Scattering from a Buried PEMC Cylinder Illuminated by a Normally Incident Plane Wave Propagating in Free Space

─ A rigorous solution is presented to the problem of scattering by a perfect electromagnetic conducting (PEMC) circular cylinder buried inside a dielectric half-space that is excited by a normally incident transverse magnetic (TM) plane wave propagating in free space. The plane wave incident on the planar interface separating the two media creates fields transmitting into the dielectric halfspace becoming the known primary incident fields for the buried cylinder. When the fields scattered by the cylinder, in response to those fields incident on it, are incident at the interface, they generate fields reflected into the dielectric half-space and fields transmitted into free space. These fields, and the fields scattered by the cylinder are expressed in terms of appropriate cylindrical waves consisting of unknown expansion coefficients which are to be determined. Imposing boundary conditions at the surface of the cylinder and at a point on the planar interface, enables the evaluation of the unknown coefficients. This procedure is then replicated, by considering multiple reflections and transmissions at the planar interface, and multiple scattering by the cylinder, till a preset accuracy is obtained for the reflection coefficient at the particular point on the interface. The refection coefficient at this point is then computed for cylinders of different sizes, to show how it varies with the PEMC admittance of the cylinder, its burial depth, and the permittivity of the dielectric half-space. Index Terms ─ Buried PEMC circular cylinder, cylindrical wave expansions, multiple reflections, planar interface, rigorous solution.


I. INTRODUCTION
Solutions to electromagnetic scattering by buried bodies may be used to detect pipes, tunnels, mines, and other like objects. Analytical and numerical solutions for locating two and three dimensional bodies are already available in the literature. For instance, solutions using various cylindrical wave expressions have been obtained in the literature by some authors, for cylinders buried in homogenous and/or nonhomogeneous ground, lossy earth, and quite a few rough surfaces [1][2][3][4][5][6][7][8][9][10]. Alternatively, numerical solutions for scattering from arbitrary shaped buried bodies have been obtained using the Method of Moments (MoM) [11][12][13]. The objective of this paper is to obtain a rigorous solution to scattering by a PEMC circular cylinder buried in a dielectric half space, when excited by a plane wave propagating in free space. The fields scattered by the PEMC cylinder due to those primary fields incident on it are expressed in terms of cylindrical waves comprising of unknown expansion coefficients. When the fields scattered by the buried cylinder are incident at the interface, they generate fields which are reflected into the dielectric medium and transmitted into free space. These fields are also expressed in terms of suitable cylindrical waves involving unknown expansion coefficients. Enforcing the boundary conditions at the surface of the buried PEMC cylinder and at a specified point on the planar interface leads to the evaluation of the unknown expansion coefficients. This procedure is then replicated, by considering multiple reflections and multiple transmissions at the planar interface, and multiple scattering by the cylinder, till a preset accuracy is obtained for the reflection coefficient at the specified point on the interface. It is worth noting that the method of finding the unknown expansion coefficients does not require any matrix inversion, making it quite computationally efficient. Details of the paper are given in the following sections.

II. FORMULATION
An infinitely long PEMC circular cylinder of radius a is buried in a dielectric half-space having a relative permittivity of r  while positioned at a distance d from the planar interface separating the dielectric half-space and the free space regions, as shown in Fig. 1. Consider a uniform plane electromagnetic wave of amplitude E 0 propagating in free space, being incident on this planar interface.
where k 0 is the wavenumber and 0  is the wave impedance in free space.
The electric and magnetic field components of the reflected and transmitted waves generated due to the plane wave incident on the planar interface (in the absence of the cylinder), are given by where R and T are the reflected and transmitted field coefficients due to the presence of the planar interface and 1  is the intrinsic wave impedance in the dielectric half-space. These coefficients can be evaluated by enforcing the boundary conditions on the planar interface at x=-d, as where k 1 is the wavenumber in the dielectric halfspace. The initial transmitted electric and magnetic fields corresponding to their components given by (5) and (6) will be incident on the PEMC cylinder. These field components may be written in terms of circular cylindrical wave functions in the dielectric medium as k  , and the prime denotes its derivative with respect to the argument. The incident field expansion coefficients are given by A n =(TE 0 )j -n . The scattered co-polar and cross polar field components due to the initial incident fields on the PEMC cylinder can be written as , 0 0 ( 2 ) ' where are the co-polar and cross polar scattered field coefficients, which can be found by applying the PEMC boundary condition on the 2 surface of the PEMC cylinder. The expressions for these coefficients are given by [14] ( 2 ) where M is the admittance of the PEMC cylinder. field components back into the dielectric halfspace as second order field components, which are incident on the cylinder. These can be written as  Fig.1 (11)-(14) and (17) and ( 2 ) 3 The reflected fields that correspond to components in (21)-(24), upon arriving at the buried cylinder, act as secondary incident fields and create higher order scattered fields. Using Graf's addition theorem, the reflected field components which act as secondary incident field components can be written as The corresponding magnetic field components can be obtained using Maxwell's equations. The field components scattered by the cylinder in response to the above secondary field components incident on it, can be written referring to (11)- (14), by changing the superscript 0 to 1. The unknown scattered field expansion coefficients are evaluated by applying the PEMC boundary conditions on the surface of the cylinder, which can be written as , 0   The z and  components of the electric field in the free space region above the interface can then be written as , The co-polar and cross polar reflection coefficients are thus given by

