### Microstructural design and working principles

To construct the mechanical Willis meta-layer, an array of thin slits is first cut out of a host plate such that one sensing and two actuating beams are formed in each of the meta-atoms (“scatterers”) as shown in Fig. 2a. One piezoelectric sensor is bonded at the center of the sensing beam and two pairs of piezoelectric actuators are attached on the actuating beams. The actuators are placed symmetrically with respect to the sensor (Fig. 2a). The sensor is used to detect the incident wave by measuring the local curvature on the sensing beam, and actuators are employed to generate desired symmetric and antisymmetric scattered fields. The sensor and actuators are connected to a digital control system. The two actuators in the left share the same output voltage *V*_{a1}, and the two actuators in the right share the same output voltage *V*_{a2} (Fig. 2a). It should be mentioned that, once actuating, the sensing signal contains information of incident waves and feedback responses produced by the actuators. To eliminate those feedback sensing components due to actuating, a transfer function, *G*, defined as feedback responses of the mechanical system is implemented to the actuating units (See Supplementary Note 1 for detailed descriptions of *G*). The output from *G* is subtract from the sensing voltage originating from the sensor to constitute a new sensing signal, *V*_{s}, representing the incident signal and as the input to the controller (Fig. 2a), creating a feedforward control loop. This is the key to system stability. The circuit design is detailed in Supplementary Note 2.

Transfer functions, *H*_{1} = *V*_{a1}/*V*_{s} and *H*_{2} = *V*_{a2}/*V*_{s}, are encoded in the controller and control relations between sensing and actuating signals of the active Willis meta-layer. Specifically, the symmetric part of actuating voltages *V*_{+} = (*V*_{a1} + *V*_{a2})/2 produces an effective bending curvature that generates the symmetric wave mode as illustrated in Fig. 2b. On the other hand, the antisymmetric part of actuating voltages *V*_{−} = (*V*_{a1} − *V*_{a2})/2 induces an effective shear strain that produces the antisymmetric wave mode (Fig. 2b). Therefore, combining the symmetric and antisymmetric parts of transfer functions provides a simple and efficient way to independently tailor transmission and reflection coefficients of flexural waves with the incidence from one side of the Willis meta-layer. In other words, transfer functions *H*_{1} and *H*_{2} can be determined for any desired transmission and reflection coefficients for the incidence from the left side, *T*_{l} and *R*_{l}, as (see Supplementary Note 3 for details)

$$H_1 = frac{{(T_l – 1){rm{e}}^{{rm{i}}ka} – R_l{rm{e}}^{ – {rm{i}}ka}}}{{kappa _skappa _a{rm{e}}^{2{rm{i}}ka} – kappa _skappa _a{rm{e}}^{ – 2{rm{i}}ka}}}, \ H_2 = frac{{(T_l – 1){rm{e}}^{ – {rm{i}}ka} – R_l{rm{e}}^{{rm{i}}ka}}}{{kappa _skappa _a{rm{e}}^{ – 2{rm{i}}ka} – kappa _skappa _a{rm{e}}^{2{rm{i}}ka}}},$$

(1)

where *a*, *k*, *κ*_{s}, and *κ*_{a} denote the horizontal distance between the sensor and the actuator, wavenumber of flexural waves in the beam, electromechanical coupling coefficients of the sensor and actuator, respectively. Note that passive scattering due to inhomogeneities in materials and geometries are ignored as they are of significantly lower amplitude than the actively scattered fields, due to the deep subwavelength geometry of sensors and actuators (see Supplementary Note 4 for details).

