1. Introduction

A systematic methodology for controlling electromagnetic and light waves, known as the transformation electromagnetics/optics technique, has been recently introduced [1, 2]. It is based on the invariance of Maxwell’s equations under coordinate transformations with the material parameter tensors in the transformed system interpreted as inhomogeneous, anisotropic stretching and compression of the original medium.

The transformation electromagnetics/optics technique has led to numerous novel device designs. The recently proposed electromagnetic and optical cloaks [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] have been the most salient application to date of the coordinate transformation technique. Another related transformation electromagnetics/optics device is the field concentrator [8], which utilizes a different transformation from that employed in cloaking.

The embedded coordinate transformations introduced in [12] have led to additional device designs such as beam shifters and beam splitters, which could not be obtained via conventional design methodologies. The embedded transformation allows discontinuities of spatial mappings on the boundary of the transformed domain. Other transformation optical devices which have been obtained include cylindrical-to-planar wavefront converters for embedded line sources [13, 14, 15], beam bends [16, 15], and beam expanders [16]. In [17], polarization splitter and polarization rotator designs were presented as the first demonstration of transformation optical device designs that manipulate the polarization state of an incoming field.

A number of far-zone focusing flat lens designs were presented in [15] for line sources located outside the transformation optical device. Although transformation from cylindrical to planar wavefronts was clearly demonstrated for the far-zone focusing lens, strongly diffracted wave components were unavoidable for the line source excitation and reflections from the lens surfaces were also present. In fact, in all the device designs that involve transformation of a curved boundary into a straight boundary, evidences of reflections can be clearly observed [13, 14, 15].

In this paper, improved designs for the far-zone focusing lens are introduced. First, a two-dimensional (2D) lens of trapezoidal cross section is presented. An improved impedance match is achieved by choosing the proper mapping from a curved boundary to a straight boundary having the same length. Secondly, an all-dielectric flat focusing lens of rectangular cross section is presented, which eliminates the need for magnetic materials and is based on a non-linear coordinate transformation for improved reflection properties. The cylindrical-to-planar wave conversion and the improved reflection properties are verified using full-wave finite-element simulations.

 

Fig. 1. (Color online) Simulation setup and incident field generation: (a) A COMSOL simulation model setup, (b) The angle-limited cylindrical wave generated by the electric surface current Js.

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2. Angle-limited cylindrical waves and simulation model

Angle-limited waves with cylindrical phasefronts are desirable when illuminating finite-sized lenses that convert cylindrical waves into planar waves without causing edge diffraction. In 2D TE-mode configurations, ẑ-polarized incident waves may be physically created either by an electric current directed in the longitudinal (ẑ) direction or by a magnetic current flowing in the transverse (contained in the x-y plane) direction. For full-wave numerical simulations of the lens designs that follow, COMSOL Multiphysics was employed.

A typical simulation model setup is illustrated in Fig. 1(a). The computational domain is a union of two regions having a rectangular and a semi-circular cross sections. Thin regions of perfectly matched layers (PMLs) are added to truncate the computational domain without causing spurious numerical reflections from the simulation boundary. To create a TE-mode incident electric field E ⁱ = ẑEⁱ_z with converging cylindrical wavefronts in a COMSOL model, a ẑ-directed electric surface current J _s is placed over a cylindrical 2D surface having the cross section of an arc (indicated by a red contour) of radius a ₀ centered at the coordinate origin O. The following ϕ-dependent Gaussian distribution is chosen for the time-harmonic surface current:

where J ₀ is the strength of the electric surface current at ϕ = ϕ ₀, and σ _ϕ is a parameter that determines the angular spread of the distribution. With the choice of ϕ ₀ = -π/2, this ϕ-limited surface current will generate cylindrical waves in the region y < 0 which converge at the coordinate origin. Furthermore, they will then become diverging cylindrical waves, propagating away from the origin in the region y > 0 with the field distribution concentrated along the direction ϕ = ϕ ₀+π. In the absence of a lens, a snapshot of the electric field distribution created by J _s is shown in Fig. 1(b), corresponding to a ₀ = 1 m and σ _ϕ = 0.316 at the time-harmonic frequency of 3 GHz. The thickness of the PML regions is equal to 0.1 m. The converging and the diverging cylindrical wavefronts are clearly observed.

