Introduction. Objects can be easily deformed by various influences such as thermal expansion, vibration, and distortion. Displacement sensing is a vital technology that measures the displacement caused by such deformation and is applied across diverse fields, including medical engineering [1], civil engineering [2], geotechnical engineering [3], etc. Numerous physical principles (e.g., resistive, piezo, electromagnetic, etc.) have been employed for displacement sensing. Optical interferometry (OI) stands out as a promising technique because it gives the highest resolution, the longest sensing distance, and the ability to measure submicron displacements without making physical contact with the targets.

In the conventional optical intensity-based OI method, displacement information on the target is recovered by acquiring the intensity of the interfering light. This intensity fluctuates over time due to the unstable optical path length difference (OPD), leading to fluctuations in sensitivity during detection [4–6]. To tackle this issue, the quadrature detection method, which utilizes both the light’s intensity and phase information to recover the displacement, has garnered significant attention in recent years. However, several challenges remain, e.g., the need for a feedback connection between the transmitter and receiver for lock-in demodulation of the interference signal [7]; potential fluctuations in the intrinsic phase difference of couplers in multiport OI setups, which may lead to a low signal-to-error ratio [8]; the doubling of noise and an additional 3-dB degradation of the signal-to-noise ratio (SNR) caused by differential operation in signal processing [9,10]; and the loss of the initial phase information of the signal in the cross-correlation technique, making real-time detection difficult [11,12].

To tackle the aforementioned challenges, we recently introduced an innovative displacement sensing approach utilizing phase-diversity optical digital coherent detection (ODCD). This method retrieves displacement information by capturing the in-phase (I) and quadrature (Q) components of the signal [13,14]. The sensing component of this approach uses the conventional ODCD technique, which includes a 90° optical hybrid and two balanced amplified photodetectors, as illustrated in Fig. 1(a). Additionally, we employ a narrow-linewidth (NLW) probe laser to reduce the phase noise in the system. The system allows us to achieve nanometer-scale displacement sensing capabilities thanks to the flexibility of high-speed digital signal processing.

 

Fig. 1. Schematic diagram of the displacement sensing schemes.

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However, there is room for streamlining system configuration and enhancing displacement sensing capabilities. Unlike communication systems, our system does not require a separation between the transmitter and receiver. On a related note, we have also implemented a straightforward self-homodyne detection scheme for multi-level signal sensing using a quadrature phase modulation technique based on alternating quadrature phase-modulated (AQPM) pilot carries [15]. Drawing inspiration from this, this paper introduces a quadrature interferometry method based on an AQPM reference light for displacement sensing, as depicted in Fig. 1(b). In contrast to the previous scheme in Fig. 1(a), we obtain the π/2 phase shift between the quadrature components in our proposed scheme using a phase modulator instead of a 90° optical hybrid that includes four couplers and a phase shifter, eliminating a redundant balanced amplified photodetector (BAP) at the same time. Moreover, to further lower the cost and enhance marketability, we employ a common distributed feedback (DFB) laser as the probe light in our experiments instead of the NLW laser, demonstrating the achievable displacement sensing capabilities with a cost-effective DFB laser.

Experimental setup. Figure 2 depicts the schematic experimental setup for the proposed scheme based on AQPM reference light. In the upper part, handling the displacement sensing, continuous light emitted by a DFB (NEL JAPAN, KELD5C0GCAA) was polarization-controlled and split into reference and probe lights for displacement sensing. The upper-arm reference light underwent alternating phase modulation, switching between 0 and π/2, carried out by a phase modulator (Sumitomo Osaka Cement, TPM1.5-20) at a modulation frequency of 250 kHz, denoted as ${E_{\textrm{ref}}}(t )$. The lower-arm probe light passed through a polarization-maintaining circulator, was then directed into the air, focused onto the sidewall of a piezo actuator (Thorlabs PA1CEW), and subsequently reflected.

 

Fig. 2. Schematic experimental setup of displacement sensing based on AQPM reference light.

