Breaking temporal symmetries for emission and absorption

Significance

Antennas, from radiofrequencies to optics, are forced to transmit and receive with the same efficiency to/from the same direction. The same constraint applies to thermophotovoltaic systems, which are forced to emit as well as they can absorb, limiting their efficiency. In this paper, we show that it is possible to efficiently overcome these bounds using temporally modulated traveling-wave circuits. Beyond the basic physics interest of our theoretical and experimental findings, we also prove that the proposed temporally modulated antenna can be efficiently used to transmit without being forced to listen to echoes and reflections, with important implications for radio-wave communications. Similar concepts may be extended to infrared frequencies, with relevant implications for energy harvesting.

Abstract

Time-reversal symmetries impose stringent constraints on emission and absorption. Antennas, from radiofrequencies to optics, are bound to transmit and receive signals equally well from the same direction, making a directive antenna prone to receive echoes and reflections. Similarly, in thermodynamics Kirchhoff’s law dictates that the absorptivity and emissivity are bound to be equal in reciprocal systems at equilibrium, $e(\omega ,\theta )=a(\omega ,\theta )$, with important consequences for thermal management and energy applications. This bound requires that a good absorber emits a portion of the absorbed energy back to the source, limiting its overall efficiency. Recent works have shown that weak time modulation or mechanical motion in suitably designed structures may largely break reciprocity and time-reversal symmetry. Here we show theoretically and experimentally that a spatiotemporally modulated device can be designed to have drastically different emission and absorption properties. The proposed concept may provide significant advances for compact and efficient radiofrequency communication systems, as well as for energy harvesting and thermal management when translated to infrared frequencies.

Thermal management and heat control is a science with a long tradition in many engineering contexts, and over the years it has become of fundamental importance to address growing challenges related to heat dissipation. In the context of energy harvesting, solar panels and thermophotovoltaic cells are tailored to be highly absorbing in the spectral range of interest, typically in the visible or infrared range (1–6). However, reciprocity and time-reversal symmetry fundamentally requires these highly absorbing structures to also be very good emitters in the same spectral range (7–9). This fundamental relationship implies that, as the panels heat up, they are required to emit a significant portion of absorbed energy in the form of thermal infrared emission toward the source, causing a reduction in efficiency, as illustrated in Fig. 1A. Similarly relevant challenges are present in heat dissipation and thermal management in other engineering contexts, connected with fundamental reciprocity limitations. Reciprocity poses also severe restrictions in radio communications: Wireless systems and antennas are bound by reciprocity to transmit and receive in the same direction, i.e., the transmission and reception radiation patterns $G_{TX}\left(\theta \right),G_{RX}\left(\theta \right)$ of an antenna are identical (10, 11). This presents challenges in complex environments in which a directive antenna is typically forced to listen to its reflected echo.

Fig. 1.

Description of the main concept. (A) Equal absorption and emission due to reciprocity leads to efficiency limitations in thermophotovoltaic systems, and similarly limits the performance of communication systems. (B) Nonreciprocity may be used to overcome these limitations.

Over the years, a few groups have pointed out that by breaking reciprocity, we may be able to overcome these challenges (12–14), as illustrated in Fig. 1B. The most established route to break reciprocity is based on biasing ferromagnetic materials or ferrites with a magnetic field (15, 16). This method requires the use of scarcely available materials, such as rare-earth metals, and bulky magnets. For instance, a nanoscale plasmonic nonreciprocal antenna was proposed in ref. 17, but its requirements on magnetic biasing make it largely impractical. Alternatively, reciprocity can be also broken with nonlinearities (18, 19); however, this leads to undesirable signal distortion and a power-dependent response.

Recently, the realization of nonreciprocal acoustic and radio-wave components that break reciprocity without the need of magnets, using widely available materials, was proposed. The concept is based on imparting a suitable form of momentum biasing that breaks time-reversal symmetry, achieved with mechanical motion or with spatiotemporal modulation (20–29). In this paper, for the first time to our knowledge, we apply these concepts to emitting/absorbing systems, showing that it is possible to realize magnetic-free nonreciprocal structures that can emit without absorbing from the same direction over a broad frequency range. More specifically, we show theoretically and experimentally that by simultaneously modulating an emitting structure in both space and time with a judicious strategy, it is possible to break reciprocity constraints in radiation, significantly altering the structure’s absorptivity and emissivity patterns, and opening exciting possibilities in the areas of thermal management, energy harvesting, and radio-wave communications.

