Optical wireless information transfer with nonlinear micromechanical resonators

Wireless transfer of information is the basis of modern communication. It includes cellular, WiFi, Bluetooth, and GPS systems, all of which use electromagnetic radio waves with frequencies ranging from typically 100 MHz to a few GHz. However, several long-standing challenges with standard radio-wave wireless transmission still exist, including keeping secure transmission of data from potential compromise. Here, we demonstrate wireless information transfer using a line-of-sight optical architecture with a micromechanical element. In this fundamentally new approach, a laser beam encoded with information impinges on a nonlinear micromechanical resonator located a distance from the laser. The force generated by the radiation pressure of the laser light on the nonlinear micromechanical resonator produces a sideband modulation signal, which carries the precise information encoded in the subtle changes in the radiation pressure. Using this, we demonstrate data and image transfer with one hundred percent fidelity with a single 96-by-270 μm silicon resonator element in an optical frequency band. This mechanical approach relies only on the momentum of the incident photons and is therefore able to use any portion of the optical frequency band—a band that is 10 000 times wider than the radio frequency band. Our line-of-sight architecture using highly scalable micromechanical resonators offers new possibilities in wireless communication. Due to their small size, these resonators can be easily arrayed while maintaining a small form factor to provide redundancy and parallelism.


FURTHER CHARACTERIZATION OF FIRST ORDER UPPER SIDEBAND
In addition to the sideband characterizations presented in our paper, we performed measurements presented below. In order to successfully transmit data, it is useful to produce the largest sideband amplitude possible, but to avoid noisy regions in the frequency spectrum as much as possible. To assist in choosing a laser modulation frequency, the results in Supplementary Figure  S2a were used.
These results show how the sideband size varies with the laser modulation frequency. As the modulation frequency increases, the sideband size decreases approximately exponentially. Evidently, a small modulation frequency should be most effective. However, as approaches the frequency of the carrier signal, the size of the noise increases rapidly, and can hide the sideband at random times. We found that the noise signal was small enough that we could consistently measure the sideband successfully when the modulation frequency was larger than 1 kHz.
In addition, we verified that that the sideband size increases with the size of the modulation signal (Supplementary Figure S2b) as well as with the size of the carrier signal (Supplementary Figure   S2c). From these results, it can be seen that the sideband is not measureable until the average laser diode current is at least 30 mA. While the results shown are only for an oscillator power of 0 dBm, this threshold was consistent for other powers as well. Based on these plots, it is obvious that the largest sideband can is obtained by maximizing the LD current and the oscillator driving power.

EFFECTIVE MODAL MASS AND MODAL STIFFNESS
Using COMSOL, the mode shown in Supplementary Figure S3 was found to be the modeshape of the resonance used in our experiments.
Using that modeshape, the effective mass was calculated using where m eff is the effective mass, ρ is the material density, and U is the modeshape, normalized such that the maximum displacement is 1. The effective mass was found to be 24.5 ng. The effective modal stiffness can then be found using where k eff is the effective stiffness and ω 0 is the angular resonance frequency.

EFFECTIVE FORCE PRODUCED BY ELECTRODES
To first order, the substrate (silicon and silicon dioxide layers) are assumed to be stationary. Using basic analysis of the AlN layer, we calculate that the location of the center of mass (COM), relative to the grounding electrode, moves according to where t 0 is the natural thickness of the AlN without applied stresses or fields, d 33 is the piezoelectric charge constant relating deformation along the thickness to the potential change along the thickness, V 0 is the peak applied voltage, f is the driving frequency, and t is time. Using this, the amplitude of the force applied to the AlN by the substrate is where m is the mass of the material expanding and contracting due to the applied potential. For our resonator, this comes to approximately 342 μN. Normalizing this by the mode shape gives us an effective force of 10.6 μN.

ESTIMATION OF FORCE PRODUCED BY RADIATION PRESSURE
The radiation pressure produced by an electromagnetic wave can be written as where R is the reflectance of the material on which the radiation is incident, I rad is the intensity of the wave, and c is the speed of light. Supplementary Equation S1 assumes that all incident radiation is either absorbed or reflected from the surface, and that none is transmitted through the resonator. The top surface of our MEMS resonator is primarily covered with molybdenum, which as a reflectivity near 60% at wavelength 520 nm.
The specific laser diode that we used in this experiment has an output of 15.0 mW when the current is 141.6 mA. The beam passes through a collimator, which produces a spot with diameter, 2 w, of 1.48 mm (using the 1/e 2 method) and which diverges at 0.029 degrees. At the surface of the resonator which is about 30 cm from the collimator, the spot size is 1.78 mm.
The beam is estimated to have a Gaussian profile, so 86.47% of the optical power is deposited within the 1.78-mm spot. It is difficult to verify that oscillator is precisely in the center of the spot, so we average the power over the spot size (so as to not overestimate the force) to find an average intensity of 5190 Wm −2 . Inserting this into Supplementary Equation S1, and multiplying it by the 270-by-96 micron top surface area of the resonator, we find an approximate radiation force of 0.718 pN. Note that the area used in calculating the modeshape should be normalized by the mode shape. However, since we use the average laser intensity instead of the maximum, we assert that these errors approximately cancel. Further, this number is used as an order-ofmagnitude estimate for comparison purposes only; the exact size of the force is not critical to the results presented in the paper.

DUFFING EQUATION FAST FOURIER TRANSFORM
To numerically verify that upper and lower sidebands can be produced by driving the duffing resonator with a large signal at resonance and a small non-resonant frequency to produce a sideband, we performed numerical simulations using Wolfram Mathematica. The first-order lower and upper sidebands are in Supplementary Figure S4. Simulations were performed using a resonance frequency normalized to 1, a Q of 1000, and softening k 3 value of − 1. The resonant signal is 10 times larger than the nonresonant signal.
The code used to generate the data in the plot above is below. ClearAll["Global`*"] F s = 200; (* Sampling Rate in Hz *) T = 10000; (* Sampling period in s *) L = T*F s ; (* Number of points in signal table*) stepSize = 0.001; Q = 1000; The results may be plotted using a list line plot.

Figure S1
(a) Frequency shift of 120.4 MHz resonance peak as a function of temperature. The shift is measured in parts per million of the frequency at 25°C. (b) Shift of the 120.4 MHz resonance peak as the laser diode current (power) is increased. As the current increases, the steady state frequency is decreased, indicating heating. Beyond 20 mA, the resonance peak shifts at an approximate rate of − 0.6 ppm mA −1 , indicating a steady-state temperature increase of 0.04°C mA −1 . Figure S2 (a) Upper sideband size as a function of laser modulation frequency for 19 dBm oscillator driving and 70 mA average LD current. As the frequency increases, the sideband size exponentially decreases. (b) Sideband size as function of laser diode current for 0 dBm oscillator driving. Near 30 mA, the LD power is high enough to begin producing measureable sidebands. After that the sideband size increases linearly with LD current. (c) Sideband size as function of oscillator driving power. As the power to the oscillator is increased and its response becomes increasingly nonlinear, the sideband grows.

Figure S3
Numerically-identified modeshape used in wireless data transfer experiments. Color scale is arbitrary, and length scale is in microns.

Figure S4
Frequency spectrum numerically generated using a Duffing resonator driven at resonance and at a smaller modulation frequency. The amplitude is on a logarithmic scale so the sidebands can be seen in the same plot as the resonance.