HE design of metasurfaces to create tunnel-like connections through space (with a finite travel time from port to port) connecting two space wave ports at distant locations is addressed in this paper (see Fig. 1). The incident space wave field is absorbed at port 1, perfectly converted into a surface wave which connects the two ports and transfers power between them, and reradiated from port 2 located at a distant location. The reradiated field from port 2 is designed with arbitrary control over its phase and amplitude in a completely passive and lossless way utilizing nearly all (over 97.5%) of the power contained in the incident field over port 1. As the metasurface transfers nearly all of the available power in the incident wave to the reradiated wave, the operation is said to be near perfect.
Jordan Budhu is with Virginia Tech, Blacksburg, VA 24060 USA. (e-mail: jbudhu@vt.edu).
The enabling technology for the presented designs is near perfect space-wave to surface-wave couplers. Although these devices have been demonstrated before for planar [1]- [4] and cylindrical surfaces [5]- [7], near perfect coupling over an arbitrarily shaped non-canonical conformal surface has not been shown. Furthermore, complex-valued field control over non-canonical conformal surfaces using passive and lossless metasurfaces has also not been shown. In this paper, near perfect space-wave to surface-wave couplers enable wave transportation of incident beams to spatially dislocated ports along any desired shape surface and with complex-valued field control of the reradiated beam from a completely passive and lossless metasurface.
The metasurface itself is a textured interface, modeled as a spatially variant homogenized purely reactive impedance sheet supported by a grounded dielectric substrate [8]. The metasurface is capable of complex-valued radiated field control and seamless conversion between guided and unguided modes in a lossless or perfect manner. It is modeled using integral equations [9]- [11], the integral equations are solved using the method of moments technique [12], and the reactances of the impedance sheet optimized using the adjoint variable method [11], [13], [14]. For an overview of the design procedure, see [11], [15]. A customized integral equation for the design of the conformal cases specific to this paper can be found in the appendix of this paper. The metasurface can be made to conform to any shape such as the corner of a building (see Fig. 10 for preview) or a general non-canonical curvilinear surface (see Fig. 15 for preview) for example. We will first present a planar design to demonstrate the near perfect wave transportation function and understand its operation.
Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org Incident Field (Plane Wave) Reradiated Field (Shaped Amplitude and Phase) Subsequently, the same functionality from conformal geometries will be shown. Similar planar transporting metasurfaces have appeared in recent scientific works. In [2], a planar wave transporting metasurface was designed by juxtaposing three separate metasurfaces, two space wave to surface wave couplers separated by a metasurface supporting a pure surface wave. The overall metasurface system laterally shifts a plane wave incident at an angle of 30° on the left-hand side of the metasurface to a transmitted wave emanating from the righthand side of the metasurface at an angle of -7.2° with respect to the normal. Due to the analytical design procedure, the metasurface junctions scatter and reduce the efficiency. Also, the coupling metasurfaces do not perfectly convert the incident fields to surface wave fields. The authors report an efficiency of only 10%, and hence the wave transportation cannot be deemed perfect. Furthermore, the approach cannot control both the phase and amplitude of the transmitted field and hence does not have the capability of complex-valued field control. The metasurfaces in the referenced work are also not conformal.
In [16], a planar teleporting metasurface is designed using the principles of 𝒫𝒯 symmetry. A reactive layer is sandwiched between an absorbing lossy Salisbury screen layer matched to free space and an active layer with a negative impedance also matched to free space. An incident plane wave is absorbed nearly completely by the lossy layer, while the inductive perforated layer allows the remaining small amount of power to couple to the active layer where it is resonantly amplified to recreate or teleport the incident plane wave to the opposite side. Although this device requires active and lossy components, the loss and gain are balanced according to 𝒫𝒯 symmetry and hence represents an overall lossless system. Nonetheless, the structure is planar, requires active layers which complicates fabrication and cannot achieve beamforming.