Results and Discussion

Consider first a conventional open waveguide, such as a dielectric slab supporting slow-wave propagation with wavenumber $\beta >k$. Because its dispersion is outside the light cone, when excited the guided modes do not couple to free space. Directional emission can be achieved when a periodic loading with periodicity $l$ is introduced, as shown in Fig. 2A (Top). An equivalent circuit model of this structure is given in Fig. 2A (Bottom) using shunt capacitors $C_0$ as loads. The introduction of periodicity folds the modal dispersion in the first Brillouin zone, as shown in Fig. 2B, and thus a portion enters into the light cone, associated with fast, radiating modes. In the corresponding circuit model, this radiation is modeled with a conductance $G_{rad}$ that depends on the frequency and wavenumber. Because the structure is reciprocal, the Brillouin dispersion is symmetric: Emission and absorption at frequency $\omega$ take place through symmetric channels with wavenumbers $\beta \left(\omega \right)$ and $-\beta \left(\omega \right)$, yielding equal receiving and transmitting properties, in compliance with time-reversal symmetry. Similar constraints apply if one looks at the absorption properties of periodic structures at infrared frequencies, with their thermal emission pattern being governed by Planck’s law.

Fig. 2.

Modal dispersion. (A) Periodically loaded open waveguide, and its equivalent circuit model. (B) The unloaded (periodically loaded) waveguide dispersion is shown in solid red (blue). The loaded waveguide dispersion enters the light cone, enabling radiation. However, the structure is reciprocal with symmetric dispersion. (C) Nonreciprocity is introduced by applying space–time modulation. (D) The modulated structure dispersion is asymmetric, indicating nonreciprocity. The wave solutions represented by green and pink lines correspond to higher-order harmonics, replicas of the dispersion of the unmodulated structure (blue line) shifted in the first Brillouin zone. The coupling between harmonics, enabled by space–time modulation, is much stronger on the right side of the diagram.

This picture breaks down once we modulate the electric characteristics of the grating simultaneously in space and time, as sketched in Fig. 2C. In the equivalent model, the capacitors are assumed to follow the temporal dispersion $C_n\left(t\right)=C_0+\mathrm{\Delta }C\cos ({\omega }_{m}t-n{\varphi }_m)$, where ${\omega }_m$ is the modulation frequency, and ${\varphi }_m$ is the phase difference between successive capacitors. A technique to calculate the dispersion of the corresponding structure and the associated eigenvectors up to the desired accuracy is described in the Supporting Information, and the calculated dispersion is shown in Fig. 2D. To explain the result, we assume first that the modulation amplitude is vanishingly small, $\mathrm{\Delta }C\to 0$. Then, applying the Bloch theorem, it is possible to show that the dispersion consists of infinite replicas of the unmodulated dispersion bands, corresponding to space–time harmonics shifted by ${\omega }_m$ and ${\varphi }_m$ along the frequency and wavenumber axes, respectively. Excitation at $\omega$ will in general allow coupling to other harmonics, based on frequency transitions $\omega \to \omega +n{\omega }_m$ with $n=0,\pm 1,\mathrm{...}$. Because we assume $\mathrm{\Delta }C\to 0$, these transitions are weak, and none of the higher-order harmonics is practically excited.

As the modulation amplitude $\mathrm{\Delta }C$ grows, coupling between harmonics takes place, and the proximity between dispersion curves in Fig. 2D determines the coupling strength. Intraband transitions, marked by red arrows in the figure, transfer energy from fundamental to higher-order Bloch harmonics, with higher efficiency if the coupling is stronger. Consider for instance the transitions $2\to 1\prime$ and $5\prime \to 4$, occurring between same frequencies and opposite branches, as shown with red arrows in Fig. 2D. For the transition $2\to 1\prime$, on the right of the band diagram, excitation of the fundamental harmonic at frequency $f_1$ will result in up-conversion of part of the energy to frequency $f_2$ ($+1$ harmonic), with large coupling strength ${\kappa }_{12}$. On the left of the band diagram, the transition $5\prime \to 4$ down-converts an excitation at frequency $f_2$ to $f_1$ ($-1$ harmonic), but with weaker coupling strength ${\kappa }_{21}<{\kappa }_{12}$, because of the larger distance between branches. In the Supporting Information, we discuss how these coupling coefficients can be analytically derived, and in Fig. S1 we plot the ratio ${\kappa }_{12}/{\kappa }_{21}\gg 1$ for the case at hand.

Fig. S1.

Ratio between coupling coefficients. The up-conversion (${\kappa }_{12}$) and down-conversion (${\kappa }_{21}$) coefficients were calculated using Eq. S15 in this article, based on the eigenvectors of the generalized ABCD matrix formulation, validating the measured nonreciprocal response shown in the paper. The measured isolation (17 dB) is smaller than the 23-dB contrast calculated using our simplified analytical model.

Based on these asymmetric transitions enabled by space–time modulation, we demonstrate the concept of nonreciprocal emission at radiofrequencies (rf). We use a space–time-modulated traveling-wave antenna consistent with the circuit model in Fig. 2C, showing that it can provide largely asymmetric transmission and reception patterns. The antenna is based on a grounded coplanar waveguide with slotted apertures with period $l=25.82\hspace{0.5em}\text{mm}$, designed to couple the guided wave into a radiation mode in the $f_{rf}\approx 3.6-4.2-\text{GHz}$ frequency range, as shown in Fig. 3A. The transmission line was loaded with voltage-tunable capacitors, enabling space–time modulation, positioned right above each slot, as in Fig. 3A. An image of the fabricated structure together with its feeding network is shown in Fig. 3B, and a detailed discussion of the experimental setup is provided in Materials and Methods. Related structures may be envisioned at infrared frequencies for thermal emission, for example using dielectric slabs periodically doped to create p-i-n junctions that can respond to a modulation signal and act as voltage-tunable capacitors.

