Terahertz time-domain derivative spectrometer using a large-aperture piezoelectric micromachined device: supplement

The engineering of optomechanical systems has exploded over the past decades, with many geometries and applications arising from the coupling of light with mechanical motion. The modulation of electromagnetic radiation in the terahertz (THz) frequency range through optomechanical systems is no exception to this research effort. However, some fundamental modulation capabilities for THz communications and/or high-speed data processing applications are yet to be established. Here, we demonstrate a THz time-domain derivative spectrometer based on a piezoelectric micromachined (PM) device. Insertion of the PM device into the THz beam path provides reference modulation for the lock-in detection unit, which in turn provides access to the nth-order derivative information of the incoming THz signal. Strikingly, the integration of the recorded derived signal leads to a recovered reference signal with an equivalent or even better signal-to-noise ratio, opening the door to a new type of highly sensitive THz measurements in the time domain.


Lock-in amplifier:
Lock-in amplifiers are essential to measure a weak signal within a noisy background. A lock-in amplifier is a system which selectively amplifies this signal at a given reference frequency. Fig.S1 shows the input signal, , which is multiplied by a reference wave, , at reference frequency which modulates the input signal. Then, after applying a low-pass filter (LPF), the final result is a DC signal, which is the signal of interest to measure [1]. In our experiments, we used a SR865A lock-in amplifier from Stanford research. The input signal is a sinusoidal signal with frequency of , at 10.82 . In general, the low LPF period is always fixed to 1/ , regardless of the harmonic order to be detected (i.e. N as a harmonic number on the lock-in amplifier) in the operation of the lock-in amplifier [1].The output signal will be averaged over N periods of the detected frequency. Then the lock-in is able to detect signals at harmonics order of the frequency of the reference signal. Simply, the lock-in multiplies the input signal with a digital sine wave with frequency of N× . Therefore, only input signals at this, Nth harmonics order, will be detected. Consequently, the other signal is not detected. In order to describe the performance of the PM device in combination with lock detection, the relationship between the moving parts of the PM device and lock detection must be explained. In this context, it is important to understand two things: (i) how the lock-in amplifier reads "device position" and (ii) how the harmonic function operates according to this first description. In our case, the lock-in reference signal is a fixed sine wave that drives the PM device, called the modulation frequency , or modulation angular frequency Ω . The angular frequency Ω of the reference signal is changed based on the selection of the number of harmonics in the lock-in, N. Therefore, for the first harmonic measurement, is 2 . The angular frequencies for the second, third, and fourth harmonic measurements are 4 , 6

Low pass filter
, and 8 , respectively. This means that to measure the first derivative of the THz beam, the lock-in detects the differential THz signal of two positions of the moving part of the PM device during a period of one cycle from 0 to 2 . By increasing the harmonic order N, the lock-in amplifier detects the signal at N times the referenced signal. Therefore, it will have N times more stop position of the PM device to read the THz field amplitude and thus N times a subdivision of the differential optical path leading to the measurement of the N-order derivative function of the THz waveform. These subdivisions are carried over periods of 4 , 6 and 8 , corresponding to the 2 , 3 and 4 ℎ harmonics respectively. As an example, the relationship between the motion and the lock-in detection phase as function of the Nth order derivative of one part of the PM device is shown in Fig.2 of the main article for the green wings (see at the top of these figures). As mentioned in the article, in Fig.2(a), (b), (c) and (d) the value of displacement with its corresponding phase value are shown for each Nth order derivative function. For measurement of the second derivative signal, as the harmonic N increases to 2, the angular frequency, Ω , of the reference signal for Lock-in detection will be 2 (2 ), resulting in detection over two distinct displacement regions, i.e., +6.33 to 0 and 0 to -6.33 also corresponding to the 0 to 2 and 2 to 4 phases. These subdivisions in displacements of the device and the corresponding phases according to the order of the harmonics N increase accordingly, see (c) and (d) in this figure for the 3 and 4 ℎ harmonics, respectively. In light of this explanation, it is important to note that in the experimental measurement, the derived information obtained from the incoming THz signal results from the combination of the complex movements of each part of the PM device, i.e., green and yellow wings as well as grey central part displacements.

