1. Introduction

In recent decades, the development of several techniques for the generation and detection of electromagnetic waves in the terahertz (THz) frequency range [1,2] has led to a wide range of applications in the fields of security [3,4], nondestructive testing [5,6], spectroscopy [7–9], telecommunications [10,11], imaging [12,13] and microscopy [14–16]. In order to improve the performance of THz systems in these various applications, the design of innovative and efficient modulators in this frequency range is the subject of intense research activity [17–19], particularly for sensing applications [17,18] and data transmission systems [19]. As in the world of optics for ultrafast signal processing [20], it would be necessary to design and implement fundamental mathematical operators, such as differentiator [21], integrator [22] and Hilbert transformer [23]. A differentiator is one of these fundamental operators that performs the mathematical action of differentiating an input signal. In the same way, an nth-order derivative type differentiator provides an nth-order derivative type output signal derived from an arbitrary input signal, which can be a function of time, frequency or wavelength [24]. Differentiators have been realized for the ultrafast processing of light in the temporal, spatial and spectral domains, which have generally been realized using special fiber-optic routing and wave-mixing devices [25]. When dealing with slowly varying optical signals, a simple method to obtain the derivative of the input signal is the use of a resistor-capacitor (RC) filter in combination with a modulation device in the optical path, such as a tachometer [26]. This latter case is a well-established method that has been used for decades is known as derivative spectroscopy [26–30]. While the ultrafast optical differentiator [20–25] and derivative spectroscopy [26–30] approaches deal with the complex envelope of the intensity profile of an input signal, THz measurement systems provide direct access to the amplitude and phase of the electric field, thanks to coherent methods of generation and detection by sampling [31,32]. This additional feature can have several advantages for signal processing in the THz range. One such advantage is the use of low-frequency electronic tools, such as a lock-in amplifier, to recover ultra-fast THz phenomena [33]. Since a lock-in amplifier is already a sophisticated RC filter, the addition of a suitable modulator only could in principle transform the traditional THz time-domain spectroscopy (THz-TDS) system [6,33] into a THz time-domain derived spectrometer (THz-TDDS). Using electronics, an accurate reading of the derived THz signal could quickly measure the slope and/or the position of the peak in the time-varying signal [34]. Derived information could also be applied to allow the precise identification of fixed points, such as local maxima and minima. The latter case could be of great interest for THz phase-contrast imaging, improving the accuracy of layer detection [35] or refractive index changes [36]. However, to date, only a few attempts have been made to demonstrate passive THz time differentiators, using a grating [37] and a silicon ring resonator [38]. More recently, we reported an active THz time differentiator with an 8$\times$8 piezoelectric micromachined ultrasonic transducer (PMUT) array [39]. Nevertheless, the latter demonstration was limited due to the small pixel size (i.e., sub mm$^{2}$) and the small motion of the PMUT unit cell relative to the THz propagation path. Additionally, the need to use a voltage driver with a MHz-scale oscillator to sustain the resonant frequency of the PMUT array made its use less convenient for the lock-in amplifier. Accordingly, we demonstrate here an nth-order THz time-domain derivative spectrometer using a piezoelectric micromachined device (PM) with a surface area that is larger than 1 mm$^{2}$ and operating in the kHz frequency range. Originally designed for energy harvesting [40], once inserted in the THz beam path as a modulator, this new device has an impressive performance to accurately measure not only the first derivative of the incoming signal, but our results show a good response up to the fourth-order derivative. This paper is organized as follows; we first present the proposed structure for PM devices and its functionality. We also explain the operating principle of the lock-in amplifier to access the multiple-order derivative of the incoming THz signal. We then describe the experimental setup and the results obtained. Finally, we show the good agreement of our experimental results with the simulations and demonstrate the ability to recover the original signal with high accuracy by integrating the information from the differentiated signal.