To examine the performance of the design on independent control of transmitted and reflected fields, numerical simulations of flexural wave interaction with a Willis meta-layer atom are conducted. Figure 2c–e shows the design of a transmission-type meta-layer (*R*_{l} = 0) with the amplitude of the transmission coefficient, *T*_{l} varied from 0 to 2 (horizontal axis) and with a full 2π phase change coverage (vertical axis). Figure 2c, d shows amplitudes and phases of electrical transfer functions, *H*_{1} and *H*_{2}, required to achieve this control (see Supplementary Note 3 for details). It is found that *H*_{1} and *H*_{2} possess the same amplitudes and π/2 phase differences. To quantitatively measure the success of the design, Fig. 2e shows differences between the numerically determined transmission, *T*_{l}^{(num)} and reflection, *R*_{l}^{(num)} coefficients and their desired values, *T*_{l} and *R*_{l}, that the meta-layer is designed to achieve (|*T*_{l}^{(num)} − *T*_{l} | and |*R*_{l}^{(num)} − *R*_{l} | ). It can be found that the Willis meta-layer can accurately reach the desired transmission and reflection coefficients with small absolute errors, and the largest errors are found at points of high transmission. Note that the meta-layer also enables manipulation of the reflected field, and changing the functions of the Willis meta-layer, i.e. from the transmission-type to reflection-type, only requires reprogramming electrical controllers.

Furthermore, we also find that the transmission and reflection coefficients for the incidence from the right side, *T*_{r} and *R*_{r}, are different from those with the incidence from the left side with the transfer functions given in Eq. (1) and their relations can be expressed as (see Supplementary Note 3 for details)

$$begin{array}{l}T_r = 1 + R_l,\ T_l = 1 + R_r.end{array}$$

(2)

The Willis meta-layer is therefore intrinsically nonreciprocal.

### Willis meta-layer on a beam

The unique properties of the Willis meta-layer can circumvent fundamental constraints of extreme flexural wave control with conventional approaches. For example, designing deep subwavelength structures for broadband one-way total absorption of flexural waves in the transmitted domain has been extremely challenging or even impossible, since high material damping properties and impedance match conditions are required at the same time^{36,37}. However, with the current approach, total absorption can be implemented through properly programming electrical transfer functions to induce *T*_{l} = *R*_{l} = 0 (or *T*_{r} = *R*_{r} = 0 by taking a mirror operation). To demonstrate this phenomenon, experiments on a beam with the Willis meta-layer are conducted. Figure 3a shows the fabricated sample and experimental setup of the test (see Supplementary Note 5 for details). The Willis meta-layer is placed in the middle of a host steel beam and two piezoelectric actuators are attached to the left and right sides of the meta-layer to generate incident flexural waves from each direction. A ten-cycle tone-burst signal with a central frequency of 10 kHz is used for this measurement. The out-of-plane velocity wave field is measured by a scanning laser Doppler vibrometer (Polytec PSV-400). Figure 3b, c shows experimentally measured wave fields with incidences from both sides, when the control circuit is switched ON and OFF, respectively. Comparing left portions of these figures where the incidence is from the left side, nearly perfect absorption of flexural waves is clearly seen, when the circuit is switched ON to induce the desired function of the Willis meta-layer. Whereas the wave can pass through the meta-layer, when the circuit is switched OFF. Passive wave scattering due to the presence of the meta-layer is small enough to be ignored. To examine nonreciprocity, we change the incident source from the left to the right side of the meta-layer (Fig. 3b, c). As can be observed from these figures, the transmitted wave fields are almost same for the cases with the circuit switched ON and OFF, indicating that the Willis meta-layer displays unitary transmission for incidence from right. In addition to the unitary transmission, we also notice that the reflected wave shown in the bottom of Fig. 3c has the same amplitude as that of the incident wave, such that the Willis meta-layer functions as a perfect transparent mirror for incidence from the right. This extreme nonreciprocal behavior is easily found using Eq. (2), where zero transmission and reflection for one side of incidence will induce unitary transmission and reflection for the other side incidence. Corresponding 3D piezoelectric-coupled numerical simulations are also performed to validate the phenomena, and good agreement is observed as shown in Fig. 3d.

