Corresponding author: D. H. Werner, Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA. (email@example.com)
 A body of revolution, finite-difference time-domain (BOR-FDTD) method is developed for rigorous analysis of axisymmetric transformation optics (TO) lens devices. For normal incidence, a one dimensional (1-D) FDTD method based on the total-field scattered-field (TFSF) technique was proposed to model the propagation of a plane wave launched from the top of a layered medium in cylindrical coordinates. The 1-D FDTD solutions were employed to efficiently inject normally incident plane waves into the BOR-FDTD method. For oblique incidence, analytical formulations were derived and presented by expanding the plane wave into a series of cylindrical modes via Fourier series expansion of the ϕ-dependent variables, which were then used to introduce obliquely incident plane waves into the TFSF formulas associated with the BOR-FDTD method. These procedures allowed for accurate simulations of BOR TO lenses embedded in layered media illuminated by obliquely incident waves. The accuracy and efficiency of the proposed method were verified by comparing numerical results with either analytical solutions or a commercial software (COMSOL) package. Thereafter, the developed BOR-FDTD code was utilized to study the imaging properties of (a) radial gradient-index (GRIN) lenses with a parabolic index profile, (b) a flat TO GRIN lens, (c) a spherical Luneburg lens, and (d) a cylindrical TO Luneburg lens both in free space and on top of a substrate. Here the TO GRIN lenses were designed by using the quasi-conformal transformation optics (QCTO) technique. It was found that the flat TO lens was able to provide identical focusing properties as a cemented doublet in both free space and over a dielectric substrate. Moreover, the numerical results demonstrated that the flattened TO Luneburg lens possessed the desired imaging properties under different illuminations for both polarizations.
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 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.
 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.
 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.
 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'.
2 The BOR-FDTD Methodology
 The EM scattering from BOR objects has been widely investigated during the past several decades [Andreasen, 1965; Chen et al., 1996; Dunn et al., 2006; Davidson and Ziolkowski, 1994; Geng and Carin, 1999; Geng et al., 1999; Gedney and Mittra, 1990; Jin, 2005; Kucharski, 2002; Mautz and Harrington, 1979; Morgan and Mei, 1979; Prather and Shi, 1999; Shi and Prather, 2001; Taflove and Hagness, 2005; Viola, 1995; Yu et al., 2006; Zhai et al., 2011]. The axial symmetry of the scatterer allows the original 3-D problem to be reduced to a 2-D problem by factoring out the ϕ variation using the Fourier series expansion technique. As a result, the 3-D volumetric computation domain can be projected onto a 2-D (ρ-z) plane as shown in Figure 1. Consequently, the memory and CPU time requirements can be reduced significantly as compared to the 3-D FDTD algorithm.
2.1 BOR-FDTD Formulations
 The BOR-FDTD method developed in this paper is based on the TFSF formulation [Taflove and Hagness, 2005] in which the TFSF boundary separates the computational grid into two regions: a total-field region (TFR) that contains the incident and scattered fields and a scattered-field region (SFR) that contains only the scattered fields. In addition, the UPML (in cylindrical coordinates) ABC is employed to truncate the infinite computational region [Gedney, 1996; and Taflove and Hagness, 2005]. Let us consider a TE-polarized plane wave launched from an upper isotropic medium upon a BOR lens with a layered medium. As shown in Figure 1a, a plane wave at oblique incidence can be introduced along the boundaries (three solid red lines) between the TFR and SFR. Figure 1b illustrates the two-dimensional (2-D) layout of the electric (black) and magnetic (red) field components in cylindrical coordinates for the BOR-FDTD algorithm.
 The BOR-FDTD method is employed to solve the pair of time-dependent Maxwell's curl equations in cylindrical coordinates. For a BOR object, the electric and magnetic fields can be expanded using an infinite Fourier series with the following equations [Taflove and Hagness, 2005],
in which the Fourier coefficients and represent electric and magnetic fields. The order of the Fourier mode is denoted by m and the subscripts e and o indicate fields with cosinusoidal and sinusoidal dependence, respectively.