This paper is organized as follows. In section II, we present a planar example. Next, in section III, two conformal examples will be presented. The first contains planar coupling regions and a conformal surface wave region. The second contains both conformal coupling regions and conformal surface wave regions. Some concluding remarks are provided in section IV. An 𝑒 𝑗𝜔𝑡 time convention is assumed and suppressed throughout the paper.
We first present a planar wave transporting metasurface to understand the wave transportation function as it pertains to metasurfaces. The metasurface geometry is shown in Fig. 1. The electromagnetics problem is 2-dimensional (out-of-plane wavenumber is zero) and hence the geometry is invariant in the 𝑧-direction. The patterned metallic cladding, described by a spatially variant homogenized sheet impedance 𝜂 𝑠 (𝑥), is supported by a grounded dielectric substrate of thickness 𝑑 = 1.27mm (50mil) and complex relative permittivity 𝜖 𝑟𝑐 = 𝜖 𝑟 (1 -𝑗𝑡𝑎𝑛𝛿) = 2.2 -𝑗0.002. The impedance sheet is broken into 200 unit cells of width 𝜆 0 /20 each at 𝑓 = 10GHz. The metasurface is therefore 𝑤 = 200(𝜆 0 /20) = 10𝜆 0 wide along the 𝑥-axis. The design of the transporting metasurface begins by specifying the desired total field tangential to the metasurface. The incident field is assumed a normally incident plane wave illuminating only the right-hand portion of the metasurface between 𝜆 0 ≤ 𝑥 ≤ 4𝜆 0 as shown in Fig. 2a. The reradiated (scattered) field is defined to have both a cosine tapered amplitude and uniform phase (complex-valued field control) exiting from only the left-hand portion between -4𝜆 0 ≤ 𝑥 ≤ -𝜆 0 . Critically, the absolute level of the amplitude in V/m of the scattered field is chosen to conserve power globally meaning the total power in the incident field, 𝑃 𝑖𝑛𝑐 = |𝐸 0 | 2 /2𝜂 0 = 0.12 mW/m for a unit strength plane wave, is equal to the total power in the scattered field. This definition will lead to the near perfect transportation, i.e., all the power is transferred from the incident field to the scattered field.
The metasurface design algorithm [11] starts from a local active/lossy (Ac/Ly) design used as a seed for the non-local passive/lossless (Pa/Ll) design. The local active/lossy design is obtained from the solution of the governing integral equations given the desired total field, 𝐸 𝑡𝑜𝑡 , associated with the wavefront transformation (see ( 16) and ( 17)). By solving the integral equations, the surface current density, 𝐽 𝑠 , on the metasurface is obtained, thereby allowing for the direct calculation of the metasurface impedances, 𝜂 𝑠 , following from the boundary condition 𝜂 𝑠 = 𝐸 𝑡𝑜𝑡 /𝐽 𝑠 (see Fig. 2b). As expected, the metasurface is lossy over the incident field region, and contains gain in the scattered field region. Furthermore, the loss and gain are balanced as a result of the choice of the scattered field amplitude. A transmission line model can be used to understand the result. Modelling the panel as a transmission line terminated in a shunt impedance representing the metasurface in parallel with an inductance representing the thin grounded dielectric substrate, the sheet impedance can be calculated as
( ) ( ) ( ) 0 0 , 0 0 tan 1 27.54 98 tan 1 1 j d s inc d e d j dj
(1) where 𝜂 𝑠,𝑖𝑛𝑐 is the sheet impedance within the illuminated portion of the metasurface, Γ is the reflection coefficient looking into the parallel load, 𝛽 is the wavenumber in the dielectric region, and 𝜂 0 and 𝜂 𝑑 are the intrinsic impedances of the free space and dielectric regions, respectively. For perfect absorption, the reflection coefficient should be zero. The resulting sheet impedance in (1) matches the numerically obtained value in Fig. 2b. In order for power conservation, the power absorbed in the lit region must exit the output region, and hence the sheet impedance in the scattered field region can be described as 𝜂 𝑠,𝑠𝑐𝑎 = -27.54 -𝑗98Ω. Note, since the shape of the amplitude differs between the two port regions, the sheet impedance tapers are different at the ends of their respective regions. Thus, the desired functionality of wave transportation can be achieved with a local balanced active/lossy metasurface described by the sheet impedances in Fig. 2b. In this case, the beam would teleport as the incident energy is not transported to the output beam but rather the output beam is created through resonant amplification given some small diffractive coupling. Its balanced loss and gain operating principle is similar to the balanced loss and gain of the 𝒫𝒯 symmetric teleporting structure in [16]. In both cases, although the incident energy itself does not teleport, a small diffractive coupling is necessary to excite the gain medium to resonantly create the teleported beam.