 

Fig. 2. (Color online) The coordinate systems associated with the lens design: (a) The original system in free space, (b) The intermediate system obtained by stretching and compressing space only in the ±ŷ directions, and (c) The final transformed system.

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3. Trapezoidal focusing lens with minimal reflection

To design a 2D far-zone focusing flat lens using transformation optics, a coordinate transformation may be defined to convert a curved equi-phase, equi-amplitude contour in the original space to a straight boundary in the transformed space. This transformation is achieved here in a two-step process illustrated in Fig. 2. Figure 2(a) shows a section of the circular domain given by $\rho ″=\sqrt{{x″}^2+{y″}^2}\le a,y″\ge g$ in the original (x″, y″, z″) system. By stretching and compressing the original system only in the ±ŷ directions, one can arrive at the rectangular-cylindrical slab of thickness l shown in Fig. 2(b). The associated transformation is given by

This set of transformations was used in [15] to design a 2D far-zone focusing lens for a line source. The arc ρ″ = a, ∣ϕ″ - π/2∣ ≤ Δϕ ″ in the original system is mapped to a straight line segment ∣x′∣ ≤ w, y′ = g+l in the transformed (x′,y′, z′) system. However, the length of the boundary is shortened from 2aΔϕ″ to 2w, and this causes reflections at the planar interface y′ = g+l. Such reflections have also been observed at planar boundaries when the transform medium is either compressed or expanded in the beam expander designs reported in [16].

To improve the impedance match characteristics at the exit boundary, it is desirable to have 1) a straight boundary that has the same length as the arc in the original system, and 2) the coordinate x equal to the arc length variable along the original contour. In other words, a straightened version of the arc of length 2aΔϕ″ without any stretching is desired after transformation. Recognizing that the straight line segment ∣x′∣ ≤ w, y′ = g must remain unchanged by the transformation, we introduce a second set of transformations from the (x′,y′, z′) system to the (x, y, z) system given by

where

In essence, x represents a mapping that transitions smoothly from x′ to a sin^-1(x′/a) as y′ is increased from g to g+l. The final transformed (x, y, z) system is shown in Fig. 2(c). Since α is a linear function of y′, constant-x′ line segments are mapped into straight line segments in the (x,y, z) system. Therefore, the shape of the final transformation optical focusing lens is trapezoidal.

 

Fig. 3. (Color online) The material parameters of the trapezoidal lens design: (a) μ_xx, (b) μ_xy, (c) μ_yy, and (d) ϵ_zz. The geometry of the lens design is given by g = 0.4 m, w = 0.6 m, and l = 0.1 m.

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The elements A_ij (i, j =1,2,3) of the Jacobi matrix corresponding to the transformation from the (x″,y″, z″) system to the (x, y, z) system are found to be

together with A ₁₃ = A ₂₃ = A ₃₁ = A ₃₂ = 0 and A ₃₃ = 1. Next, the material parameter tensors ϵ̿ and μ̿ are obtained from the relation [4] ϵ̿ = μ̿ = AA ^T/det(A). The expressions for the material tensor elements are not shown for brevity. However, the material parameter values should be expressed in terms of the transformed coordinates x, y, and z. The values of x″, y″, and z″ can be obtained in terms of the transformed coordinates from Eqs. (2) –(4), which involves solving the non-linear equation for x′ in Eq. (3).

The material parameters for an example lens design are plotted in Fig. 3. The performance of the design is shown in Fig. 4, where a snapshot and a magnitude distribution of the total electric field for the trapezoidal lens case are compared with those of the rectangular lens case obtained from the transformation shown in Fig. 2(b) [15]. In terms of the free-space wavelength λ = 0.1 m at the time harmonic frequency of 3 GHz, the dimensions of the lens are specified by w = 6λ and l = λ. The width of the lens at the exit interface where z = 0.5 m is equal to 0.709 m. The full-wave simulation results were obtained using COMSOL Multiphysics. The incident electric field was generated by the surface current of Eq. (1) with a ₀ = 1m, J ₀ =1 A/m, and σ ϕ = 0.316.