 It is important to note that for a given mode m, equations ((1a)) and ((1b)) can be decomposed into two sets of electric and magnetic fields which are independent and mutually orthogonal with respect to each other in isotropic materials and can be solved separately. For TE polarized (ϕ-polarized) incident plane waves, the fields are given by
For TM polarized (θ-polarized) incident plane waves, they may be written as
In equations. ((2a)), ((2b)) ((3a)) and ((3b)), , , , and , , are the unknown Fourier coefficients, which follow the space-time-mode dependence as illustrated in Figure 1b. By substituting ((2a)), ((2b)), ((3a)),and ((3b)) into Maxwell's equations, six coupled equations can be derived in cylindrical coordinates as follows,
where the upper and lower signs in equations ((4a)) and ((4b)) correspond to the choice of basis functions in (2) and (3), respectively. (ερ,εϕ,εz) and (μρ,μϕ,μz) represent the relative permittivity and permeability in cylindrical coordinates. The BOR FDTD update equations in the time domain can be derived from equations ((4a)) and ((4b)) for all field components in the ρ-z plane, which are provided in equations ((A1))–((A6)) for non-magnetic materials.
2.2 Singularity Issues of Electric and Magnetic Fields on the Axis of Symmetry
 Due to the singularity of the field components (, and ) located on the axis of symmetry, special boundary conditions are required for the update equations in the BOR-FDTD method. For instance, the component is zero for any mode with m > 0, while and are zero for any mode with m ≠ 1. The update equations for the on-axis field components are derived and provided in Appendix B. The fields solved for equations ((A1))–((A6)) can be substituted into equations ((2a)), ((2b)) ((3a)) or ((3b)) to reconstruct the full 3-D field components. For the TE case, equations ((2a)) and ((2b)) can be used; otherwise, equations ((3a)) and ((3b)) should be applied.
2.3 Absorbing Boundary Condition
 To efficiently truncate the infinite computational domain, proper ABCs are needed. The PML ABC techniques [Berenger, 1994; Chew and Weedon, 1994; Gedney, 1996; Taflove and Hagness, 2005; Teixeira and Chew, 1997; Yu et al., 2006] have been the focus of many research efforts and were found to be robust and effective. Here a generalized un-split field PML technique [Gedney, 1996; Taflove and Hagness, 2005] was employed to maximize the absorption of the outgoing waves in the PML regions in cylindrical coordinates for efficiently modeling BOR lenses within or on top of a layered medium. The detailed information and the derived formulas can be found in the reference [Taflove and Hagness, 2005]. Note that the choice of parameters used in the PML region depends on the local material properties.
2.4 Introduction of Incident Plane Wave into the TFSF Formulations for BOR Lenses with a Layered Medium
 A TFSF plane-wave source for the 3-D FDTD analysis (in Cartesian coordinates) of layered media has been introduced in the literature [Capoglu and Smith, 2008; Demarest et al., 1995; Hsu and Carin, 1996; Winton et al., 2005; Wong et al., 1996; Yi et al., 2005; Zhang and Seideman, 2010]. However, these techniques cannot be directly applied to the BOR-FDTD method. In this paper, two approaches have been proposed to rigorously introduce incident fields into the TFSF formulas associated with the BOR-FDTD method in the presence of a layered medium. Here the entire computational domain is divided into the TFR, the SFR, and the PML region as illustrated in Figure 1. The incident fields can be efficiently and accurately injected into the TFSF formulas at the interface between the SFR and TFR by using the proposed method.
2.4.1 1-D FDTD Based Implementation for Normal Incidence: BOR Lens Embedded in a Layered Media
 For a TE polarized incident plane wave in the (ρ-z) plane as shown on the right hand side of Figure 2, we have
where E0 is the electric-field magnitude and f(t,ki,r) is a time- and space-dependent function. For a normally incident plane wave, E0 = 1 and f(t,ki,r) = exp(jωt − jk0z). In this case, only the m = 1 mode in (1) needs to be calculated, which results in a fast solver for simulation of large and complex BOR targets embedded in a layered medium.
