{o} (as per equation (2)), and

equation (89) reduces to

where all the involved quantities do not depend on *k*_{o}, *Z*_{o} and *Y*_{o}. That is to say that the normalized WH equation (96) does not change for a fictitious medium that has the same normalized matrices , , , and , but different values of *k*_{o}, *Z*_{o} and *Y*_{o}. As a matter of fact, values of *k*_{o} with a vanishing or a very small imaginary part make the evaluation of the eigenvalues and the numerical factorization of **G**_{e} difficult; for this reason it is convenient to introduce a fictitious lossy medium, and in our numerical simulations we have used

In the following, to avoid any confusion, the quantities computed for the fictitious medium are indicated by a tilde. The eigenvalue _{i} = *j*_{o}_{i} computed for the fictitious medium permits one to evaluate the corresponding physical eigenvalue *γ*_{i} = *k*_{o}_{i}/_{o} of the real medium.

[38] In the fictitious medium, equation (89) now reads

where numerical evaluation of _{+}(*η*) by the Fredholm method given by *Daniele* [2004a] is now possible. The physical solution **X**_{+}(*η*) is obtained by noticing that

which yields

This permits one to solve the WH problem (89) and to obtain the functions **V**_{+}(*η*), **I**_{a}(*η*) = **V**_{+}, and **I**_{b}(*η*) = **V**_{+}.

#### 5.2. Field Evaluation by Inverse Fourier Transformation

[39] In the following, without loss of generality, we consider only the evaluation of the transverse component of the electric field for the PEC half plane problem where, to simplify the notation, we set _{V} = , _{V} = . The transverse component of the magnetic field can be similarly obtained. The longitudinal field components *E*_{y}(*z*, *x*, *y*) and *H*_{y}(*z*, *x*, *y*) are obtained from the transverse ones by Maxwell's equations [see *Daniele*, 2006].

[40] The plus function **V**_{+}(*η*) is obtained by solving equation (84). The transverse field **E**_{t}(*z*, *y*, *x*) is then obtained by inverse Fourier transformation

with = for *y* > 0, and = in the region *y* < 0, and where the integration path *B*_{+} is a horizontal straight line located above all the singularities of **V**_{+}(*η*). The primary field **E**_{t}^{p}(*z*, *y*, *x*) = **E**_{t}^{i}(*z*, *y*, *x*) + **E**_{t}^{r}(*z*, *y*, *x*) represents the contribution of the incident plus the reflected field; the primary field is evaluated by assuming an entire PEC plane at *y* = 0. Notice that **E**_{t}^{p} is equal to zero for *y* < 0 if the incident wave impinging of the half plane propagates in the negative *y* direction. Vice versa, the primary field is zero for *y* > 0 whenever the incident wave propagates in the positive *y* direction.

[41] Let us now assume that the incident plane wave propagates in the negative *y* direction. By using the Cayley representation for the exponential factor exp(−*y*) in (101) one gets

for *y* > 0, and

for *y* < 0. By further assuming an incident plane wave with a propagation factor equal to exp[−*jα*_{o}*z* − *jη*_{o}*x* − *γ*_{3}(*η*_{o})*y*] (without loss of generality, the case of an incident plane wave with propagation factor equal to exp[−*jα*_{o}*z* − *jη*_{o}*x* − *γ*_{4}(*η*_{o})*y*] can be omitted) one gets

where **e**_{0t} is the eigenvector of (*η*_{o}) associated with the eigenvalue *γ*_{3}(*η*_{o}), whereas the amplitude vectors **E**_{t1}^{r} and **E**_{t2}^{r} can be obtained by geometrical optics considerations. For example, while dealing with a PEC plane, one has **E**_{t1}^{r} = *c*_{1}**e**_{t1} and **E**_{t2}^{r} = *c*_{2}**e**_{t2}, where **e**_{t1} and **e**_{t2} are the eigenvectors of (*η*_{o}) associated with the eigenvalue *γ*_{1}(*η*_{o}) and *γ*_{2}(*η*_{o}); the scalar coefficients *c*_{1} and *c*_{2} are simply obtained by enforcing **E**_{t}^{p} = 0 on the PEC plane.