As a baseline for comparison, a snapshot and the magnitude distribution of the total electric field for the rectangular lens under a circular cylindrical wave illumination produced by an electric line source at the origin are shown in Figs. 4(a)–4(b). It can be observed in Fig. 4(a) that the circular wavefronts that are captured by the lens are converted to planar wavefronts upon exit. However, illumination of the lens edges by the line source causes the incident wave to be diffracted in every direction. The effect of diffraction is clearly visible in the magnitude distribution plotted in Fig. 4(b). The presence of these diffracted waves makes it difficult to separate out reflections from the lens interfaces, which is the main focus of this study. Ideally, the angle-limited Gaussian beam does not illuminate the lens edges, and thus the total field consists of the illumination and the fields reflected from the lens interfaces only.

 

Fig. 4. (Color online) Comparison of the rectangular lens of [15] and the new trapezoidal lens: (a) A snapshot and (b) the magnitude distribution of the total electric field for the rectangular lens due to the cylindrical wave illumination from a line source at the coordinate origin. (c) A snapshot and (d) the magnitude of the electric field for the rectangular lens under an angle-limited Gaussian beam illumination. (e) A snapshot and (f) the magnitude distribution of the electric field for the trapezoidal lens under an angle-limited Gaussian beam illumination. The boundaries of the lenses are indicated by the black contours.

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Figures 4(c) and 4(e) clearly show the planar wavefronts emerging from the exit boundaries for the two lens designs. The minimized reflection property of the trapezoidal lens compared with the rectangular lens is demonstrated by the field magnitude plots shown in Figs. 4(d) and 4(f). For the rectangular lens, reflections from the exit boundary cause ripples in the magnitude of the total electric field in the range y≤g+l. In comparison, no reflection is observed from the exit boundary of the trapezoidal lens. The incident cylindrical waves are converted into planar waves by the lens without reflection from both the entry and the exit boundaries.

The reduction in the amount of reflection can be quantified by comparing the field distributions. The total ẑ-directed electric fields for the two designs are compared along the lens axis (x = 0) in Fig. 5(a). Ripples in the total field for the rectangular lens indicate reflections from the device. Let the scattered field E ^s = ẑE^s_z be defined by the difference between the total field E = ẑE_z and the incident field E ⁱ = ẑEⁱ_z, i.e.,

 

Fig. 5. (Color online) Electric fields along the lens axis for the rectangular and the trapezoidal lenses: (a) The magnitudes of the total fields, (b) The ratio of the scattered field to the incident field.

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Except for a small amount of diffraction from the lens edges, E ^s represents the reflected field. Therefore, the ratio between the magnitudes of the scattered and the incident fields can be defined as a measure to assess the amount of reflection from the lens. This quantity is analogous to the reflection coefficient commonly used in transmission line theory. The normalized magnitude of the scattered field ∣E ^s/E ⁱ∣ is plotted in Fig. 5(b) with respect to the variable y along the lens axis. It is observed that the trapezoidal lens has an amount of reflection which is smaller by a factor of 2 to 5, compared with the rectangular lens.

4. TM-mode all-dielectric rectangular flat lens with reduced reflections

Both the original transformation optical rectangular lens [15] and the trapezoidal lens designs require magnetic materials. Synthesizing magnetic responses can be challenging especially at optical frequencies because of the higher losses and the weaker resonances associated with metallic structures that are typically employed [18]. Due to this reason, all-dielectric (nonmagnetic) device designs are desirable.

Nonlinear coordinate transformations provide an additional degree of flexibility for designing transformation devices [18, 19]. In 2D circular cylindrical cloak designs [18], it was found that the impedance at the cloak’s outer boundary depends on the derivative of the radial variable. In a similar fashion, a nonlinear transformation can be applied to arrive at non-magnetic flat focusing lens designs that feature reduced reflections from the outer boundary. Unlike in non-magnetic cloak designs, impedance matches should be provided by the nonlinear transformation at two different interfaces (the entry and the exit planes) for the lens designs.

