Electromagnetic Scattering from a Buried Cylinder using T-Matrix and Signal-Flow-Graph Approach Ayman S. Negm 1 , Islam A. Eshrah 1 , Ragia I. Badr 1 , 1 Department of Electronics and Electrical Communications, Cairo University, Giza, Egypt Abstract—A fast analytical technique based on T-matrix ap- proach is formulated to solve the problem of direct electro- magnetic scattering by an infinite circular cylinder buried in a dielectric half-space and illuminated by a normally incident transverse magnetic (TM) plane wave. The technique employs the signal-flow graph (SFG) model to include all multiple reflections that take place during the scattering process. The selection of the truncation order of the obtained T-matrix is also discussed. Index Terms—scattering, buried cylinder, signal-flow graph. I. I NTRODUCTION T He study of scattering from buried objects has gained a lot of attention in the past few decades; due to its wide and many applications in several fields like detection of landmines, non-invasive biomedical investigations and remote sensing. Among the different approaches proposed to solve this problem [1, and references therein], the T-matrix approach stands out as a powerful and widely used method [2]–[4]. The main idea of the T-Matrix approach is to expand both the incident and scattered field in terms of spherical or cylindrical wave series. Given that the coefficients of incident and scattered field expansions are put in vector forms, the objective is to find a matrix that relates these two vectors to each other, which is called Transition Matrix or T-Matrix. The process of finding the T-Matrix is based on enforcing boundary conditions on surfaces in the involved scattering system. Many modifications were proposed to the T-matrix method using fast Fourier Transform [5], local shape function [6], recursion based on method of moments [7], extended boundary condition method (EBCM) [8], stabilized EBCM [9] and iterative technique for large aspect ratios [10]. In [11], an iterative analytical approach is introduced to quickly solve the scattering problem of a buried cylinder where the scattered field is measured at the symmetry point on the interface. The proposed solution in [11] has two drawbacks. In this work, a T-matrix/SFG approach is proposed to address the problems encountered in [11]. In Section II, the problem formulation is revisited for completeness. Section III details the SFG model of the problem, which results in the sought T-matrix. Using the SFG reduction rules allows for summing the multiple reflections to infinity, then the T-matrix formulation is used to determine the truncation order related to the expansion modes as illustrated in Section IV. Results and discussion followed by conclusions are reported in Sections V and VI respectively. II. PROBLEM FORMULATION The geometry of the problem under investigation is shown in Fig. 1. A homogenuous circular cylinder of complex relative permittivity ε r2 , radius a and of infinite length along z direction is buried at depth d in a homogenuous dielectric half-space of complex relative permittivity ε r1 . The magnetic permeabilities of the dielectric half-space and the cylinder are assumed to be that of free space. A harmonic time dependence e jωt is assumed for all fields arising in the next derivation. For a TM plane wave of electric field amplitude E 0 prop- agating in the positive x-direction in free space, the incident electric field can be written as: E inc = E 0 e -jk0x ,x< -d, (1) where k 0 is the wave number in free space. The electric field can be written using cylindrical wave expansion [12] as: E i z (r, φ)= ∞  n=-∞ u n J n (k 0 r) e jnφ ,x< -d, (2) where u n = E 0 j -n , J n is the Bessel function of order n. The field reflected from the interface back to free space is given by: E r z = R 0 E 0 e jk0x ,x< -d, (3) where R 0 = η 1 - η 0 η 0 + η 1 e j2k0d , (4) and the field transmitted through the dielectric is given by: E t z (r, φ)= ∞  n=-∞ a n J n (k 1 r) e jnφ ,x> -d, (5) where a n = T 0 u n ,T 0 = 2η 1 η 0 + η 1 e j(k0-k1)d , (6) k 1 is the wave number in dielectric half-space, and η 0 and η 1 are the intrinsic impedances of free space and dielectric respectively. The field reflected from cylinder is given by: E r z (r, φ)= ∞  n=-∞ b n J n (k 1 r) e jnφ ,x> -d, r > a, (7)