Abstract
Nonreciprocal components, which are essential to many modern communication systems, are almost exclusively based on magnetooptical materials, severely limiting their applicability. A practical and inexpensive route to magneticfree nonreciprocity could revolutionize radiofrequency and nanophotonic communication networks. Angularmomentum biasing was recently proposed as a means of realizing isolation for sound waves travelling in a rotating medium^{1}, and envisaged as a path towards compact, linear integrated nonreciprocal electromagnetic components^{2,3}. Inspired by this concept, here we demonstrate a subwavelength, linear radiofrequency nonreciprocal circulator free from magnetic materials and bias. The scheme is based on the parametric modulation of three identical, strongly and symmetrically coupled resonators. Their resonant frequencies are modulated by external signals with the same amplitude and a relative phase difference of 120°, imparting an effective electronic angular momentum to the system. We observe giant nonreciprocity, with up to six orders of magnitude difference in transmission for opposite directions. Furthermore, the device topology is tunable in real time, and can be directly embedded in a conventional integrated circuit.
Main
Early attempts to realize magneticfree nonreciprocity were based on the nonreciprocal properties of transistors at microwave frequencies^{4}, and on networks of electrooptical modulators at optical frequencies^{5,6,7,8}. However, such approaches traded the absence of magnetic bias with other significant drawbacks, such as the strong nonlinearities and poor noiseperformance of transistors, or the large size and complexity of the required electrooptical networks. More recently, nonreciprocity has been achieved in transistorloaded metamaterials^{9,10} and nonlinear devices^{11,12,13,14}. Also these solutions impose severe restrictions on the input power levels, generally degrading the signal quality because of noise or signal distortion. Another interesting approach to magneticfree nonreciprocity has been introduced^{15}, using asymmetric mode conversion in spatiotemporally modulated waveguides. This concept is especially attractive for integrated optical networks, as it may be fully realized in silicon photonics^{16}. However, this technique and its variants^{17,18,19,20,21,22} lead to structures much larger than the wavelength, owing to the weak electrooptic or acoustooptic effects on which they rely, and require complex modulation schemes. In a similar context, the concept of a nonreciprocal device based on parametrically coupled resonators has been theoretically explored^{23}.
An approach that can lead to compact, magneticfree nonreciprocal devices with relaxed implementation complexity was recently introduced ^{1,2,3}, based on angularmomentum biasing of a resonant ring. Angular momentum can be applied either by mechanically spinning a fluid, as proved for acoustic waves^{1}, or, more conveniently for electromagnetic waves, by spatiotemporal modulation with a travelling wave, realizing an effective electronic spin^{2,3}, as illustrated in Fig. 1a. The resonant nature of the modulated ring can substantially boost the otherwise weak electrooptic effects through which spatiotemporal modulation is typically achieved^{24,25}, allowing the design of largely nonreciprocal devices with dimensions of the order of, or even smaller than the wavelength. Furthermore, in contrast to other approaches based on spatiotemporal modulation^{15,19}, angularmomentum biasing is based on uniform modulation across the ring crosssection, thus significantly simplifying the fabrication process.
Inspired by these ideas, here we propose a device that provides strong magneticfree nonreciprocity at the subwavelength scale by dynamically modulating three identical resonant circuit tanks arranged in a loop and strongly coupled to each other, as in Fig. 1b. The resonant frequencies of the individual tanks are temporally modulated in a circularly rotating fashion as ω_{1}(t) = ω_{0} + δω_{m} cos(ω_{m}t), ω_{2}(t) = ω_{0} + δω_{m} cos(ω_{m}t + 2π/3) and ω_{3}(t) = ω_{0} + δω_{m} cos(ω_{m}t + 4π/3), where ω_{0} is the static value of the resonant frequency, δω_{m} is the modulation amplitude and ω_{m} is the modulation frequency, so that an effective electronic spin is imparted to the system. Without modulation, the loop supports two degenerate counterrotating modes, similar to the uniform ring of Fig. 1a. However, when the modulation is switched on, this degeneracy is lifted and nonreciprocity is induced. As we show in the following, this solution markedly boosts the modulation efficiency of the device compared to the case of a single modulated resonant ring, as considered in refs 2, 3, largely relaxing the requirements in terms of the modulation intensity and subsequently improving the overall efficiency of the structure.