A purely passive/lossless metasurface eases fabrication and avoids unnecessary complexity. To that end, the local active/lossy metasurface is used as a seed design to obtain a non-local passive/lossless design with the same performance. Since the power is balanced, a surface wave can carry the power from the lossy region to the active region. In this case, the metasurface transports the energy to the output port rather than teleports it. However, from an outside perspective, the beam is seen to disappear and reappear somewhere else in space.
The integral equation solver is coupled with an adjoint variable optimizer to obtain the non-local design [11]. To this end, the 𝑁 = 200 reactances of the unit cells comprising the metasurface are arranged in an 𝑁-dimensional space with each reactance varying along an orthogonal axis. A surface is defined in this space as 𝑔(𝑥) = 𝑓(𝑋 1 , 𝑋 2 , 𝑋 3 , … , 𝑋 𝑁 ) and represents the response of the metasurface as a function of its reactances 𝑋 𝑁 . The cost function 𝑔 is designed such that its minimum represents the optimal solution
where † denotes the conjugate transpose and
The 𝑀 observation points 𝑟 ⃗ 𝑚 of the electric near field 𝐸 ⃗⃗ 𝑛𝑓 are defined along a contour the same shape and dimension as the metasurface and displaced one wavelength above it. The optimizer is seeded with the reactive part of Fig. 2b; and then the gradient of (3), defined in the appendix of [11], is used to descend along 𝑔 to its minimum. The optimizer converged to a value of 𝑔 = 0.05 in 160 iterations (see Fig. 3). The optimal non-local metasurface sheet impedance is shown in Fig. 2c and the near fields computed from the non-local design are shown in Fig. 4. Three distinct regions are evident, a spatially modulated input and output port connected by a nearly constant surface wave region. Sharp (a) (b) 2c excite a number of auxiliary surface waves in the input and output port regions (in addition to the prominent surface wave within the port regions and within the tunnel region connecting the two ports) responsible for redistributing power transversally within the port region facilitating passivity and losslessness [11], [17]. These perturbations also aid in obtaining a seamless transition region between the ports and the connecting surface wave region increasing the overall port-to-port power transfer efficiency.
The surface waves can be visualized in Fig. 5a. Figure 5a shows the amplitude of the plane wave spectrum of the scattered electric field evaluated on the metasurface. The tunnel surface wave responsible for transporting power between the ports is evident at 𝛽 𝑠𝑤,𝑡 = -7.95𝑘 0 . This wavenumber is in agreement with the sheet impedance of 𝜂 𝑠 = -𝑗22Ω in the tunnel region in Fig. 2c since a sheet of this impedance supports a surface wave of wavenumber 𝛽 𝑠𝑤,𝑡 = -7.95𝑘 0 . This can be verified by viewing the dispersion curves plotted in Fig. 5d. In Fig. 5d, a plot of the surface wavenumber and wavelength versus the sheet reactance of the metasurface is shown. The plot is obtained using the Transverse Resonance Technique [11]. A reactive sheet impedance of 𝜂 𝑠 = -𝑗22Ω is seen to support a surface wave of wavenumber 𝛽 𝑠𝑤,𝑡 = -7.95𝑘 0 . The surface wave wavelength can also be verified to agree with the curves of Fig. 5d. In Fig. 5b, a zoomed in view of the surface wave in the tunnel region is shown. The measured wavelength agrees with the dispersion curves.
