Single-Layer, All-Metallic Metasurface Filter With Nearly 90° Angularly Stable Resonance

We present a method to design a single-layer, all-metallic angular stable metasurface (ASM) filter unit cell for nearly the entire angular spectrum. We formulate the optimization criterion using the energy balance condition derived from Poynting’s Theorem. The proposed unit cell has a “cloverleaf” geometry, and we show that by adjusting the properties of the cloverleaf, we manipulate the in-plane spatial dispersion, thus realizing an angularly stable resonance that extends to nearly 90° incidence. After optimization, we revisit the energy balance criterion and show, using a spatially dispersive admittance expansion, how optimization at three points is enough to obtain the required balance for nearly the entire angular spectrum. The proposed ASM was fabricated from a bare aluminum sheet and measured for transverse electric (TE) and transverse magnetic (TM) polarization, with angularly stable performance that matches the theory and simulation. The proposed design offers new capabilities for radar and antenna design applications, with the simple, single-layer, all-metallic structure being particularly useful in aerospace and satellite applications.


Single-Layer, All-Metallic Metasurface Filter With
Nearly 90 • Angularly Stable Resonance Nadav Goshen , Student Member, IEEE, and Yarden Mazor , Senior Member, IEEE Abstract-We present a method to design a single-layer, allmetallic angular stable metasurface (ASM) filter unit cell for nearly the entire angular spectrum.We formulate the optimization criterion using the energy balance condition derived from Poynting's Theorem.The proposed unit cell has a "cloverleaf" geometry, and we show that by adjusting the properties of the cloverleaf, we manipulate the in-plane spatial dispersion, thus realizing an angularly stable resonance that extends to nearly 90 • incidence.After optimization, we revisit the energy balance criterion and show, using a spatially dispersive admittance expansion, how optimization at three points is enough to obtain the required balance for nearly the entire angular spectrum.The proposed ASM was fabricated from a bare aluminum sheet and measured for transverse electric (TE) and transverse magnetic (TM) polarization, with angularly stable performance that matches the theory and simulation.The proposed design offers new capabilities for radar and antenna design applications, with the simple, single-layer, all-metallic structure being particularly useful in aerospace and satellite applications.
From their very nature, the response of metasurfaces also depends on the polarization and incidence angle (θ inc ) of the impinging electromagnetic wave, on top of all other parameters [14].From an equivalent circuit point of view, for different incidence angles, the metasurface exhibits a different effective surface impedance, resulting in a dependence both on frequency ω and θ inc , i.e., Z M S (ω) = Z M S (ω, θ inc ).This angular dispersion in metamaterials and metasurfaces highlights an important subclass-angular stable, or spatially stable metasurfaces.In general, angular (spatial) dispersion limits the operation angles in which the required frequency response can be achieved, and therefore, improving the angular stability has important practical implications.
The vast majority of studies related to angular stable metasurfaces (ASMs) are focused on miniaturizing the unit cell (reducing the nonlocal effects, whose magnitude depends on period/λ) and using multilayer designs (compensating for the in-plane dispersion using the out-of-plane structure) [15], [16], [17], [18], [19], [20], [21], [22], [23].These approaches rely on taking the geometries and materials that construct the metasurface unit cell and using numerical or analytical methods to extract the equivalent circuit model parameters for the structure.While they provide an accurate equivalent circuit modeling and improve the angular stability of the metasurface, they require a combination of metal and dielectric layers (at least two layers), resulting in a thick and more elaborate structure.Moreover, the requirement of miniaturized unit cell elements, usually smaller than 0.1λ 0 , will eventually limit the operational frequencies for practical metasurface design due to the incapability to fabricate reliable micro-scale features of the unit cells.Hence, further miniaturization for modern applications across the electromagnetic spectrum, like X-, K-, and the Ka-bands, can prove challenging.Another interesting approach presents a 3-D frequency-selective structure [2], [24], [25].Three-dimensional structures consist of a 2-D periodic array of multimode cavities when the desired frequency response is controlled by the number of propagating modes and their couplings with the surrounding medium.However, these structures are synthesized for normal incidence, leaving the oblique case not carefully weighed.
