Highly-Linearized Heterodyne Self-Mixing Vibrometer

Vibration meters based on self-mixing interferometry are generally made in baseband, without modulations, because it is very difficult to obtain a linear modulation of the wavelength by controlling the supply current. In this paper, it is shown a multi-frequency modulation strategy for a heterodyne self-mixing vibrometer, which allows us to overcome the limits of frequency estimation algorithms and can work on a diffusing target up to a few meters away.

The monitor photodiode directly measures the power emitted by the laser diode, denoted as P(f), which exhibits periodic variations with the back-injected phase, f = 4π s /λ, where λ represents the laser's wavelength, and s denotes the distance from the laser to the target [2].
To ascertain the absolute distance to a target, the traditional approach involves modulating the laser wavelength and assessing the period of fringes [5].The initial signal processing method proposed relied on a basic fringe-counting approach [6].A more efficient approach to determine absolute distance involves measuring the fringe period in the frequency domain [4].As there is an approximately linear correlation between beat frequency and distance, various techniques can be employed to calculate the beat frequency of fringes and subsequently estimate the distance.When considering the balance between accuracy and processing time, the interpolated Fast Fourier Transform (FFT) [34] emerges as a viable method for calculating the beat frequency of fringes and, consequently, estimating the distance [5].However, interpolated FFT has limitations when dealing with signals that are not perfectly sinusoidal, rendering the interpolation formula invalid.These interpolation errors are systematic and cannot be reduced through averaging operations.
Numerous strategies have been suggested in the literature to improve the precision of fringe frequency estimation.One notable advancement is the all-phase FFT [36], crafted to minimize the impact of spectrum leakage and signal noise.Nevertheless, its implementation necessitates an extended processing window, leading to an augmentation in execution time.Additionally, this technique is unable to rectify errors stemming from signals with a frequency that is not consistently constant, as observed in real self-mixing signals.An alternative proposition involves utilizing an algorithm based on Multiple Signal Classification (MUSIC) for frequency estimation in distance and velocity sensing systems based on self-mixing interference [37].The MUSIC method assumes that a signal vector consists of a known number of complex exponentials with unknown frequencies.By considering the covariance matrix and applying eigenvalue decomposition, dimensions containing signals exhibit larger eigenvalues, while smaller eigenvalues belong to noise dimensions.When the signal dimension is projected into the noise subspace, it results in sharp peaks at the signal frequencies, providing a frequency estimation function for MUSIC.Although MUSIC offers notable performance advantages, it demands substantial computational effort for real-time execution.Another approach, based on Genetic Algorithm (GA), is presented in [38].A cost function is established based on the variation in emitted power from a linearly modulated laser diode.To address premature convergence and the time complexity of GA, an improved GA algorithm is proposed.Despite the high resolution in distance measurement and improvements in selection and exploration range compared to the original GA, the speckle effect is not considered: it diminishes the robustness of the GA-based SMI sensing system, as the self-mixing signal with amplitude fading strongly impacts the cost function and can lead to incorrect convergence of distance values.This paper describes the development of a laser instrument for measuring vibrations on a diffusive target placed at a few meters of distance.The proposed sensor is based on a heterodyne self-mixing interferometer (SMI), following the scheme typically used for absolute distance measurement [5][6][7][8].

Heterodyne Self-mixing interferometer
The measurement of absolute distance through a self-mixing interferometer requires wavelength l modulation, easily achieved by changing the pump current I.The distance measurement s in this case is given by the evaluation of the fringe frequency ffringe, during the modulation: The simplest modulation shape is triangular, and the distance value is obtained by averaging the frequencies of the ascendant and descendant phases of the triangular wave.Instead, the instantaneous target speed  is given by the Doppler shift between rising and falling edges ( !"#$% and  !"#$' ) in a sort of heterodyne detection [5]: This approach exhibits two primary drawbacks: the non-linearity inherent in wavelength modulation and the limitations in resolution/accuracy of frequency measurement techniques.Introducing pre-distortion to the triangular-wave modulation could partially offset the nonlinearity of (∂λ/∂I) and its frequency dependence [39,40].To derive the signal frequency, proportional to the target distance, we opted for the implementation of the Interpolated Fast Fourier Transform (IFFT), chosen for its optimal balance between accuracy and processing time [5].However, its efficacy is constrained by acquisition time, signal-to-noise ratio, and the fact that the signal shape is not perfectly sinusoidal.
To enhance performance, we propose mitigating systematic errors in IFFT by modulating the LD current with slightly varied frequencies during consecutive modulation periods.This approach causes the measured SMI frequency to shift to different positions relative to the FFT bins, resulting in improved resolution through averaging.