The amplitude spectrum in Fig. 5a also shows another peak at 𝛽 𝑠𝑤,𝑝 = -2.58𝑘 0 , which is the surface wave generated from the incident plane wave within the input port region. Figure 5d shows this surface wavenumber is associated with a sheet impedance of 𝜂 𝑠 = -𝑗53Ω in agreement with the average of the sheet impedances shown in Fig. 2c within the input port region. The sheet impedance modulation within the input port region can be understood by noting that for broadside radiation of the 𝑛 = -1 harmonic from a surface wave of wavenumber 𝛽 𝑠𝑤,𝑝 = -2.58𝑘 0 , the period of the modulation should be 𝑘 𝑥𝑛 = 𝛽 𝑠𝑤,𝑝 -2𝜋/𝑝 ⇒ 0 = 𝛽 𝑠𝑤,𝑝 -2𝜋/𝑝 or 𝑝 = 2𝜋/𝛽 𝑠𝑤,𝑝 = 𝜆 0 / 2.58. Shown in Fig. 2d, a sinusoidal sheet impedance modulation function with this period is fit to the non-local passive/lossless metasurface sheet reactances. As can be seen, the modulation period corresponding to the 𝑛 = -1 harmonic for broadside radiation fits the optimized sheet reactances well. The perturbations of the reactances around this analytic result excite the auxiliary surface waves and lead to the near perfect coupling. No other spatial harmonics fall within the light cone. A zoomed in view of the port surface wave within the input port region is shown in Fig. 5c. The measured surface wave wavelength is also in agreement with the dispersion curves in Fig. 5d.
Finally, a remark on the spectrum limits. The spectrum has a cut-off at 𝛽 𝑠𝑤 = 10𝑘 0 which is the highest wavenumber possible as the onset of a stop-band at 𝛽 𝑠𝑤 = 𝜋/𝑑 = 0.1𝑘 0 for the chosen unit cell discretization of 𝑑 = 𝜆 0 /20 occurs at this wavenumber. This corresponds to a maximum sheet impedance of -𝑗20Ω according to Fig. 5d. For this reason, hard limits of -𝑗20Ω on the impedances during the optimization phase were set, and is why the tunnel sheet impedance is approximately -𝑗20Ω. The remaining evanescent spectrum is due to the sharp Tunnel SW (a) (b) (c) Port SW (d) 2c. These surface waves are responsible for redistributing power transversally within the port regions and at their transitions with the surface wave region in order to achieve passivity and losslessness.
The power in the surface wave can be seen to grow approximately linearly in agreement with the conclusions in [3] in Fig. 6a, although here the spectrum (Fig. 5a) contains many spatial harmonics rather than the single harmonic considered in [3] Fourier transform of 𝐸 ̃𝑧𝑠𝑐𝑎 𝑘 𝑥 /𝜂 0 𝑘 0 , where 𝐸 ̃𝑧𝑠𝑐𝑎 is the electric field spectrum at the plane of the metasurface (the amplitude of 𝐸 ̃𝑧𝑠𝑐𝑎 is shown in Fig. 5a). The power density in the surface wave is shown to increase from zero within the input port region approximately linearly as more of the power in the plane wave is absorbed, then become constant through the tunnel region as the power is carried to the output port region, and finally decay approximately linearly in the output port region to zero as the power is shed into the scattered beam. The oscillations in the power density profile occur due to the interference between the similarly polarized incident and surface wave fields. Next, to show the metasurface perfectly converts the incident plane wave at port 1 to the complex-valued scattered field at port 2, the near electric field was calculated along a horizontal line one wavelength above the metasurface. In Fig. 6b, the stipulated scattered field amplitude (replicated from Fig. 2a), the directly calculated (from the induced surface currents) scattered near field amplitude, and the backprojected far fields are all shown compared. It is evident that the non-local metasurface nearly perfectly creates the stipulated near field amplitude, and hence transfers all power in the incident plane wave to the output scattered field. Integrating the power density (a) (b) (c) (d) (e) Lastly, to provide an independent verification of the transporting metasurface, the design was imported into COMSOL Multiphysics and a full-wave simulation performed. The results are compared to the MoM results in Fig. 7. As can be seen, the independent full-wave verification corroborates our results.
