## 1 Introduction

[2] The transformation optics/electromagnetics (TO/TE) technique has provided device designers with unprecedented flexibility and has enabled many novel applications, such as a variety of antenna-related devices for radio frequencies and compact lenses for optical wavelengths [*Chen et al*., 2010; *Kwon and Werner*, 2010; *Li and Pendry*, 2008; *Pendry et al*., 2006]. However, the implementations of most TO designs have been restricted by the complexity of the material requirements, which generally call for anisotropic and inhomogeneous materials with extremely large refractive index variation [*Chen et al*., 2010; *Cai et al*., 2007; *Kwon and Werner*, 2008, 2009, 2010; *Li and Pendry*, 2008; *Leonhardt*, 2006; *Narimanov and Kildishev*, 2009; *Pendry et al*., 2006; *Roberts et al*., 2009; *Valentine et al*., 2009; *Schurig et al*., 2006; *Tang et al*., 2010; *Yang et al*., 2011]. Despite recent developments on metamaterials, the associated resonant losses and narrow bandwidth hinders the practical application and degrades the performance of many TO devices. In order to simplify the material requirements, the quasi-conformal (QC) mapping technique [*Li and Pendry*, 2008] has been proposed and successfully employed to minimize the anisotropy of the constitutive materials, resulting in nearly isotropic gradient-index (GRIN) devices with low losses and broad bandwidth [*Hunt et al*., 2012; *Kundtz and Smith*, 2010; *Li and Pendry*, 2008; *Ma and Cui*, 2010; *Smith et al*., 2010]. As a result, functional QCTO devices, such as transformed Luneburg lenses, have been demonstrated in both the microwave and optical wavelength regimes [*Hunt et al*., 2012; *Ma and Cui*, 2010]. This type of flat TO lenses hold extraordinary promise, especially in optics and imaging applications. In particular, they offer the potential of dramatically simplifying the optical assemblies and reducing the size, weight and power of optical subsystems by replacing bulky and heavy curved-surface lenses with flat lenses that provide equivalent or improved imaging performance.

[3] The rapid progress on developing various types of TO devices calls for accurate and efficient numerical methods for evaluating their performance. However, it remains a challenging task to perform 3D full-wave simulations on electrically large TO designs using commercially available packages, due to the huge memory requirements and long computation times. It has been demonstrated that different body-of-revolution (BOR)-based techniques may be used for analysis of 3D objects that possess rotational symmetry in free space, such as the method of moments (MoM) [*Andreasen*, 1965; *Gedney and Mittra*, 1990; *Mautz and Harrington*, 1979], the finite-element method (FEM) [*Dunn et al*., 2006; *Morgan and Mei*, 1979], the finite-difference time-domain (FDTD) method [*Chen et al*., 1996; *Davidson and Ziolkowski*, 1994; *Prather and Shi*, 1999], and hybrid methods (e.g., FEM/MoM) [*Jin*, 2005]. Some of these BOR-based methods have been extended to efficiently analyze 3-D rotationally symmetric targets embedded in a layered medium. Such techniques include (a) frequency domain based algorithms (e.g., MoM and FEM) developed for analysis of electromagnetic (EM) wave scattering from BOR targets [*Geng and Carin*, 1999; *Geng et al*., 1999; *Kucharski*, 2002; *Viola*, 1995; *Zhai et al*., 2011] and 3-D axisymmetric invisibility cloaks with arbitrary shapes [*Zhai and Cui*, 2011] buried in a layered media background; and (b) the BOR-FDTD approach implemented for characterizing the focusing performance of axially symmetric diffractive optical elements (DOE) with a substrate [*Shi and Prather*, 2001]. Compared to the MoM and FEM methods, the BOR-FDTD algorithm does not require solving a large system of equations. Instead, it exhibits a linear computational complexity which depends on the number of solution points within the computational domain. Furthermore, wideband responses can be obtained by employing a transient excitation in a single simulation. Therefore, the BOR-FDTD method is an efficient and well-suited method for modeling 3-D axisymmetric TO devices, such as lenses. However, there have apparently been no reports in the literature that address how to rigorously inject obliquely incident plane waves into the BOR-FDTD formulation for an object embedded in layered media.

[4] In this paper, we propose a one dimensional (1-D) FDTD-based method to inject normally incident plane waves into the TFSF formulas. For oblique incidence, the analytical closed-form formulations were derived and presented by expanding the incident, reflected and transmitted plane waves into a series of cylindrical modes. These approaches have been successfully utilized for accurate analysis of 3D BOR TO GRIN lenses in free space and with a substrate. The infinite computational region was truncated with the help of an unsplit-field perfectly matched layer (UPML) absorbing boundary condition (ABC) [*Gedney*, 1996; *Taflove and Hagness*, 2005] in cylindrical coordinates.

[5] The remainder of the paper is organized as follows: the BOR-FDTD methodology is introduced in section 'The BOR-FDTD Methodology', which includes: (a) the BOR-FDTD formulations, (b) singularity issues of the electric and magnetic fields on axis, (c) the PML ABC technique, and (d) the introduction of an incident plane wave into the TFSF formulas. In section 'Numerical Results and Discussion', the proposed method is first validated by comparison of the BOR-FDTD simulation results with analytical solutions for plane wave interaction with a layered medium. The developed solver is also employed to study the imaging properties of several TO GRIN lenses. Finally, conclusions are presented in section 'Conclusions'.