Fig. 3.

Nonreciprocal antenna. (A) Coplanar transmission line periodically interrupted to enable radiation. (Top) Thin radiation apertures slots. (Bottom) Voltage-dependent capacitors, located right below the slots to control the propagation phase. (B) An image of the fabricated structure and its feeding network. The modulation control is based on a simple scheme, achieved using a diplexer that combines rf and modulation signals in a single port connected to the antenna via a Bias-T to superimpose the dc bias.

Fig. 4 shows the verification of nonreciprocal radiation based on the intraband transitions described in Fig. 2. Fig. 4A sketches a model of the system under analysis, with antenna A transmitting (Top) and receiving (Bottom), and antenna B receiving (Top) and transmitting (Bottom). Due to modulation, the left antenna may be described as fed through a mixer with mixing frequency $f_m={\omega }_m/\left(2\pi \right)$. Frequency conversion takes place between frequencies $f_1$ and $f_2=f_1+f_m$ due to the intraband coupling shown by the arrows in Fig. 2D, with conversion coefficients denoted by ${\kappa }_{12}$ and ${\kappa }_{21}$. Different from a conventional mixer, the nonreciprocal nature of our system and the asymmetry in Fig. 2 requires the conversion coefficients to be different, opening to the possibility of highly nonreciprocal radiation properties, consistent with the previous discussion.

Fig. 4.

Nonreciprocal radiation properties with frequency conversion. (A) Schematics of the frequency conversion and radiation system. (Top) Transmit mode. (Bottom) Receive mode. The device is reciprocal if ${\kappa }_{12}={\kappa }_{21}$, i.e., for a symmetric mixer. (B) Response without modulation, ${\kappa }_{12}={\kappa }_{21}=0$. The antenna is reciprocal with identical RX and TX patterns. (C) Response with modulation, normalized with respect to the maximum in B, which reveals dramatic difference between transmit and receive operation, indicating strong nonreciprocity.

In the absence of dynamic modulation, only a static bias voltage (with no modulation signal) is applied to set the varying capacitors at their nominal operation point. The antenna is reciprocal with dispersion similar to Fig. 2B, and identical radiation patterns in transmission and reception, as shown in the measurements of Fig. 4B. However, this picture breaks down as we inject a very weak modulation signal at frequency $f_m=600\hspace{0.5em}\text{MHz}\ll f_{rf}$ with amplitude such that $\mathrm{\Delta }C/C_0\approx 0.045$ through the Mod-in port in Fig. 3B. The modulation signal propagates along the transmission line and modulates the antenna in space and time, by varying the voltage-dependent capacitors. Consequently, asymmetric transitions take place as in Fig. 2D, yielding nonreciprocal frequency conversion. Thanks to the carefully designed dispersion, this weak modulation is sufficient to largely break reciprocity.

Fig. 4C shows the measured radiation patterns in this modulated scenario in transmit and receive mode (blue and red, respectively), with $f_1=3.495\hspace{0.5em}\text{GHz}$, $f_2=4.095\hspace{0.5em}\text{GHz}$. The transmit pattern radiates directively toward $\theta =50°$, as typical for a well-designed leaky-wave antenna, whereas the receive pattern is 17 dB lower. This large contrast is in agreement with Fig. 2D: In transmit operation we feed at point 2, and efficiently up-convert to the $+1$ harmonic $f_2$ at point $1\prime$, with ${\beta }_{1\prime }=k_2\cos \hspace{0.5em}\theta =53.8\hspace{0.5em}{\text{m}}^{-1}$ ($k_2=2\pi f_2/c$). This transition was designed to efficiently couple energy from outside the light cone (point 2) to inside the light cone (point 1′). Therefore, despite the fact that typically the coupling coefficients ${\kappa }_{12},{\kappa }_{21}\ll 1$ for weak modulation, radiation at the up-converted frequency is dominant. Even though the antenna used in this proof-of-concept experiment is relatively short, with an effective aperture of just about $1.2\lambda$ at frequency $f_1$, the peak intensity in Fig. 4B is of similar magnitude as in Fig. 4C after frequency conversion. With a longer antenna, the directivity of the leaky wave beam in Fig. 4C is expected to become significantly larger, leading to much stronger peak intensity compared with the beam at $f_1$ in Fig. 4B. In receive operation, on the other hand, the incoming wave corresponds to point $5\prime$, which is weakly coupled to the $-1$ harmonic, leading to very poor reception (see also the discussion around Fig. S1). These results demonstrate that it is possible to have a basic radiating structure, made of conventional materials and modulated with a weak traveling signal, which efficiently emits a directive beam toward a specific direction, while it receives poorly from all directions and thereby it is not sensitive to echoes from any angle.