A linear transformation in the y direction given by Eq. (2) was used for the rectangular flat lens design in [15]. If this transformation from y″ to y′ in Eq. (2) is designed to be nonlinear with the condition ∂ y′/∂ y″ = 1 at the two interfaces located at y′ = g and y′ = g+l, the diagonal entries of the Jacobi matrix A will be equal to unity at both boundaries. When the ideal material parameters are reduced to non-magnetic values, the feature of having ε _x′x′ = μ _z′z′ = 1 will significantly reduce the amount of reflections from the device, compared with non-magnetic designs based on a linear transformation.

There are numerous nonlinear transformations that possess the aforementioned requirement on ∂ y′/∂ y″. Here we choose a 3rd-order polynomial of the following form for this transformation from y″ to y′:

 

Fig. 6. (Color online) The coordinate transformation for the non-magnetic flat focusing lens design: (a) The original coordinate system, (b) the transformed system in the (x′, y′, z′) system.

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where ${y"}_c=(g+\sqrt{a^2-{x"}^2})/2$ and y′_c = g+l/2 denote the center values of their respective ranges under the transformation for a fixed value of x″. It is noted that y′-y′_c is an odd function of y″-y″_c. Enforcing the conditions y′(y″=g)=g, y′($\left(y"=\sqrt{a^2-{x"}^2}\right)=g+l$, and ∂ y′/∂ y″ = 1 at y′ = g, g+l leads to the values of the constants A and B which are given by

It can be easily seen that the lens design associated with the linear transformation of Eq. (2) [15] results if we choose the following values for A and B:

The original (x″,y″, z″) and the transformed (x′,y′, z′) coordinate systems are illustrated in Fig. 6, where both the original and the transformed geometries are limited to the range ∣x″,x′∣ ≤ $w=\sqrt{a^2-{\left(g+l\right)}^2}$. In this case, the spatial compression only occurs in the ŷ″ direction. At any fixed value of x′, Fig. 6(b) shows that neighboring contours associated with constant-y″ lines are spaced non-uniformly, unlike the uniformly-spaced curves found in Fig. 2(b) associated with a linear transformation.

The material parameters of the ideal lens (with magnetic material specifications) in the (x′,y′, z′) system can be obtained from ϵ̿ ′ = μ̿′ = AA ^T/det(A) using Eq. (10) together with x′ = x″, z′ = z″. In the TM mode case, where only ϵ x′x′, ϵ x′y′, ϵ y′x′, ϵ y′y′, and μ z′z′ interact with the fields, a reduced set of material parameters for a non-magnetic design are obtained by the substitutions ϵ _x′,x′, → ϵ _x′,x′,μ z′z′, ϵ _x′,y′ → ϵ _x′,y′ μ z′z′, ϵ y′x′, → ϵ y′x′,μ z′z′, ϵ y′y′ → ϵ y′y′μ z′z′ , and μ z′z′ → 1. Performing these substitutions, the material parameters for the TM-mode non-magnetic flat rectangular lens are obtained as

 

Fig. 7. (Color online) The material parameters for an example non-magnetic lens design for the TM mode: (a) ϵ _x′,x′, (b) ϵ _x′,y′, (c) ϵ y′y′, and (d) μ z′z′. The dimensions of the non-magnetic lens are given by w = 0.489 m and l = 0.13 m.

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where

The value of y″ can be obtained in terms of y′ from the inverse relation of Eq. (10).

The derived material parameters for an example design are plotted in Fig. 7. The geometrical parameters of the lens configurations are given by a = 0.721 m, w = 0.489 m, l = 0.13 m, and g = 0.4 m. From Eq. (13), one finds the maximum required value of ϵ _x′,x′, is equal to

at the center of the lens (x′,y′) = (0,g+l/2). This value turns out to be 86.95 for the design shown in Fig. 7, which may be considered excessively large for practical realizations at optical wavelengths. However, it is noted that the value of ϵ _{x′,x′,max} is reduced as the thickness of the lens l is increased. Therefore, the lens thickness can be properly chosen to limit the maximum required permittivity value depending on the fabrication technique.