Importantly, the programmable Willis meta-layer is not limited to narrowband operation, rather, it is a reconfigurable system to tailor the desired reflection, transmission, and absorption response for broadband applications. To achieve this, we first study the frequency responses of electromechanical coupling coefficients of the sensor and actuators by performing numerical and experimental tests and transfer functions are then formulated based on Eq. (1). Note that these transfer functions are derived from frequency-domain analyses, and therefore causality may not be guaranteed in time domain. The Supplementary Note 3 provides details of the causal transfer functions implemented in experiments reported here. As an illustration, we reconfigure the controller to achieve the one-way perfect absorber from 9 to 18 kHz. Figure 3e shows the transmittance, reflectance and absorption of the Willis meta-layer for incidence from left and right sides for both numerical simulations and experiments. It can be clearly seen that for incidence from the left side, both the transmittance and reflectance are almost zero and absorption is around one in the designed frequency band. The Willis meta-layer behaves as a perfect absorber for this case. Whereas, when incidence is from the right, both the transmittance and reflectance approaches unity and absorption approaches negative one in the designed frequency band. For this direction of incidence, the Willis meta-layer behaves as a perfect transparent mirror. Furthermore, it is important to mention that functionalities provided by the programmable Willis meta-layer can be tuned in real time by coding the controller, such that switching between perfect absorption, total transmission and perfect transparent mirror can be executed in seconds and even adjusted due to changing conditions. The reconfigurable broadband response and tunability are both significant improvements over what are possible using passive Wills elements.

### Polarizability tensors for flexural waves

To understand the behavior observed in the experiments described above and facilitate interpretation of the meta-layer as a Willis scatterer, we develop an analytical model based on polarizability tensors. The polarizability tensors describe general scattering properties of the Willis meta-layer in the beam, which is considered as a point scatterer for flexural waves. Analogous to previous works in electromagnetism and acoustics^{22}, we formulate the polarizability tensor, **β**, for flexural waves as

$${tilde{mathbf{Q}}} = {mathbf{beta F}}_{{mathbf{loc}}},$$

(3)

where ({mathbf{F}}_{{mathbf{loc}}} = left[ {begin{array}{*{20}{c}} {psi _{{rm{loc}}}} & {w_{{rm{loc}}}} & {F_{{rm{loc}}}} & {M_{{rm{loc}}}} end{array}} right]^{mathrm{T}}) is the local wave field vector at the scatterer location, with (psi _{{rm{loc}}}), (w_{{rm{loc}}}), (F_{{rm{loc}}}), and (M_{{rm{loc}}}) denoting local rotational angle, transverse displacement, shear force and bending moment, respectively. The vector ({tilde{mathbf{Q}}} = left[ {begin{array}{*{20}{c}} {tilde q_0} & {tilde f_0} & {tilde s_0} & {tilde p_0} end{array}} right]^{mathrm{T}}) is the multipole vector, representing the “excited” (scattered) field caused by the interaction of the point scatterer with the local field, with (tilde q_0), (tilde f_0), (tilde s_0), and (tilde p_0) representing the body torque, transverse body force, shear strain and bending curvature, respectively. The Supplementary Note 6 provides a discussion and visualization of the scattered fields associated with the multipole vector. In particular, (tilde f_0) is a monopole quantity of order zero in the multipole expansion and (tilde q_0) and (tilde s_0) are dipole quantities of order one. Similarly, (psi _{{rm{loc}}}), (w_{{rm{loc}}}) and (F_{{rm{loc}}}) are dipole, monopole, and dipole quantities, respectively. Whereas, (tilde p_0) and (M_{{rm{loc}}}) are longitudinal quadrupole quantities. In a beam, (w_{{rm{loc}}}), (M_{{rm{loc}}}), (tilde f_0) and (tilde varphi _0) are related to symmetric modes, where outward propagating waves are in-phase when traveling in opposing directions, and (theta _{{rm{loc}}}), (F_{{rm{loc}}}), (tilde m_0) and (tilde psi _0) are related to antisymmetric modes, where outward propagating waves are out-of-phase when traveling in opposing directions. It is also important to mention that the scattering properties of the Willis scatterer presented here cannot be properly captured using the simple Euler beam theory because it lacks the degrees of freedom associated shearing motion within the beam. We therefore employ Timoshenko beam theory.