In this work, we use a physically oriented optimization procedure to design a single-layer, all-metallic metasurface with nearly 90 • angular stability.When studying applications that require increased angular stability, this type of architecture is perhaps the most minimalist design one can achieve.Therefore, it could be a key component in more advanced and complex metasurface application designs.The chosen unit cell for the all-metallic thin metasurface has smoothly varying boundaries and is termed the cloverleaf element [26].
We derive a general resonance condition in terms of reactive energies using Poynting's Theorem.Then, we define the proper integration region to quantify the energy balance around the metasurface and use the energy balance condition as a constraint in a genetic optimization procedure performed on our designed unit cell geometry.This method proves particularly insightful since it gives the designer a physical intuition through reactive power distribution around the metasurface.The generality of the energy-balance condition, along with our unit cell degrees of freedom, allows us to achieve angular stability up to nearly 90 • with only two optimization iterations.Thanks to the curvature of the "leaves" and the area of the central junction, it is possible to maintain the resonance response for the effective surface admittance for a broad angular spectrum in a single metallic sheet.
This article is organized as follows.In Section II, we present the unit cell geometry and define the design degrees of freedom.In Section III, we present and study the power and energy balance using Poynting's Theorem.We use the relationship between the total ASM admittance and the electric and magnetic reactive energies accumulated around the ASM surface and derive the required reactive energy balance condition.Using a Floquet-Bloch analysis, we estimate the proper integration region in which the main contribution of accumulated reactive energies is stored.A full-wave frequency-domain simulation is performed, demonstrating and verifying those assumptions.In Section IV, we optimize the ASM for transverse magnetic (TM) incidence using the energy balance optimization objective, achieving angularly stable ASM for the entire angular spectrum.In Section V, the second optimized cloverleaf element ASM is fabricated and measured to verify the proposed method and design, To highlight the proposed method's advantages, we compare it to the state-of-the-art in Table II.Finally, in Section VI, the concluding remarks are drawn.

II. GEOMETRY DESIGN OF ASM UNIT CELL
Fig. 1(a) and (b) shows the structure of our cloverleaf element and initial unit cells, both embedded as a single layer of a perfect electric conductor (PEC).The cloverleaf element was obtained by a physically oriented optimization procedure, as will be discussed later in this article.We found that the smooth boundaries, specifically the exponential ones used here, greatly improve the angular stability and the operational bandwidth.Physically, the optimization of the cloverleaf element relies on controlling the electric and magnetic reactive energy distribution around the surface plane (z = 0).Each parameter gives the designer an additional degree of freedom, which is able to slightly tune the reactive energies imbalance, resulting in the wanted goal of broad angular stability.
The geometrical design comprises a circular slot with the radius r and four identical metallic, exponentially tapered leaves.The taper is described by the curves T 1 and T 2 where t is a coordinate parameter, t ∈ [s, D x /2] and D x is the unit cell x-periodicity.The parameter s is a scaling parameter determining the intersection of the curves with the axes.These curves need to be replicated with a 90 • rotation around the center of the circular slot to obtain the outlines of all four "leaves."For practical reasons, each edged corner in Fig. 1(a) was filet with a radius of 0.4 mm, to consider the tolerances of the milling process that was used in fabrication.An additional parameter w was defined to control the "junction" width of the cloverleaf element.
The overall size of the unit cell is D y × D x × D z = 0.45λ 0 × 0.45λ 0 × 0.01λ 0 , where λ 0 is the free-space wavelength at 13 GHz, and D z is the thickness of the metallic sheet (∼0.23 [mm]).The configuration of Floquet port excitation is shown in Fig. 1(c).The unit cell is excited by a plane wave with transverse electric (TE)/TM polarization, propagating from the Z max plane to the Z min plane or vice versa, along the z-axis.The total distance between Z max and Z min is 6λ 0 .The parameters of the initial unit cell are reported in the caption of Fig. 1.

A. Energy Balance
Examining the energy accumulated in the vicinity of the surface is a useful tool to characterize the dominant behavior of our structure.It can be used as a guideline to optimize the wanted behavior in terms of the angular stability of the resonance.Let us consider a planar metasurface unit cell shown in Fig. 1(b), surrounded by isotropic and lossless media ϵ 0 .