 Since the z-components of the electric and magnetic fields are zero, two sets of decoupled differential equations can be derived from equations ((4a)) and ((4b)) for a TE-polarized (ϕ-polarized) incident plane wave. They are given as follows and can be used to introduce a plane-wave injector:
 Due to the duality between equations ((6a)) and ((6b)), only one set of ( and ) incident fields given in ((6a)) needs to be solved to propagate the incident plane wave in the presence of planar isotropic dielectric layered media. The remaining fields ( and ) can be determined through the relationship and . Therefore, the 1-D FDTD based method can be implemented, while the solution is used to set up the incident field exactly at all the nodes adjacent to the TFSF boundary based on the 1-D auxiliary grid as shown on the left of Figure 2. Similar formulations can be derived for the TM polarized incidence case.
2.4.2 The Closed-Form Formulation for Oblique Incidence: BOR Lens Within a Layered Medium
 For an obliquely incident plane wave, the 1-D FDTD based approach described above cannot be employed, since the decoupled 1-D differential equations, such as those given in equations ((6a)) and ((6b)), cannot be easily derived. Hence, a set of analytical formulations are derived to model the propagation of obliquely incident plane waves within the planar, stratified media in cylindrical coordinates.
 Consider a TE-polarized incident plane wave launched from an upper isotropic medium upon a stack of N + 1 homogenous layered media as illustrated in Figure 3. θi is the angle of incidence. Each layer has a thickness of dn and the constitutive parameters (μn,εn). Since each interface is planar, the time-domain total fields within each region consist of the forward and backward propagating plane waves that can be written in the forms given by equations ((7a))–((7c)) in Cartesian coordinates. These equations were derived from their corresponding spectral domain forms [Kong, 1986].
where An and Bn are the amplitudes of the forward and backward propagating waves, respectively. The forward and backward propagating wave vectors and are written as follows:
Here, θn is the angle that the phase propagation vector makes with the z axis in the nth layer, which is related to the incident angle θi by the equation: kn sin θn = k0 sin θi. The wave number and transverse wave impedance in the nth layer are given by and , respectively.
 The coefficients of An and Bn in equations ((7a))–((7c)) can be derived by enforcing boundary conditions of continuous tangential electric and magnetic fields at the interface of the layered media. The derived recursive formulas are given by [Kong1986, Demarest, et al., 1995, Tan and Tan, 1998]
where kzn = kn cos θn, and Ln indicates the location of the interface between the nth layer and the (n + 1)th layer. We recall that A0 = 1 for an incident field with unit amplitude. Once all the An have been determined, the corresponding Bn can be found by using equation ((9a)). Here Zn is the input impedance looking downward between the nth layer and the (n + 1)th layer, which can be obtained using the following recursive formula:
 Due to the axisymmetric nature of the BOR-FDTD method, an obliquely incident plane wave in each layer cannot be directly introduced to the BOR-FDTD's update equations. The formulas which are expressed in equations ((7a))–((7c)) should be first converted into the equivalent forms in cylindrical coordinates. Thereafter, the incident forward and backward plane waves in the time-domain for each layer must be expanded into a series of cylindrical modes in cylindrical coordinates. Finally, the derived expressions are given by
where c0 = 0.5, cm = 1.0 for m ≠ 0, sn = knρ sin θn, and the remaining variables are defined as follows:
Hence, the total electric and magnetic fields propagating in each layer (e.g., the forward and backward propagating waves) can be expressed as a sum of mutually orthogonal cylindrical modes. Each mode, denoted by m, of the incident cylindrical waves can be introduced into the TFSF formulas in cylindrical coordinates and analyzed by the BOR-FDTD method separately. Upon achieving a steady-state solution in the BOR-FDTD solver, a discrete Fourier transform (DFT) is performed on each cylindrical mode. Consequently, the corresponding spectral domain electric and magnetic fields can be obtained by conducting a coherent summation of all the mode components.
 For numerical implementations, the infinite series of Bessel functions in equations ((11a))–((11e)) can be truncated [Andreasen, 1965] at M = kn(max)ρmax sin θi + 6, where ρmax is the maximum radial dimension of the BOR object being modeled, and kn(max) is the maximum wave number at the highest frequency of interest within the computational range for the GRIN lens and the layered media.