2. PM device material and its characteristics

Figure 1(a) shows the PM device with its electrodes for controlling and activating displacements. The PM device has been fabricated using the commercial PiezoMUMPS process from MEMSCAP, detailed in [41]. The design, fabrication and characterization of such PM are detailed in [40]. The surface area covers 2.1 $\times$ 2.1 mm$^{2}$ on a 0.4-mm-thick silicon-on-insulator (SOI) substrate. This metallic multilayer device [see Fig. 1(b)] consists of a stack from top to bottom: 1-$\mu$m-thick aluminum (Al), 0.02-$\mu$m-thick chromium (Cr), and a 0.5-$\mu$m-thick piezoelectric material (aluminum nitride (ALN)) deposited on the 10-$\mu$m-thick top silicon (Si) layer of the SOI substrate. As shown in Fig. 1(a), the metallic multilayer acts as an upper electrode layer and the Si layer acts as the lower electrode to piezoelectrically control the displacement of the structure. To avoid short circuits where needed, an additional insulating layer of silicon dioxide with 0.2 $\mu$m is deposited between the Si layer and the metallic multilayer stack. To allow large vertical movements around its resting position, the structure is suspended above an empty cavity with an area of 1.7 $\times$ 1.7 mm$^{2}$, as illustrated in Fig. 1(c). Inside this cavity, the maximum round-trip displacements perpendicular to the plane of the substrate for the green wing, the yellow wing and the central (grey) part of the device are 12.67 $\mu$m, 2.45 $\mu$m and 1.42 $\mu$m, respectively. Indeed, these values are based on the simulation and measurement of the mode shape of the PM device as presented in [40]. The PM device is a resonator, and therefore it is possible to excite it with a sinusoidal signal, in which case each moving part of the PM will oscillate at the same frequency. Thereafter, this frequency will be referred to as the modulation frequency ($f_\textrm {mod}$). To achieve the maximum displacement at $f_\textrm {mod}$, an external sinusoidal voltage amplitude 20 $V$ peak-to-peak at 10.82 $kHz$ is applied across the electrodes.

 

Fig. 1. Illustration (a) of the PM device, (b) with its layers of aluminum chromium (Al Cr), aluminum nitride (AlN), silicon (Si), and on a silicon-on-insulator substrate. (c) A graphical representation of the complex motion of the three separate areas of the PM surface; the center, yellow wings, and green wings, with lengths of 12.67 $\mu$m, 2.45 $\mu$m, and 1.42 $\mu$m, respectively.

Download Full Size | PDF

3. Principle of operation

To understand the difference between a typical THz wave measurement and the use of our device, it is important to recall the basic concept of lock-in detection. As illustrated inSupplement 1, Fig. S1, a lock-in amplifier is a kind of alternating current (AC) voltmeter that mixes an input signal with a periodic reference signal [42]. For THz-TDS measurements, the electro-optical sampling method is commonly used and requires a lock-in amplifier [33]. This method involves either a mechanical chopper or a control of the bias voltage of a photoconductive antenna (PCA), the latter acting as an ON-OFF switch of the input signal. Unlike the electronic modulation of a PCA or the mechanical chopping of the THz signal, our device does not modulate the input signal ON and OFF. In fact, the PM device induces a small change in the path length of the THz beam [39]. This modulation causes the physical system we are interested in (i.e., the THz signal) to respond with a frequency $f_\textrm {mod}$, whereas the detector and lock$-$in amplifier convert its response into a voltage. For a lock-in amplifier, a stimulus $x(t)$ of amplitude A and oscillating with an angular frequency $\Omega _\textrm {mod}$, which is $2\pi f_\textrm {mod}$, around an average value $\bar {x}$ can be expressed by:

Upon application of the stimulus, the detector reads a time-varying signal $V(t)$:

Under the assumption that the modulation A is sufficiently small, we can approximate V(x(t)) by a Taylor-series expansion about $\bar {x}$. Passing through the lock-in amplifier provides an output that is proportional not only to the modulation $A$, but also to the derivative of the system’s response to the stimulus, evaluated at $x=\bar {x}$ [42]:

According to Eq. (3), changing the ON-OFF modulation for a change in THz path length converts a THz-TDS system into a THz derivative spectrometer. Furthermore, as with derivative spectroscopy [28], reading the $N$th-order harmonic signal from the lock-in should, in principle, provide access to the Nth-order derived function of the modulated signal [42]. To describe the functionality of a THz derivative spectrometer obtained by combining a PM device with a lock-in amplifier, we show in Fig. 2 the relationship between the motion of a part of the PM device (see the green wings at the top of the figure) as a function of the lock-in detection phase. In Fig. 2(a), assuming an excitation frequency at $f_\textrm {mod}$ with 20 V peak-to-peak, a phase change of $2\pi$ corresponds to a complete excursion of about 12.66 $\mu$m on the wing edge. In such a condition, the PM motion still induces a sufficiently small differential response $\Delta E / \Delta t$, i.e., greater than the detected bandwidth of the measured THz pulses [39], to recover a derivative measurement. In Figs. 2(b)–2(d), we show the value of the phases with respect to the excitation frequency $f_\textrm {mod}$ corresponding to the second, third and fourth harmonic functions of the lock-in amplifier, respectively. In these cases, the harmonic function of the lock-in amplifier allows the slope of the slope to be determined, and so on as the harmonic value is increased (for more information, see Supplement 1).

