Frequency-hopping wave engineering with metasurfaces

Wave phenomena can be artificially engineered by scattering from metasurfaces, which aids in the design of radio-frequency and optical devices for wireless communication, sensing, imaging, wireless power transfer and bio/medical applications. Scattering responses vary with changing frequency; conversely, they remain unchanged at a constant frequency, which has been a long-standing limitation in the design of devices leveraging wave scattering phenomena. Here, we present metasurfaces that can scatter incident waves according to two variables—the frequency and pulse width—in multiple bands. Significantly, these scattering profiles are characterized by how the frequencies are used in different time windows due to transient circuits. In particular, by using more than one frequency with coupled transient circuits, we demonstrate variable scattering profiles in response to unique frequency sequences, which can break a conventional linear frequency concept and markedly increase the available frequency channels in accordance with a factorial number of frequencies used. Our proposed concept, which is analogous to frequency hopping in wireless communication, advances wave engineering in electromagnetics and related fields.


Supplementary Note 1: Frequency selectivity of conventional electromagnetic media
In this Supplementary Note, we explain how metasurfaces are theoretically approximated by equivalent circuit models that are characterized by the frequency and that remain unchanged if the incoming frequency spectra are fixed.The characteristic impedance Z of a metasurface may be approximated by a simple equivalent circuit model.For instance, the conducting geometry of the slit structure shown in Fig. 2a can be represented by a parallel circuit: where ω is the angular frequency associated with frequency f through ω = 2πf, j 2 = -1 and L and C denote the entire inductive and capacitive components of the metasurface, respectively.
Assuming a two-port network that includes the above structure, the reflection coefficient S11 and transmission coefficient S21 are obtained by 21 () = () •  0 () +  0 , where Z0 represents the characteristic impedance of the medium next to the metasurface (vacuum in this study).Note that in Eqs.(1) to (3), the impedance of the dielectric substrate behind the conducting pattern is omitted for simplicity, although it can be represented by an additional transmission line.Moreover, the reflected energy RPLS and transmitted energy TPLS of the incident pulse are derived from 1 where b + and b -are the magnitudes of the incident and reflected waves, respectively.Therefore, Eqs. ( 4) to (5) indicate that RPLS and TPLS remain the same if the frequency spectrum of the incident pulse is the same.Moreover, this study used pulsed sine waves, the electric field intensity E of which followed the function below: 2 () =  2 � ( −  0 )  0.5( −  0 ) + ( +  0 )  0.5( +  0 ) � , (6 where ω0 is the oscillating angular frequency and a and TPW are the magnitude and pulse width of the incident pulse, respectively.In this study, the pulse width is set to 50 ns or longer in the range of a few GHz, which ensures that the pulse spectrum is almost the same as the oscillating frequency. 3Hence, it is also clear from Eqs. ( 4) to (6) that under these circumstances, the scattering response essentially remains the same if the incoming frequency spectra remain unchanged.

