IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 66, NO. 1, JANUARY 2018 347 Scattering by a Dielectric Sphere Buried in a Half-Space With a Slightly Rough Interface Hasan Zamani , Ahad Tavakoli, and Mojtaba Dehmollaian, Senior Member, IEEE Abstract— Analytical expressions for the scattering coefficients of a dielectric sphere buried under a rough interface are presented. The proposed method combines the small perturbation method (SPM) and the Mie solution by using the expansion of plane waves in terms of vector spherical functions (VSFs) and vice versa. First, using SPM, the zeroth- and the first-order perturbative scattered fields of a rough interface for illuminations from above and below are derived. Using these solutions, the field transmitted to the lower half-space is determined as a spectrum of down-going plane waves. The scattered fields from the sphere are then calculated using the vector Mie solution. Subsequently, the VSFs are expanded in terms of up-going plane waves. These plane waves illuminate the interface, and using SPM, the scattered fields in the upper and lower regions are determined as infinite summations of plane waves. The reflected plane waves are once again scattered by the sphere and the scenario repeats. By inspecting the form of the fields resulting from the few first interactions of the sphere and the rough interface, a recursive form is obtained for the scattered fields. This recursive form is then used to rewrite the system of equations in a form containing all interactions in a single-step formulation. Accordingly, the zeroth- and the first-order closed-form scattered fields are obtained. The derived expressions are analytically and numerically validated. Finally, the numerical results for the case of the rough interface with sinusoidal profile are presented and briefly discussed. Index Terms— Buried objects, electromagnetic scattering, rough surfaces, small perturbation method (SPM), vector spher- ical functions (VSFs). I. I NTRODUCTION S CATTERING from objects in presence of an interface has been investigated due to applications in various fields such as remote sensing and optics [1]. Novel analytical approaches for 2-D [2], [3] and 3-D [4], [5] problems with canonical targets, and improved numerical techniques for scattering from PEC [6]–[8] and dielectric [9], [10] objects near a flat or rough interface are examples of new relevant theoretical studies. Making use of superstrates for increasing the detectability of buried targets [11], time reversal imaging of deeply buried Manuscript received September 11, 2016; revised October 9, 2017; accepted October 31, 2017. Date of publication November 9, 2017; date of current version January 2, 2018. (Corresponding author: Hasan Zamani.) H. Zamani and A. Tavakoli are with the Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 15875-4413, Iran (e-mail: h.zamani@aut.ac.ir; tavakoli@aut.ac.ir). M. Dehmollaian is with the Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, Tehran 14395-515, Iran (e-mail: m.dehmollaian@ece.ut.ac.ir). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2017.2772038 targets [12], and scattering from objects buried in underwater sediments [13] are examples of new pertinent applications. In recent years, a large number of numerical and semian- alytical techniques are proposed to solve the scattering from objects under a flat [14], [15] or rough [16], [17] interface. However, analytical solutions [18], [19], while limited to canonical objects, are still sought for due to the following reasons. First, they are much faster and this makes them applicable both in problems involving a large number of buried objects and in inverse problems. Second, they could provide deeper insight in the scattering mechanisms between the object and the interface. Finally, they could serve as reference solutions for validating the numerical ones. In this paper, we are focused on the analytical solutions and hence, in what follows, only the corresponding studies are addressed. The 2-D problem of analytical solution of scattering from buried objects is thoroughly investigated in [19]–[21]. For instance, Di Vico et al. [20] have considered the problem of scattering from a set of conducting cylinders buried in a half-space, using the spectral domain method. Then, the case of conducting cylinders buried in a dielectric slab was studied in [19], utilizing the cylindrical-wave approach. Next, the same approach was applied to the case of dielectric cylinders buried in a dielectric slab [21]. In the 3-D case, Green’s function method in conjugation with the spherical wave expansion of the dyadic Green func- tions is applied to the problem of scattering from a sphere buried in a half-space in [22], where the sphere is at an arbitrary distance with respect to the interface. In [22], the near fields are presented for different incident fields and the effect of multiple interactions between the sphere and the interface is briefly addressed. The earlier studies were mostly investigated under special approximations [23]–[25]; an excellent review of these and other relevant techniques is provided in [18]. Recently, the same problem is solved using the spectral domain method [18]. In the spectral domain technique, below the interface, the transmitted field, the scattered field from the sphere, and the scattered-reflected wave are considered and the boundary conditions are satisfied on the surface of the sphere. The same method is then extended to the case of multilayered sphere buried in a multilayered half-space [5]. Finally, the spectral domain method is applied to the case of the lossy ground [4] using the expansion of inhomogeneous plane waves in terms of vector spherical functions (VSFs) [26]. The effect of the roughness on the scattering from targets in presence of an interface is rarely considered [1], [27]–[29]. 0018-926X © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.