 

Fig. 2. Illustration of the displacement of a PM device in the $y$ direction relative to its $x$ position, all in relation to the phase interpretation of the lock-in amplifier for the measurement of (a) 1st-order, (b) 2nd-order, (c) 3rd-order, and (d) 4th-order derivatives.

Download Full Size | PDF

4. Experimental setup and results

Terahertz spectroscopy [43] is well-known for the characterization of THz devices, such as modulators [44], filters [45] and artificial metamaterials [46], to name a few applications. Similar to these experiments, we employed a THz-TDS system based on the transmission and detection of THz waves by two PCAs from TeraVil Ltd. The PCAs were pumped using a Ti:Sapphire laser oscillator providing 810 nm, 40 fs laser pulses at a repetition rate of 80 MHz and an average power of 20 mW on each antenna. The hemispherical silicon lens at the back of the transmitting antenna collimates the THz output beam with a diameter of about 10 mm. As shown in Fig. 3(a), the collimated THz beam is guided by a gold-coated flat mirror, followed by a concave mirror 3 inches in diameter and 3 inches in focal length. The PM device is located exactly at the focus of the concave mirror, and the reflected beam returns symmetrically to the opposite side of the concave mirror, where the THz beam is recollimated. Finally, another gold-coated flat mirror guides the THz beam to the high resistivity hemispherical silicon lens of the PCA detector. The latter focuses the THz beam on the micrometer gap of the antenna. In order to study the behavior of our PM device, we first measured the THz signal without any external electric field applied to the device, as shown in Fig. 3(b). In this case, the PM device is considered at rest, behaves like a metallic mirror, and only the THz transmitting antenna is biased with a voltage of 40 V modulated at a repetition frequency of 30 kHz. This ON-OFF modulation on the emitted field of the transmitting antenna is the typical lock-in detection operation used for the standard THz-TDS configuration [43], hereinafter referred to as the THz reference measurement. Figures 3(b) and 3(c) show the THz pulse reflected from the PM device (i.e., in the state of rest) and its fast Fourier transform (FFT) spectra, which reveal a bandwidth of 0 to 5 THz with the central frequency located at 0.7 THz. To study the modulation effect of the PM device on the THz signal, the lock-in detection is referenced to the 10.82 kHz excitation frequency obtained using an AC signal generator and the time steps used for all measurements are set to 66 fs. Figure 4 shows the time traces (in red) obtained with the lock-in amplifier and the PM device activated for (a) the first harmonic, (b) second harmonic, (c) third harmonic and (d) fourth harmonic. The black trace represents the reference signal, i.e., the one obtained with the emitter modulated ON and OFF. The traces in blue represent (a) the first derivative, (b) second derivative, (c) third derivative and (d) fourth derivative of the reference signal obtained using the mathematical derivation function. Figure 4 also shows that the signal modulated by the PM device (in red) is slightly different from the one calculated (in blue). This difference is mainly observed for the first (a) and second derivative (b), while a great similarity exists between the experimental and calculated data for the third (c) and fourth derivatives (d). Nevertheless, as shown in Fig. 4, the slope of the reference signal is recovered by the odd-order derivatives (1st and 3rd), while the peak of the reference signal is found by the even-order derivatives (2nd and 4th), as expected for the derived function.

 

Fig. 3. (a) Experimental setup in reflection geometry using a gold-coated concave mirror (in position of $M_{2}$) of 3-in focal distance and a signal generator to drive the PM device to modulate the THz signals. (b) and (c) are the measured THz pulse and its spectrum, respectively, when the modulation is applied to the antenna. The inset in (c) shows the measured reference spectra.