Supplementary Note 2: Frequency conversion by full wave rectification and time constants
This Supplementary Note explains how the diode bridge and the internal circuit elements of a unit cell of a waveform-selective metasurface can effectively be approximated by the DC circuit at the bottom of Fig. 2b. 4 First, due to the diode bridge, the waveform of an incident sine wave is converted to the waveform based on a modulus of the sine function.The frequency spectra of the rectified waveform can readily be obtained by using Fourier series expansion: where T and n denote the period of the incident sine wave (i.e., T = 1/f) and the natural number, respectively.Suppose that the rectified waveform is a modulus of a sine function, namely, an and bn are rearranged as follows: According to Eqs. ( 11) and ( 12), the energy of the incident frequency f is converted to those of other frequencies, as seen in Table S1.This table indicates that a large portion of the energy appears at zero frequency, which enables us to exploit the transient phenomena well known in DC circuits even if an alternating current (AC) signal comes in.
Next, as seen above, the internal circuit configuration of a unit cell of the waveform-selective metasurface is found to effectively respond to a DC signal.Under this assumption, it is well known that the transient voltage across the inductor  L can be readily estimated by  L =  0  −/ L , where  0 and  are the DC voltage applied and time, respectively, while the time constant  L is determined by the circuit components used 76 In Eq. ( 13),  and  L are the inductance and the resistance inside the diode bridges, respectively. d represents the resistive component of the diodes used.Note that  d plays an important role in determining the time constant since  L becomes totally different without the use of  d . 4Eq.( 13) indicates that the waveform-selective transient responses can be readily tailored by changing the circuit values.
metasurface is independent of the incoming waveform with a fixed frequency if the incident power is small enough (e.g., -20 dBm), as the diodes used are not turned on yet.However, by sufficiently increasing the input power to 10 dBm, the difference between the transmittance for a short pulse and that for a CW is maximized.Additionally, Figure S3 shows that with a further increment in the input power, the transmittance for a short pulse starts decreasing because the voltage across the diodes becomes larger than the breakdown voltage.Therefore, it is important to properly design both the turn-on voltage and the breakdown voltage of diodes.
Figure S1: Supercell model used for Fig. 2. The design parameters and circuit parameters are shown in Table S2, Table S3 and Table S4.The substrate is Rogers3003.Table S3 and Table S4).However, part of the circuit values is changed, as seen in Table S5.
The frequency-domain profile of the simulation model in Fig. 3a     Regarding the recovery time to restore the inductor voltage, more information is provided in Figure S6.As seen in Figure S6a, we consider only one unit cell with periodic boundaries to readily evaluate the relationship between the recovery time and the circuit values used within the diode bridge.With an increase in the inductance from 10 μH to 10 mH, the transient transmittance for 100-ns pulses is enhanced near 4.9 GHz (Figure S6b and Figure S6c).This is because when a large inductance value is applied, the electromotive force of the inductor is maintained for short pulses, which leads to the intrinsic resonant mechanism of the slit structure being maintained.However, the inductor voltage is found to be slowly restored to zero voltage if a large inductance is used (see Figure S6d for the result at 4.9 GHz).
Therefore, there is a trade-off between the recovery time and transient transmittance level in terms of the design of the inductance value.Moreover, the frequency-domain transmittance profile is also related to the resistor deployed within the diode bridge (Figure S6e).
Specifically, the transient transmittance decreases with a reduction in the resistance, as the related time constant is also increased (see Figure S6f and Supplementary Note 2).In the time domain, the smaller the resistance is, the longer the recovery time becomes (Figure S6g).
Importantly, however, increasing the resistance value reduces the difference between the transient transmittance during the initial period and that in the steady state (Figure S6f); thus, the waveform-selective characteristics disappear.Therefore, there is another trade-off between the transient transmittance level and waveform-selective performance in terms of the design of the resistance value.Furthermore, to see how the recovery time and the transmittance are related to the pulse period, additional simulations are performed in Figure S6h.Here, the pulse width is fixed at 100 ns, while the duty cycle is set to either 0.25, 0.5 or 0.75 (i.e., the pulse period of 400 ns, 200 ns or 150 ns).As a result, the transient transmittance is maximized when the duty cycle is 0.25 (or the pulse period of 400 ns).This is because the inductor voltage requires a recovery time of about 300 ns to approach its original voltage value as seen in Figure S6i (see that the voltage becomes zero in approximately 300 ns after the pulse duration of 100 ns).Therefore, when the pulse period is smaller than 400 ns, the metasurface's transient transmittance cannot be maximized since the inductor voltage is not fully restored yet.For the multi-resonance case, as seen in Fig. 3d, the transient transmittance is enhanced by increasing the number of incident frequencies since it extends the pulse period and the recovery time to restore the inductor voltage.Based on these relationships, one may need to optimize the circuit values used for waveform-selective metasurfaces.In contrast to Fig. 3, Fig. 4 shows transient transmittance varying in response to a particular frequency sequence.Such selectivity is not obtained by the simulation model of Fig.