A consequence of the designed intraband transitions is also the generation of an asymmetry at the fundamental frequency, between forward (4–6) and backward (1–3) modes. This in turn ensures that the same structure is also nonreciprocal when analyzed at the fundamental frequency, without considering frequency conversion, as shown in Fig. 5. In this regime, the designed antenna operates as a traveling wave, without supporting directive leaky radiation, consistent with the unmodulated radiation pattern shown in Fig. 4B. Once we inject the weak modulation signal at frequency $f_m$, the radiation patterns in transmit and receive modes are altered and become asymmetric, as seen in Fig. 5A. In this example, the major effect is observed in the angular range between $\theta =60°$ and $\theta =90°$, where absorption ($RX$) and emission ($TX$) peak in different directions. The nonreciprocal response is stronger in the proximity of a sidelobe associated with higher spatial frequencies of the current distribution on the antenna, which are more affected by small perturbations associated with the weak modulation. In Fig. 5 B–D, we show measured absorption and emission spectra for three different directions $\theta =75°,80°,85°$ as a function of frequency, normalized with respect to the maximal signal intensity measured in Fig. 4B. Even though this was not the tailored operation of the antenna, which was designed for operation with frequency conversion as discussed in Fig. 4, significant (10–15-dB) difference is demonstrated also in this scenario over a reasonably broad frequency band. Absorption can be made much larger than emission, and vice versa, and the reciprocity bound is clearly violated also at the same frequency. Large isolation is achieved with this design, especially around the nulls of radiation of the unmodulated case, which are shifted by the applied modulation. Better performance in this operation without frequency conversion is expected for more directive beamwidths, which may be achieved with a longer line, and using leaky-wave antennas that are based on zeroth-order diffraction, such as composite right-/left-handed transmission lines (30–32).

Fig. 5.

Nonreciprocal radiation at the fundamental frequency. (A) Radiation pattern of the spatiotemporally modulated antenna at the feeding frequency, normalized with respect to the maximum in Fig. 4B. Signal reciprocity is violated also at the fundamental frequency, leading to different patterns in transmit and receive. The most significant effect is seen around 60–90°. (B–D) Measured radiation intensity spectra, normalized as in A, shown at $\theta =75°,80°,$ and $85°$, and demonstrating an isolation of about 15 dB (∼30-fold) between receive and transmit patterns.

To conclude, in this article we have theoretically and experimentally introduced a basic device enabling largely nonreciprocal emission/absorption properties, based on space–time modulation of the radiation aperture of a leaky-wave antenna. We have shown, by a remarkably simple scheme, that it is possible to overcome common yet stringent limitations in radiating/emitting systems, with direct applications in compact and efficient rf communication systems, as well as energy harvesting and thermal management when translated to infrared frequencies. The modulation scheme proposed here, for which the same structure guides both the modulation and transmitted/received signals, avoids the use of additional circuitry that typically limits the maximum available modulation frequency. In a similar fashion, the use of p-i-n junctions, acoustooptic or nonlinearity-based modulation may be envisioned to realize these concepts at infrared/optical frequencies. In this context, particularly relevant are the recent developments in state-of-the-art graphene-based modulators that demonstrate an intrinsic device modulation frequency of 150 GHz (33), which is about one or two orders of magnitude smaller than the thermal emission frequency at 500–1,500 K––therefore making our proposal realistic also in the thermal infrared frequency range. In this context, our group has recently proposed a theoretical design for a nonreciprocal radiation setup based on a dual-mode waveguide operating at infrared frequencies based on modulated graphene layers (34). More broadly, our results also show that time-varying emitters and antennas may provide a fertile ground for future communication systems. Temporal modulation in antennas has recently been explored to enhance near-field communication channel capacity (35, 36), and here we have proven that, combined with spatial modulation, it may also largely break reciprocity constraints in a simple and compact setup.

Materials and Methods

Device Fabrication.

The antenna prototype was implemented over an FR-4 substrate with 1 oz copper foil, with details provided in Fig. S2. The lossy dielectric properties of FR-4 are $\epsilon ={\epsilon }_r\left(1-j\tan \delta \right)$, where ${\epsilon }_r=4.3$ and $\tan \delta =0.025$, measured at 10 GHz, and the substrate is nonmagnetic. The copper foil has a nonconductive adhesive, which is applied directly to the bare FR-4 board, and was cut using a Roland vinyl cutter from artwork files generated by CST Microwave Studio (www.cst.com). A grounded coplanar waveguide (GCPWG) was realized with 54 vias and 14 solid copper wires soldered between the radiation- (Fig. S2B) and ground-plane (Fig. S2C) sides. Proper matching to standard 50-$\mathrm{\Omega }$ lines was designed based on the width of the track line and the gaps between it and the ground plane (37). Vias were placed to allow space for the radiating slots and good local ground continuity for the reverse-biased diodes.

Fig. S2.