In the absence of modulation (δω_{m} = 0), the loop of Fig. 1b supports three resonant states: a common state with state vector c〉 = [1 1 1]^{T} and frequency ω_{c} = ω_{0} + 2κ, and two degenerate right and lefthanded states with state vectors ±〉 = [1 e^{±i2π/3} e^{±i4π/3}]^{T} and frequencies ω_{±} = ω_{0} − κ, where κ is the coupling coefficient. The applied modulation mixes right and lefthanded states, producing two new hybrid states
where , ω_{R} = ω_{±} − Δω/2 and ω_{L} = ω_{±} + Δω/2, as analytically derived in the Methods for ω_{m} ≪ ω_{c} − ω_{±}.
It is interesting to observe that both R〉 and L〉 consist of a dominant state at ω_{R, L} and a secondary state, red or blueshifted by ω_{m}, as in the case of the uniformly modulated ring of refs 2, 3. However, despite this apparent resemblance in the form of R〉 and L〉, the mechanism that creates the frequency separation Δω between resonant states in the geometry analysed here is significantly different from the uniformly modulated ring of refs 2, 3. For a uniform resonant ring (Fig. 1a), Δω is the result of an azimuthally travelling wave modulation Δɛ_{m} cos(ω_{m}t − l_{m}ϕ), where Δɛ_{m} and l_{m} are the modulation amplitude and azimuthal order, respectively. If l_{m} = 2l, where l is the resonant order of the ring, δω_{m} = ω_{l}Δɛ_{m}/(2ɛ) (refs 2, 3), where ω_{l} is the resonant frequency and ɛ the static permittivity. The ideal continuous modulation assumed in Fig. 1a (top) is difficult, if not impossible, to realize, and an angular discretization is typically required. A discrete modulation profile with N different modulation regions, as sketched in Fig. 1a (bottom), is equivalent to a continuous effective modulation with reduced effective amplitude Δɛ_{m, eff} = Δɛ_{m}sinc(2l/N) (ref. 3), revealing a fundamental tradeoff between fabrication complexity, proportional to N, and nonreciprocal response. As an example, for N = 3—that is, the minimum value for which Δɛ_{m, eff} ≠ 0—and l = 50—a typical value in realistic highQ microring resonators at 1.55 μm—Δɛ_{m, eff} = 0.008Δɛ_{m}, implying that only 0.8% of the modulation signal is effectively used to generate nonreciprocity.
In contrast, in the composite resonant loop of Fig. 1b the frequency splitting is achieved by modulating the frequency of each resonator, which can be obtained by applying a uniform permittivity modulation with amplitude Δɛ_{m} all across each resonator^{24}, leading to δω_{m} = ω_{0}Δɛ_{m}/(2ɛ), as shown in the Methods. Remarkably, in this topology the entire modulation signal is used to produce the frequency separation Δω, indicating that maximum (unitary) modulation efficiency is achieved in a simple fabrication scheme requiring only three independent modulation regions.
We realized the structure of Fig. 1b at RF using three basic L–C circuit tanks, as in Fig. 2a, where the capacitance C is equally distributed on both sides of the inductance L to maintain a symmetric structure (see Supplementary Methods for a full schematic of the realized circuit). The resonance frequency modulation is achieved by means of capacitance modulation, commonly obtained in RF with varactor diodes. These diodes are biased by two signals, a static signal V_{d.c.}, which provides the required reverse bias and controls the static capacitance, and a RF signal ν_{m} with frequency ω_{m} and amplitude V_{m}, providing the time modulation. Assuming that the resonators are coupled to each other through capacitances C_{c}, as in Fig. 2b, the frequencies of the common and rotating states are and , respectively, where is the static resonance frequency of each tank. Then, if the amplitude of the capacitance modulation is ΔC_{m}, the frequency modulation amplitude is found as δω_{m} = ω_{±}ΔC_{m}/(2C). The frequency ω_{c} should be designed to be as far as possible from ω_{±} in order for the common mode not to affect the operation of the structure at ω_{±}, at which nonreciprocity occurs. In the lumpedelement circuit of Fig. 2b this condition is satisfied by taking C_{c} → ∞, or equivalently by coupling the tanks through short circuits, yielding ω_{c} = 0 and .