3 Numerical Results and Discussion
3.1 Validation and Efficiency of the Proposed Method
 To validate the proposed method, the results calculated by BOR-FDTD were compared with analytical solutions. The efficiency of the BOR-FDTD method was demonstrated by comparing the performance of the newly developed solver with that of a commercial software (COMSOL) package. If not specified otherwise, the following parameters are used in all simulations: the working wavelength is λ = 1.0 µm, while the mesh size in the BOR-FDTD computational domain is λ/50. In addition, all the simulation results shown here are in the frequency domain, which are obtained by performing a DFT on the corresponding steady state time domain results.
3.1.1 BOR-FDTD Method Versus Analytical Solutions
 The BOR-FDTD code was first employed to analyze a planar dielectric interface (with a dielectric constant of 2.5) illuminated by TE-polarized plane waves. The reflection (R) and transmission (T) coefficients calculated by the BOR-FDTD solver are in good agreement with analytical solutions as shown in Table 1.
Table 1. BOR-FDTD Method Versus Analytical Solutions
R/T Coeff. (0º) [m = 1]
R/T Coeff. (10º) [m = 11]
 The wave scattering from a four-layer structure was also calculated, as shown in Figure 4a, which is illuminated by a TM polarized plane wave propagating in the −z direction. Figures 4b, 4c, and 4d present the total electric field intensity distributions in the plane of ϕ = 0°, ϕ = 45°, and ϕ = 90° as simulated by the BOR-FDTD code (m = 1). As expected, the field distributions are identical in different planes. In addition, the transmission coefficient (T = 0.589) calculated by the BOR-FDTD method is in good agreement with the analytical result (T = 0.59).
3.1.2 BOR-FDTD Method Versus Commercial Software
 To further investigate the efficiency and accuracy of the developed BOR-FDTD solver, we simulated an inhomogeneous dielectric BOR lens in free space illuminated by a normally incident TE polarized plane wave (λ = 2.0 µm) using the BOR-FDTD solver (m = 1) and the FEM-based software package COMSOL (3-D solver). The geometric parameters of the BOR lens are presented in Figure 5a. The electric field distributions in the XOZ and the YOZ plane cuts are plotted in Figures 6a and 6b for the COMSOL simulation results, and in Figure 6c and 6d for the BOR-FDTD simulation results, respectively. Excellent agreement can be observed between these numerical simulation results, validating the accuracy of the developed BOR-FDTD code.
 To demonstrate the efficiency of the proposed method, the memory requirements and the CPU times used for the two solvers are compared in Figure 5b. Even though much smaller meshes are employed by the 2-D BOR-FDTD solver, it outperforms the COMSOL software by reducing the CPU time and memory requirements by ~52% and 99.43%, respectively. The distinct advantage of the BOR-FDTD solver as compared with the FEM-based COMSOL package is that wideband responses can be obtained by employing a transient excitation in a single simulation.
3.2 Application of the Developed BOR-FDTD Solver
 In this section, the developed BOR-FDTD code was first employed to investigate the imaging properties of radial GRIN lenses with a parabolic index profile. Afterwards, it was used to explore the imaging characteristics of a flat TO GRIN lens, a spherical Luneburg lens, and a flattened TO Luneburg lens in free space and with a substrate.
3.2.1 Normal Incidence Cases: Radial GRIN Lenses with a Parabolic Index Profile
 Light propagation in GRIN lenses with a wide variety of gradient profiles has been analyzed with emphasis on the focusing and collimation properties of importance for integrated optics, micro-optics, and optical sensing [Iga et al., 1984; Gómez-Reino et al., 2002]. In designing a GRIN lens, one goal is to select an index profile that provides the desired focusing and collimating effect. Let us consider a GRIN material characterized, for expositional convenience and without loss of generality, by a radial parabolic index profile [Flores-Arias et al., 2006; Gómez-Reino et al., 2002, 2008]. Here the BOR-FDTD method is employed to explore how the gradient parameter can affect the imaging properties of a GRIN lens with a radial parabolic index profile. The dielectric constant profile of such BOR GRIN lenses is described as follows [Flores-Arias et al., 2006; Gomez-Reino et al., 2008]:
where n(ρ) is the refractive index of the GRIN material at given transverse position ρ, n0 is the highest refractive index at the center axis of the GRIN material, and α is the gradient parameter.
