Metasurface magnetless specular isolator

We present a (nongyrotropic) metasurface magnetless specular isolator. This device reflects as a mirror a wave incident under a specified angle in one direction and absorbs it in the opposite direction. The metasurface is synthesized in terms of bianisotropic susceptibility tensors, whose nonreciprocity resides in normal components and exhibits a hybrid electric, magneto-electric nature. The metaparticle is implemented in the form of a U-shaped conducting structure loaded by a transistor. The operation principle of the specular isolator is demonstrated by both full-wave simulation and experiment, with isolation levels reaching 41 and 38 dB respectively. This system represents the first realization of a metasurface involving nonreciprocal normal susceptibilities and features a previously unreported type of nonreciprocity.


Scientific Reports
| (2022) 12:5652 | https://doi.org/10.1038/s41598-022-09576-0 www.nature.com/scientificreports/ In addition, we assume that the metasurface is nongyrotropic, i.e., that it does not rotate the polarization of the incident wave upon reflection. Moreover, we shall consider only the p-polarized problem with incidence in the xz-plane, whose nonzero electromagnetic field components are E x , E z and H y , the s-polarized problem being solvable in an analog fashion.
Required susceptibility components. Given its deeply subwavelength thickness, a metasurface can be conveniently modeled by the so-called Generalized Sheet Transition Conditions (GSTCs) 42,43 . The GSTCs are a generalization of the conventional boundary conditions via the addition of surface polarization currents. These conditions were originally derived by Idemen 42 , next applied to metasurfaces by Kuester et al. 43 , and finally incorporated in a general metasurface synthesis technique devolopped by Achouri et al. 39 . The GSTCs read where the symbol represents the difference of the fields on both sides of the metasurface, and where P and M are the induced electric and magnetic surface polarization densities, which may be expressed as where the "av" subscript represents the difference of the field on both sides of the metasurface and χ ee , χ em , χ me and χ mm are the bianisotropic surface susceptibility tensors characterizing the metasurface 37,39 . Assuming that the metasurface is placed in the xy-plane at z = 0 , the field differences and averages are where denotes either the electric or magnetic field.
In general, each of the susceptibility tensors in (2) includes 3 × 3 = 9 components, which leads to a total of 4 × 9 = 36 independent parameters. In the case of the proposed specular isolator, these components should be independent on x and y to satisfy the metasurface uniformity condition. Moreover, the required nonreciprocity implies the global condition 6,39 Finally, the nongyrotropy and p-polarized incidence assumptions eliminate 24 out of the 36 susceptibility components, simplifying the bianisotropic susceptibility tensors to  , and the corresponding nonreciprocity condition reads χ xz mm = χ zx mm , χ yx em = −χ xy me or χ yz em = −χ zy me . Note that this polarization, involving different susceptibilities, would necessarily imply a different metaparticle design than the design that will be presented for p-polarization in the next section.
Inserting (2) into (1) with (5) leads the following explicit scalar GSTC equations where ∂ x denotes the spatial derivative versus x. Assuming plane wave incidence, which allows the substitution ∂ x → −jk x , where k x = k sin θ , reduces then (7) to which are the final GSTC equations for the problem at hand. In these relations, the field differences and averages are found from (3) in terms of the reflection and transmission coefficients, R and T. Assuming incidence in the +z direction, theses quantities read where k z = k cos θ . Substituting (9) into (8), and solving for R gives 40 where Realizing the specular isolation operation (see Fig. 1) requires breaking the symmetry of the reflection coefficient with respect to x or, equivalently, with respect to k x . In other words, the reflection coefficient (10) must be a non-even function of k x , i.e., (7a) www.nature.com/scientificreports/ Inspecting (10) reveals that this condition requires which correspond to the first and third relations in (6), respectively. Thus, breaking reciprocity in reflection can be accomplished only via normal susceptibilities (under the prevailing nongyrotropy assumption 26 ). It can be shown that the second relation in (6), involving tangential susceptibilities, breaks reciprocity in the z-direction 40 , which would be useful for transmission-type nonreciprocity.