The nonreciprocal response of the circuit of Fig. 2b is demonstrated by capacitively coupling it to three microstrip transmission lines, realizing a threeport device. Exciting the structure from, for example, port 1 at frequency ω_{±} results in the excitation of R〉 and L〉 with the same amplitude and opposite phase φ_{R} = − φ_{L}, owing to the symmetrical distribution of these states around ω_{±}. Then, the signals at ports 2 and 3 are proportional to ${\text{e}}^{i2\pi /3}{\text{e}}^{i{\varphi}_{R}}{\text{+e}}^{i4\pi /3}{\text{e}}^{i{\varphi}_{L}}$ and ${\text{e}}^{i2\pi /3}{\text{e}}^{i{\varphi}_{R}}{\text{+e}}^{i4\pi /3}{\text{e}}^{i{\varphi}_{L}}$, respectively, as the superposition of R〉 and L〉 at these ports. If δω_{m} and ω_{m} are selected so that φ_{R} = − φ_{L} = π/6, the signal at port 3 is identically zero, while the signal at port 2 is nonzero, routing the incident power from port 1 to port 2. Owing to the symmetry of the structure with respect to its ports, incident power from ports 2 and 3 is similarly routed to ports 3 and 1, thus realizing the functionality of a nonreciprocal circulator with infinite isolation. Notice that the above description assumes a weak excitation of the common state, which makes clear the importance of choosing its resonance frequency as far as possible from the resonance frequency of the rotating states.
The realized device was designed to resonate at 170 MHz with a Qfactor of about 10 for V_{d.c.} = 1.99 V and V_{m} = 0. The modulation frequency was set to 15 MHz, in order for the intermodulation byproducts at frequencies ω ± ω_{m}, created by the secondary substates of R〉 and L〉, to fall outside the resonance band, whose bandwidth is here around 10 MHz. The exact values of the circuit components and its full topology are provided in the Methods and Supplementary Methods. Figure 2c shows a photograph of the fabricated prototype. We underline here the deeply subwavelength size of the realized device (∼λ/75), simply based on three lumped resonant circuit tanks.
Without modulation, the signal is equally split at the two output ports, as expected from symmetry, and the system is fully reciprocal (Fig. 3a). When the modulation signal is switched on, the symmetry is broken and power is unequally split. By varying the modulation amplitude it is possible to find a value for which the signal entering port 1 is routed exclusively to port 2, corresponding to φ_{R} = − φ_{L} = π/6. This condition is satisfied for V_{m} = 0.6 V, as can be seen in Fig. 3b: at the resonance frequency of 170 MHz, power incident to ports 1, 2 and 3 is routed to ports 2, 3 and 1, respectively, demonstrating the operation of an ideal, magneticfree, deeply subwavelength linear circulator. For comparison, Fig. 3c shows the Sparameters obtained using fullwave and circuit simulations: the agreement with the measurement is excellent.
To get a deeper insight into the effect of V_{m} on the device operation, Fig. 4a shows the transmission between ports 1 and 2 at resonance versus V_{m}. For V_{m} = 0, S_{21} = S_{12}, as expected. Increasing V_{m} results in an increase of S_{21} and a decrease of S_{12} until V_{m} = 0.6 V, where S_{12} = 0. Past this point, S_{21} and S_{12} get closer, as expected when we depart from the destructive interference condition. For very large values of V_{m}, S_{21} and S_{12} both tend to zero, because the counterrotating states move far from ω_{±} and, therefore, are weakly excited at ω_{±}. The magnitude of the asymmetry between S_{21} and S_{12} is measured by the isolation S_{21}/S_{12}, plotted in Fig. 4b on a logarithmic scale versus V_{m}. At the optimum modulation voltage V_{m} = 0.6 V, S_{21} is over four orders of magnitude larger than S_{12}, indicating giant nonreciprocity, well above the levels of any commercial magneticbased device.