To obtain all the 16 polarizability coefficients of the Willis meta-layer in the 4×4 matrix, **β**, we have formulated the polarizability-retrieval method with Timoshenko beam assumptions by considering both propagated (far-field) and evanescent (near-field) wave solutions (see Supplementary Note 7 for details). Figure 4a shows the amplitudes of those retrieved normalized complex polarizabilities of the Willis meta-layer demonstrated in Fig. 3d. We find (left| {beta ^{prime}_{34}} right|) and (left| {beta ^{prime}_{44}} right|) are close to 2, and other polarizabilities are near zero, making the polarizability tensor asymmetric (see Supplementary Note 7 for definitions of (beta ^{prime}_{34}) and (beta ^{prime}_{44})). In particular, (beta ^{prime}_{34}) describes the coupling between the quadrupole bending moment and the dipole shear strain, and (beta ^{prime}_{44}) represents the coupling between the quadrupole bending moment and the quadrupole bending curvature. Thus, (beta ^{prime}_{34}) and (beta ^{prime}_{44}) can be termed as quadrupole-to-dipole and quadrupole-to-quadrupole polarizabilities, respectively. The retrieved polarizabilities are also validated by numerical simulations where an effective scatterer characterized by those retrieved polarizabilities is implemented into a beam (Fig. 4b). It is clearly seen that the wave fields calculated are almost the same with those in Fig. 3c, d where the meta-layer is physically implemented, demonstrating validity of the polarizability model. Figure 4c, d show amplitudes and phase angles of the two nonzero normalized polarizabilities, (beta ^{prime}_{34}) and (beta ^{prime}_{44}), of the Willis meta-layer demonstrated in Fig. 3e for the frequencies from 9 to 18 kHz, respectively. It is found that amplitudes of (beta ^{prime}_{34}) and (beta ^{prime}_{44}) are around 2, and phase angles of (beta ^{prime}_{34}) and (beta ^{prime}_{44}) are near 0 and −π/2, respectively. Furthermore, it should be noted that Willis coupling reported in acoustics and elastodynamics generally only considers monopole-dipole coupling. However, the Willis coupling in the meta-layer presented in this work exploits coupling between higher order terms, specifically quadrupole-to-dipole and quadrupole-to-quadrupole couplings.

To find insights of the two nonzero polarizabilities, we rewrite transmission and reflection coefficients in terms of (beta ^{prime}_{34}) and (beta ^{prime}_{44}) as

$$T_l approx 1 – frac{{beta ^{prime}_{34} + ibeta ^{prime}_{44}}}{4},;R_l approx frac{{beta ^{prime}_{34} – ibeta ^{prime}_{44}}}{4},\ T_r approx 1 – frac{{ – beta ^{prime}_{34} + ibeta ^{prime}_{44}}}{4},;R_r approx – frac{{beta ^{prime}_{34} + ibeta ^{prime}_{44}}}{4}.$$

(4)

Inserting values of (beta ^{prime}_{34}) and (beta ^{prime}_{44}) from Fig. 4c, d into Eq. (4), the derived transmission and reflection coefficients coincide with the wave phenomenon in Fig. 3e. In addition, we notice that, in the presence of (beta ^{prime}_{34}), reciprocity is broken since *T*_{l} is no longer equal to *T*_{r}. This is because the asymmetry of the polarizability tensor for the active meta-layer only permits symmetric-to-antisymmetric scattering and not the converse, i.e. (beta ^{prime}_{34} , ne , 0) but (beta ^{prime}_{43} = 0). We also note that the local bending moments are identical no matter the direction of incidence, which results in the same induced shear strain, (tilde s_0 = beta _{34}M_{loc}). This shear strain excites antisymmetric modes propagating to left and right, such that the final wave interference patterns are different for incidences from the left or right sides. Furthermore, for a given direction of incidence, the corresponding transmission and reflection coefficients can be arbitrarily and precisely tailored using Eq. (4) by properly designing polarizabilities, (beta ^{prime}_{34}) and (beta ^{prime}_{44}). One can therefore combine the scattered symmetric and antisymmetric modes excited by the bending curvature and shear strain to produce any wave interference pattern in transmitted and reflected domains. The reflected and transmitted fields generated using this approach can be controlled independently and simultaneously keeping in mind that the transmission and reflection coefficients for the incidence from the other direction will be automatically determined, as indicated by Eqs. (2) and (4). In fact, electrical transfer functions and mechanical polarizabilities are intrinsically related by the simplified expressions (beta _{44} = frac{{chi _qleft( {H_1 + H_2} right)}}{2}) and (beta _{34} = frac{{chi _fleft( {H_1 – H_2} right)}}{2}), where (chi _q) and (chi _f) denote factors describing the electromechanical coupling of symmetric and antisymmetric modes, respectively.