Using full-wave simulations, the performances of the non-magnetic focusing lens designs utilizing linear and nonlinear transformations are evaluated and compared. The geometrical parameters are the same as the example shown in Fig. 7. Snapshots of the total ẑ′-directed magnetic field and the magnitude distributions are plotted in Fig. 8. In the COMSOL model, an angle-limited TM-mode beam created by a ẑ′-directed magnetic surface current of the same form as described by Eq. (1) with a ₀ =1m, σ _ϕ =0.25, and M ₀ =1 V/m is used as the excitation for both lenses at the time-harmonic frequency of 3 GHz. Material specifications associated with the linear transformation are found from Eqs. (13)–(16) using the values of A and B in Eq. (12). In this case, the value of A′₂₁ is equal to

The field snapshot and the magnitude distribution for the non-magnetic lens based on the linear transformation show strong reflections from the exit boundary at y′ = g+l. As a result, a weak transmitted field results in the region of space past the exit boundary. A strong interference pattern between the incident and the reflected fields can be observed in the range y′ < g+l, which includes the lens medium itself as well as the free space region in front of the entry plane. When the nonlinear transformation is employed, the resulting non-magnetic lens results in a significantly reduced amount of reflections as can be seen from Figs. 8(c)–8(d). However, there are some reflections visible in the range y′ < g+l. These are not caused at the device boundaries, but rather from inside the device. A large gradient in y′ with respect to y″ along x′ = 0 near the center of the device causes a rapid variation in ε _x′,x′. The associated large variation of the impedance within the device causes reflections. However, they can be reduced by increasing the lens thickness l. The performance of the non-magnetic focusing lens is improved due to a proper non-linear transformation in the ŷ′ direction over that of the non-magnetic lens based on the linear transformation.

 

Fig. 8. (Color online) Rectangular non-magnetic lens designs with linear and nonlinear coordinate transformations under a TM-mode beam illumination: (a) A snapshot and (b) the magnitude distribution of the total magnetic field for the design based on a linear transformation. (c) A snapshot and (d) the magnitude distribution of the magnetic field for the design based on a nonlinear coordinate transformation. The dimensions of these lenses are specified by w = 0.489 m, l = 0.13 m.

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The improvement in the reflection property of the design based on the nonlinear transformation is examined by comparing ∣H_z∣ for the two designs along x′, = 0 in Fig. 9(a). The presence of ripples in front of the entry surface at y′ = 0.4 m indicate reflections for both designs. However, smaller magnitudes in the ripples for the lens based on the nonlinear transformation indicate a smaller amount of reflections. This is confirmed in Fig. 9(b), where the magnitudes of the scattered magnetic fields H ^s = H-H ⁱ are compared relative to the magnitude of the incident field. It is observed that the amount of reflection is reduced by a factor of 6 to 14 when a nonlinear transformation is employed. As a result, the strength of the transmitted magnetic field is seen in Fig. 9(a) to be higher for the nonlinear-transformation-based design in the range y′ < 0.53 m.

5. Conclusion

Two far-zone focusing flat lens designs with improved reflection characteristics were presented based on the transformation optics technique. For the trapezoidal lens design, full-wave simulation results indicated that no noticeable reflection occurs at the exit boundary of the lens, beyond which point the emerging wavefronts are completely planar. A specially constructed angle-limited cylindrical wave illumination was employed to avoid diffractions from the lens edges. The key to minimizing reflections lies in mapping the circular arc centered at the line source location in the original space into a straight line in the transformed space without any stretching or compression. A non-magnetic rectangular focusing lens based on a nonlinear coordinate transformation was also presented. The amount of reflections caused by employing the non-magnetic material parameter values was significantly reduced from the conventional design employing a linear coordinate transformation.

 

Fig. 9. (Color online) Comparison of the magnetic field distributions along x′, = 0 between the lens designs obtained based on linear and non-linear transformations: (a) The magnitude of the total magnetic field, (b) The ratio of the scattered field to the incident field.

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