Although it leads to intuitive results, the finite-width nondiffracting plane wave used to illuminate port 1 in the previous section is non-physical. This is evident in the shadow region behind the metasurface where the non-diffracting incident field does not totally cancel the diffracting scattered field. To model a more physical excitation, the same metasurface is illuminated with a Gaussian beam [18] with a waist radius of 𝑤 0 = 𝜆 0 at the metasurface plane. The center of the waist is located at 𝑥 = 2.5𝜆 0 and 𝑦 = 0. For this example, both the MATLAB design codes and the COMSOL Multiphysics verification models are available through the IEEE DataPort at [19].
The design procedure was repeated with the Gaussian beam illumination in place of the plane wave. All other parameters were kept the same as the previous design. The resultant sheet impedances of the active/lossy design and the passive/lossless design are shown in Fig. 8a and Fig. 8b, respectively. As can be seen, the boundary of the input port is now not sharply defined, but rather spread out as the Gaussian beam illumination does not have sharp boundaries. Nonetheless, we see the same features, sinusoidal like modulation within both the input and output port regions. The real part of the total electric field for the passive/lossless design of Fig. 8b is obtained using COMSOL Multiphysics and is plotted in Fig. 8c and Fig. 8d. The transverse power density (Fig. 8e) and scattered near field obtained along a line one wavelength above the metasurface (Fig. 8f) show excellent agreement with COMSOL Multiphysics. The ratio of the integrated scattered and incident field power densities produces a result of 97.5% for the port-toport transfer efficiency.
By incorporating conformal geometry modelling capabilities into the integral equation/moment method algorithm (see Appendix), transporting metasurfaces connecting two distant non-colinear ports in space can be accomplished (see Fig. 9). These types of transporting metasurfaces can be useful for channel optimization in urban environments where the windowpane sized metasurface conforms to the corner of a building for example (see Fig. 10).
Optimization In Fig. 9, the geometry of a conformal transporting metasurface which routes the surface wave around a 90° bend is shown. Surface Wave Routing Around Building Corner (a) The parameterization is also shown graphically in Fig. 11. The parameters 𝑎 and 𝑏 control the aperture length along the 𝑥-axis and 𝑦-axis, respectively, and the parameter 𝑑 controls the substrate thickness. The parameter 𝑝 controls the metasurface shape and radius of curvature at the bend. For 𝑝 = 2, for example, (4) defines a quadrant of a circular annulus in the 𝑥𝑦plane. As 𝑝 → ∞, the parameterization approaches a quadrant of a square ring with thickness 𝑑. When 𝑣 = 1, the superquadric has the largest radius (the curve 𝑔 in Fig. 11). The impedance sheet will be placed along this arc. When 𝑣 = 1 -𝑑/𝑎, the superquadric has the smallest radius (the curve 𝑒 in Fig. 11). This is where the perfectly conducting ground plane will be placed. ∀𝑣 between these two values, the space between is filled (the dielectric material of the substrate will fill this area). For the parameters in (4), the greatest radius of curvature calculated using (11) at the 90° bend point is 𝑅 = 0.587𝜆 0 . The metasurface has length 𝑎 = 8𝜆 0 along the 𝑥-axis, 𝑏 = 8𝜆 0 along the 𝑦-axis, and thickness 𝑑 = 1.27mm (50mil). It is constructed from the same three layer stack: a patterned metallic cladding represented as a spatially variant homogenized impedance sheet, a dielectric spacer, and a ground plane. The incident field is a Gaussian beam again with a waist radius of 𝑤 0 = 𝜆 0 . For this case, the center of the waist is located at 𝑥 = -3.5𝜆 0 and 𝑦 = 𝑏. The Gaussian beam will be absorbed at this space wave port and converted into a surface wave. The surface wave will travel around the bend delivering the power to port 2 defined along 𝜆 0 ≤ 𝑦 ≤ 4𝜆 0 , where it will be reformed into a collimated beam corresponding to an aperture field with uniform amplitude and phase. Both the MATLAB design codes and the COMSOL Multiphysics verification models for this design are available through the IEEE DataPort at [19].