S16
in Figure S8, where the frequency sequence of an incoming wave is changed.Figure S8 shows only a minor difference among different frequency sequences, which results from small coupling between unit cells.Note that this variation is much smaller than the one seen in Fig. 4d.Additionally, we numerically tested the quadband waveform-selective metasurface of Fig. 3a in free space to show that the use of our metasurfaces is not limited to one-dimensional systems.In this simulation we used a pair of standard horn antennas (Schwarzbeck BBHA 9120 D) as a transmitter and a receiver, as shown in Figure S9a and Figure S9b.The metasurface tested was composed of 6 × 6 unit cells (i.e., 3 × 3 supercells) and surrounded by a PEC wall to ensure that the incident wave transmitted through the metasurface.Other design parameters are given in Table S6.Under this circumstance, the metasurface continued to exhibit enhanced transient transmittance if the incident frequency was regularly switched, as seen in Figure S9c.This simulation result supports that the concept of our metasurfaces can be exploited in free space as well, which is important to find applications in, for instance, wireless power transfer and communications.
Figure S9: Simulation of the metasurface of Fig. 3a using the free-space configuration.(a) A pair of two horn antennas are used as a transmitter and a receiver in the electromagnetic simulation.The metasurface is deployed in a metallic conductor plate to narrow the metasurface area (lower the computational resources needed) while preventing the direction propagation of the signal between the transmitter and the receiver.The design parameters are shown in Table S6.(b) Circuit schematic integrating the electromagnetic simulation result.Two terminals are connected to the transmitter and receiver ports, and the remaining terminals are connected to diode bridges including inductors and resistors with parallel capacitors.(c) Simulation results of the cosimulation method.The top panel shows the transmitted power at the receiver when the transmitter generates a signal of 20 dBm.The bottom panel shows a transient transmittance that is obtained after normalization to the transmitted power when the metasurface is absent.Comparison to the waveguide simulation (the blue line and the same as Fig. 3d) shows a good agreement between both configurations, indicating the possibility of extending the proposed metasurface to free-space application scenarios.In addition, another sample is introduced in Figure S12, where the supercells are composed of three types of cells.These cells are designed to operate at different frequencies by introducing  S9 and Table S10.The substrate is Rogers3003.In the actual measurement sample, only inductors are used inside diode bridges, as the inductors contain resistive components equal to RL.   S10f.In Figure S10g and Figure S10h, the incident frequency was regularly changed to obtain transient transmittance similar to Fig. 3d, in which the utilization of a lower Q factor is more favourable for a higher transient transmittance.Therefore, the transient transmittance can potentially be improved by minimizing the lossy elements of the metasurfaces and lowering the resonance's Q factor.
To validate this circuit analysis results, we have conducted an experimental investigation employing a metasurface composed of metallic bars interconnected through lumped LC resonant circuits.The schematic is shown from Figure S14a to Figure S14c.This adjustment from the original metallic slit geometry to an LC interconnection scheme allows for convenient control of the Q factor of the resonant structure to achieve higher transmittance.
To test the Q factor variation, the meta-atom periodicity in the x axis is altered as seen from Figure S14a to Figure S14b.Within the rectangular waveguide, the metasurface is configured to have either three metallic rows (Figure S14d) or four metallic rows (Figure S14e).The geometrical parameters for the meta-atom as well as circuit parameters are detailed in Table S12 and Table S13.In the measurement results, we observed a reduction in the Q factor as depicted in Figure S14f (see the broadened operating bandwidths).Consequently, the lowered Q factor resulted in increasing transient transmittance, as shown from Figure S14g to Figure S14h, where the transmittance increased from 0.09 in the slit structure to 0.36 for the threemetallic-row structure, which indicates that the performance was four times greater than that of the previous slit structure design.The slit structure measurement results are detailed in Figure S12, and a similar dual-band equivalent measurement sample having frequency combination dependency is presented in Fig. 3d.Note that the average transmittance of this structure was also approximately 0.09 only.
Importantly, note that the time-varying and frequency-hopping mechanism of our structures comes from "transients in electric circuits", namely, DC circuit mechanisms widely known to require not only imaginary part of impedance (reactance) but also the real part (resistance) potentially leading to energy loss.This fact is clarified in Eq. ( 13) of Supplementary Note 2, where time constant is no longer obtained if resistance is not used.Nonetheless, we showed that our new design successfully enhanced the transmittance performance by properly designing Q factors and related circuit values.S12 and Table S13.(f) Frequency-domain profiles of the transmittance for 50-ns pulses.The result is compared to the slit configuration of Figure S12c.(g) Measured transient transmittance for switched-frequency cases.Here, the three frequencies used, i.e., f1, f2 and f3, are changed in each structure to maximize transient transmittance in three different frequency bands.Specifically, for the 3-metallic-row case, f1, f2 and f3 are 2.69, 3.12 and 3.82 GHz, respectively, while for the 4-metallic-row case, f1, f2 and f3 are 2.83, 3.31 and 3.91 GHz, respectively.(h) Average transmittance for entire pulse period.In these measurements, the input power is set to 10 dBm.S14, Table S15

