Deterministic approach to design passive anomalous-diffraction metasurfaces with nearly 100% efficiency

: Designing perfect anomalous reﬂectors is crucial for achieving many metasurface-based applications, but available design approaches for the cases of extremely large bending angles either require unrealistic gain–loss materials or rely on brute-force optimizations lacking physical guidance. Here, we propose a deterministic approach to design passive metasurfaces that can reﬂect impinging light to arbitrary nonspecular directions with almost 100% efficiencies. With both incident and out-going far-ﬁeld waves given, we can retrieve the surface-impedance proﬁle of the target metadevice by matching boundary conditions with all allowed near-ﬁeld modes added self-consistently and then construct the metadevices deterministically based on passive meta-atoms exhibiting local responses. We

Anomalous deflectors that can perfectly redirect incident light to predesigned nonspecular directions are the simplest type of metasurfaces and are the basis for realizing metadevices with more sophisticated functionalities. Unfortunately, designing perfect anomalous deflectors is still challenging despite of many approaches proposed. In early years, anomalous deflectors are designed to exhibit linearly varying phase profiles based upon Huygens' principle. However, such a design principle becomes less valid as the deflection angle increases, where parasitic scatterings inevitably appear (see Figure 1(a)) [38][39][40]. In 2016, Alu et al. [41] proposed a new strategy to design wave-bending metasurfaces with perfect efficiencies. The key idea is to retrieve the surface impedance of target device by matching boundary conditions on the metasurface with both incident and out-going waves considered (see Figure 1(b)). However, while such a surface-impedance solution is in principle rigorous, metasurfaces thus designed inevitably exhibit certain gain and lossy responses varying in space, being extremely difficult to realize in practice. Later, some modified design approaches were proposed, based upon adding limited auxiliary surface modes in matching the boundary conditions and/or exploiting the nonlocal responses of the meta-structures (Figure 1(c)) [42,43]. However, many of them still need brute-force optimizations to assist finding the desired realistic structures of meta-atoms [44][45][46] or can only be applied to certain restricted cases [47], since the nonlocal response of a certain meta-atom may also affect the designs of its adjacent meta-atoms and thus a self-consistent numerical optimization is needed. A deterministic design approach for realizing perfect-efficiency metadevices with diversified functionalities is highly desired.
In this paper, we establish a deterministic approach to design nearly perfect anomalous-diffraction metasurfaces constructed with passive materials exhibiting local responses. Different from previously proposed approaches, here we purposely add all allowed surface modes into the process of boundary-condition matching, with amplitudes and phases of different surface modes determined self-consistently requiring the device being purely passive. Retrieving the surface-impedance profile of the target device, we then construct the device with meta-atoms exhibiting purely local responses based on the profile without further structural optimizations. To illustrate the powerfulness of our approach, we design and fabricate two microwave metasurfaces and experimentally demonstrate that the first one can redirect a normally incident wave to a 70 • reflection angle with an efficiency 98%, and the second one can generate two reflection beams with desired bending angles (23 • and −50 • ) and power flux allocation (60% and 40%). Consider a generic metasurface placed on the xoy plane (with z = 0), which, as shined by an incident EM wave with inc = (E inc , H inc ), can generate a reflected beam with predesigned EM field tar = (E tar , H tar ). Then the question is: what surface admittance distribution Y(r ‖ ) should the metasurface exhibit?
To determine Y(r ‖ ), we note that tar only contains farfield (FF) information. Suppose that we can find a series of surface modes [SMs, SM ] in the space above the metasurface, which exhibit the following characteristics: (1) they are eigen solutions of Maxwell equations in this space region; (2) they do not radiate to the FF but are bounded on the surface; and (3) they exhibit certain symmetry in accordance with the metasurface and the incident light. Thus, adding these nonradiative SMs into the total field in this space does not affect our final goal of generating the desired FF radiation but can change the near-field (NF) EM environment where the metasurface is located. Therefore, we can write the total field as tot = inc + bar + We now illustrate the design strategy, taking the anomalous-reflection metasurface as a specific example. As shown in Figure 2(a), suppose that the incident and reflected light beams are both transverse-magnetic (TM)polarized plane waves, propagating along i and r directions, respectively. Thus, inc and tar can be explicitly written as It is interesting to note that Y(x) is generally a complex function, which consists of both real and imaginary parts.