Another unique property of the proposed device consists in its realtime tunability features. The biasing voltage V_{d.c.}, which provides the reverse biasing condition for the varactor diodes, determines their static capacitance. Therefore, V_{d.c.} can be used to actively control the static resonance frequency of the L–C tanks, and consequently the frequency band over which nonreciprocity occurs. Figure 4c shows the measured isolation versus frequency for V_{d.c.} varied between 1.73 V and 4.5 V. The nonreciprocal response of our device can be efficiently tuned between 150 MHz and 210 MHz, corresponding to a relative bandwidth of over 30%. Across all this range, our measured isolation is above 40 dB. This strong tuning capability is an additional advantage of the proposed device compared to conventional magneticbased microwave circulators, and it may be exploited in scenarios requiring dynamic tuning to balance changes in temperature or in the environment. The electronic spin applied to the proposed coupledresonator loop realizes the equivalent of a dynamically tunable, strongly biased ferromagnetic metamaterial substrate.
In addition to being an ideal replacement for microwave nonreciprocal components, with significant advantages in terms of size, integration, cost, linearity, tunability and noise reduction, our findings may become even more important when applied to different frequencies or other types of waves, such as light or sound. For instance, this concept may be disruptive for integrated nanophotonic technology, for which optical nonreciprocal components are critical for laser protection and signal routing. At visible frequencies, electrooptic modulation in siliconbased components is typically achieved by means of carrier injection/depletion^{24,25}, which can provide relatively strong permittivity modulation, but is typically accompanied by significant loss and low modulation frequencies for large modulation amplitudes. These side effects impose limitations on the applicability of the principle of angularmomentum biasing in uniform microring resonators^{2,3}, as discussed before. In contrast, the concept presented here ensures maximum modulation efficiency, significantly relaxing the requirements in terms of modulation amplitude. This in turn allows large quality factors and large modulation frequencies, which translate into strong nonreciprocal response in deeply subwavelength devices. We also envisage the realization of the proposed rotating modulation of coupledresonator loops in photonic crystal technology, for which highQfactor coupled cavities may be implemented and efficiently modulated^{26}. Our study also represents a new demonstration of the exciting possibilities offered by dynamic modulation of coupledresonator networks, with unique control over the flow of light, in the context of recently presented concepts of photonic topological edge states and effective magnetic fields for photons^{27,28,29}.
Methods
Modes of the coupledresonators loop.
The coupledmode equations of the system of Fig. 1 read
where a_{1}, a_{2} and a_{3} are the complex amplitudes of the three resonators. Equation (2) can be written in the more compact form
where is the state vector of the system and
In the absence of modulation, ω_{1}, ω_{2} and ω_{3} are equal to ω_{0}. Then, the eigenfrequencies and the corresponding state vectors of the system are found by the eigenvalues and eigenvectors of Ω, respectively. In particular, it can be shown that there are three states, a common state with resonant frequencyω_{c} = ω_{0} + 2κ and state vector , a righthanded rotating state with resonant frequency ω_{+} = ω_{0} − κ and state vector , and a lefthanded rotating state with resonant frequency ω_{−} = ω_{+} and state vector . Note that because the eigenvalues ω_{+} and ω_{−} are degenerate, the eigenvectors +〉 and −〉 are not the only ones corresponding to these eigenvalues. As a matter of fact, any linear combination of +〉 and −〉, such as the vectors and , are also valid eigenvectors of ω_{+} and ω_{−}. However, hereafter, we will use +〉 and −〉, because they bear an immediate physical meaning as counterrotating states of the coupledresonators loop, and they significantly simplify the mathematical analysis.
In the presence of modulation ω_{1}(t) = ω_{0} + δω_{m} cos(ω_{m}t), ω_{2}(t) = ω_{0} + δω_{m} cos(ω_{m}t + 2π/3) and ω_{3}(t) = ω_{0} + δω_{m} cos(ω_{m}t + 4π/3) result in a timedependent Ω. In this case, the eigenstates of the system cannot be found by the eigenvalues and eigenvectors of Ω, and a full solution of equation (3) is necessary. Because modulation constitutes a perturbation of the coupled system, it is convenient to express the eigenstates of the modulated system in terms of the eigenstates of the coupled nonmodulated system. To this end, equation (3) is expressed in the basis of c〉, +〉 and −〉, by multiplying ψ〉 with the conjugate transpose of the matrix
with columns the state vectors c〉, +〉 and −〉. In the new basis, , where a_{c}, a_{+} and a_{−} are the complex amplitudes of the common, right and lefthanded states, respectively, and
where
If ω_{m} ≪  ω_{c} − ω_{±}, coupling between the common and rotating states, corresponding to the first row and column of δΩ, can be neglected and equation (3) simplifies to
It can be seen that equation (4) is satisfied if
where A_{+}, A_{−} and ω are constants. Then, equation (4) becomes
which is an eigenvalue problem with respect to ω. The eigensolutions of this problem yield the eigenstates of the modulated system, as given by equation (1).