S34
The design parameters and circuit values used for Fig. 4c are provided in Figure S16, Table S17, Table S18 and Table S19.The power dependence of the frequency-domain and timedomain profiles of the sample can be seen in Figure S17.As shown in this figure, the measurement sample is designed to operate near 2.5, 3.3 and 3.9 GHz and that its optimum power level is approximately 20 dBm.In addition, Figure S18 shows how the time-domain profiles change by adjusting the oscillation frequencies.Moreover, Figure S19 provides additional information on the transmittance shown on the right of Fig. 4d.S17, Table S18 and Table S19.The substrate is Rogers3003.

S38
As a design strategy for the preferable sequence of frequencies, one may change connection lines among unit cells.As depicted in Figure S20a, even with the same unit cell alignment including three different operating frequencies, connection lines can be arranged to determine the frequency sequence that maximizes the transmittance.In this instance, by altering the physical connection between unit cells, the preferable sequence is changed from f1f2f3 to f1f3f2.
However, some conducting geometry designs may show nonnegligible coupling between unit cells, which lowers the transmittance performance.In this case, alternatively the capacitance values can be changed as shown in Figure S20b, where the capacitance values used for two unit cells are swapped to change the preferable frequency sequence from f1f2f3 to f1f3f2.

Figure S2 :
Figure S2: Transmittances of conventional single-band waveform-selective metasurfaces.C1 and C2 are set to the same values, as opposed to Fig. 2c, where the additional capacitances are different to achieve a dual-band operation.
is shown in Figure S4.As seen in this figure, the simulation model used in Fig. 3a exhibits four distinctive transmittance peaks at 2.0, 2.5, 3.3 and 4.3 GHz.However, to maximize the transient transmittance for an incoming signal whose oscillation frequency changes regularly, it may be necessary to further optimize the frequencies used.For instance, in Figure S5, 4.3 GHz (i.e., one of the four frequencies maximizing the frequency-domain transmittance) is changed between 4.1 and 4.4 GHz.The time-domain transient transmittance increases more at 4.1 GHz than at 4.3 GHz.This presumably occurs because with switching between four frequencies (2.0, 2.5, 3.3 and 4.3 GHz), which possibly leads to coupling between different cells, the voltage across diodes changes in the time domain compared to that in the frequency-domain result.Therefore, the related resistive component within the diodes is influenced, which varies the time constant and the recovery time of the entire circuit.

Figure S4 :
Figure S4: Frequency-domain profile of the simulation model of Fig. 3a.The pulse width and input power are set to 100 ns and 10 dBm, respectively.

Figure S5 :
Figure S5: Simulated transient transmittance of Fig. 3a using various frequencies for f4.Compared to Fig. 3d, the fourth frequency used is changed between 4.1 GHz and 4.4 GHz.Note that in Figure S4, the fourth transmittance peak does not appear at 4.1 GHz but at 4.3 GHz.

Figure S6 :
Figure S6: Relationship between circuit values and recovery time to restore inductor voltage.(a) Simulation model.Basically, the design parameters and circuit values are the same as those applied to Figure S1 (i.e., those applied to Fig. 2).(b) Frequency-domain profile with various inductances and 10-dBm input power and(c, d) corresponding time-domain profiles at 4.9 GHz.RL is fixed at 10 Ω. (e) Frequency-domain profile with various resistances and 10-dBm input power and (f, g) corresponding time-domain profiles.L is fixed at 10 μH.In the time-domain results, incident waves are excited until 100 ns as 100-ns short pulses.(h) Timedomain profile of transient transmittance at 4.9 GHz with repeated pulses.The pulse width is fixed at 100 ns, while the duty cycle is set to either 0.25, 0.5 or 0.75.(i) Inductor voltage

Figure S7 :
Figure S7: The transient transmittance of the simulation model depicted in Fig. 3a with various pulse duration scenarios.(a-i) From #1 to #9, the oscillation duration of each frequency is changed from 50 ns to 200 ns.(j) Average transmittance.