If we define a certain volume V around the unit cell, bounded by a surface and apply Poynting's Theorem, we obtain [27] where S is the complex Poynting vector defined as S = E × H ⋆ , W e and W m are the electric and magnetic energies, respectively (the volume integral of the corresponding energy densities w e and w m ), and P d and P s are the dissipation and source power, and the symbol ( •) denotes the time-averaging of the quantity over a full period τ = 2π/ω.Equation (3) has four main terms, each having physical significance.This equation can first be split into real parts (indicating real power flow) and imaginary parts.If the real part of the integral of S is positive, then power leaves the region, corresponding to the source injected power within the region overcoming the dissipated power.The imaginary parts of (3) are the imaginary part of Poynting's vector flux and can provide some intuition about the metasurface system we examine.The balance between W e and W m can characterize the system as mainly capacitive, W e > W m , or mainly inductive W e < W m .While in resonance, the system is perfectly balanced, satisfying [W e − W m ] ω r esonance = 0. Let us define this contribution by U = [W e − W m ].This can serve as a very useful guideline on which behavior should be emphasized to achieve better performance.This term is strongly related to the effective impedance of the surface.It has been shown that the equivalent circuit that represents a slot-like PEC unit cell is a parallel LC circuit [28].The total admittance for such a circuit is given by where Y , L, and C are the admittance, inductance, and capacitance of the circuit, respectively.If we write the energy balance on the effective surface impedance, use a simplified transmission line model to represent the incident and scattered waves, and then incorporate it into (3), we will obtain where Z T E/T M represents the impinging wave impedance, and Ṽ corresponds to the tangential component of the incident electric field on the unit cell plane.When the effective surface admittance satisfies Y → 0, the reflection coefficient will vanish, corresponding to full transmission.From ( 5), we see that this condition also corresponds to the traditional resonance condition-the electric and magnetic energy accumulated around the surface will be equal The fourth term in (3), P d , represents the energy dissipated due to losses in the metal.
The transmission function (both phase and amplitude) of the ASM varies with θ inc since the energy balance (and therefore the effective impedance) is a function of θ inc .This angular dispersion can have strong adverse effects on the behavior and cause the resonance frequency to "drift," losing the wanted performance in the operation frequency.Therefore, improving the angular stability of the metasurface response becomes essential.Using (3), we can examine the changes in the reactive energy balance around the unit cell U versus the incidence angle θ inc where the desired situation is such that W e and W m remain in perfect balance ( U = 0) in the largest possible angular range.

B. Floquet-Bloch Analysis
When designing and optimizing our ASM unit cell, U is calculated numerically and used as an indicator for the matching of the ASM.Since the physical origin of this reactive energy is the excited evanescent spectrum, to correctly evaluate U , we need to properly define the integration region , which includes the reactive contribution surrounding the unit cell.The impinging plane wave is given by where H 0 , k x , and k are the complex magnetic field amplitude, x component, and magnitude of the free-space wave vector, respectively.Since the ASM which lies on the plane z = 0 is d-periodic (D x = D y = d), we can write the Floquet-Bloch (FB) expansion of the scattered fields H n e − jk x,n x e − jk z,n z ŷ ( where H n , k x,n , and k z,n are the complex magnetic field amplitude, modal transverse wavenumber, and longitudinal wavenumber, respectively, given by and a similar expression for z < 0, with the proper choice of the √ () branch, I m{k z } < 0. Only a finite number of these waves will be propagating, satisfying k z,n ∈ R. All other values of n lead to purely imaginary values of k z,n .These are evanescent waves corresponding to reactive energy stored in surface wave fields closely bound to the surface.For purely imaginary k z,n , the condition |k x,n | > k is satisfied, and the decay coefficient of the n'th harmonic α n can be written as As the values of |n| become larger in the summation in (7), the harmonics decay faster, and therefore the slowest decaying mode can be used to evaluate the region where most of the reactive energy will be stored.Consequently, from (10), we can define the integration region ′ taking our unit cell geometry with d = 0.45λ at the operation frequency of f 0 = 13 GHz.Substituting these into (8) and ( 9) yields For our parameters, all non-fundamental FB harmonics (n ̸ = 0) are decaying with k z,n being purely imaginary for any angle of incidence θ inc ∈ [0, π/2] with the decay coefficients Therefore, the slowest decaying mode will be n = −1.