According to the Poynting theorem, we find that the real part of surface admittance Re[Y(x)] can be expressed as where P tot denotes the z component of Poynting vector measured on a plane right above the metasurface. Equation (2) indicates that the presence of Re[Y(x)] signifies the imbalance of energy flow at the local x position on the surface, which further dictates that the local constitutional material must be either lossy (with P tot z < 0) or active (with P tot z > 0). Hence, the criterion that our metasurface can be constructed by purely passive and lossless materials is simply at every point on the surface. In general, Equation (3) cannot be satisfied without auxiliary fields. To demonstrate this point, we take only the propagating waves into account and get A r e ik 0 sin r x + e ik 0 sin i x cos r A r e ik 0 sin r x − cos i e ik 0 sin i x ⋅ (4) Equation (4) shows that Y(x) is a complex function as long as r ≠ i , except for the special case of r = − i (i.e., the retro-reflection case) where a purely imaginary Y(x) function can be found without the help of auxiliary fields (see more details in Section I of Supporting Information (SI)). Figure 1(b) plots the Re[Y(x)] distribution of a metasurface for a case of i = 0 • and r = 70 • , calculated with Equation (4). The periodic oscillations in Re[Y(x)] already imply that the local constitutional material constructing the metasurface must be either gain or lossy materials, which is indeed the case predicted by Ref. [41].
We now discuss how to make Equation (3) satisfied with appropriate auxiliary modes added. Assuming that our metasurface is a periodic structure with lattice constant P, we understand that the reflected light beam must be a linear combination of diffracted modes dictated by Bloch's theorem. Here, we set the incident wave as the 0 th -order mode while the reflected one is set as the 1st-order of the whole periodic system. The periodicity and two angles must satisfy the constraint 2 ∕P = k 0 sin r − k 0 sin i . In this case, SMs allowed in the system are all high-order evanescent waves (k (m) Figure 2(b)). Therefore, we get the exact field distribution of each evanescent modes as: w h e r e (m) add into the total field and obtain the total field near the metasurface, thus deriving the admittance distribution: Obviously, varying the values of {C m } can significantly change the form of Y(x), and our task is to find a solution Practically, Equation (3) is difficult to solve since the profile has infinite degrees of freedom. To make the criterion more tractable, we define: (6) to measure the root-mean-square (RMS) value of the imbalanced energy follow within a super cell. Therefore, Equation (3)     at four representative cases with N = 1, 2, 3, 5, respectively. Without auxiliary SMs added (i.e., N = 1), we find P tot z (x) fluctuates over x and its absolute values are large in certain regions, indicating that the metasurface must be constituted by lossy materials (in the region where P tot z (x) < 0) or gain materials (in the regions where P tot z (x) > 0). However, with more auxiliary SMs (with amplitudes {C m } appropriately adjusted) added, we find that fluctuations in P tot z (x) are significantly suppressed. In particular, in the cases of N > 3, we find that P tot z (x) ≈ 0 nearly everywhere, indicating that the metasurface can be constructed by purely passive materials with local responses determined by the corresponding Y(x) functions. The fact that Y(x) is (nearly) purely imaginary is also consistent with our notion that the metasurface can be constructed by local and passive materials. To validate our design approach, we perform full-wave simulations to study the scattering of normally incident light by metasurfaces exhibiting Figure 2(e) plots the H-field distributions of scattered waves on the xoz plane in different cases. Obviously, the working efficiency of the anomalous reflector increases as a function of N and approaches 100% as N ≥ 3. In addition, the wavefront of the reflected beam becomes more flattened as N increases, in consistency with the enhanced anomalous-reflection efficiency. Since the complexity of the corresponding realistic structure is decided by the number of SMs included in theoretical calculation according to the Shannon sampling theorem [48-50], we decide to choose for final practical realization, after balancing the requirements on efficiency and simplicity.
It is interesting to compare our metasurface design scheme with the one based on conventional Huygens' principle scheme [6,7], in which the metasurface under design should exhibit the following surface admittance: at each position x (see detailed derivation in Section II of SI). Comparing Equations (4) and (5) with Equation (7), we find that the admittance profile retrieved in Ref. [41] (Equation (4)) has correctly considered the local-field correction due to the desired radiated FF, and our newly developed formula (Equation (5)) further considers the local-field corrections due to NFs contributed by all possible SMs allowed by the system. With more possible modes added, it is natural to expect that our scheme can yield better results than previous trials.
