In this paper, the inherent error as well as the robustness of a previously published displacement retrieval technique called the phase unwrapping method (PUM) is analyzed. This analysis, based on a detailed study of laser feedback phase behavior, results in a new algorithm that removes the PUM inherent error while maintaining its robustness. The said algorithm has been successfully tested on simulated and experimental Self-Mixing (SM) interferometric signals. Simulations in weak and moderate feedback regimes demonstrate that the said algorithm can reach a subnanometric precision compared to approximately 25nm for PUM. For experimental SM signals affected by noise, the mesured rms displacement error and the maximum absolute error is approximately 14nm and 37nm respectively for the proposed algorithm and 34nm and 123nm for the PUM, which indicates a three fold displacement precision improvement over the PUM. Finally, it is explained that the precision can be further improved by a reduction of the noise level of experimental SM signals.

A self-mixing (SM) micro-interferometer laser displacement sensor coupled with an adaptive liquid lens (ALL) system is proposed and implemented. This has been made possible by a new method of real-time estimation of the optical feedback coupling factor C. It is shown that such an estimation of C combined with an appropriate amplification of the SM signal Gain allows the ALL system to seek and maintain the SM signal in the moderate optical feedback regime in spite of variations in the optical feedback. The ALL system thus enables robust real-time displacement sensing in an unmanned autonomous manner. The implemented system has provided measurement precision better than 90 nm for different target surfaces and distances. The paper also investigates the impact of the weighting attributed to C and Gain on the retrieved displacement precision. As this autofocus is presently only performed once during the sensor initialization, so maximum displacement span after achieving optical feedback regime locking has also been investigated and tabulated. This proof of concept, thus paves the way for the deployment of autonomous SM sensors.

Laser self-mixing (SM) sensors are successfully used to measure displacement in the absence of speckle. However, speckle deforms the SM signal rendering it unusable for standard displacement extraction techniques. This article proposes a new signal-processing technique, based on tracking the signal envelope, to remedy this problem. Algorithm was successfully employed to measure long-range displacements (25 mm), in the presence of speckle and the lateral movement of the target, both causing severe corruption of the SM signal. It therefore enabled the use of the sensor on noncooperative targets without the need for sensor positioning and/or alignment. The results have been obtained for SM signals in which the envelope amplitude is varied by a factor of 28, without the loss of interferometric fringes. The use of this technique effectively removes the need for opto/electromechanical components traditionally used to measure long-range displacement in the presence of speckle.

A new algorithm for self-mixing (SM) sensors has been developed to perform displacement measurements. It is able to differentiate between the different SM regimes (very weak, weak, moderate, and strong) and thus converges automatically to the optimum threshold level required to detect all the SM fringes, independently of the shape of the signal. Displacement reconstructions based on this algorithm have been validated with counter measuring commercial sensors for both weak and moderate regime acquisitions which are most frequently encountered under experimental conditions. Index Terms-Displacement measurement, optical feedback, optical interferometry.

This paper presents a different approach to classify self-mixing (SM) signals operating in the moderate feedback regime. A total of six distinct classes of SM signals can be defined based on the SM inherent shapes, which depend on both the feedback factor C and the linewidth enhancement factor α. This classification allows clear identification of SM signals for which normalization issues can arise and thus for which displacement precision is inherently reduced due to the very nature of the signal itself. Finally, it is shown that phase unwrapping approaches can theoretically retrieve displacement with subnanometer precision for usual laser diodes in the moderate feedback regime, in the absence of noise, only for α values greater than approximately 4.3.

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A self-mixing laser diode vibrometer including an adaptive optical element in the form of a liquid lens (LL) has been implemented and its benefits demonstrated. The LL arrangement is able to control the feedback level of the self-mixing phenomenon, keeping it in the moderate feedback regime, particularly suitable for displacement measurements. This control capability has enabled a remarkable increase in the sensor-to-target distance range where measurements are feasible. Target vibration signal reconstructions present a maximum error of lambda radical16 as compared with a commercial sensor, thus providing an improved working range of 6.5 cm to 265 cm.

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract-An innovative signal processing method based on custom-made wavelet transform (WT) is presented for robust detection of fringes contained in the interferometric signal of Self-Mixing (SM) laser diode sensors. It enables the measurement of arbitrarily-shaped vibrations even in the corruptive presence of speckle. Our algorithm is based on the pattern recognition capability of bespoke WTs for detecting SM fringes. Once the fringes have been correctly detected, phase unwrapping methods can be applied to retrieve the complete instantaneous phase of the SM signals. Here, the novelty consists in using two distinct mother wavelets Ψr(t) and Ψ d (t) specifically designed to distinguish SM patterns as well as the displacement direction. The peaks, i.e. maxima modulus of WT, then allow the detection of the fringes. Permanent repository link