Schematic views of the realized TWA. (A) The TWA is built on an FR-4 substrate (shown in blue), which is excited by two SMA ports at either end. Shown in purple, the ground-plane side of the antenna is loaded periodically by varactor diodes to impart the required space–time modulation, centered at each radiation slot. (B) The radiation side of the antenna is implemented with apertures etched out in between vias of period 25.8 mm. (C) The ground-plane side of the antenna shows the feeding and spatial modulation schematics of the structure.

External Biasing and Experimental Setup.

Fig. S3 illustrates the biasing and matching circuit implementation applied to the traveling-wave antenna. A Minicircuits LDPW-162–242+ diplexer was used to combine the modulation (low-frequency) and carrier (high-frequency) signals. After combining these two signals, a Minicircuits TCBT-14+ bias-T was used to add the reverse dc bias to set the capacitance–voltage modulation bias point across the varactors. Five reverse-biased Skyworks SMV2019 varactor diodes were soldered from the center conductor (cathode) to the ground plane (anode) of the GCPWG, placed directly below the radiation slots. Finally, a broadband Minicircuits BLK-89+ dc block and 50-$\mathrm{\Omega }$ load was applied to P2.

Fig. S3.

Experimental setup and circuit connections for the frequency conversion case. The SGS100A provides the modulation signal of 1-V amplitude at $f_m=600\hspace{0.17em}\text{MHz}$, which is the same modulation used in both the frequency conversion (Fig. 4) and fundamental frequency (Fig. 5) cases. For the frequency conversion scenario shown here, the SMB100A injects a much higher carrier frequency; here $f_{rf}=3.495\hspace{0.5em}\text{GHz}$ of 0-dBm input power. To apply these two separate frequencies to the same rf line, a high isolation diplexer (LDPW-162–242+) is used, which is shown here as a parallel connected multiorder low-pass ($f_m$) and high-pass ($f_{rf}$) filter. A wideband Bias-T (TCBT-14+) is then used to couple a tunable dc reverse bias, for the varactor diodes (SMV2019), to a coaxial line with the injected rf signals. Finally, the leaky-wave antenna (LWA) is rf-coupled (BLK-89+) to a standard broadband 50-Ω load. Please note for the fundamental frequency case (Fig. 5), the SMB100A is replaced by port 1 of the vector network analyzer; otherwise, the setup shown here is the same.

With an 8-V reverse bias, we were able to obtain good matching in the carrier port band, with a moderate capacitive modulation of $\mathrm{\Delta }{C}_m\approx 0.02\hspace{0.17em}\text{pF}$ centered at $C_0=0.44\hspace{0.17em}\text{pF}$, $f_m=600\hspace{0.17em}\text{MHz}$. Under these testing conditions, Fig. S4 shows the matching and isolation characteristics measured at the carrier frequency, whereas Fig. S5 shows the matching and isolation for the modulation network. The diplexer used in Fig. S3 allowed us to inject the two signals into the antenna, while isolating them from each other. We stress that the antenna was hand-built, and the performance can easily be improved with professional fabrication. Still the matching is more than sufficient, and the antenna is well matched within the modulation band, so that efficient space–time modulation of the structure may take place.

Fig. S4.

Carrier band matching and isolation performance with an 8-V reverse bias. The LWA matching performance as seen at the high-pass port (consider (Fig. S3) under the reverse-bias setting used throughout this work. Here we also see the diplexer isolation in the lower modulation band. Moderate transmission between the fundamental port and the LWA termination port indicates a noticeable portion of the injected power is absorbed by the termination, due to the short aperture of the LWA prototype.

Fig. S5.

Modulation band matching and isolation performance with an 8-V reverse bias. Similar to (Fig. S4, but now for the modulation port in (Fig. S3. Notice the good modulation band matching and high isolation performance by the circuit in (Fig. S3.

Far-Field Measurements.

Measurement setup for Fig. 4.

Here we provide details of the measurement setup used to prove the nonreciprocal mixing response of the proposed antenna, as shown in Fig. 4. To prove nonreciprocal mixing, we have shown that the TX and RX patterns are vastly different through the conversion process $f_{rf}\to f_{rf}\pm f_m$. To demonstrate this nonreciprocal conversion, a signal generator is used to transmit power from either the traveling-wave (modulated) antenna or the testing antenna at $f_{rf}$. Independently, a spectrum analyzer is used to measure the received powers at $f_{rf}$ and $f_{rf}\pm f_m$ at the testing antenna (TX mode), or the modulated antenna (RX mode). As schematically shown in Fig. S6, to calculate the radiation patterns $G_{TX}$ and $G_{RX}$, the two antennas are separated by a distance $R=1.1\hspace{0.17em}\text{m}\approx 13.7\lambda$ at $f=3.7\hspace{0.5em}\text{GHz}$, more than twice the Fraunhofer distance, and therefore in the far field of each other. The test antenna remains at a constant position throughout each measurement, while the traveling-wave antenna (TWA) is rotated about the $x$ axis sweeping $0°\le \theta \le 180°$ in steps of $5°$, for both TX and RX measurements.

Fig. S6.