We now discuss the performance of our design scheme for different { i , r } combinations. Figure 3 depicts how the theoretically calculated RMS varies against i and r with different number of SMs added. In each case, we always find that RMS = 0 in the case of specular reflection ( i = r , dashed line), or retro-reflection ( i = − r , solid line), as expected. Meanwhile, the convergence area (i.e., with RMS < ) in the ir phase diagrams continuously expands as N increases, indicating that adding more SMs can indeed help speed up obtaining convergent solutions for highly mismatched incidence/deflection angles. Practically, we find that setting N = 5 in our scheme is enough to ensure finding appropriate admittance profiles for metasurfaces with arbitrary anomalous-reflection abilities.

Meta-atom designs
We now design a set of meta-atoms that are suitable to construct our metasurfaces with admittance distributions given in Equation (5). Since our scheme has ensured that the final Re[Y(x)] function is negligible, we only consider the Im[Y(x)] function in matching the boundary conditions. In practical designs, it is more convenient to sort out metaatoms from the required reflection properties (including reflection phase and amplitude). Connecting the admittance Im(Y) with reflection coefficient r of a given surface and assuming that the meta-atoms exhibit local responses, we find that the meta-atom located at the position x should exhibit the following reflection coefficient determined by the admittance value required at this very position x. Equation (8) can be further re-written as r(x) = e i (x) with reflection amplitude being 100% and reflection phase given by (x) = −2 arctan[i 0 Y(x)]. Therefore, our task is to find a series of meta-atoms that can perfectly reflect EM waves with different reflection phases and exhibit local responses. We find that the groove structure, where a groove is etched on a metallic plate (see inset to Figure 4(a)), is the suitable meta-atom satisfying all above requirements. The continuous metallic film on the back can ensure total reflection of incident EM wave, while changing the height h of the groove can modify the resonant mode supported by the structure, which in turn, changes the reflection phase drastically [51]. More importantly, compared with the metal-insulator-metal (MIM) meta-atoms (see Figure 4(b)) that can also perfectly reflect EM waves with tailored reflection phases [52], the groove meta-atoms exhibit more localized responses to external illuminations, manifested by much reduced mutual couplings between neighboring meta-atoms. To illustrate this property, we numerically examine how the resonant modes supported by the groove or MIM meta-atoms, both being repeated in xy-planes forming periodic metasurfaces, evolve as a function of the incident angle i (Figure 4(c) and (e)) or the width of the metaatoms w ( m ) for the periodic structure (Figure 4(d) and (f)), respectively. As either i or w ( m ) changes, while the resonant mode is hardly changed in the groove structure case (Figure 4(c) and (e)), the same thing is not true for the MIM case (Figure 4(d) and (f)). Noting that angular dispersion of a metasurface is ultimately determined by the coupling between neighboring meta-atoms [14,53], we reach the conclusion that mutual-coupling issue is weaker in the groove structure case than in the MIM case. These results confirm that the groove meta-atoms exhibit highly localized optical responses under external excitations, which are highly desired for constructing our metasurfaces in a deterministic way. Figure 4(a) depicts the numerically computed spectra of reflection amplitude and phase of a periodic array of a typical meta-atom with = 6 mm and h = 3.05 mm.
Clearly, we find that the reflection amplitude is always 100%, while the reflection phase changes from − to continuously as frequency increases, due to the cavity resonant mode at f = 13.3 GHz supported by the groove.
Fixing the working frequency at 13.3 GHz, we employ FEM simulations to establish the relation between the reflection phase of the meta-atom and the parameter h, which facilitates our designs in the remaining part of this paper (see more details in Section III of SI).