 Figure 7 shows the dielectric constant profile in the XOZ plane cut for the radial parabolic GRIN lenses on top of a substrate. The corresponding geometric parameters of the GRIN lenses are also presented. Here the gradient parameters are chosen to be α = 0.24 in Figure 7a and α = 0.34 in Figure 7b. In order to minimize the reflection from the substrate, we assume that the dielectric constant of the substrate is 4.0, which matches the maximum value of dielectric constant of the BOR GRIN lenses.
 The imaging properties of these radial GRIN lenses in free space and on a substrate are analyzed using the BOR-FDTD method. The simulation results for a TM-polarized normally incident plane wave (e.g., the electric field intensity distributions at the XOZ plane cuts) are plotted in Figures 8a [α = 0.24] and 8b [α = 0.34] for the free space case, and in Figures 8c [α = 0.24] and 8d [α = 0.34] for the substrate case, respectively. These results clearly demonstrate that lenses with a smaller gradient parameter α [e.g., α = 0.24] can produce a longer focal length. Therefore, changing the value of the gradient parameter can effectively tune the imaging properties of the radial GRIN lenses. Compared to a GRIN lens in free space, the lens on the substrate possesses a longer focal length. In addition, the refractive index profile of the lens along the z-direction can be varied to achieve the desired imaging property with improved performance, such as reduced reflection. This type of GRIN lens design can be created by combining the fast BOR-FDTD simulation tool with suitable optimization techniques.
3.2.2 Normal Incidence Cases: Flat TO GRIN Lens
 In imaging applications, multiple elements are usually required to correct for optical aberrations. For example, an achromatic lens is commonly used to correct for spherical and chromatic aberrations due to the refractive index dispersion. On the other hand, reducing the number of optical components is desired, as it can not only decrease complexity and avoid misalignment issues in an imaging system, but also miniaturize the optical system and reduce its size and weight. For instance, Figure 9c shows a cemented achromatic doublet, which consists of two individual lenses made from crown (n1 = 1.517) and flint (n2 = 1.649) glasses with different amounts of dispersion [Kingslake and Johnson, 2010]. Alternatively, Figure 9d presents a flat TO-GRIN lens design with a maximum Δn of 0.5, which can provide the same imaging properties. Here the flat TO lens was designed by using the QCTO technique, which converts the doublet lens with curved surfaces in the original space (Figure 9a) to a flat bulk TO lens in the transformed space (Figure 9b). It has been demonstrated that the QCTO technique can be employed to effectively manipulate light propagation by modifying the spatial distribution of dielectric materials [Li and Pendry, 2008; Ma and Cui, 2010]. Since this technique does not rely on resonant structures, it can create TO devices with low-loss and broadband performance [Landy et al., 2010].
 The BOR FDTD method was employed to characterize the performance of the 3D achromatic lens with curved surfaces and the corresponding flat TO GRIN lens both in free space and on a substrate with a dielectric constant of 2.5. A TM-polarized incident plane wave is launched at the top of the simulation domain and propagates in the −z direction. The dielectric constant profiles in the XOZ plane cut for the achromatic lens and the flat TO-GRIN lens are plotted in Figures 10a and 10b for the free space case; and in Figures 10c and 10d for the substrate case, respectively.
 For the free space case, the calculated electric-field intensity distributions generated by the BOR-FDTD solver (m = 1) in the XOZ plane cut are plotted in Figures 11a and 11b for the achromatic lens and flat TO GRIN lens illuminated by a TM-polarized normally incident plane wave, respectively. The corresponding electric-field intensity distributions at the focal (XOY) plane cuts are plotted in Figures 11c and 11d. Nearly identical focusing properties are observed for the two lenses, which demonstrate the accuracy and flexibility of the QCTO design method. Both lenses exhibit a focal length of about 40 µm under plane wave illumination. From the focused beam patterns at the imaging plane of each lens shown in Figures 11c and 11d, it is found that the Airy disk diameter is around 5.2λ0 for both lenses, demonstrating their diffraction-limited imaging properties with minimum aberrations.