Figure 12a shows the metasurface sheet impedances for both the local active/lossy metasurface design and the non-local passive/lossless metasurface design. Figure 12b shows the near field amplitude taken along a contour following the metasurface and one wavelength above the metasurface. As can be seen, the non-local passive/lossless metasurface performs identically to the local active/lossy design. Finally, the real part of the total near electric field is shown in Fig. 10b. As in the planar case, ground plane metasurface dielectric spacer Patterned Metallic Cladding Ground Plane Dielectric Spacer (a) (b) the metasurface is performing the function of near perfect space wave transportation, only in this case, the space wave beam is seen to transport around the corner of a building.
The final example is a near perfect conformal wave transporting metasurface where the coupling regions are not planar. The geometry and its parameterization are shown in Fig. 13a and in Fig. 14, respectively. The parameterization can be described as a sinusoidally modulated exponential
The parameter 𝑤 controls the aperture length along the 𝑥-axis, and the parameter 𝑑 controls the substrate thickness. The parameters 𝑐 and 𝑎 controls the amplitude and the growth rate of the exponential function, which acts as a fundamental term of which the sinusoid is added to. The parameters 𝑝 and 𝑏 control the amplitude and period of the sinusoidal term. The impedance sheet will be placed along the curve resulting from 𝑣 = 0, The perfectly conducting ground plane will be placed along the curve at 𝑣 = -𝑑. ∀𝑣 between these two values, the space between is filled (the dielectric material of the substrate will fill this area). The metasurface width, as projected onto the 𝑥-axis, is 𝑤 = 10𝜆 0 . The incident field is again a Gaussian beam with a waist radius of 𝑤 0 = 𝜆 0 . For this case, the center of the waist is located at 𝑥 = -3.5𝜆 0 and 𝑦 = 𝜆 0 . Defining the scattered field as 𝐸 𝑧 𝑠𝑐𝑎 = 𝑒 -𝑗𝑘𝑦 , for 𝜆 0 ≤ 𝑥 ≤ 4𝜆 0 , and solving the governing integral equation, results in the active/lossy design impedances shown in Fig. 13b. The corresponding passive/lossless design's reactances after optimization are also shown in Fig. 13b. Finally, the COMSOL Multiphysics simulation results of the real part of the total near electric field for the Gaussian beam excited passive/lossless design is shown in Fig. 15. Wave transportation is again observed, as well as near perfect coupling of a normally incident plane wave to a surface wave over a conformal surface. In Fig. 13c, the magnitude of the electric near field is plotted along a contour the same shape as the metasurface displaced one wavelength above it. The figure shows some imperfect coupling in the COMSOL Multiphysics result and/or impedance matching between the port region and the tunnel region as some scattered electric field is present over the input port region. This may be due to the paraxial approximation for the Gaussian beam used in the MoM whereas in COMSOL Multiphysics, a more accurate representation based on a plane wave decomposition is used. Nonetheless, the full-wave results again corroborate our results.
Note, both the MATLAB design codes and the COMSOL Multiphysics verification models for this design are also available through the IEEE DataPort at [19].
