ASYMPTOTIC EVALUATION OF SCATTERING OF INHOMOGENEOUS PLANE WAVES BY A PERFECTLY ELECTRIC CONDUCTING HALF PLANE

Abstract: In the present study, reflected and diffracted fields of an inhomogeneous plane wave obliquely incident on the surface of a perfectly electric conducting (PEC) half plane are evaluated for a two-dimensional case asymptotically and reexamined by considering the different aspects. Obtained results are plotted by Matlab numerically. Relationships between the complex angle of incidence and field intensities, and phase shifts in the fields are noted. Matlab plots are interpreted and compared to the theory for consistence.


INTRODUCTION
Inhomogeneous waves, along with the Gaussian beams, are considered in the category of evanescent waves that have been under investigation for decades. Noteworthy feature of these waves is they attenuate in the direction perpendicular to that of propagation. Due to their rapid fading with respect to the distance, they are negligible in the far field but they are taken into account in the near field because of their contribution. Ronchi et al. (1961) examined the scattering of evanescent waves by using complex angles to express an inhomogeneous wave. Keller and Streifer (1971) worked on complex ray application for the Gaussian beams. Choudhary and Felsen (1973) studied on the asymptotic theory for inhomogeneous waves. Wang and Deschamps (1974) applied complex ray tracing to scattering problems. Shevernev (1976) analyzed the diffraction of an inhomogeneous plane wave by a wedge. Later, Felsen (1976) examined evanescent waves. Bertoni et al. (1978) studied on the shadowing of an inhomogeneous plane wave by a wedge. Deschamps et al. (1979) investigated the diffraction of an evanescent plane wave by a half plane. Kouyoumjian et al. (1996) analyzed the diffraction of an inhomogeneous plane wave. Manara et al. (1998) worked on the diffraction of an inhomogeneous plane wave by an impedance in lossy medium in their study where they asymptotically evaluated the rigorous integral representation of the field by considering the uniform geometrical theory of diffraction (UTD). Kouyoumjian et al. (2007) examined inhomogeneous electromagnetic plane wave diffraction by a perfectly electric conducting wedge at oblique incidence.  worked on the uniform theory for the diffraction of evanescent plane waves, and he studied the diffraction of homogeneous and inhomogeneous plane waves by a planar junction between perfectly electric conducting (PEC) and impedance half planes . Diffraction of evanescent plane waves by a resistive half-plane  and Scattering of a line source by a cylindrical parabolic impedance surface (Umul, 2008), Scattering of a plane wave by a cylindrical parabolic perfectly electric conducting reflector (Kara, 2017), and scattering of inhomogeneous plane waves by a resistive half-screen (Umul, 2013) are also studied. In the present study we reexamine the scattering of an inhomogeneous plane wave for a two-dimensional case by a PEC half plane.

REFLECTION OF AN INHOMOGENEOUS PLANE WAVE BY A PEC HALF PLANE
Assuming that complex incident angle α is given as the combination of real and imaginary components as , The expression of incident inhomogeneous plane with a complex angle α can be given as, , which can be rewritten as, where , , k is the wave number which is equal to 2π/λ where λ is the wavelength. Scattered field can be expressed as, If the surface is a perfect electric conductor, electric field along the surface becomes zero. In that case scattered field can be rewritten as,

Figure 1: Geometry of the problem
where n denotes the surface normal which is in y-direction for our geometry. Partial derivative of with respect to y is obtained as, .
At y=0 plane, . ∬ where √ For the solution of Eq.(14) let, (12) and (13) Then the scattered field is obtained as In the asymptotic evaluation of scattered field, reflected field can be found by the method of stationary phase point. For this method phase function is written as, Since the first derivative of the phase function at the stationary phase point is zero, we write . (20) Its second derivative is found as, .
By employing Taylor expansion, phase function is concluded as, Reflected field is written as or , where the first exponential term represents the plane wave, and the second term denotes the amplitude of the plane wave. The following figures, Fig. 3 through Fig. 6, illustrate the reflection behavior of the inhomogeneous wave for the geometry given in Fig. 2. For the Matlab plots, the wavelengths λ is taken as 0.1 meter and the observation distance ρ=λ is considered   Fig. 4 shows the incident and reflected fields of the inhomogeneous plane wave with incident angle α which is complex conjugate of the one used for Fig. 3. In this case propagation direction is in -60 degrees, and incident electric field is in 210-degree direction. As a result, reflected field lies in the 150 degree direction.

Figure 5: Intensity variations of incident and reflected fields with respect to positive φ i
In Fig. 5 the variation of field intensities with respect to imaginary parts of the incident angle α are given. It can be observed that field intensities and φ i values are directly proportional. Thus, imaginary component of complex incident angle determine the amplitude of the incident or reflected field. In Fig.  5, real part of the incident angle is 60˚, and the direction perpendicular to this angle is -330˚ which is the field direction. In this direction, amplitude of the field is attenuates by the decrease in φ i . In Fig. 6, this time the field direction perpendicular to 60˚ is 150˚. Attenuation of the field can be implemented by decreasing φ i again. In either case, attenuation occurs in the way that is perpendicular to the propagation direction. In a similar manner, this phenomenon takes place for the reflected field as well.

DIFFRACTION OF AN INHOMUGENEOUS PLANE WAVE BY A PEC HALF PLANE
For the scattered field expression given in Eq. (26), diffracted field can be evaluated by the edge point method given by Eq. (27).
where the amplitude function f(x) is, From Fig. 2 it can be seen that, at the edge point, and .
At and Finally diffracted field becomes, Which can be written as, Ud can be divided into incident-diffracted and reflected-diffracted parts as and where the detour parameters are given as, and For complex parameters sign(x)F[abs(x)] does not yield correct results. In that case complex detour parameter decomposition [12] is employed. Letting, it can be written that, Using the identities of, and and Letting, and We write, In a similar manner by letting, and and using the same steps it is found that, Finally incident diffracted and reflected diffracted fields are respectively obtained as, and The following Matlab plots depict the incident diffracted and reflected diffracted field variations according to some parameters. The wavelengths λ is taken as 0.1 meter and the observation distance ρ=λ is considered.

Figure 7: Total diffracted field variations for some φ i values
In Fig. 7, it is observed that field intensity and the imaginary component φ i of the incident angle α are directly proportional. Peak points of the diffracted field shift due to the change in φ i .

CONCLUSION
In this study firstly, for the asymptotic evaluation scattering integral, reflection behavior of an inhomogeneous plane wave by a perfectly electric conducting half plane is investigated by the method of stationary phase point. As a second step diffracted field expression is obtained through the edge point method. To obtain correct results for the uniform fields, obtained field expressions are transformed into another form eliminating incorrect results. Matlab plots of the derived fields are obtained for some parameters, and the results are interpreted. It is concluded that intensity of the reflected fields and imaginary component of the incident angle are directly proportional, but for the diffracted fields they are inversely proportional. Also it is observed that any phase change in the real component of the incident angle resulted in the same amount of phase shift in the reflected and diffracted fields.