Experimental demonstration of an anomalous deflector with large bending angle and nearly perfect efficiency
We first employ the proposed scheme to design an anomalous reflector that can deflect a normally incident TMpolarized wave to the angel of 70 • with nearly perfect efficiency at the frequency 13.3 GHz. Retrieving the admittance profile Y(x) of the target metadevice with 4 SMs added self-consistently (N = 3), we employ Equation (8) to obtain the required phase distribution (x) of the metasurface (red curve in Figure 5(b)). Calculations on a model system possessing the retrieved Y(x) profile show that the device exhibits a working efficiency = 99.9%. Set the working frequency at 13.3 GHz, we follow the design strategy described in Section 3 to sort out 4 groove meta-atoms with different h based on the discretized reflection phases (see blue curves in Figure 5(b)) and use them to form our metadevice. Figure 5(a) shows the photograph of a fabricated sample containing 20 meta-atoms in total, with a side-view picture shown in the upper panel of Figure 5(b). It is interesting to note that the desired (x) profile is quite different from the gradient phase profile according to Huygens' principle, due to the local-field corrections discussed in Section II of SI. We then experimentally characterize the performance of the fabricated metadevice. In our measurements, we adopt a horn antenna as a source to normally illuminate the fabricated sample and put another one on a circular track with a radius of 1 m to measure the angular distribution of scattered power flux. Both antennas are connected to a vector network analyzer. Figure 5(c) plots the measured scattered power flux as a function of frequency and reflection angle. Here, the reference is defined as the power flux reflected by a flat metallic mirror (with the same size as our metadevice) under the same excitation condition. Clearly, the anomalous-reflection mode takes nearly all scattered power, while parasitic scatterings are significantly suppressed within the whole frequency band (12.5-15.0 GHz). Meanwhile, the measured peak deflection angle r is a decreasing function of frequency f , following the diffraction law r = arcsin( c 0 f p ). We quantitatively estimate the working efficiency of the fabricated metadevice, defined as the ratio between the integrated power carried by the anomalous reflection beam and that by the impinging beam. Insets to Figure 5(c) depict the measured angular distributions of the power flux of scattered beam at two particular frequencies 13.3 GHz and 14.5 GHz, from which we can quantitatively evaluate the working efficiencies based on the power integrations. We depict the evaluated working efficiencies of our metadevice at different frequencies as solid circles in Figure 5(d). We also employ FEM simulations to study the scattering patterns of the metadevice, from which we retrieve the working efficiencies of the device at different frequencies, and depict them as a solid line in Figure 5(d). Experimentally retrieved working efficiencies are generally in excellent agreement with numerically retrieved ones. In particular, the experimental efficiencies of our metadevice are found to exceed 95% within 13.3-14.5 GHz, unambiguously demonstrating the broadband high-efficiency performance of the device. We note that the imperfect working efficiency obtained in our experiment is due to imperfections in our experimental characterizations, inclu-ding the finite-size effect of the fabricated metasurface and the nonideal incident wave. Such issues are generally irrelevant with the design strategy to realize the metasurfaces but rather solvable via improving the experimental conditions.
To illustrate its powerfulness, we further design anomalous reflectors with bending angle changing from 50 • to 80 • under normal incidence and then numerically estimate their working efficiencies. It should be noted that the working frequencies of these metadevices are tied with their target bending angle r as long as we fix the value of super periodicity P. Solid stars in Figure 5(e) are the calculated efficiencies of these metadevices, which are all very close to 100%. In comparison, we also use the groove meta-atoms to construct metadevices exhibiting different deflection angles, designed with the conventional Huygens' principle scheme (see Section IV of SI for details). Numerical evaluations on these realistic structures show that their working efficiencies (black hollow triangles) follow nicely with theoretical prediction (purple solid line) based on Equation (8), deviating quickly from 100% as the target deflection angle increases. This reinforces our notation that metadevices designed with the conventional Huygens' principle scheme inevitably exhibit low working efficiencies for large bending angles, since such a design approach neglects local field corrections. We emphasize that our design scheme can also be adopted to realize meta-deflectors for arbitrary bending angles (e.g., even near the grazing angle) with almost perfect efficiency.
