Nonreciprocal Radio Frequency Transduction in a Parametric Mechanical Artificial Lattice

Generating nonreciprocal radio frequency transduction plays important roles in a wide range of research and applications, and an aspiration is to integrate this functionality into microcircuits without introducing a magnetic field, which, however, remains challenging. By designing a 1D artificial lattice structure with a neighbor interaction engineered parametrically, we predicted a nonreciprocity transduction with a large unidirectionality. We then experimentally demonstrated the phenomenon on a nanoelectromechanical chip fabricated by conventional complementary metal-silicon processing. A unidirectionality with isolation as high as 24 dB is achieved, and several different transduction schemes are realized by programing the control voltage topology. Apart from being used as a radio frequency isolator, the system provides a way to build a practical on-chip programmable device for broad research and applications in the radio frequency domain.

Figure 1

Concept of the mechanical artificial lattice. (a) A chain of harmonic oscillators with adjacent ones being parametrically coupled, with coupling strength ${\eta }_i\cos ({\omega }_i^{p}t)d_i(t)$ between oscillator $i$ and $i+1$; each oscillator has a decay rate ${\gamma }_i$ due to its interaction with the environment. Coherent oscillation waves propagate forward (purple arrow) or backward (yellow arrow) unidirectionally in such a lattice by the design of the coupling topology. (b) The parametric coupling frequency ${\omega }_i^p$ is set to satisfy the frequency conversion condition ${\omega }_i^p=|{\omega }_i-{\omega }_{i+1}|$. (c) The periodic modulation $d_i(t)=\cos (\mathrm{\Omega }t-i\mathrm{\Theta })$ can be taken as a delay $\tau =\mathrm{\Theta }/\mathrm{\Omega }$ of oscillator ($i+1$) relative to oscillator $i$. (d) The results of such a lattice can be expressed as a three-port device, with one port connected to the environment. For positive delay ($\tau >0$), forward coherent signal input at port 1 could transport to port 2 with only weak decay into the environment. For negative delay, only a backward signal passes. For continuous coupling, i.e., $d_i(t)=1$, the device is bidirectional in that both forward and backward signals pass.

Figure 2

Numerical simulation of the model. (a) Dependence of the isolation of forward transduction $I_{\text{for}}$ as a function of coupling strength $\eta$ (or, alternatively, the corresponding normalized coherence length $L_c$) and the modulation frequency $\mathrm{\Omega }$. The modulation takes the form $d_i(t)=\cos (\mathrm{\Omega }t-i\mathrm{\Theta })$, with $\mathrm{\Theta }=\pi /8$ and the lattice number $N=9$. Insets are the profiles indicated by the arrows. (b) Same as panel (a) but as a function of $\mathrm{\Theta }$ and $\mathrm{\Omega }$, with coupling strength $\eta =0.001\mathrm{N}/\mathrm{m}$. (c) Transduction losses for forward (blue points) and backward (yellow points) propagations as a function of lattice number $N$. Positive delays are employed and the parameters are set for each $N$ so that the forward transduction loss is always approximately $-6\mathrm{dB}$ and, at the same time, the isolation is optimized (indicated by the light blue regime). (d) The system’s frequency responses to forward (right panel) and backward (left panel) transductions, with the inset being the zoom-in of the panel. The modulation is $d_i(t)=\cos (\mathrm{\Omega }t-i\mathrm{\Theta })$ with $\mathrm{\Omega }/2\pi =50\mathrm{Hz}$, $\mathrm{\Theta }=\pi /8$, and $\eta =0.001\mathrm{N}/\mathrm{m}$. The responses are normalized to the peak values when $d_i(t)=1$. (e) Same as panel (d) but for the time-reversal case, i.e., $\mathrm{\Omega }\to -\mathrm{\Omega }$. (f) Same as panel (e) but under continuous coupling $d_i(t)=1$. The peak values equal to 1 because of normalization.

Figure 3

Experimental device and scheme. (a) Photograph of the device chip, whose size is approximately $3\times 3\mathrm{mm}$. The mechanical artificial lattice is located in the middle, while the effective device is less than $100\mu \mathrm{m}$, or $1/(1\times {10}^6)$ of the wavelength of the electromagnetic field of the same characteristic frequency. (b) Scanning electron microscopy of the artificial lattice in false color. The oscillators are clamped silicon beams with a thin layer of gold deposited on top of them. The first oscillation mode of each beam vibrates along the $x$ axis. The magnetic field is applied along the $z$ axis for excitation (input) and detection (output), as indicated by the leftmost and rightmost arrows. Parametric couplings used to program the functionality of the lattice are realized by voltage $V_i(t)$ applied between adjacent beams. (c) The pulse scheme for realization. Parametric coupling between beams $i$ and $i+1$ is realized by a continuous pumping wave ${\eta }_i\cos ({\omega }_i^{p}t)$ modulated by the plotted squared wave. The repetition frequencies for all channels are the same $\mathrm{\Omega }$, while a delay $\tau$ is set between adjacent channels.

Figure 4

Results of programmable unidirectionality transduction. (a) A represented frequency response when only two adjacent beams are coupled by continuous pump $d_i(t)=1$. The splitting $\mathrm{\Delta }{\omega }_i$ is used to calibrate the coupling strength (data from beam 7 with input to the coupled beam 6). Dots are measured data and the solid curve is a Lorentz fitting to guide eyes. (b),(c) Same as panel (a) under the same pump strength but with three beams (beams 5, 6, and 7) and four beams (beams 4, 5, 6, and 7) coupled, respectively. (d) The transduction of the lattice with $N=7$ under time-dependent modulation with positive delay; parameters used are $\mathrm{\Theta }=\pi /6$, $\mathrm{\Omega }/2\pi =50\mathrm{Hz}$, and ${\eta }_i\approx 0.001\mathrm{N}/\mathrm{m}$. The backward result is shown on the left, with excitation applied on beam 9 and amplitude measured on beam 3, normalized to response of continuous pump. (e) The same as panel (d) but with negative delay. The isolation is calculated to be 24 dB by using the data shadowed. (f) The same as panel (d) but with continuous drive $d_i(t)=1$.