# Breaking temporal symmetries for emission and absorption

See allHide authors and affiliations

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved February 17, 2016 (received for review August 31, 2015)

## 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,

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. 1*A*. 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

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. 1*B*. 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 *A* (*Top*). An equivalent circuit model of this structure is given in Fig. 2*A* (*Bottom*) using shunt capacitors *B*, 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

This picture breaks down once we modulate the electric characteristics of the grating simultaneously in space and time, as sketched in Fig. 2*C*. In the equivalent model, the capacitors are assumed to follow the temporal dispersion *Supporting Information*, and the calculated dispersion is shown in Fig. 2*D*. To explain the result, we assume first that the modulation amplitude is vanishingly small,

As the modulation amplitude *D* 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 *D*. For the transition *Supporting Information*, we discuss how these coupling coefficients can be analytically derived, and in Fig. S1 we plot the ratio

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. 2*C*, 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 *A*. The transmission line was loaded with voltage-tunable capacitors, enabling space–time modulation, positioned right above each slot, as in Fig. 3*A*. An image of the fabricated structure together with its feeding network is shown in Fig. 3*B*, 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. 4 shows the verification of nonreciprocal radiation based on the intraband transitions described in Fig. 2. Fig. 4*A* 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 *D*, with conversion coefficients denoted by

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. 2*B*, and identical radiation patterns in transmission and reception, as shown in the measurements of Fig. 4*B*. However, this picture breaks down as we inject a very weak modulation signal at frequency *B*. 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. 2*D*, yielding nonreciprocal frequency conversion. Thanks to the carefully designed dispersion, this weak modulation is sufficient to largely break reciprocity.

Fig. 4*C* shows the measured radiation patterns in this modulated scenario in transmit and receive mode (blue and red, respectively), with *D*: In transmit operation we feed at point 2, and efficiently up-convert to the *B* is of similar magnitude as in Fig. 4*C* after frequency conversion. With a longer antenna, the directivity of the leaky wave beam in Fig. 4*C* is expected to become significantly larger, leading to much stronger peak intensity compared with the beam at *B*. In receive operation, on the other hand, the incoming wave corresponds to point

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. 4*B*. Once we inject the weak modulation signal at frequency *A*. In this example, the major effect is observed in the angular range between *B*–*D*, we show measured absorption and emission spectra for three different directions *B*. 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).

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 *B*) and ground-plane (Fig. S2*C*) sides. Proper matching to standard 50-

### 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-

With an 8-V reverse bias, we were able to obtain good matching in the carrier port band, with a moderate capacitive modulation of

### 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

In the example shown in Fig. S6, the modulated antenna is in TX mode to calculate

#### 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

## 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. 2*C*, 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

## 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. S8*A*. Each unit cell is loaded by a modulated capacitance**S1**, *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.

### Generalized ABCD Matrix for a Shunt R C n ( t ) Element.

Consider a shunt network formed by a resistor and a time-modulated capacitor *B*. We denote by **S2** states that the voltage and current can be represented by a column vector of coefficients **S2** into Eq. **S3**, we obtain the following relation between current and voltage space–time harmonics:**S4** can be truncated and written in the following 3 × 3 matrix form:

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 **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 *C*. 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 **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 *n*th unit cell, we can relate *A*) by**S9**, we have**S12** is shift invariant; therefore, its solution can be written as a Bloch function*B* and *D*. The eigenvectors can be used to calculate the coupling coefficients shown in Fig. 2*D* in the main text by

## Acknowledgments

This work was supported by the Office of Naval Research with Grant N00014-15-1-2685, the National Academy of Engineering Frontiers of Engineering Program, and The Grainger Foundation.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: alu{at}mail.utexas.edu.

Author contributions: Y.H., J.C.S., and A.A. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1517363113/-/DCSupplemental.

## References

- ↵.
- Campbell P,
- Green MA

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Planck M

- ↵.
- Siegel R,
- Houell J

- ↵.
- Collin RE,
- Zucker FJ

- ↵.
- Balanis CA

- ↵
- ↵
- ↵
- ↵.
- Jackson JD

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Lin Q,
- Fan S

- ↵
- ↵
- ↵.
- Fleury R,
- Sounas DL,
- Sieck CF,
- Haberman MR,
- Alù A

- ↵.
- Sounas DL,
- Alu A

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Yao W,
- Wang Y

*Proceedings of IEEE MTT-S International Microwave Symposium Digest*(IEEE, Fort Worth, TX), Vol 2, pp 1273–1276 - ↵.
- Azad U,
- Wang YE

- ↵.
- Wadell BC

- ↵.
- Pozar DM

## References

- ↵.
- Campbell P,
- Green MA

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Planck M

- ↵.
- Siegel R,
- Houell J

- ↵.
- Collin RE,
- Zucker FJ

- ↵.
- Balanis CA

- ↵
- ↵
- ↵
- ↵.
- Jackson JD

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Lin Q,
- Fan S

- ↵
- ↵
- ↵.
- Fleury R,
- Sounas DL,
- Sieck CF,
- Haberman MR,
- Alù A

- ↵.
- Sounas DL,
- Alu A

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Yao W,
- Wang Y

*Proceedings of IEEE MTT-S International Microwave Symposium Digest*(IEEE, Fort Worth, TX), Vol 2, pp 1273–1276 - ↵.
- Azad U,
- Wang YE

- ↵.
- Wadell BC

- ↵.
- Pozar DM

## Citation Manager Formats

## Article Classifications

- Physical Sciences
- Engineering