III. NUMERICAL RESULTS
Results are presented to show how the co-polar and cross polar reflection coefficient magnitudes at the point P in Fig. 1, vary with the size of the cylinder, for PEMC cylinders of different PEMC admittances and burial depths, as well as for when each cylinder with a given PEMC admittance is buried at different depths and in different types of dielectric media. For the sake of convenience, the normalized PEMC admittance has been expressed in the dimensionless form Mη 1 = tan ν.
To validate the analysis and the software used for obtaining results, |Γ cp | has been calculated at the point P in Fig. 1, for ν=90 0 , r  =2.0, and d=2.25 cm, when the electrical radius k 0 a of the cylinder increases from 0.0 to 1.6. These results have been compared in Fig. 2, with the corresponding results for a buried PEC circular cylinder considered in [10], and are in excellent agreement, verifying the accuracy of the analysis and the software used for obtaining the results. Figure 3 shows the variation of | Γ cp | and |Γ xp | at P for a PEMC cylinder buried in a dielectric medium having a relative permittivity of 2.0, at a depth of 2.25 cm from the planar interface, for 5 different PEMC admittances, when the electrical radius of the cylinder varies from 0 to 1.6. In Fig. 3(a), the minima move towards the right as the value of ν is increased and the sharpness of the first minimum also decreases with increasing ν. In Fig. 3(b), we observe that the curves for ν=15 0 and ν=75 0 are essentially the same and so are those for ν=30 0 and ν=60 0 . Also, variations of | Γ cp | with k 0 a are quite rapid as compared to those of |Γ xp |.  Figure 4 shows the variations of |Γ cp | and |Γ xp | at P for the PEMC cylinders of Fig. 3, buried at a depth of 5 cm from the planar interface. |Γ cp | in Fig. 4(a) are more oscillatory when compared with the same in Fig. 3(a). However, the variations of |Γ xp | in Fig.  4(b) are not too different to those in Fig. 3(b). Yet, the values of |Γ cp | and |Γ xp | in Fig. 4 are smaller, when compared with the corresponding values in Fig. 3. Variations of |Γ cp | against the electrical radius k 0 a of each buried PEMC cylinder, for ν=45 o , r  = 2, and for different burial depths d are shown in Fig.  6. As the value of d increases from 2.25 to 6.25, the oscillations in the plots also increase. Also, the corresponding minima and maxima move to the left as d increases.

IV. CONCLUSION
A rigorous solution that does not require inversion of matrices and based on the multiple reflection of waves has been obtained for scattering by a PEMC cylinder buried in a dielectric half-space, when it is illuminated by a plane wave propagating in free space incident normally on the interface separating the two media. Numerical results have been given in the form of co-polar and cross polar free space reflection coefficient magnitudes, to exhibit how they vary with the size, burial depth, and PEMC admittance of the buried cylinder, as well as with the permitttivity of the burial medium. Results for PEC and PMC cylinders can be obtained from this analysis, as special cases. The solution process can also be used for solving related inverse scattering problems.

ACKNOWLEDGMENT
Prof. A-K. Hamid wishes to acknowledge support provided by the University of Sharjah, U.A.E. 6