To reveal the underlying physics, we compare the power flux distributions in the space above the metasurfaces designed with different approaches, all under illuminations of the same normally incident EM waves (see Figure 6(a)). We consider three different systems, which are the model metasurfaces with admittance profiles given by the gain & loss scheme and our newly proposed theory, and the realistic metasurface composed by groove metaatoms. All these metasurfaces are designed for 70 • bending angle under normal incidence. Figure 6(b)-(d) compare the distributions of P tot z (x) on three planes with different z above the metasurfaces. We find that the P tot z (x) distribution only exhibits a lateral shift as z varies in the case of gain & loss metasurface (blue lines in Figure 6(b)), caused by the interference between incident wave and anomalously reflected wave, which possesses a phase factor e ik x x . In contrast, P tot z (x) changes significantly as z varies in the case of our metasurface (black lines in Figure 6(b)). Such an obvious difference is induced by the additional SMs added in our scheme, which significantly changes the total wave pattern and thus the final P tot z (x) distribution. As z decreases, auxiliary modes considered in our scheme exhibit stronger fields due to their evanescent natures and thus the modification on P tot z (x) becomes more dramatic. In particular, on the z = 0 plane right above the metasurface, P tot z (x) is nearly 0 everywhere in the case of our metasurface, consistent with the proposed design criterion Equation (3). The evolution of P tot z (x) over z well reveals the crucial role played by the additional SMs added in our design scheme, which help match the boundary conditions on the metasurface yet keep the far fields as desired. Finally, we also compare P tot z (x) in the cases of model metasurface and its corresponding realistic structure. These two P tot z (x) distributions essentially exhibit the same behaviors except for certain fluctuations in the realistic case, due to strong near fields induced around the corners of realistic meta-atoms. Nevertheless, we find that the average of P tot z (x) over a single meta-atom in the realistic-sample case (red line) is close to the value of P tot z (x) in the model case (black line), both being nearly zero, in consistency with the effective medium theory.

Application: metasurface enabling multiple beam generation with predesigned property
Our scheme is so general that it can also be extended to design other metadevices for realizing more complex wavefront controls with nearly 100% efficiencies. As an illustration, we design a metasurface that can reflect normally incident EM wave to two different directions ( r1 = 23 • and r2 = −50 • ) with efficiencies 60% and 40%, respectively, as schematically shown in Figure 7(a). We purposely set the super periodicity as P = 0 ∕ | | sin r1 − sin i| | so that the deflection angles for +1 and −2 diffraction modes are just r1 = 23 • and r2 = −50 • , respectively, according to Bloch's theorem. With the target scattering pattern known, we can thus write out the total EM fields in the region above the metasurface. Specifically, the parallel components of E and H fields are: where the expansion coefficients of two desired scattering , respectively, dictated by the pre-conditions and the law of energy conservation, and {C m } are a set of coefficients of the NF SMs added in our scheme. We then follow the design strategy described in Section 2 to determine the expansion coefficients {C m } to complete our design. Setting N = 8, we obtain a metasurface design exhibiting satisfactory accuracy, with reflection phase profile shown as a red line in Figure 7(b). We then truncate the profile phase into 32 sub cells (blue line in Figure 7(b)) and sort out the corresponding groove metaatoms based on the desired phases of each subcells. We next fabricate out the sample according to the design (see Section III of SI for the detailed parameters), and a sideview picture is shown in the upper inset to Figure 6(b). With the fabricated metasurface at hand, we then experimentally measure its reflection patterns at different frequencies under normal incidence, following the characterization scheme described in last section. Figure 7(c) depicts the measured normalized power flux as the function of reflection angle and frequency, with the inset showing the angle distribution measured at the frequency 12.2 GHz corresponding to the dashed line. From the measured data, we can easily retrieve the power flux taken by two scattered modes and calculate their efficiencies with the reference signal taken as that reflected by a flat metallic surface. We find that at the working frequency 12.2 GHz, these two scattered modes take the power efficiencies 59.7% and 39.7%, respectively, in nice agreement with the predesigned conditions. The sum of these efficiencies is generally close to 100% and reaches 99.4% at 12.2 GHz in particular, indicating that the parasitic scatterings are negligible. We also note that the working efficiencies oscillate with the frequency of input wave, which is due to our frequency-dependent design strategy and the intrinsic dispersion properties of our constitutional meta-atoms. Figure 7(e) compares the normalized scattering patterns of our metadevice under normalincidence illumination at 12.2 GHz, obtained by experiments and FEM simulations. Nice agreement is noted between these two patterns.

Conclusions
To summarize, we established a deterministic approach to design purely passive metasurfaces that can reflect incident light to arbitrary nonspecular directions with nearly perfect efficiencies. With both incident and out-going farfield waves known, we purposely add all allowed surface waves into the process of boundary-condition matching and determine their coefficients under the requirement that the device remains purely passive and local. Retrieving the surface-impedance distribution of the target device, we employ groove meta-atoms to construct realistic metasurfaces according to the impedance profiles. Two metadevices are designed/fabricated and experimentally characterized, one enabling perfect anomalous reflection to a large bending angle and another splitting normally incident beam to two anomalous reflection channels with predesigned efficiencies. Our results open the door to realize high-efficiency wave-control metadevices with diversified functionalities.