Radiation pattern measurement nonreciprocal setup. Schematics showing the setup to measure nonreciprocal radiation/reception properties at the mixing frequencies.

In the example shown in Fig. S6, the modulated antenna is in TX mode to calculate $G_{TX}$. To complete the TX-mode/RX-mode pattern measurement pair, the TWA was then placed in RX mode (shown in dashed green line connections) to calculate $G_{RX}$. Notice here the bias block simply indicates the circuit in Fig. S3, where only the carrier frequency $f_{rf}$ is injected and the modulation and dc bias are assumed.

Measurement setup for Fig. 5.

To demonstrate nonreciprocity without frequency conversion, we may use the S-parameter data directly extracted from the vector network analyzer connected to the antenna port in this configuration. Here we note that $G_{TX}={\left|s_{21}\right|}^2$ and $G_{RX}={\left|s_{12}\right|}^2$, where $s_{21}=b_2/a_1$ and $s_{12}=b_1/a_2$, and $a_n,b_n$ are the absolute transmitted and received signal levels at port $n$, respectively. The corresponding setup scheme is shown in Fig. S7. A calibration was performed in this case to remove cable loss, and the measurement phase plane was placed directly at each of the antenna ports.

Fig. S7.

Radiation pattern measurement nonreciprocal setup at the fundamental frequency. Schematics showing the setup to measure nonreciprocal radiation/reception properties at the fundamental frequency.

Derivation of the Dispersion Equation for Modulated Periodic Structures

In Fig. 2 A and B of the main text, a periodic waveguide and its dispersion diagram are shown. The dispersion in this case is a classic result that can be obtained in several ways, for instance, by solving the eigenvalue problem associated with the ABCD transmission matrix of a single unit cell (38). When the periodic structure is modulated in space and time, as shown in Fig. 2C, it is not strictly periodic, and the derivation of the dispersion equation is not so straightforward. In the following, we describe the procedure we developed to calculate the dispersion diagrams.

We assume that in the absence of periodic loading, the waveguide/transmission-line dispersion is $\gamma \left(\omega \right)$, and its characteristic line impedance is given by $Z_0\left(\omega \right)$ (with corresponding line admittance $Y_0=1/Z_0$). The waveguide is loaded at period locations every $z_n=nl$ with shunt resistors of conductance $G_{rad}$, connected to shunt capacitors with capacitance $C_n\left(t\right)$.

Generalized ABCD Matrix Formalism for Space–Time-Modulated Loads

In the conventional time-harmonic case, an ABCD matrix of a two-port linear network relates the harmonic voltage–current vector at port 1 (P1) to its counterpart at port 2 (P2). However, if one of the network components is time modulated, then even if the signal injected to the input port of the network is time harmonic, the reflected signal at the input port and the transmitted signal from the output port will in general contain also other frequency components. Therefore, a relation between the input and output ports of time modulated network has to relate the various harmonics.

The structure that we analyze is shown in Fig. 2 of the main text. It consists of an infinite cascade of unit cells as in Fig. S8A. Each unit cell is loaded by a modulated capacitance

$$C_n\left(t\right)={\text{C}}_0+\mathrm{\Delta }C\cos ({\omega }_{m}t-n{\varphi }_m).$$ [S1]

In Eq. S1, $C_0$ is the nominal capacitance value (assigned by the dc bias on the varactors), $\mathrm{\Delta }C$ is the modulation amplitude, assumed to be much smaller than $C_0$ (in our experiment $\mathrm{\Delta }C/C_0=0.02/0.44\approx 4.5\%$; Materials and Methods). The latter assumption is not essential for the technique proposed below, and it can be removed easily, yielding a calculation procedure that can be accurate up to any desired accuracy. ${\omega }_m$, and ${\varphi }_m$ are the modulation frequency and phase, respectively. The following derivation consists of developing a generalized ABCD matrix for a shunt resistor–capacitor (RC) connection with capacitance (38); then, we derive a generalized transmission matrix for a finite transmission-line section and we use these matrices to write the desired dispersion equation.

Fig. S8.

Unit cell of the modulated periodic waveguide. (A) The entire time-modulated unit cell consists of shunt resistor and time-modulated capacitor sandwiched between two transmission-line sections. (B) Shunt $R{C}_n\left(t\right)$; the conductance represents radiation loss. (C) A finite transmission-line section with wavenumber dispersion $\beta =\gamma \left(\omega \right)>k$ and characteristic impedance $Z_0\left(\omega \right)$.

Generalized ABCD Matrix for a Shunt $R{C}_n\left(t\right)$ Element.