Realized vibrometer
The realized prototype of vibrometer consists of two parts: the analog electronics required to process the fringe signal and introduce a specific current waveform into the laser diode, along with a commercially available data acquisition card (DAQ, Analog Discovery II).The DAQ card was used to generate the modulating wave and capture the signal, with a conceptual arrangement similar to what is depicted in [7].The schematic representation of the instrument's components is illustrated in Figure 1.The realized instrument, with real-time digital processing, is able to get 2000 measurements per second, working on a diffusing target up to 3 m of distance, with a simple collimating lens.In the realized prototype, the laser diode is a distributed-feedback (DFB) laser at 1550 nm (WSLD-1550-020m-1-PD).
Using the DAQ, a customized waveform is generated for modulating the laser and simultaneously capture the interferometric signal.To precisely assess the non-linearity introduced by wavelength modulation, a real-time software is developed to measure locally the frequency of the fringes.This software relies on an interpolated FFT computation performed on a sliding window of 32 samples, following the methodology outlined in [41].At the start of any positive and negative slope of the modulation signal, it is challenging to keep constant the period of the fringes.However, for subsequent data processing, it is enough to achieve a sufficiently long flat interval.Special care should be taken into account while setting the interval size to ensure an optimal number of samples, such as 256 or 512, for efficient FFT processing and measurement speed improvement.In the realized prototype, the measured signal is acquired with a sampling rate 8 MSa/s, and IFFT is evaluated on 256 points.Figure 2 shows an example of pre-distortion of the modulating wave, able to realize an almost constant modulation frequency, with maximum relative frequency variation limited to 10 -3 .

Multiple modulations
Even after effectively linearizing the wavelength modulation to better than 10 -3 , periodic systematic errors with distance persist, contingent on the signal-to-noise ratio.These errors are deterministic in nature and cannot be mitigated through averaging.Following a thorough analysis of error sources, it became apparent that these errors were primarily linked to the limitations of interpolated FFT when dealing with signals that deviated from perfect sinusoids and included noise.In response to this issue, we explored a more intricate modulation scheme involving multiple modulating waveforms at slightly varied frequencies.These modulating waves were designed to position the fringe frequencies differently in relation to the FFT bins.This means that when a generated fringe in one modulation signal period aligns precisely with an integer bin in the spectrum, the frequencies of the generated fringes in other modulation periods are guaranteed to be on non-integer bin positions due to the selected modulation frequencies.This approach introduced deterministic errors from interpolated FFT that were distinct for each waveform and uncorrelated.Moreover, it allowed for the exclusion of measurements from waveforms corresponding to unfavorable frequency positions, where interpolated FFT errors were maximal.The least favorable position for this purpose was directly over a bin, as a two-bin interpolated FFT on a real signal demonstrated the highest error in that position.The proposed technique effectively decorrelates individual measurements, resulting in improvements through averaging procedures.However, a drawback of this approach is the time-consuming process of manually determining the correct distortion since individual waveforms influenced each other, and the pre-emphasis procedure had to be applied simultaneously to the entire modulating signal.Consequently, the same shape could not be used for all modulating waves at different frequencies.To streamline this process, we implemented a recursive algorithm using LabVIEW to automatically optimize the modulation shape.The use of several modulation periods also allows signal fading problems to be better overcome, as it is easy to select only the periods with sufficiently high amplitude in the average of the measurements.The only drawback of this technique is the reduction of the measurement frequency, by the number of realized modulations.In our realization, three modulation waves at 9 kHz, 9,5 kHz, and 10 kHz are realized, for a total measurement time of about 300 μs, allowing, in theory, 3 kSa/s of measurement frequency.
The error correction process of the interpolated FFT is crucial for this type of measurement, particularly in the context of low-speed movements.In vibration measurements, even a minimal imbalance error between the ascending and descending phases of the modulating wave can result in a persistent speed offset, leading to a drift in the measured displacement.Experimentally, there has been a noteworthy enhancement in the quality of vibration measurements achieved with the proposed technique.Figure 3 illustrates an instance of measuring a target vibrating at 25 Hz with an amplitude of approximately 4 μm, positioned half a meter from the instrument.The upper panel displays the speed calculated through the Doppler shift, while the lower panel depicts the target displacement derived from the integration of speed.

Conclusions
The presented work sets the stage for a low-cost vibrometer with high resolution.Thanks to the multiple modulation technique, it is able to overcome the limits of this kind of instrument, while targeting diffusive surfaces at a few meters of distance.It could therefore be the right solution for different applications, for example on human skin.The actual main limit of the realized prototype is the acquisition rate, about 2 kSa/s, suitable for not-too-fast vibration measurements.

Figure 1 .
Figure 1.Block scheme of the realized interferometer.

Figure 2 .
Figure 2. Top panel: Modulation of the laser diode with triangular and pre-distorted waveforms.Bottom panels: Corresponding self-mixing signals with a consistent modulation frequency.