Modulation amplitude of the coupledresonators system.
We assume a resonator whose modal distribution fully resides in materials with permittivity ɛ, permeability μ, and resonant frequency ω_{0}. It can be shown from perturbation theory that a small change Δɛ of the permittivity results in the following change of the resonance frequency^{30}
where E_{0} and H_{0} are the resonant electric and magnetic field, respectively, and integration is performed all over the volume of the resonator. Furthermore, it is known that at resonance the electric and magnetic energies are equal, hence ∫ ɛ E_{0} ^{2}dV = ∫ μ H_{0} ^{2}dV and
If, in addition, ɛ and Δɛ are uniform over the volume where most of the resonator’s energy is concentrated, ∫ Δɛ E_{0} ^{2}dV ≍ Δɛ∫ E_{0} ^{2}dV and ∫ ɛ E_{0} ^{2}dV ≍ ɛ∫ E_{0} ^{2}dV, showing that the frequency modulation, δω_{m}, for a permittivity modulation amplitude Δɛ_{m} is equal to δω_{m} = ω_{0}Δɛ/(2ɛ).
Description of the experimental setup.
The complete experimental setup is shown in Supplementary Fig. 2 and a list of the associated equipment is provided in Supplementary Table 2. A waveform generator provides the modulation signal, which is split into three equal parts by means of a power divider. The output signals are then led to three phase shifters, which provide the required phase difference of 120° for the modulation signals of the three coupled resonators. The phase shifters are powered by a d.c. source and their phase shift is controlled by potentiometers. The outputs of the phase shifters are connected to the lowpass ports of three diplexers, whose outputs are connected to the RF/modulation ports of the ring. The highpass ports of two of the diplexers are connected to the VNA ports while the highpass port of the third diplexer is terminated to a matched load. The diplexers combine the modulation and RF signals and at the same time provide infinite isolation between the RF and modulation paths. By rotating the diplexers, which are connected to the VNA ports, it is possible to measure all the Sparameters of the circuit. The d.c. signal for biasing of the varactors is provided by a d.c. source connected to ports 4, 5 and 6 of the ring.
References
 1
Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alù, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).
 2
Sounas, D. L., Caloz, C. & Alù, A. Giant nonreciprocity at the subwavelength scale using angular momentumbiased metamaterials. Nature Commun. 4, 2407 (2013).
 3
Sounas, D. L. & Alù, A. Angularmomentumbiased nanorings to realize magneticfree integrated optical isolation. ACS Photon. 1, 198–204 (2014).
 4
Tanaka, S., Shimimura, N. & Ohtake, K. Active circulators—The realization of circulators using transistors. Proc. IEEE 53, 260–267 (1965).
 5
Hwang, I. K., Yun, S. H. & Kim, B. Y. Allfiberoptic nonreciprocal modulator. Opt. Lett. 22, 507–509 (1997).
 6
Bhandare, S. et al. Novel nonmagnetic 30dB travelingwave singlesideband optical isolator integrated in III/V material. IEEE J. Sel. Top. Quantum Electron. 11, 417–421 (2005).
 7
Galland, C., Ding, R., Harris, N. C., BaehrJones, T. & Hochberg, M. Broadband onchip optical nonreciprocity using phase modulators. Opt. Express 21, 14500–14511 (2013).
 8
Doerr, C. R., Chen, L. & Vermeulen, D. Silicon photonics broadband modulationbased isolator. Opt. Express 22, 4493–4498 (2014).
 9
Kodera, T., Sounas, D. L. & Caloz, C. Artificial Faraday rotation using a ring metamaterial structure without static magnetic field. Appl. Phys. Lett. 99, 03114 (2011).
 10
Wang, Z. et al. Gyrotropic response in the absence of a bias field. Proc. Natl Acad. Sci. USA 109, 13194–13197 (2012).
 11
Soljačić, M., Luo, C., Joannopoulos, J. D. & Fan, S. Nonlinear photonic crystal microdevices for optical integration. Opt. Lett. 28, 637–639 (2003).