Consider a shunt network formed by a resistor and a time-modulated capacitor $C_n\left(t\right)$, as shown in Fig. S8B. We denote by $V_1,I_1$, and $V_2,I_2$ the voltage–current pairs at the RC network ports 1 and 2, respectively. Our goal is to write a relation between the voltage–current at port 1 and the voltage–current at port 2, similar to the relation at the basis of a conventional ABCD matrix. The current flows into port 1 and out of port 2. We define $V=V_1=V_2$, $I=I_1-I_2$. Anticipating Bloch-type solutions in space–time, we write

$$V={\displaystyle \sum _{r=-\infty }^{\infty }{V}^{\left(r\right)}}{e}^{j({\omega }_{r}t-rn{\varphi }_m)},\hspace{0.17em}I={\displaystyle \sum _{r=-\infty }^{\infty }{I}^{\left(r\right)}}{e}^{j({\omega }_{r}t-rn{\varphi }_m)},$$ [S2]

with ${\omega }_r=\omega +r{\omega }_m$, where $\omega$ denotes the frequency of the harmonic input signal. Eq. S2 states that the voltage and current can be represented by a column vector of coefficients $\overline{V}=\left\{V^{\left(r\right)}\right\},\overline{I}=\left\{I^{\left(r\right)}\right\}$. At the network, the voltage $V$ and current $I$ are related through

$$I=G_{rad}V+C\left(t\right)\frac{dV}{dt}.$$ [S3]

By plugging Eq. S2 into Eq. S3, we obtain the following relation between current and voltage space–time harmonics:

$$I^{\left(r\right)}=Y^{\left(r\right)}{V}^{\left(r\right)}+\frac{\mathrm{\Delta }{Y}^{\left(r\right)}}{2}\left[V^{(r-1)}+V^{(r+1)}\right],$$ [S4]

with $Y^{\left(r\right)}=G_{rad}+j{\omega }_r{C}_0$ and $\mathrm{\Delta }{Y}^{\left(r\right)}=j{\omega }_r\mathrm{\Delta }C$. For weak modulation $\mathrm{\Delta }C\ll C_0$, it is reasonable to assume that the lowest-order harmonics $r=-\mathrm{1,0,1}$ are dominant. Then, Eq. S4 can be truncated and written in the following 3 × 3 matrix form:

$$\overline{I}=\mathbf{Y}\overline{V},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{Y}=\left(\begin{array}{ccc}{Y}^{(-1)}& \mathrm{\Delta }{Y}^{(-1)}/2& 0\\ \mathrm{\Delta }{Y}^{\left(0\right)}/2& Y^{\left(0\right)}& \mathrm{\Delta }{Y}^{\left(0\right)}/2\\ 0& \mathrm{\Delta }{Y}^{\left(1\right)}/2& Y^{\left(1\right)}\end{array}\right),$$ [S5]

where $\overline{I}={\left[I^{(-1)},I^{\left(0\right)},I^{\left(1\right)}\right]}^T$ and $\overline{V}={\left[V^{(-1)},V^{\left(0\right)},V^{\left(1\right)}\right]}^T$.

We can always take more harmonics than the first three fundamental ones, simply increasing the matrix rank and therein increasing the solution accuracy, and address problems with stronger modulation amplitude.

Now it is possible to define a “generalized” voltage–current vector ${\overline{X}}_i={[{\overline{V}}_i^T,{\overline{I}}_i^T]}^T,i=\mathrm{1,2}$ for each of the network ports. We also have $\overline{V}={\overline{V}}_1={\overline{V}}_2$ and $\overline{I}={\overline{I}}_1-{\overline{I}}_2$. Therefore, we can write

$${\overline{X}}_1=\mathbf{B}{\overline{X}}_2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathbf{B}=\left(\begin{array}{cc}\mathbf{I}& \mathbf{0}\\ \mathbf{Y}& \mathbf{I}\end{array}\right).$$ [S6]

The matrix $\mathbf{B}$ in Eq. S6 is the generalized ABCD matrix representation of a shunt RC connection with time-modulated capacitance. This is the nonharmonic equivalent to the matrix that can be found in classical textbooks, such as ‎ref. 38.

Generalized ABCD Matrix for a Finite Transmission-Line Section.

Next we develop the generalized ABCD matrix of a finite transmission-line section with length $l/2$, characteristic impedance $Z_0\left(\omega \right)$, and wavenumber dispersion $\gamma \left(\omega \right)$, as shown in Fig. S8C. We emphasize that although this section is not modulated, and therefore no frequency mixing takes place on it, we still have to develop an appropriate ABCD matrix representation that considers the frequency mixing in the rest of the network. We define a propagator matrix and characteristic impedance matrix for $2N+1$ (to be consistent with previous derivation, we assume $N=1$) harmonics by

$$\begin{array}{l}{\mathbf{P}}_N=\exp \left\{-j\text{diag}\left[\gamma \left({\omega }_{-N}\right),\mathrm{...},\gamma \left({\omega }_{-1}\right),\gamma \left({\omega }_0\right),\gamma \left({\omega }_1\right),\mathrm{...},\gamma \left({\omega }_N\right)\right]l/2\right\}\\ {\mathbf{Z}}_{0N}=\text{diag}\left[Z_0\left({\omega }_{-N}\right),\mathrm{...},Z_0\left({\omega }_{-1}\right),Z_0\left({\omega }_0\right),Z_0\left({\omega }_1\right),\mathrm{...},Z_0\left({\omega }_N\right)\right].\end{array}$$ [S7]