Figure S8 :
Figure S8: Transient transmittances of the simulation model drawn in Fig. 3a with different frequency sequence scenarios.(a) From #1 to #6, the frequency sequence used is changed.(b) Average transmittance.

Figure S11 :
Figure S11: Frequency-domain profile of the transmittance of the measurement sample shown in Fig. 3f.

Figure S12 :
Figure S12: Measurement of the triband waveform-selective metasurface.(a, b) Measurement sample and its dimensions.The design parameters and circuit values used are shown in TableS9and TableS10.The substrate is Rogers3003.In the actual measurement sample, only inductors are used inside diode bridges, as the inductors contain resistive components equal to RL. (c) Frequency-domain profiles of the transmittance for 50-ns pulses and CWs.f1 = 2.96 GHz, f2 = 3.45 GHz and f3 = 3.92 GHz.Time-domain profiles of the transient transmittances for (d) single-frequency cases and (e) switched-frequency case.(f) Transient transmittance for the switched-frequency case using different frequency sequences.
Figure S12: Measurement of the triband waveform-selective metasurface.(a, b) Measurement sample and its dimensions.The design parameters and circuit values used are shown in TableS9and TableS10.The substrate is Rogers3003.In the actual measurement sample, only inductors are used inside diode bridges, as the inductors contain resistive components equal to RL. (c) Frequency-domain profiles of the transmittance for 50-ns pulses and CWs.f1 = 2.96 GHz, f2 = 3.45 GHz and f3 = 3.92 GHz.Time-domain profiles of the transient transmittances for (d) single-frequency cases and (e) switched-frequency case.(f) Transient transmittance for the switched-frequency case using different frequency sequences.

Figure S13 :
Figure S13: Simplified equivalent transmission line model to represent the metasurface of Fig. 3a for analysing the contribution of the Q factor.(a) Equivalent circuit model connected to the transmission line for the single slit scenario (one unit cell).C0 and L0 represent the capacitive and inductive components of the metallic slit structure.Ladd and Radd represent the inductive and resistive components contributed by the dielectric substrate.The circuit parameters used for the diode and the series inductor and resistor (inside the diode bridge) are the same as those used in Fig. 3. (b, c) Simulated transmittance from linear network analysis for the circuit shown in (a) with varying Q factors while considering (b) zero parasitic resistance Radd = 0 Ω and (c) nonzero parasitic resistance Radd = 3 Ω (see Table S11 for variation of C1).(d) Equivalent circuit model connected to the transmission line for the slit scenario (one super cell).(e, f) Simulated transmittance from linear network analysis for the circuit shown in (d) with varying Q factors while considering (e) zero parasitic resistance Radd = 0 Ω, and (f) nonzero parasitic resistance Radd = 3 Ω.(g, h) Nonlinear analysis of transient transmittance for four parallel resonance scenarios with switched frequency.Here, the input power is set to 10 dBm.f1, f2, f3 and f4 are 2.0, 2.5, 3.3 and 4.1 GHz, respectively.Simulated transient transmittance with varying Q factor while considering (g) zero parasitic resistance Radd = 0 Ω and (h) nonzero parasitic resistance Radd = 3 Ω.