Additionally, when the dependence of θ inc in ( 12) is examined, it is revealed that for higher incidence angles θ inc , the value of α −1 becomes smaller.For this reason, the proper choice is taking which renders the integration region as z ∈ [−(1/α −1,min ), (1/α −1,min )], assuring that the accumulated energy density has decayed by a factor of 1/e 2 .This choice balances between not having a too large region that will overburden the computation and incorporate undesired contributions in non-optimal cases and including most of the reactive energy in the calculation.
To verify the definition of region ′ , let us define two lines Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Both w e and w m will be numerically evaluated at f 0 = 13 GHz, for θ inc = 0 • , 20 • , 40 • along l 1 and l 2 , respectively.Fig. 2(a) and (b) shows the intersection of lines l 1 and l 2 with the unit cell plane z = 0, respectively.The points where lines l 1 and l 2 intersect with the unit cell plane are chosen where each of w e and w m are dominant with respect to the other.
The value of reactive electric energy w e found in the unit cell central "junction" is mainly determined by the parameter w (which controls the capacitive properties), while the value of the reactive magnetic energy w m , dominant in the inter-element regions (around l 2 ), is mainly determined by parameters r and q (which control the inductive properties).Fig. 2(c) and (d) presents the electric and magnetic energy densities in a cross section of the simulation domain around the cell, respectively.We see that the energy is mostly accumulated in the vicinity of the ASM, in the region we defined as ′ (dashed lines), stemming from the evanescent spectrum bounded in this region.Fig. 2(e) and (f) presents the normalized electric and magnetic energy densities w e and w m along the ASM z-axis, respectively.It can be clearly seen that as we move away from the ASM plane, both w e and w m decrease rapidly.Furthermore, the transition point in which w e and w m decrease to the level of the background field occurs around z ∈ [−(1/α −1 ), (1/α −1 )]) for all θ inc examined.

IV. ANGULAR STABILITY AND UNIT CELL OPTIMIZATION
To demonstrate the optimization method and our angularly stable unit cell design, let us first look at U as a function of the incidence angle θ inc for the initial unit cell, as shown in Fig. 1(b).We evaluate the initial unit cell response using a CST MWS full-wave frequency-domain simulation, under periodic boundary conditions with Floquet port excitation (TM).Then, we numerically evaluate U using the integration in (3) at f 0 = 13 GHz, with ′ as our integration region.
Fig. 3(a) shows that U depends on the incidence angle θ inc , and as θ inc approaches 20 • it starts to deviate from the balance condition ( U = 0), indicating some angular instability in our initial metasurface at higher incidence angles, which will cause a drift in the resonance frequency.Indeed, in Fig. 3(b), we see the transmission curve |S 21 | for TM polarization at two different incidence angles θ inc , demonstrating the resonance shift to higher frequencies.We define a metric for this resonance drift-the frequency deviation f dev (θ inc )-as the ratio between the deviation of the transmission maximum (termed | f |) and the wanted resonance frequency f 0 at a given θ inc .In this example, at θ inc = 60 • , the frequency deviation for the initial unit cell is f dev (60 • ) = 6.5%.This correlation between the angular stability and the dynamics of U provides an indicator that can be used as our optimization goal.Based on this correlation, we used the genetic optimization of CST MWS over the geometric parameters q, s, w, and r of the initial unit cell.The objective of the optimization is to bring U → 0 at the frequency f 0 = 13 GHz, and the angle of optimization is θ opt = 60 • .The algorithm was allowed to run a maximal number of 993 evaluations with a maximum number of iterations and population size of 61 and 32, respectively, and a mutation rate of 60%.The algorithm reached the objective after 31 iterations.It is important to choose the initial optimization angle at a point where the spatial dispersion is significant, manifesting via the relatively large value of f dev , since this is the feedback required for optimization.To further increase the angular stability performance, we conduct a second iteration of the optimization scheme, intended to achieve a stable response over the angle range of 0 • -89 • .Here, the optimization objective remains similar to the first iteration, with the only change being the optimization angle set to θ opt = 89 • .The algorithm successfully achieved the objective after seven iterations.Fig. 5 showcases the performance of our optimized ASM unit cell from a broader point of view.Fig. 5(a) presents a comparison of U as a function of θ inc between the initial (unoptimized), the first, and the second optimized unit cells.It can be noted that the optimization substantially balanced the electric and magnetic reactive energies ([W e − W m ]) across the entire incidence angle spectrum.Consequently, the frequency deviation values of the initial, first, and second optimized unit cells, within the incidence angle range of 0 • -89 • , have been significantly improved with the maximum deviation of f dev = 0.69% and f dev = 0.6% for the first and second optimized unit cells, respectively, as shown in Fig. 5  showing how the peak transmission remains very stable at f = 13 GHz.Table I shows the geometric parameters for all three unit cells.