Then the relation between ${\overline{X}}_i={[{\overline{V}}_i^T,{\overline{I}}_i^T]}^T,i=\mathrm{1,2}$ will be given by

$${\overline{X}}_1=\mathbf{A}{\overline{X}}_2,\hspace{0.17em}\hspace{0.17em}\mathbf{A}=\left(\begin{array}{cc}\left[{\mathbf{P}}_N+{\mathbf{P}}_N^{-1}\right]& -{\mathbf{Z}}_{0N}\left[{\mathbf{P}}_N-{\mathbf{P}}_N^{-1}\right]\\ -{\mathbf{Z}}_{0N}^{-1}\left[{\mathbf{P}}_N-{\mathbf{P}}_N^{-1}\right]& \left[{\mathbf{P}}_N+{\mathbf{P}}_N^{-1}\right]\end{array}\right),$$ [S8]

where port 1 (2) is the port of the transmission line, from the left (right) side. The matrix $\mathbf{A}$ in Eq. S8 is the time-modulated generalization of the one that may be found in ‎ref. 38 for the unmodulated case.

Derivation of the Eigenvalue Problem.

Our next goal is to derive an eigenvalue equation, and use the eigenvalues to obtain the dispersion equation. In the nth unit cell, we can relate ${\overline{X}}_{1,n}$ and ${\overline{X}}_{2,n}$ (Fig. S8A) by

$${\overline{X}}_{1,n}=\mathbf{A}\mathbf{B}\mathbf{A}{\overline{X}}_{2,n}.$$ [S9]

Although $X_{2,n}=X_{1,n+1}$, it is not possible to write ${\overline{X}}_{2,n}={\overline{X}}_{1,n+1}$ because these two coefficient vectors are used to expand $X_{2,n}$ and $X_{1,n+1}$ with different expansion functions. Therefore, to get the correct relation between the two vectors we should write

$$X_{2,n}=X_{1,n+1}\to {\displaystyle \sum _{r=-\infty }^{\infty }{X}_{2,n}^{\left(r\right)}}{e}^{j({\omega }_{r}t-rn{\varphi }_m)}={\displaystyle \sum _{r=-\infty }^{\infty }{X}_{1,n+1}^{\left(r\right)}}{e}^{j({\omega }_{r}t-r(n+1){\varphi }_m)},$$ [S10]

concluding that $X_{2,n}^{\left(r\right)}=X_{1,n+1}^{\left(r\right)}{e}^{-jr{\varphi }_m}$. Alternatively, in matrix form

$${\overline{X}}_{2,n}=\mathbf{Q}{\overline{X}}_{1,n+1},\hspace{0.17em}\hspace{0.17em}\mathbf{Q}=\text{diag}\left[e^{jN{\varphi }_m},\mathrm{..},e^{j{\varphi }_m},1,e^{-j{\varphi }_m},\mathrm{..},e^{-jN{\varphi }_m},e^{jN{\varphi }_m},\mathrm{..},e^{j{\varphi }_m},1,e^{-j{\varphi }_m},\mathrm{..},e^{-jN{\varphi }_m}\right].$$ [S11]

Finally, together with Eq. S9, we have

$${\overline{X}}_{1,n}=\mathbf{A}\mathbf{B}\mathbf{A}\mathbf{Q}{\overline{X}}_{1,n+1}.$$ [S12]

Note that the matrix $\mathbf{A}\mathbf{B}\mathbf{A}\mathbf{Q}$ does not depend on the unit cell index, despite the fact that the capacitance modulation does. This property leads to the important observation that Eq. S12 is shift invariant; therefore, its solution can be written as a Bloch function

$${\overline{X}}_{1,n}={\overline{X}}_{\mathrm{1,0}}{e}^{-jn\beta l},$$ [S13]

and the propagation wavenumber $\beta$ and the mode amplitude ${\overline{X}}_{\mathrm{1,0}}$ can be obtained from the solution of the eigenvalue problem

$$\mathbf{A}\mathbf{B}\mathbf{A}\mathbf{Q}{\overline{X}}_{\mathrm{1,0}}=\lambda {\overline{X}}_{\mathrm{1,0}},\hspace{0.17em}\hspace{0.17em}\beta l=-j\ln \lambda .$$ [S14]

We have used this procedure to calculate the dispersion diagrams in Fig. 2 B and D. The eigenvectors can be used to calculate the coupling coefficients shown in Fig. 2D in the main text by

$${\kappa }_{12}={\left|\frac{V^{\left(1\right)}}{V^{\left(0\right)}}\right|}^2@{\beta }_2,\hspace{0.17em}{\kappa }_{21}={\left|\frac{V^{(-1)}}{V^{\left(0\right)}}\right|}^2@{\beta }_{5\prime }.$$ [S15]

The coupling coefficients calculated for the experiment parameters are shown in Fig. S1 and clearly explain the high contrast between radiation patterns shown in Fig. 4 in the main text.