Figure S14 :
Figure S14: Experimental demonstration of metasurface configuration based on lumped LC circuit resonators for increased transmittance.(a) Schematic of the metasurface with three metallic rows inside the rectangular waveguide (6 unit cells).(b) Schematic of the metasurface with four metallic rows inside the rectangular waveguide (9 unit cells).(c) The corresponding unit cell designs.(d, e) Measurement samples and their dimensions for (d) the three-metallic-row case and (e) the four-metallic-row case.The design parameters and circuit values used are shown in TableS12and TableS13.(f) Frequency-domain profiles of the transmittance for 50-ns pulses.The result is compared to the slit configuration of FigureS12c.(g) Measured transient transmittance for switched-frequency cases.Here, the three frequencies used, i.e., f1, f2 and f3, are changed in each structure to maximize transient transmittance in three different frequency bands.Specifically, for the 3-metallic-row case, f1, f2 and f3 are 2.69, 3.12 and 3.82 GHz, respectively, while for the 4-metallic-row case, f1, f2 and f3 are 2.83, 3.31 and 3.91 GHz, respectively.(h) Average transmittance for entire pulse period.In these measurements, the input power is set to 10 dBm.

Figure S15 :
Figure S15: Demonstration of the simplified concept of Fig. 4. (a) Simplified circuit system.Compared to Fig. 4b, only two circuit configurations (i.e., two frequencies) are considered.VL1 and RJFET vary in accordance with the incoming frequency.(b) Front and (c) back designs of (d) the measurement sample.The design parameters and circuit values are given in TableS14, TableS15and TableS16.The substrate is Rogers3003.(e) Frequency-domain profiles of transmittance with various input powers.(f, g) Time-domain profiles of transient transmittance using different frequency sequences.f1 = 2.7 GHz and f2 = 3.6 GHz.The time-

Figure S16 :
Figure S16: Design of the measurement sample presented in Fig. 4c.(a) Front and (b) back designs.The design parameters and circuit values are given in TableS17, TableS18and TableS19.The substrate is Rogers3003.

Figure S18 :
Figure S18: Time-domain profiles of the measurement sample demonstrated in Fig. 4c with adjusted oscillation frequencies.The results with various (a) f1, (b) f2 and (c) f3 values.As the default values, f1, f2 and f3 are set to 3.3, 3.9 and 2.5 GHz, respectively.

Figure S19 :
Figure S19: Average transmittance of the measurement sample demonstrated in Fig. 4c for each frequency.(a) The transmittance during the initial period.(b) The transmittance in the steady state.f1, f2 and f3 are set to 3.3, 3.9 and 2.5 GHz, respectively.

Figure S20 :
Figure S20: Design strategy for the preferable sequence of frequencies within a supercell configuration.(a) The same supercell with different physical connections.(b) The same supercell with different capacitance values.

Figure S21 :
Figure S21: Example of data transmission (logo mark of Nagoya Institute of Technology) using the diagram block shown in Fig.5a.The S/N is varied from -20 to 0 dB.In these measurements, the input power is set to 20 dBm.Two frequency sequences are used for the carrier, similar to the scenario explained in Fig.5c.

Table S1 :
Energy at each frequency component after full wave rectification.

Table S7 :
Design parameters used for the unit cell model drawn in Fig.3fand FigureS10.

Table S8 :
Circuit values used for the unit cell model drawn in Fig.3fand FigureS10.The value in parentheses represents a self-resonant frequency.

Table S9 :
Design parameters used for the measurement sample shown in FigureS12b.

Table S11 :
Circuit values used for FigureS13.

Table S12 :
Design parameters used for the measurement sample shown in FigureS14d to

Table S13 :
Circuit values used for the measurement sample shown in FigureS14dto FigureS14e.The number in parentheses represents a self-resonant frequency.

Table S14 :
Design parameters used for the measurement sample shown in FigureS15b to and TableS16.The substrate is Rogers3003.(e) Frequency-domain profiles of transmittance with various input powers.(f, g) Time-domain profiles of transient transmittance using different frequency sequences.f1 = 2.7 GHz and f2 = 3.6 GHz.The time-

Table S15 :
Circuit values used for the measurement sample shown in FigureS15bto FigureS15d.The number in parentheses represents a self-resonant frequency.

Table S17 :
Design parameters used for the measurement sample shown in FigureS16(i.e.,

Table S18 :
Circuit values used for the measurement sample shown in FigureS16(i.e., Fig.4c).The number in parentheses represents a self-resonant frequency.

Table S19 :
SPICE parameters used for the JFETs of the measurement sample shown in FigureS16