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The reflected light, denoted as ${E_{\textrm{sig}}}(t )$, carried the displacement information $\Delta D(t )$ of the piezo actuator and interferes with the reference light ${E_{\textrm{ref}}}(t )$ through a polarization maintaining coupler. This process alternately acquired the in-phase (I) and quadrature (Q) components of the displacement signal symbolically. Following optical-to-electrical conversion by a BAP (Thorlabs PDB425), a sequence of serial symbols containing alternating quadrature components was obtained by a digital storage oscilloscope (DSO, Tektronix TDS6154C) and was later processed offline.

In the displacement generation part on the lower side, the piezo actuator experienced displacement due to a sinusoidal waveform drive voltage of 25 kHz from a signal generator (NF Wave Factory 1942). Consequently, the light reflected from the sidewall of the piezo actuator contained a phase-modulated signal with a repetition rate of 25 kHz. The change in phase was directly proportional to the amount of displacement and was detected using the proposed sensing scheme.

Note that to ensure sufficient symbol samples of I and Q components and to facilitate a more effective and efficient separation of the I, and Q components in digital signal processing (DSP), as well as to adhere to the Nyquist theorem, the frequency of the phase modulator was set to be 10 times that of the actuator’s driving signal repetition rate. Additionally, the sampling rate of the DSO was set at twice the frequency of the phase modulation, 500 kilo Samples per second (kS/s). This arrangement implies that the I and Q components were sampled ten times each in an interleaved manner.

Experimental results and discussion. Figure 3 illustrates a data processing example featuring a series of raw signal waveforms at a drive signal of 3 V. Figure 3(a) shows the output waveform of the BAP, where the I and Q components are displayed in a time-interleaved manner. After being separated by DSP, they are represented in blue and red, respectively, in Fig. 3(b). Fig. 3(c) presents the corresponding raw displacement development, calculated as $\Delta \textrm{D}(t )= \textrm{arctan}({Q(t )/I(t )} )/2\kappa $ with an unwrap operation. Here, $\mathrm{\kappa }$ represents the wave number, and the multiplier 2 accounts for the round trip propagation of the probe light hitting the sidewall of the piezo stage. As shown in Fig. 3(c), the raw displacement development varies slowly over long term due to the OPD fluctuations between the two paths. To observe the details of the signal, Figs. 3(d), (e), and (f) give partially enlarged view of Figs. 3(a), (b), and (c), respectively. The graph in Fig. 3(f) displays a periodic-like waveform, indicating the periodical back-and-forth movement pattern of the piezo actuator. The magnitude spectrum of Fig. 3(f) is shown in Fig. 3(h), where a frequency component of 25 kHz is evident, matching the repetition rate of the piezo actuator’s movement driven by the 25 kHz drive voltage. To obtain a more accurate absolute value of the displacement, a bandpass filter (BPF) [Fig. 3 (i)] was applied to filter out unnecessary frequency components beyond the displacement signal, including DC components. Figure 3(j) demonstrates the displacement change after band-pass filtering. Due to the removal of the DC component, the displacement change centers around 0, and the absolute value of the displacement change can be determined by measuring the difference between the peaks and valleys. In this example, the mean value of the displacement change was approximately 31 nm.

 

Fig. 3. Example of restoring displacement change when drive voltage was at 3 V. (a) is raw data of alternating quadrature components output from the BAP. (b) shows separated I, Q components by DSP. (c) is calculated raw displacement development. (d), (e), and (f) are zoomed-in portions of (a), (b), and (c), respectively. (g) is zoomed-in portions of (d). (h) is the sideband spectrum of the raw displacement in (f). (i) is characteristic of BPF. (j) is restored displacement change.

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For statistical analysis, we computed the mean ΔD and standard deviation σ of displacement over 100 periods (40 ms) within a drive voltage range of 0.1–10 V, as depicted in Fig. 4. In this context, we defined the mean value as the displacement at a specific drive voltage and the standard deviation as an indicator of resolution. The coefficient of determination of the linear regression of displacement, denoted as ${R^2},$ reached 99.79%, indicating a nearly linear relationship. However, it is worth mentioning that the piezoelectric effect of the actuator inherently exhibits slight non-linear behavior due to hysteresis and creep. Determine a displacement as “detectable” when it falls within the error bars or one standard deviation of the linear regression. The minimum detectable value was approximately 1.6 nm, with a resolution of 0.6 nm, as represented in the inset of Fig. 4.

 

Fig. 4. Detection linearity and resolution.