Simulation study of PM device:
In order to simulate the effect of the PM device motions on the THz field, we used the Finite-Difference-Time-Domain (FDTD) software from Lumerical. As the simulations produce discrete information, we used the differentiation function to calculate numerically the ℎ order derivative that expressed as an analytical expression in the form of a set of discrete points [2].We first simulated the response of the PM device for discreet displacement points in the FDTD simulation and then we calculated the Nth order derivative of corresponding data from simulation using the numerical differentiation function. Table S1 shows the formula of the central difference approximation of the derivatives [2] that we used to calculate the first-, second, third, and fourth-order derivatives from simulation data using various set of discrete data . The discrete Table S1. shows the finite central difference formulas for numerically calculating the first-, second-, third-, and fourth-order derivatives, where h is the time interval difference between points, is the center point of origin, and + is the displacement for each device motion.
Derivative order Definition Eq. set of points  THz beam is assumed to be Gaussian with a central frequency of 0.83 THz and a bandwidth of 2.5 THz. Finally, the incident beam impinges normal to the surface of the PM device and then the reflected THz signal is recorded by a monitor. In (a), the entire device undergoes a round-trip motion of ±6. 33 . In (b), the green and yellow wings and the central part of the PM device undergo a motion of 12.66 , 2.45 , and 1.42 , respectively. To better understand how to use the numerical differentiation function for the flat surface, Fig.S3 shows a set of (a) two, (b) three, (c) six, and (d) five discrete simulation conditions. The positions chosen correspond to the data needed to compute the finite central difference formulas for recovering the first-, second-, third-, and forth-order derivative of the THz signal, respectively. For example, Fig.S3 (e) illustrates how to position the device at a set of three locations in the simulation to obtain the second order derivative of the THz signal from the motion by using Eq.(S2) in table S1. The temporal derivative traces obtained by simulations are shown in Fig.S4 with (a) the reflection on a flat surface with uniform displacements and (b) on an angular surface with complex displacements. The subfigures show the 4 signals derived from the simulation data using the numerical differentiation function (in red) and from the simulated THz reference signal using the derivation function (in blue). An excellent match is obtained between the simulation results and the calculation from the reference signal when the PM device acts as a flat surface with uniform displacements. In (b), when the PM device operates with complex angular motions, we observed a very good agreement for the 1 and 2 order derivatives with the calculations. However, the simulations of the 3 and 4 ℎ order derivatives produce a significant difference to the response of the calculations of these same derivatives. We translate this difference into the fact that the numerical method uses simulation data with discrete points and the signal reflected from the non-moving parts of the device has a magnitude that far exceeds the signal reflected from the moving parts of the device.

Difference between measurement and calculation of derivatives:
To explain the discrepancy between the experimental and calculated derived THz signal, as shown in Fig. 4(e) in the main article, we compared the THz signal reflected from all elements of the PM device and that reflected from the moving parts of the PM device, i.e. by FDTD simulation. Fig.S5(a) shows the spectra of the reflected THz signals from these two cases. It is clear that the amplitude of the signal reflected by all the elements (blue curves) is greater than that of the signal reflected by only three parts (red curve) of the device, which is explained by the smaller surface covered by the moving parts of the device. More interestingly, in Fig.S5(b), the normalization between the two spectra reveals a significant shift for frequencies below 0.7 THz, as was also observed experimentally in Fig. 4(e) in the main article. Therefore, we suggest that the main difference between the experimental and calculated results comes from the way we measured our reference spectrum. This result is important because it also suggests the importance of using a uniform moving structure to accurately realize the derived function over a wide range of frequencies.