 For the lenses on a substrate, the corresponding BOR-FDTD simulation (m = 1) results are presented in Figure 12 for a normally incident TM-polarized plane wave. Electric field intensity distributions in axial (XOZ) plane cuts are plotted in Figures 12a and 12b for the original lens and the flat TO GRIN lens, respectively. Figures 12c and 12d illustrates the electric field intensity distributions at the focal (XOY) plane cuts. Similar to the free space case, nearly identical focusing properties are found for the original lens and the flat TO lens with a substrate. In addition, the substrate environment greatly influences the performance of the lenses by producing a longer focal length.
 In the following two sections, the developed BOR-FDTD method was employed to characterize the imaging properties of a 3D spherical Luneburg lens and a flattened TO Luneburg GRIN lens under both normal and oblique incidence. First, a spherical Luneburg lens in free space was investigated, whose dielectric constant distribution in an axial (XOZ) plane cut is plotted in Figure 13. It possesses a spherical index profile according to the equation , where r ≤ rmax, and rmax = 6 µm represents the radius of the lens [Luneburg, 1944]. Such a spherical Luneburg lens can focus incident light with a wide field of view onto the spherical surface at r = rmax with no aberrations. Using the BOR-FDTD code, the Luneburg lens was illuminated by normally and obliquely incident TE-polarized and TM-polarized plane waves with incident angles of 0º, 5º, 10º, 15º, and 20º. The modes required in the BOR-FDTD simulation are given by 1, 10, 14, 18, and 22, respectively. The corresponding imaging properties of the lens are demonstrated by the electric field intensity distributions in the XOZ plane cut, as shown in Figures 14a–14e for the TE case, and in Figures 15a–15e for the TM case. The field intensity distributions in the focal (XOY) plane cut are plotted in Figures 14f–14j for the TE case, and in Figures 15f–15j for the TM case. As expected, the focal point is located on the surface of the spherical lens with equal field intensity for different incident angles, which further confirms the accuracy of the developed BOR-FDTD code. Comparison between simulation results for the two polarizations reveals that the full width at half maximum (FWHM) of the focusing spot along the x axis in the focal plane is larger for the TM-polarized wave than that of the TE case. It was also observed that the shape of the focusing spot is an ellipse instead of a circle. This effect can be attributed to the linearly polarized incident waves, whose electric field is parallel to the major axis of the elliptical focusing spot. By measuring the spot size with the FWHM along the major axis of the ellipse, the same focusing performance was found for both TE and TM polarized incident plane waves.
3.2.4 Oblique Incidence Cases: Flat TO Luneburg Lenses
 The conventional spherical Luneburg lens was then transformed into a flat TO lens by using the QC mapping technique [Hunt et al., 2012; Kundtz and Smith, 2010; Li and Pendry, 2008; Ma and Cui, 2010; Smith et al., 2010]. This type of transformation yields TO designs comprised of isotropic, dielectric-only materials, facilitating their applications at optical wavelengths. For this design, a geometrical mapping was performed to convert a rectangle with a circular protuberance in the original space to a rectangular region in the transformed space by using the QCTO technique [Kundtz and Smith, 2010]. Figure 16 illustrates the spherical Luneburg lens in the original space and the transformed flattened Luneburg lens in the physical space. Such a transformation maintains the imaging capability of a conventional spherical Luneburg lens, while providing a flat focal plane and facilitating its integration with planar detector arrays. The relative location of the circular protuberance with respect to the rectangular region determines the resulting GRIN profile of the TO lens. Instead of having the center of the circular protuberance on the edge of the rectangle (i.e., t = R) [Kundtz and Smith, 2010], the circular protuberance was placed farther into the rectangle (i.e., t = 0.6R). As a result, this TO lens possesses a reduced field of view of 130° but a much lower Δn of about 1.5 compared to the TO design in [Kundtz and Smith, 2010]. A corresponding 3D cylindrical TO Luneburg lens can be generated by rotating the 2D GRIN profile about the z-axis [Ma and Cui, 2010]. The dielectric constant profile of this TO Luneburg lens in the XOZ plane cut is demonstrated in Figure 17, where the flat lens was placed either in free space or on top of a substrate (not shown here). For the flattened TO lens with a substrate, the permittivity of the substrate is the same as the maximum permittivity of the TO Luneburg lens.