One intriguing phenomenon we see in Fig. 5(a) is the single point in which U = 0 for the initial, unoptimized unit cell that occurs roughly around θ inc = 65 • .According to Section IV, such an outcome is obtained at Poynting's resonance condition, resulting in f dev (θ inc • ) = 0%.However, in Fig. 5(b), the frequency deviation f dev (65 • ) = 6.5%, which is clearly not 0%.Such result can be explained as follows: at the lower range of incidence angles (θ inc < 20 • ) where the unit cell is resonant, the dominant contribution of the reactive accumulated energy within the integration region ′ comes from the evanescent modes, which store energy close to the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
unit cell surfaces, as described in Section III-B and shown in Fig. 2(e) and (f).On the other hand, as the incidence angle increases, the ′ region "accumulates" energy in the partial standing wave pattern due to reflection at z < 0. Furthermore, the reflected energy contribution can be shifted from mainly inductive to capacitive and vice versa, depending on the angle, consequently shifting U from negative to positive inside ′ .This creates artificial points in which U = 0.However, as we demonstrated in the previous section, this does not affect our ability to optimize the unit cell and can be neglected by adding an additional optimization objective, bringing the frequency deviation near zero.Such an objective is uncorrelated to the integration region ′ and can only improve the optimization process.It is also worth taking a closer look at the optimization process.Intuitively, it is not clear why the second optimized ASM, with the last optimization step at θ opt = 89 • , would maintain its angular stability performance for the lower incident angles.The sheet impedance concept, Z s , is used to relate the effective surface current as a function of the averaged electric field where E s,avg is the averaged tangential electric field (which is expected to be continuous from both sides of the sheet), and J s,avg is the averaged surface current density on the metasurface.The sheet impedance of wire mesh screens in free space has been investigated extensively throughout the literature.
Based on [29], [30] applied the resulting mesh model and found the equivalent sheet impedance of the grid [31].The results found the Z s depends upon the tangential wavenumber k t,n = k 0 sin θ inc .This implies that the wire mesh is spatially dispersive, i.e., Z s = Z s (ω, θ ).The ASM structure we discuss is, in principle, a more elaborate version of a mesh grid surface.Indeed, these structures effectively act as parallel LC circuits [32], [33], characterized by their effective admittance Y s .A particularly helpful presentation of the dispersive profile of the effective admittance Y s is obtained by expanding it into a power of series in k t [34], [35] Y Equation ( 17) reveals some fundamental features about the effective admittance Y s .First, at normal incidence (θ inc = 0 • ), the only contribution is given by Y 0 (ω) implying that at the resonance frequency the 0th-order admittance satisfies Y s (ω r es , 0 • ) = Y 0 (ω r es ) = 0.This result is correlated with our initial unit cell performance, chosen to resonate at a given ω r es .At oblique incidence, higher order terms (mostly the first terms, m = 2, 4) in ( 17) contribute to the total effective admittance.Since the initial unit cell is matched only for the 0th-order admittance, the resonance drifts to a higher frequency.Therefore, the first optimization is performed in the operation frequency ω r es , in a moderate angle.This optimization step mostly affects the Y 2 , since Y 0 (ω r es ) = 0, and the contribution of higher orders is still quite small.In that sense, the second optimization is performed at ω r es for a large angle, tailoring the higher order contributions (mostly the fourth, since we see that these two optimization steps  are enough) and achieving the presented angular stability for almost the entire angular spectrum.