### Willis meta-layer in plates

We next investigate an active Willis meta-layer consisting of 10 unit cells on a steel plate as shown in Fig. 5a to demonstrate the ability to independently control flexural wave fields in the transmitted and reflected domains. The design presented here considers the case where each unit cell is connected to their own unique circuit and is controlled independently. An array of piezoelectric actuators is bonded to the left of the meta-layer to generate incident plane flexural waves and we again use a ten-cycle tone-burst signal with the center frequency of 10 kHz for the experimental demonstration. The unit cells of the meta-layer are first programmed to achieve absorption for incidence from the left (Fig. 5b). The measured out-of-plane velocity fields for this case are shown in Fig. 5c, where near total absorption of flexural waves is clearly observed. Numerical simulations using the same geometric and material parameters show good agreement (Fig. 5d). Note that the performance of the meta-layer is not restricted to plane-wave incidence. For example, excellent absorption performance is observed for cylindrically spreading wave incidence due to point excitation (See Supplementary Fig. 9).

Furthermore, the programmable Willis meta-layer is also able to support wavefront redirection. By programming a linear phase profile for the transmission coefficient, *T*_{l}, along the meta-layer and enforcing *R*_{l} = 0, the transmitted wave can be steered in any desired direction while negating the reflected field (Fig. 5e–g). Similarly, by enforcing *T*_{l} = 0, the reflected wavefront can be controlled by specifying the phase profile applied to the meta-layer to generate a position-dependent reflection coefficient, *R*_{l} (See Supplementary Fig. 10). Beyond the conventional applications such as perfect absorption or arbitrary control of reflected or transmitted wavefronts, the programmable Willis meta-layer can transform both the transmitted and reflected wavefronts at the same time by prescribing phase profiles on *T*_{l} and *R*_{l} simultaneously (Fig. 5h–j). The example shown in the figure is the result of encoding a hyperbolic phase profile in *T*_{l} to focus the transmitted wave, whereas a linear profile is encoded in *R*_{l} to steer the reflected wave to a specified direction (24° from normal). The out-of-plane wave field shown in this figure demonstrates the successful operation of the meta-layer to display independent control of transmissions and reflections.

### Homogenization of a beam with periodic active scatterers—a Willis beam

In addition to the Willis meta-layer studied above, it is also of scientific importance to construct a new 1D Willis medium by using a periodic arrangement of Willis meta-layers (scatterers) on a beam^{27} (see Supplementary Notes 9 and 10 for details). Effective properties of the Willis medium are analytically formulated as a function of local polarizabilities. It is interesting to note that the effective mass density, rotational inertia and shear compliance of the beam with the scatterers presented here are left unchanged, and only one Willis coupling coefficient is nonzero. Applying conservation of translational and rotational momentum, we find that the nonzero coupling coefficient will induce nonreciprocal wave propagation in the periodic Willis beam considered in this work. On the other hand, the effective bending stiffness of the Willis beam is modified by the polarizability, (beta _{44}), which is reciprocal. This description provides a theoretical foundation for the creation of nonreciprocal 1D and 2D Willis media to provide unprecedented control of elastic waves in structural elements like beams and plates by using active scatterers.