 BOR-FDTD simulations were performed to evaluate the performance of the flat TO Luneburg lens in free space. Figures 18a–18e illustrate the resulting electric field intensity distributions in the XOZ plane cut illuminated by TE-polarized plane waves with incident angles of 0º, 5º, 10º, 15º, and 20º, respectively. The corresponding modes required in the BOR-FDTD solver are recorded in Table 2. With the transformed GRIN profile, incident waves at various angles are focused onto a flattened image plane. The lateral distance from the focusing point to the optical axis is presented in Table 2 for different incidence angles. To further characterize the flattened TO Luneburg lens, the focusing spots formed in the imaging plane are illustrated by Figures 18f–18j. The FWHMs of the focusing spot along the x and y axes in the imaging plane are measured and reported in Table 2. The corresponding spot sizes of the focusing beam can be defined as the FWHM along the major axis of the ellipse. These results demonstrate the nearly diffraction limited imaging properties produced by the TO Luneburg lens. They also clearly demonstrate that the developed BOR-FDTD code allows for precise computation of all the field components within the entire 3D domain, providing an efficient method for investigation of the imaging properties of cylindrical TO GRIN lenses.
Table 2. The Lateral Distance and the FWHM of the Focusing Spots Along the x and y Axes for Free Space Case
0º [m = 1]
5º [m = 10]
10º [m = 14]
15º [m = 18]
20º [m = 22]
FWHM x-axis (µm)
FWHM y-axis (µm)
FWHM x-axis (µm)
FWHM y-axis (µm)
 With TM-polarized incidence plane waves, the corresponding simulation results are plotted in Figures 19a–19e for the electric field intensity distributions in the XOZ plane cut. The modes required in the BOR-FDTD solver and lateral distance from the focusing point to the optical axis are also reported in Table 2 for different incident angles. The electric field intensity distributions in the imaging (XOY) plane cut are plotted in Figures 19f–19j for incident angles of 0º, 5º, 10º, 15º, and 20º, respectively. The correspondingly FWHM of the focusing spots along the x and y axes are reported in Table 2. For both TE and TM polarization states, the focusing spot size increases as a function of incident angle. This is mainly due to the reduced permittivity of the TO Luneburg lens at the position which is farther away from the symmetric axis. Furthermore, comparison of the TE and TM cases reveals that the focusing spot size under both polarizations are nearly the same, while the focusing spot formed under the TM incident wave possesses a slightly larger size when the angle of incidence increases to 20º.
 Finally, we investigated the imaging characteristics of the designed flat TO Luneburg lens on top of a substrate, which is illuminated by TE-polarized and TM-polarized plane waves with different incident angles. The required modes in the BOR-FDTD simulation for different incidence angles are presented in Table 3. In order to minimize reflections, the dielectric constant of the substrate was chosen to be 6.463, which is close to the maximum permittivity of the TO Luneburg lens. For the TE case, Figures 20a–20e presented the BOR-FDTD simulation results for electric field intensity distributions in the XOZ plane cut at the incident angles of 0º, 5º, 10º, 15º and 20º, respectively. The corresponding focusing spots at the focal (XOY) plane cut were demonstrated in Figures 20f–20j. Similar focusing performance to the lens in free space was found at the flat imaging plane between the TO Luneburg lens and the substrate. Note that the substrate produces a slightly longer focal length, which causes the focusing points to be located in the substrate. This numerical example also demonstrates the capability of the BOR-FDTD code for modeling cylindrical lenses mounted on a substrate. Due to the reduced index contrast between the TO GRIN lens and the substrate, less reflection was observed at the back surface of the lens.
Table 3. The Lateral Distance and the FWHM of the Focusing Spots Along the x and y Axes for Half Space Case
0º [m = 1]
5º [m = 10]
10º [m = 14]
15º [m = 39]
20º [m = 49]
FWHM x-axis (µm)
FWHM y-axis (µm)
FWHM x-axis (µm)
FWHM y-axis (µm)
 For the TM case, the simulation results were reported in Figures 21a–21e for electric field intensity distributions in the XOZ plane cut. The corresponding imaging properties at the (XOY) plane cut are plotted in Figures 21f–21j. It is clearly seen that this TO Luneburg lens provides nearly the same imaging properties for both TM and TE-polarized waves, making it a polarization-independent device.