V. CONCLUSION Conformal, metasurface-based, space-wave to surface-wave couplers were designed to transport a space wave from one location in space to another via surface wave and reradiate it as a newly formed beam. The reradiated space wave can be formed into an arbitrary beam with control over its amplitude and phase. The metasurfaces can conform to any shape, are fully passive and lossless, and hence require no powered connections. The metasurfaces are deemed near-perfect, as over 97.5% of the available power in the incident space wave is transported to the output wave in all cases. These types of metasurfaces can be useful to fill in hard to reach shadow zones in new high-frequency communications systems in an efficient, power-free manner. The conformal metasurfaces are designed using a coupled system of integral equations formulated for arbitrarily shaped geometries defined parametrically. The integral equations are solved via the method of moments. All ground plane metasurface dielectric Fig. 15. COMSOL Multiphysics simulation results of a conformal transporting metasurface shaped using a sinusoid added to an exponential function. Finally, note the primary purpose of this paper is to show that near perfect transportation is possible and what the sheet impedances look like. Support for dielectric materials is currently being added to the unit cell design process required to realize these metasurfaces outlined in [20]. Once this is complete, follow-on work involves translating the optimized impedance sheets for all designs in this paper to patterned metallic claddings.
The conformal upgrade of the integral equation in [9]- [11] involves both line and area integrations over the boundary and domain of the region 𝑅 in Fig. 11 and Fig. 14, respectively. Single integrals over curves bounding the region 𝑅 (curve 𝑔 or 𝑒 in Fig. 11 for example) can be evaluated by integrating over curves bounding the parameter space region 𝑆 (curve ℎ or 𝑓 in Fig. 11 for example) from
00 22 00 ,,
)
where 𝑣 0 = 1 or 1 -𝑑/𝑎 for integration over the metasurface layer (curves 𝑔 or ℎ) or ground plane layer (curves 𝑒 or 𝑓), respectively. To evaluate the double integrals over the region 𝑅, the integrations can be done over the region 𝑆 in the parameter space using ( , ) ( ( , ), ( , ))
Care must be taken to ensure the absolute value of the Jacobian determinant appearing in (7) is always taken (note, the Jacobian determinant associated with the mapping in Fig. 11 turns out to be negative so the absolute value must be taken).
The integral equations will be derived for the parametric geometry in Fig. 11. Those for any other geometry can be found analogously. The metasurface can be modeled as consisting of three layers, denoted by different values of 𝑣 0 . Layer 1, at 𝑣 0 = 1 denotes the impedance sheet. Layer 2, at 𝑣 0 = 𝑣, denotes the dielectric spacer. Layer 3, at 𝑣 0 = 1 -𝑑/𝑎, denotes the perfectly conducting ground plane. An integral equation can be constructed for each layer, and hence there are three total integral equations, one for each choice of 𝑣 0 in the following
1 11 cos sin sin cos sin 1 cos cos sin 1 sin cos sin p p pp pp p pp p u u u u u a a b b x u v uu ab y u v uu ab
The derivatives of (5) are are 1, cos , 0, 1 au x y x y cae pb bu u u v v = = + = = (10) Also the curvature, 𝜅, of the impedance sheet layer can be computed from k Z H k x u x u d a y u y u d a
The self terms (𝑚 = 𝑛) in ( 14) are calculated using the procedure outlined in [21]. For completeness and since those formulas are adapted to the conformal parametric geometry case, the formulas for calculating the self-terms are provided in the next section. Following the same procedure for the remaining layers (𝑣 0 = 𝑣 for layer 2 and 𝑣 0 = 1 -𝑑/𝑎 for layer 3) leads to the block matrix equation