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Note that the results in Fig. 4 were generated by the experimental system utilizing the AQPM sensing scheme and DFB probe laser, as illustrated in Fig. 2. To assess the differences in detection capabilities between the proposed AQPM sensing scheme and the previous 90° hybrid-based scheme, as well as between the broadband DFB laser and the NLW laser, we conducted experiments using three different combinations of the experimental system, as detailed in Table 1. The resulting data are illustrated in Fig. 5.

 

Fig. 5. Comparison of the experimental results of three systems. (a) Displacement change was consistent for different systems, while the detectable displacement degraded for the DFB probe laser. (b) Estimated SNR indicated that the signal degraded by phase noise due to the wide bandwidth of the DFB laser.

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Table 1. Experimental Systems in Three Different Combinations

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Figure 5(a) depicts the variation in displacement with respect to the drive voltage. The displacements remain consistent across different systems, as indicated in the inset. However, it is worth noting that the NLW probe laser exhibits a detectable displacement of approximately 0.6 nm with a resolution of 0.2 nm, whereas the DFB probe laser shows a degraded detectable displacement of 1.6 nm with a resolution of 0.6 nm. This scheme senses displacement by extracting the phase information of the signal. Therefore, the noise of the phase affects both the minimum detectable displacement and detection resolution. According to Ref. [16], the energy of steady-state phase noise can be described as $\sigma _n^2(\tau )= 2\pi \cdot \tau \cdot \mathrm{\Delta }f$, where, $\tau $ is time delay due to OPD between the two paths and Δf is the laser linewidth. In both the NLW-AQPM and DFB-AQPM systems, aside from the laser, the same experimental setup was utilized, with different $\mathrm{\Delta }f$ and identical $\tau $. Therefore, it can be inferred that the degradation in the sensing performance of the DFB-AQPM system is attributed to the broader linewidth of the laser.

Subsequently, we conducted SNR analysis [14,16] for the different systems, as depicted in Fig. 5(b). The dashed lines represent logistic regression for each case. Notably, the NLW-AQPM and NLW-90° hybrid systems yield similar results, differing from the DFB-AQPM system. This suggests that the phase noise of the probe laser has the most significant impact on SNR. At a drive voltage of 6 V, the SNR difference between the NLW-AQPM and DFB-AQPM systems is approximately 8.6 dBm. Considering the typical linewidth of the NLW and DFB lasers, which are around $\mathrm{\Delta }{f_1} = $100 kHz and $\mathrm{\Delta }{f_2} = $1 MHz, respectively, the theoretical SNR degradation due to the linewidth can be calculated as $\textrm{SN}{\textrm{R}_{\textrm{De}}} = 10\textrm{lo}{\textrm{g}_{10}}({\mathrm{\Delta }{f_1}/\mathrm{\Delta }{f_2}} )= 10\; \textrm{dBm}$ [14]. These theoretical calculations align with our experimental findings.

In this system, the influences of noise extend beyond the previously mentioned laser linewidth $\mathrm{\Delta }f$. These factors include the OPD, the modulation accuracy of the π/2 phase difference in the phase modulator, the consistent maintenance of the polarization state of light among different components in the system, and phase vibrations induced by factors such as bending, pressure, and mechanical stresses on the optical fiber. Through improvements in these aspects, achieving smaller displacements and higher detection resolution is desirable.

Conclusion. We upgraded a non-contact displacement sensing system by introducing quadrature interferometry based on AQPM reference light. This innovative approach replaced the complex traditional ODCD scheme, and utilizing an ordinary DFB probe laser instead of a costly NLW laser. According to the experimental findings, the AQPM scheme demonstrates comparable sensing capabilities to the traditional ODCD scheme but with a significantly simplified configuration. Conversely, the system using the DFB laser, which has a broader linewidth, achieved displacement sensing down to 1.6 nm with a resolution of 0.6 nm. However, it exhibited degradation in performance compared to the NLW laser with a narrow linewidth, which detected displacements down to 0.6 nm with a resolution of 0.2 nm. This degradation was attributed to the DFB laser’s more pronounced phase noise, which affected both the signal quality and the minimum detectable displacement. Nevertheless, given its cost-effectiveness and straightforward configuration, the DFB-AQPM system holds substantial potential for nanometer-scale sensing applications.