Download Full Size | PDF

 

Fig. 4. The reference signal (black) when the PM device is not operating, the temporal measured THz waves (red) using PM device as a differentiator with lock-in amplifier set to (a) 1st, (b) 2nd, (c) 3rd and (d) 4th harmonic measurement and the calculated derivative signal of the reference (blue) which are normalized separately. In (e)-(h) are the corresponding retrieved THz spectra from (a)-(d), respectively.

Download Full Size | PDF

5. Analysis and discussion

As THz measurements are performed on the electric field as a function of time, the FFT allows us to move into the frequency domain to analyze the information in amplitude and phase. Figure 4 also shows the normalized amplitude spectra in a linear scale of the reference signal (black curves), and in red the signals derived from (e) 1st, (f) 2nd, (g) 3rd and (h) 4th order recovered by the PM device. The blue curves represent (e) the first-order, (f) second-order, (g) third-order and (h) fourth-order derived signals calculated from the reference signal. As for the information in the time domain, there is some difference in the low-frequency part between the calculated and experimental spectra, but mainly for the first-order derivative [see Fig. 4(e)]. In this figure, as highlighted with green color, it can be seen that frequencies around 0.7 THz are modulated differently from their higher frequency counterpart [i.e., frequencies shorter than 1 THz or wavelengths longer than 300 $\mu$m]. The increase in the order of the derivatives produces a closer agreement between the experimental information and the calculated derived information [see Figs. 4(f)–4(h)].

These discrepancies between the low and high frequencies of the first derivative signal are partly due to the complex motion of the PM and the wave diffraction principle [47]. Therefore, the spatial distribution is dictated by the diffraction-limited condition at focus and is not uniform with frequency. In other words, only longer wavelengths are reflected from the entire substrate (2.1 x 2.1 mm$^{2}$), in contrast to shorter wavelengths, which are mainly reflected at its center. In this case, it is reasonable to assume that the low-frequency part of the spectrum is more affected by the complex geometry of the device, while the higher frequencies are mainly located in a uniform area, the central part. Moreover, as mentioned previously, the reference signal is taken for a complete reflection on the whole substrate and the whole device simultaneously. However, the modulated signal only comes from the reflection of the moving complex geometry of the device, as it shown in Supplement 1, Fig. S5. Only this "complex" signal should be used for comparison purposes, as a reference. Unfortunately, this reference information is inaccessible to obtain experimentally with this device. Fortunately, we have to stress that since the derivative function acts as a high-pass filter [48], the transition to higher-order derivative information becomes less sensitive to the complex geometry of the device, or to the discrepancy observed at low frequencies. The shift of the peak position of the spectra towards higher frequencies, up to 2.5 THz for the 4th derivative order, confirms this high-pass filtering effect and is in a very good agreement between the calculated and measured high-order derivative [see Figs. 4(f)–4(h)].

 

Fig. 5. (a) Shows the transfer function of the PM device and (b) presents the ratio of normalized amplitude spectra between the experimental data and the calculated signal to evaluate the PM device as a THz differentiator for four order derivative measured signals that are first, second, third and four derivatives.

Download Full Size | PDF

To confirm the quality of the derivative measurements, Fig. 5(a) shows the calculated transfer function of the PM device obtained from the integral of the measured first derivative signal over the measured reference signal. The majority of the spectral range, i.e., from 0.8 to 4 THz, shows a flat response while complex variations appear for frequencies below this range. As mentioned earlier, the difference in beam size at the focus position as a function of frequency [15] accounts for this non-uniform frequency response and can now be viewed as a new spectral response function for this system. In Fig. 5(b), we also show the normalized amplitude between the experimental data and the expected signal, i.e., those calculated from the reference signal. Normalization is expressed by the ratio between the experimental signal and the calculated derivative signal depends on the frequency, called $\alpha (\omega )$, as follows:

 

Fig. 6. (a) and (b) show the integral of the first and second derivative measured signals in time domain (red) with the reference (black). (c) and (d) show the corresponding spectra of these two signals compared with the reference spectra and the insets highlight the water absorption lines for selected frequency ranges.