To validate this discussion, the values of I m{Y 0 }, I m{Y 2 }, I m{Y 4 } for the initial, unoptimized unit cell (solid), and the second optimized unit cell (dashed) are shown in Fig. 5(d).We notice that the initial unit cell has its high-order admittance terms (Y 2 , Y 4 ) resonate in different frequencies than the predesigned 13 GHz (solid line, notice that their zeros are around 14 GHz).On the other hand, after the optimization (dashed line), all terms resonate around the same frequency, ∼13 GHz.This result aligns with the initial analysis and also nicely corresponds to the condition given in (5).This analysis also highlights another key difference with respect to previous works.Here, we essentially design the in-plane spatial dispersion such that every consecutive term in the expansion resonates in the same prescribed frequency.

V. MEASURED RESULTS
To verify the proposed approach and design, the second optimized ASM design was fabricated from a 1.02 [mm] thick (0.044λ 0 ) bare aluminum sheet, as shown in Fig. 6(a).The fabricated surface is 26 × 26 [cm] (which is approximately 11.2λ 0 × 11.2λ 0 in the operation frequency).
The ASM S parameters were measured using a compass focused beam system (FBS) [36].The measurement setup Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
is shown in Fig. 6(b).The FBS is composed of two open boundary quad ridge feed horns (dual polarization that operate in the 2-32 GHz frequency range), two RF cables (operating in the frequency range of dc to 50 GHz), two 24 ′′ diameter lenses (each lens is two half lenses that can be reconfigured for focusing or collimated beams), reconfigurable specimen holder with elevation and azimuthal tilts axis, and a power network analyzer (PNA).All of these components are programmable and controlled via the control computer.The system uses the lenses to focus the feed horn radiation, with the ASM placed at its beam waist, where phase fronts are planar to mimic a plane-wave incidence.Moreover, to measure the transmission curves |S 21 | for the oblique cases, we can adjust the incidence angle θ inc of the plane-wave radiation via the specimen holder azimuthal axis.
Due to practical fabrication constraints, the fabricated design has a different thickness than the second optimized ASM we simulated, slightly increasing the capacitance and red-shifting the resonance.Therefore, we have performed simulations of the same second optimized ASM with the fabrication thickness to correctly compare and verify our design.The simulated and measured |S 21 | for TM and TE polarization for the practical ASM are shown in Fig. 7(a) and (b), respectively, for several incidence angles.Due to the change in thickness, the resonance frequency f 0 at θ inc = 0 is shifted from 13 to 12.82 GHz.For angles up to 70 • , we see a very good match between the simulation and the fabricated model S 21 response, exhibiting excellent angular stability.Beyond that, we start seeing small deviations from the wanted response.This is caused mainly by the fact that for shallow incidence angles, we observe various edge diffraction effects.In addition, the surface projection along the propagation axis becomes on the order of the Rayleigh length of the focused source beam.Therefore, the incident field phase fronts are no longer planar over the entire surface.
To quantify the angular stability, in Fig. 7(b), we present a comparison of the frequency deviation ( f dev ) between the simulated and fabricated designs, where we consider the frequency deviation with respect to the practical design resonance frequency f 0 = 12.82 GHz.Again, we get a good agreement, with f dev < 1% and f dev < 4% for the TM and TE polarization, respectively, throughout the angular spectrum.This is expected since the ASM was optimized for TM incidence.

VI. CONCLUSION
Achieving angular stability for a broad angular spectrum using only a single layer is challenging, due to the limited degrees of freedom the designer has to match all terms of Y s (ω, θ ), presented in (17).In this article, we have presented a design for a single-layer, all-metallic angularly stable metasurface filter based on smooth boundary elements without the need for dielectric layers.The optimization objective was derived from the Poynting Theorem, and a proper integration region ′ was defined through a Floquet-Bloch analysis.Based on the analysis and the optimization scheme that was performed for the TM polarization, we designed a metallic single-layer ASM that covers nearly 90 • of the angular spectrum with a maximum frequency deviation of f dev = 0.6% for TM polarization.The proposed design was fabricated, and the angularly stable performance was demonstrated using transmission measurements for both TM and TE polarization.In Table II, we present a comparison of our structure with various recent designs of angularly stable metasurfaces, showing the advantages of the proposed metasurface.Angularly stable metasurfaces have relevance for many applications.For example, designing spatial filters on curved surfaces, where the proposed ASM can be used as an FSS that maintains the frequency response for each θ inc of the propagating wave.Furthermore, for applications in nano-satellite communication, the all-metallic design can be superior [37], [38] since the absence of dielectric layers avoids the dielectric losses, reduces the overall weight, and significantly improves the ability to withstand extreme environmental conditions [39], [40].