1 11 1 12 13 1 2 21 22 23 2 3 31 32 33 3
Note, for the dielectric layer, 𝑍 𝑣 = 𝑒𝑦𝑒([𝑗𝜔𝜖 0 (𝜖 𝑟 -1)] -1 ), where eye( ) indicates an identity matrix with the argument appearing along the diagonal. Also note, 𝜂 𝑠 = 0 for layer 3 and hence 𝑍 𝑠3 does not appear in (15). Finally, note in our implementation, each of the voltage vector and impedance matrix elements in ( 14) are normalized to their own arc length (for 1D) or area (for 2D). The matrix equation ( 15) can be solved by either knowing the sheet impedances 𝑍 𝑠1 (as in each iteration of the optimization phase leading to the passive/lossless design) or by knowing the desired total field and making the substitution 𝑊 1 = 𝑍 𝑠1 𝐼 1 following from the boundary condition 𝐸 𝑡𝑜𝑡 = 𝜂 𝑠 𝐽 1 (as in the initial solve phase leading to the active/lossy design)
1 11 12 13 1 2 21 22 23 2 3 31 32 33 3 v
Since, in our case, the desired total field, 𝐸 𝑡 , is known, the 𝑊 1 vector can be found from
2 22 11 12 m m u u u u u u vv ss xy du uu II
The inner integral of 𝐼 𝑠2 (in brackets) to integrate analytically becomes ( )
22 '2 2, 11 '2 ln ' '' n n u u u s in vv u u u xy I u P du uu
With reference to Fig. 16, the coordinate origin is denoted by 𝑂. The observation point position vector is denoted by 𝜌 ⃗ = [𝑥(𝑢, 1), 𝑦(𝑢, 1)]. The source point position vector is denoted by 𝜌 ⃗ ′ = [𝑥(𝑢 ′ , 1), 𝑦(𝑢 ′ , 1)]. The source line segment, 𝐶, is colored red in Fig. 16 and is parameterized by the arc length variable 𝑙′ measured from the line perpendicular to the extension of 𝐶 and which passes through the point located by 𝜌 ⃗ . The endpoints of the source segment are pointed to by the vectors 𝜌 ⃗ ′-and 𝜌 ⃗ ′+ . The coordinates 𝑃 0 and 𝑙′ can be considered a pair of rectangular coordinates with origin at 𝜌 ⃗, locating points on the line segment 𝐶. Thus, the endpoints of source segment are located at a distance of ( ) ( ) the integration is done in the 𝑥𝑦-space using the above paradigm rather than over the parametric 𝑢𝑣-space since the vectors in (24) are constant (see Fig. 11). Thus, (21) becomes 1 lim ln ' ' 22 lim ln ' ' v in A s AA A I Pdx dy P P P P dx dy Pdx dy → -→ = = - + (30)
where 𝐴 𝜖 is a small circular region of radius 𝜖 enclosing the observation point included to make the integrand continuously differentiable. For more information, see [21]. A divergence theorem can be applied to (30) to express the surface integral in terms of a flux integral around the boundary enclosing the area 𝐴. Following the result derived in [21], (30) is evaluated as
0 2, 1 0 1 1 00 1 ( , ) ln ln 2 3 tan tan 2 i i v in i i i i i ii i i i ii I u v P u l P l P ll P l l PP
where 𝑃 ⃗⃗ 𝑖 0 = 𝑃 𝑖 0 𝑃 ̂𝑖0 , 𝑃 ̂= (𝜌 ⃗ ′ -𝜌 ⃗)/𝑃, and 𝑢 ̂ is the outward normal vector (ensure 𝑧̂• (𝑢 ̂𝑖 × 𝑙 ̂𝑖) > 0, else set 𝑢 ̂𝑖 = -𝑢 ̂𝑖). Note, the mapping in (4) negates the signed area (Jacobian determinant is negative), and thus, going clockwise around the perimeter of the quadrilateral patch in the 𝑢𝑣-space maps to a counterclockwise path around the perimeter of 𝐴 in the native 𝑥𝑦-space (see Fig. 17). Index 𝑖 denotes one of the four sides of the quadrilateral patch in 𝑅 bounding the surface area 𝐴, and thus each side 𝑖 is represented by the geometry in Fig. 16. Note, although the quadrilateral patch in the 𝑢𝑣-space is rectangular (is constructed from straight line segments), its image under the mapping 𝑟 ⃗ may have curved bounding line segments. However, if the discretization is fine enough, the perimeter of 𝐴 can be approximated as constructed from straight line segments, and the results of Fig. 16 will hold. With this result, (28) is found as 2 2 ' 2 ' 2 00 0 2 2 ' 2 ' 2 2 00 2, 22 2 1 ln 42 '' '' '' ( , ) 2