Download Full Size | PDF

where $N$ is the derivative order, Ref. and Exp are the reference and experimental modulated THz signal, respectively. In order to qualitatively evaluate the purity of the experimentally derived signals, a ratio $\alpha$ equal or close to 1 would mean perfect agreement. In Fig. 5(b), the results of this calculation show that the ratio $\alpha$ for the frequency range between 0.7 THz to 3.5 THz is almost flat around 1, which clearly indicates a good performance of the PM device with a lock-in amplifier as a THz derivative spectrometer. Only for frequencies below 0.7 THz, the performance of the PM device did not allow accurate recovery of the "readable" original signal. A second indication of the good performance of our THz derivative spectrometer is the achievable signal-to-noise ratio (SNR). In general, derivative spectroscopy has the disadvantage of poor SNR [49]. To evaluate if the THz derivative spectroscopy with this method suffers from the same problem, we were interested in comparing the SNR of the original reference signal with those obtained from the derivative measurements. An interesting way to carry out this comparison on a comparable scale is to recover the reference signal from the integration of the measured derivative signals. For example, the integral of the 1st order derivative measurement leads to the reference signal. Similarly, the double integral of the measured signal of the 2nd order derivative also leads to the reference signal, and so on. However, we found that by increasing the number of integration on the signal, the DC component in the generated FFT signal was amplified. As a result, a succession of integral calculations on the higher-order derivative signals introduces a non-negligible offset in the recovered signal. On the other hand, the positive result of the integration is its low-pass filtering action, which nicely eliminates the high-frequency noise level. Figures 6(a) and 6(b) show the 1st and 2nd order integration of the detected modulated THz signal in the time domain and Figs. 6(c) and 6(d) their respective frequency spectra. By way of comparison, the black curves of each figure are the reference signal in the time and frequency domains. For a fair comparison, it is important to note that all scans were done with the same scale and integration time. As expected, the signals of the first-order integral in the time and frequency domain are similar to the measured reference signal, except for the low-frequency part. In Fig. 6(c), we can also see that the SNR between the reference signal and the integrated signal are surprisingly comparable, without any sign of SNR degradation. Even more strikingly, compared to its reference, the double-integrated signal in Fig. 6(d) has an order of magnitude better dynamic range in the frequency range of 1 to 3 THz, i.e., in the operational range of the device. To ensure that this setup does not introduce errors for future spectroscopic applications, we show in the insets of Figs. 6(c) and 6(d) a zoomed-in view of the water absorption lines. From these results, it is clear that no negative effect exists on the spectral shape of the water lines, only an increase in signal with respect to the reference spectrum. Therefore, the device can be used for practical THz derivative spectroscopy experiments in the time domain.

6. Simulation results

In order to simulate the effect of the PM device on the THz field, we used Lumerical’s finite-difference-time-domain (FDTD) software. As the simulations produce discrete information, the differentiation function must be expressed as an analytical expression in the form of a set of discrete points (see Supplement 1, Table S1 for more information) [50]. As detailed in the complementary information S2, we have simulated two configurations of the PM device with a time step of 1fs: first for a uniform displacement such as a plane mirror [see Supplement 1, Fig. S2(a)] and second for a complex angular displacement [see Supplement 1, Fig. S2(b)], closer to the experimental reality of the PM device. The formula of the central difference approximation of the derivatives was used [50], see Supplement 1, Fig. S3 for the discretization of the points according to the position of the simulated structure. This derivative function allows to estimate the derivation from points on both sides of the central point at which the derivative is calculated. For example: Two-point and three-point central difference formula for the first and second derivative is derived using [50],

 

Fig. 7. (a) and (b) show the integral of the first and second derivative measured signals in time domain (red) with the reference (black). (c) and (d) show the corresponding spectra of these two signals compared with the reference spectra.

Download Full Size | PDF

7. Conclusion

Using a large aperture PM device, we have investigated a simple strategy to convert the traditional THz-TDS system into a derivative THz spectrometer. Our results confirm that the THz signal modulated by the PM device provides access to the $n$th-order derivation of THz pulses. We have experimentally demonstrated up to the 4th derivative and found good agreement with the expected derivative information of the reference signal. Moreover, after the integration of the nth-order derivative signal, the recovered information shows an increase of the SNR compared to its reference signal. This demonstration could be very useful for fine THz spectroscopy where the phase changes on a signal is the critical parameter to improve. In the near future, this differentiator can be implemented as a two-dimensional uniform motion matrix that would allow the development of new methods for computational imaging and advanced manipulation of THz pulses.