 The lateral distance from the focusing point to the optical axis and corresponding FWHM size of the focusing spots along the x and y axes in the imaging plane are reported in Table 3 for both TE and TM cases. Similar to the free space case, the flattened TO Luneburg lens provides the desired imaging properties and functions for both polarizations when placed on top of a substrate. We also note that the electric field intensity at the focal point of the TO Luneburg lens on a substrate becomes weaker compared to that of the free space case, which is due to the higher permittivity of the dielectric substrate material.
 Finally, a summary of the pros and cons of the QCTO approach for the flattened BOR-TO-GRIN lens designs is presented in Table 4 by comparison with the conventional curve-shaped homogeneous isotropic and the metamaterials-based anisotropic lens design approaches.
Table 4. The Pros and Cons of Different Lens Designs
The Flat TO Lens Versus a Cemented Achromatic Lens
TO Luneburg Lens Versus Spherical Luneburg Lens
The QCTO Approach Versus a Metamaterials-Based Design
(a) Reduce the number of lens elements from 2 to 1;
(a) Flat profile with flat imaging plane;
(a) Eliminates resonant type of metamaterials and only requires a GRIN medium;
(b) Flat profile makes it easy for assembly;
(b) Makes it possible for integration into a planar detector (CCD);
(b) By using this type of TO method, the use of non-resonant metamaterials confirms its focusing and extreme-angle broadband behavior.
(c) Potentially thinner and lighter weight design.
(c) In a sense this type of imaging properties are superior to the original Luneburg lens.
Challenge in gradient-index material implementation and fabrication
 We have developed a BOR-FDTD solver for efficient analysis of EM scattering from 3D rotationally symmetric TO lenses. The 1-D FDTD based method and the derived closed-form formulations have been proposed and presented to rigorously inject normally and obliquely incident plane waves into the TFSF formulas at the interfaces between the SFR and TFR. The accuracy and efficiency of the proposed method are demonstrated by the presented numerical results as compared with analytical solutions and a commercial software (COMSOL) package. For oblique incidence with increasing angles, high-order cylindrical modes must be calculated for a complete analysis. To further improve the performance, parallel techniques, such as the recently developed GPU based parallel processing can be employed to accelerate the BOR-FDTD solver.
 The developed BOR-FDTD code has been employed to investigate the imaging characteristics of various 3D axisymmetric TO GRIN lenses, both in free space and on a substrate. Using the QCTO technique, (a) a curved achromatic lens was converted into a flat bulk TO GRIN lens; and (b) a flattened TO Luneburg lens composed of isotropic, dielectric-only materials was designed. The BOR-FDTD simulation results demonstrate that (a) flat TO GRIN lenses provide nearly the same imaging performance as compared to the original curve-shaped lenses, which demonstrate the design flexibility of the QC mapping technique; (b) the substrate environment greatly influences the focal length of the lenses and produces a longer focal length; and (c) the flattened TO Luneburg lens has the desired planar imaging properties under both normal and oblique incidences for TE and TM polarizations.
 In this work, the developed BOR-FDTD solver has only been employed to model continuous waves with a monochromatic frequency. However, the algorithm can be extended to handle transient excitation. In addition, the BOR-FDTD formulations presented in this paper are derived for BOR objects with lossless materials in a layered, lossless medium. However, the proposed method can be extended to a more general model containing lossy and dispersive materials.
 This work was supported by the NSF-MRSEC Center for Nanoscale Science under grant DMR-0820404.
BOR FDTD Update Equations
 We can obtain the BOR FDTD update equations from equations ((4a)) and ((4b)) in the time domain for all field components in the ρ-z plane, which are given as follows for lossless and non-magnetic dielectric material:
Note that the ϕ-dependence of the field has been removed in these equations.
Update Equations on the Axis
 Due to the singularity of the field components located on the axis of the BOR devices, special boundary conditions are required for the update equations in this case. The derived update equations on the axis are given as follows for the field components of (, , and ):