Fig. 1 .
Fig. 1.(a) Basic single cell of cloverleaf element.(b) Single cell of the initial, unoptimized element.The geometrical dimensions are r = 5.00, s = 0.4, w = 0.1 (all in [mm]), and the exponential parameter q = 0.2 (in [1/mm]).(c) Simulation configuration.Floquet port excitation Z max in green and Z min in red, and periodic boundary in purple.The total distance between the excitation ports is 6λ 0 (symmetric).

Fig. 2 .
Fig. 2. (a) and (b) Cross section at z = 0 of the electric and magnetic reactive energies distributions (dB[J/m 3 ]) for the initial unit cell, blue dots represent l 1 and l 2 1-D curves.(c) and (d) Cross section of the electric and magnetic reactive energy distributions (dB[J/m 3 ]) in .(e) and (f) Normalized electric and magnetic reactive energy densities (dB) along l 1 and l 2 , respectively, the red horizontal dashed line represents the normalized incident electric and magnetic energy for θ = 0 • .
Fig. 4(a) presents a comparison of U as a function of the incidence angle θ inc between the initial and first optimized ASM unit cells, showing that the optimization balanced between the electric and magnetic reactive energies, minimizing (| U | = [|W e − W m |]) across a broad range of incidence angles.Consequently, the angular stability performance of the optimized unit cell, within the incidence angle range of 0 • -60 • , is significantly improved.Fig. 4(b) shows the frequency response of the first ASM for various incidence angles, showing that the resonance remains at f 0 = ∼13 GHz, even for a broader range of incidence angle 0 • -80 • .The corresponding values of frequency deviation are f dev (60 • ) = 0.26% and f dev (80 • ) = 0.69%.To further increase the angular stability performance, we conduct a second iteration of the optimization scheme, intended to achieve a stable response over the angle range of 0 • -89 • .Here, the optimization objective remains similar to the first iteration, with the only change being the optimization angle set to θ opt = 89 • .The algorithm successfully achieved the objective after seven iterations.Fig.5showcases the performance of our optimized ASM unit cell from a broader point of view.Fig.5(a) presents a comparison of U as a function of θ inc between the initial (unoptimized), the first, and the second optimized unit cells.It can be noted that the optimization substantially balanced the electric and magnetic reactive energies ([W e − W m ]) across the entire incidence angle spectrum.Consequently, the frequency deviation values of the initial, first, and second optimized unit cells, within the incidence angle range of 0 • -89 • , have been significantly improved with the maximum deviation of f dev = 0.69% and f dev = 0.6% for the first and second optimized unit cells, respectively, as shown in Fig.5(b).In panel 5(c), we see a 2-D map of |S 21 | versus the incidence angle and frequency,

Fig. 5 .
Fig. 5. (a) Comparison of U as a function of θ i nc, between the different unit cells discussed ([pJ]).(b) Comparison of the frequency deviation ([%]).(c) |S 21 | map versus the incidence angle and frequency.(d) Imaginary part of the admittance orders in (17) ([S]) for the initial, unoptimized unit cell in Fig. 1(b) (solid) and the optimized unit cell in Fig. 1(a) (dashed).The vertical green dashed line around 14.1 GHz corresponds to the emergence of a grating lobe which is associated with an angle of incidence of 80 • presented in Fig. 4(b).

Fig. 7 .
Fig. 7. (a) and (b) Comparisons of simulated and measured transmission curves |S 21 |(dB) for TM and TE polarization, respectively.(c) and (d) Comparison of simulated and measured frequency deviation ([%]) for TM and TE polarization, respectively.All for the practical ASM configuration.