[42] The far-field contributions are evaluated by applying the saddle point method [*Felsen and Marcuvitz*, 1973] to each integral of (102) and (103). For each given observation point with azimuthal angle , this method requires the determination of the saddle points _{s} of the function

with

and the determination of the steepest descent paths (SDP) that cross the saddle points. In fact, the integration path *B*_{+} of (102) and (103) is warped into a SDP, and each saddle point has is own SDP. Occasionally, more saddle points may occur for *i* = 1, 2, 3, or 4; in these cases, application of the saddle point method is more difficult. For the sake of simplicity we do not discuss these cases, and we assume that each integral has only one significant saddle point. We also observe that the _{s} are real for a lossless medium. Since the observation angle appearing in (106) depends on the location of the observation point, it may happen that the SDP captures the pole *η* = *η*_{o} of the function **V**_{+}. A detailed study, not reported here, shows that the pole is captured only if _{s} > _{o}; when this happens, the pole *η* = *η*_{o} is always captured clockwise. A careful study of the residues of the integrands and use of the boundary condition for the PEC half plane permit one to write, for *y* > 0

whereas for *y* < 0, one gets

The previous results show that the total field does not present any reflected wave contribution in the region *y* > 0 if the saddle points _{si} satisfy the condition _{si} > _{o}, for *i* = 1, 2. Furthermore, for *y* < 0, the total field does not present any geometrical optics contribution (in particular, the incident wave contribution) if the saddle point _{s3} satisfies the condition _{s3} < _{o}.

[43] As far as the evaluation of the SDP integrals is concerned, after some algebraic manipulations one may prove that all the transverse field components, including the magnetic field ones, can be expressed in terms of scalar integrals of the form

where *ρ* is the radial distance from the observation point to the edge of the half plane, and *q* is given in (106). The saddle points _{s} are obtained by evaluating the zeroes of the derivative of the phase term

with

The saddle points are therefore given by the solutions, with respect to , of the following system of algebraic equations

which does not change by changing ϕ into (ϕ − *π*). As a matter of fact, for lossless media, the saddle points can be obtained graphically [*Felsen and Marcuvitz*, 1973, p. 110] by considering the dispersion curves *f*(, ) and the vector distance *ρ* = *ρ*( cos ϕ + sin ϕ) from the observation point to the edge of the half plane, as shown in Figure 3. In Figure 3, at the saddle points _{s1} and _{s2}, the straight lines tangent to the dispersion curve () are orthogonal to **ρ**, and this happens only and for all the saddle points.

[44] More generally, since equations (113) are algebraic, the saddle points _{s} are defined to be the zeroes of the polynomial in obtained from the resultant

of the two equations (113) with respect to [*Bliss*, 2004].

[45] For lossless media, the contribution of each real saddle point to the integral (110) is

where the plus or minus sign in the exponential factor is chosen according to the sign of *q*″_{s}) [*Felsen and Marcuvitz*, 1973, p. 387].

[46] In this connection we recall that the function **V**_{+}(*η*) has one pole in *η* = *η*_{o}, with residue **T** = *jE*_{o}**e**_{0t}. The function **T**/(*η* − *η*_{o}) is the Fourier transform of the geometrical optic field over the aperture (*x* > 0, *y* = 0). The Physical Optics (PO) contribution to the diffracted field is calculated by approximating *A*(*k*_{o}) with the value obtained by using **V**_{+}^{PO} = **T**/(*η* − *η*_{o}), **I**_{a}^{PO} = **T**/(*η* − *η*_{o}), or **I**_{b}^{PO} = **T**/(*η* − *η*_{o}) in (108) and (109). For this reason, the quantity

evaluated at the saddle points is able to represent the difference between the true diffracted field and the PO contribution to the diffracted field. Notice that the superscript *d* in (116) stays for *difference*. This difference is considered in the numerical case study that follows.

#### 5.3. Numerical Results for a PEC Half Plane in a Gyrotropic Medium

[47] Let us consider a lossless gyrotropic medium with = **1**, = = **0** and

By choosing _{o} = 1/2, from equations (48) and (56), we obtain

with

These data yield 12 normalized branch points in the normalized plane; four of them are real

and eight are complex

[48] The values at the four real branch points are

Recall that the real branch points individuate the four branch points of the characteristic impedance and admittance. The dispersion curves for this case are shown in Figure 2, where it is evident that two different kinds of waves can propagate in this gyrotropic medium.

[50] The incident wave originates two diffracted waves, one of the first and one of the second kind, which are computed very accurately by numerically solving, with the technique explained by *Daniele* [2004a], the Fredholm equation associated with the factorization of the WH kernel. The results of Figure 4 show the real and the imaginary part of the two normalized components of the quantity _{+}^{d}() = *k*_{o}**V**_{+}^{d}() given in (116). As previously explained, in order to obtain the far diffracted field according to (115) it is enough to know the saddle points and the value of **V**_{+}^{d}(*η*) at the saddle points. The location *η*_{s} of the saddle points is obtained from (114), and the values of *η*_{s} in the observation angular region {−180° < ϕ ≤ 180°} are reported in Figure 5. Tables 1 and 2 report the values of the diffracted field components

relative to the first and second wave for different observation angles. The values for the first kind wave are given in Table 1, whereas Table 2 shows the values relative to the wave of the second kind. Notice from equation (124) that in general, the phase of the cylindrical waves 1 and 2, with respect to the phase one has for an isotropic medium, is corrected by a θ_{c} factor which depends on the azimuthal observation angle ϕ as well as on the kind (1 or 2) of the diffracted wave.