 12
Shadrivov, I. V., Fedotov, V. A., Powell, D. A., Kivshar, Y. S. & Zheludev, N. I. Electromagnetic wave analogue of an electronic diode. New J. Phys. 13, 033025 (2011).
 13
Fan, L. et al. An allsilicon passive optical diode. Science 335, 447–450 (2012).
 14
Peng, P. et al. Paritytimesymmetric whisperinggallery microcavities. Nature Phys. 10, 394–398 (2014).
 15
Yu, Z. & Fan, S. Complete optical isolation created by indirect interband photonic transitions. Nature Photon. 3, 91–94 (2009).
 16
Lira, H., Yu, Z., Fan, S. & Lipson, M. Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip. Phys. Rev. Lett. 109, 033901 (2012).
 17
Huang, X. & Fan, S. Complete alloptical silica fiber isolator via stimulated Brillouin scattering. J. Lightwave Technol. 29, 2267–2274 (2011).
 18
Kang, M. S., Butsch, A. & Russell, P. St. J. Reconfigurable lightdriven optoacoustic isolators in photonic crystal fibre. Nature Photon. 5, 549–553 (2011).
 19
Fang, K., Yu, Z. & Fan, S. Photonic Aharonov–Bohm effect based on dynamic modulation. Phys. Rev. Lett. 108, 153901 (2012).
 20
Hafezi, M. & Rabl, P. Optomechanically induced nonreciprocity in microring resonators. Opt. Express 20, 7672–7684 (2012).
 21
Poulton, C. G. et al. Design for broadband onchip isolator using stimulated Brillouin scattering in dispersionengineered chalcogenide waveguides. Opt. Express 20, 21235–21246 (2012).
 22
Wang, DW. et al. Optical diode made from a moving photonic crystal. Phys. Rev. Lett. 110, 093901 (2013).
 23
Kamal, A., Clarke, J. & Devoret, M. H. Noiseless nonreciprocity in a parametric active device. Nature Phys. 7, 311–315 (2011).
 24
Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. Micrometerscale silicon electrooptic modulator. Nature 435, 325–327 (2005).
 25
Xu, Q., Manipatruni, S., Schmidt, B., Shakya, J. & Lispon, M. 12.5 Gbits/s carrierinjectionbased silicon microring silicon modulators. Opt. Express 15, 430–436 (2007).
 26
Shambat, G. et al. Ultralow power fibercoupled gallium arsenide photonic crystal cavity electrooptic modulator. Opt. Express 19, 7530–7536 (2011).
 27
Hafezi, M., Demler, E., Lukin, M. & Taylor, J. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011).
 28
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. Imaging topological edge states in silicon photonics. Nature Photon. 7, 1001–1005 (2013).
 29
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photon. 6, 782–787 (2012).
 30
Haus, H. A. Waves and Fields in Optoelectronics (Prentice Hall, 1984).
Acknowledgements
The work was supported in part by AFOSR with YIP award No. FA95501110009 and DTRA with YIP award No. HDTRA11210022.
Author information
Affiliations
Contributions
N.A.E. performed the experiment. D.L.S. and N.A.E. designed the structure and conducted the numerical calculations and theoretical modelling. J.S. helped in the selection and modelling of the modulation varactors. A.A. directed and supervised the project. All authors have read and commented on the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Information (PDF 536 kb)
Rights and permissions
About this article
Cite this article
Estep, N., Sounas, D., Soric, J. et al. Magneticfree nonreciprocity and isolation based on parametrically modulated coupledresonator loops. Nature Phys 10, 923–927 (2014). https://doi.org/10.1038/nphys3134
Received:
Accepted:
Published:
Issue Date:
Further reading

PhaseInduced Frequency Conversion and Doppler Effect With TimeModulated Metasurfaces
IEEE Transactions on Antennas and Propagation (2020)

Effective medium concept in temporal metamaterials
Nanophotonics (2020)

Design and modulation of the plasmoninduced transparency based on terahertz metamaterials
Infrared Physics & Technology (2020)

Serrodyne Frequency Translation Using TimeModulated Metasurfaces
IEEE Transactions on Antennas and Propagation (2020)

Linear isolators using wavelength conversion
Optica (2020)