Metasurface design. Susceptibility derivation. The specular isolator metasurface may be designed in the following three steps: (i) define the fields related to the desired wave transformations; (ii) insert these fields into (3) to determine the appropriate field differences and averages; (iii) insert these last quantities into (8), and solve the resulting equations for the susceptibility components. According to the analysis performed in "Required susceptibility components" section, the susceptibility components obtained by this procedure should automatically respect the condition (12). The field definitions in (i) correspond here to the two field transformations represented in Fig. 1. The first field transformation is the specular reflection of the wave incident in the +z-direction at the operation angle θ 0 (Fig. 1a). The related fields are where φ is the reflection phase induced by the metasurface. The second transformation is the absorption of the wave incident at the angle −θ 0 (Fig. 1b). The related fields are Successively substituting both (13) and (14) into (3), according to (ii), and inserting the resulting expressions into (8), according to (iii), leads a system of 2 × 2 = 4 scalar equations with 9 unknowns. This is an underdetermined system with an infinite number of possible sets of susceptibilities. Since the operation of the metasurface has been completely determined at θ 0 , these sets correspond to different responses at other (unspecified) angles of incidence, and represent therefore degrees of freedom, which may be generally leveraged in the design of the metaparticle. Among these degrees of freedom, the parameters χ xy em and χ yx me correspond to structural asymmetry along the z-direction 40,44 , which would imply considerable complexity in the metaparticle design. Therefore, we heuristically set these parameters to zero ( χ xy em = χ yx me = 0 ). This reduces the number of unknowns to 7, which we shall maintain as degrees of freedom at this point. The resulting system of equations leads to the 2 explicit susceptibility solutions and to the 2 constraint relations between the remaining 5 susceptibilities.
Transistor-loaded metaparticle. The metaparticle structure satisfying the condition (12) (nonreciprocity along the x-direction for p-polarization) and the relations (15) (reflection and absorption at ±θ 0 ) may be devised as follows. Let us start by enforcing the first nonreciprocity condition in (12), namely χ xz ee = χ zx ee . This condition implies the existence of nonreciprocally related electric dipole responses along x and z, which immediately suggests an L-shape conducting structure loaded by a transistor, operating as a unilateral element (e.g., commonsource configuration in the case of a FET), in the xz-plane; this configuration is incidentally consistent with (15a) and (15b). Such a structure implies in particular a χ zz ee response, which generally implies in turn a χ yy mm response according to (15c). The latter corresponds to a y-directed magnetic dipole moment, which prompts us to close the L-shape into a loop in the xz-plane. We shall leave the loop open, as is customarily done for compactness in ring resonators, and we shall terminate the opened ends of the resulting U-shaped loop by T-shaped strips to reduce the size of the metaparticle. All these considerations lead to the metaparticle structure represented in (12) χ yy mm = − 2j cos θ 0 + kχ zz ee sin 2 θ 0 + e jφ k sin θ 0 (χ zx ee cos θ 0 + χ zz ee sin θ 0 ) (1 + e jφ )k , www.nature.com/scientificreports/ Fig. 2, which is composed of conducting strips in the three directions of space, with the spacing between the two xy-plane metallization planes being much smaller than the wavelength ( v ≪ l < , figure not to scale). We shall next analyze this metaparticle in details to verify that it indeed satisfies all the required conditions and to fully characterize it. Figure 2 decomposes the excitations (incident fields) and responses (dipole moments) in order to determine how the metaparticle realizes the sought after nonreciprocal susceptibility components, although all of the excitations and responses naturally occur simultaneously. Using this approach, we shall next examine the polarizability responses of the isolated metaparticle, which are directly related to the susceptibilities of the metasurface formed by its periodic repetition 39 . Figure 2a depicts the operation principle of the metaparticle realizing a response of the type χ xz ee = χ zx ee . On the left, the x-directed incident electric field induces in-phase ( v ≪ ) currents in the two x-directed strips. When they reach the z-directed strip, these currents cancel out, which implies that the electric response along z to the electric excitation along x is zero ( p zx ee = 0 → χ zx ee = 0 ). On the right, the z-directed incident electric field induces a current in the z-directed strip. This induces a current only in one of the x-directed strips, given the orientation of the transistor, which implies that the electric response along x to the electric field along z is nonzero ( p xz ee = 0 → χ xz ee = 0 ). Thus, the metaparticle corresponds to a specific set of solutions of (15c) and (15d) that is characterized by χ zx ee = 0 , which simplifies these constraint equations to x y (a) www.nature.com/scientificreports/ and Equation (16b) reveals that the metaparticle must also satisfy the second nonreciprocity condition in (12). Let us see whether this is indeed the case with the help of Fig. 2b. On the left, the y-directed incident magnetic field induces a current in the metaparticle loop. The current flowing in the z-directed strip implies an electric response along z due to the magnetic excitation along y ( p zy em → χ zy em ). On the right, the z-directed incident electric field induces a current in z-directed strip. This can induce a current only in one of the two x-directed strips given the to the orientation of the transistor, which produces only a weak magnetic loop along y ( p yz me → χ yz me ). This implies that χ zy em = −χ yz me , which is consistent with the requirement of (16b). We have thus found that the metasurface constituted of the heuristic metaparticle shown in Fig. 2 breaks reciprocity in two distinct fashions, through χ zx ee = χ xz ee and χ yz me = χ zy em . These two types of nonreciprocity represent, both in isolation and in combination, novel metasurface nonreciprocal responses. Moreover, these responses, involving normal susceptibility components, were deemed particularly difficult to realize in practice 40 . The asymmetry of the electric susceptibility tensor, χ ee = χ T ee , also appears in magnetized plasmas, but conjunctly with gyrotropy, while the non-antisymmetry between the magneto-electric susceptibility tensors, χ em = −χ T me , also appears in the transmissive nonreciprocal metasurface in 23 , but in terms of tangential nonreciprocal components. Figure 3 shows the metasurface unit cell of our experimental prototype. This unit cell corresponds to the metaparticle in Fig. 2, except for the additional supporting substrate, backing ground plane and conducting front frame (connected to the ground plane). The ground plane ensures impenetrability of the structure for all angles of incidence (extra specification) and the front frame both isolates the unit cells from each other (hence ensuring direct correspondence between the polarizabilities and the susceptibilities) and provides DC return to the ground for the transistor. The parameters of the unit cell were optimized to satisfy (15a) and (15b) and one of the possible solutions of (16). Fig. 3 was simulated with periodic boundary conditions using a full-wave electromagnetic simulator (CST Microwave Studio) and the unilateral transistor circuit was modelled as an ideal isolator with a phase shifter. An FR4 slab with ǫ r = 4.5 was used as the substrate and the geometrical parameters of the metasurface were optimized to realize the specular isolation operation. The metasurface was designed to provide specular isolation between the angles ±18 • at the frequency of f sim 0 = 6.56 GHz for p-polarization. Figure 4 presents the simulation results. Figure 4a shows the isolation response versus frequency, with the isolation R(−18 • )/R(+18 • ) (see Fig. 1) reaching 41.75 dB at f sim 0 . Note that the asymmetry in the curve of the isolation versus frequency is not due to a Fano resonance, despite its very similar asymmetric appearance. It rather results from taking the (mathematical) ratio of the two frequency responses of the structure for incidences at θ i = ±18 • , which are themselves symmetric but different due to the nonreciprocal nature of the structure (see supplementary material). Figure 4b shows the angular response of the reflection coefficient at the operating frequency of f sim 0 , whose strong asymmetry with respect to broadside ( θ i = 0 ) is the expected signature of the device. Figure 5 presents the experimental results. Figure 5a shows the prototype, composed of 2 × 3 unit cells. It includes two FR4 substrates of thickness 1.6 mm glued together. The device was measured by a bistatic measurement system with two horn antennas symmetrically aiming (under the same angle with respect to the normal   Figure 5b shows the measured isolation ( |S 12 |/|S 21 | ) versus frequency for the incidence angle of θ i = ±20 • . An isolation of more than 38 dB is observed at the frequency of f exp 0 = 6.797 GHz, whose discrepancy (0.217 GHz, i.e., 3.3% ) may be explained by the small gap between the two substrates that was not taken into account in the simulation and by the difference between the actual transistor circuit response and    www.nature.com/scientificreports/ the ideal isolator model used in simulation. Figure 5c shows the measured angular reflection coefficient at the operating frequency of f exp 0 . Here, the discrepancy translates into an angular difference ( 2 • ).

Discussion
We reported a metasurface magnetless specular isolator. We derived, under the assumption of nongyrotropy, the corresponding bianisotropic susceptibility tensors, which include unusual, normal components, and which represent a novel type of nonreciprocity. We designed a metaparticle realizing these susceptibility tensors under the form of a U-shaped conducting structure loaded by a transistor, and demonstrated the overall metasurface by full-wave simulation and experimental results in the microwave regime. Potential applications of this device include nonreciprocal reflectors, nonreciprocal waveguide walls, nonreciprocal quantum state mediation, advanced analog processing, as well as more sophisticated nonreciprocal electromagnetic wave transformations.