1 Abstract— The disappearance of self-mixing fringes in the moderate feedback regime decreases the displacement measurement accuracy. The proposed method detects and compensates the fringe-loss, to limit the error to around 40 nm for micrometer range harmonic amplitude displacements. Moreover, it can also treat arbitrary displacements without any time- consuming optimization procedure and is suitable for implementation in a real-time displacement sensor. Index Terms— Displacement measurement, Optical feedback, Optical interferometry, Vibration measurement. I. INTRODUCTION HE self-mixing (SM) effect in laser diodes (LD) has been widely investigated for the last decades for vibration [1] and displacement measurements [2]. A major advantage of the SM sensing scheme is that no optical interferometer external to the source is needed, resulting in a very simple and compact set-up. The behavior of a laser operating as a SM vibrometer can be depicted through a phase equation presenting different regimes for increasing optical feedback levels, described by C a feedback coupling factor varying in particular with the linewidth enhancement factor α, the surface reflectivity of the target and with the external round-trip time delay τ D =(2D 0 /c), where D 0 is the distance to the target and c the speed of light. Displacement measurements are generally performed in the moderate feedback regime where C > 1 [1]. We report in this paper an algorithm which enables the reconstruction of the displacement by using a simple, single lens sensor based on the variations of P(t), the optical output power (OOP) i.e. the SM signal. These variations can be expressed as [1]: ( ( 29 ( ( 29 [ ] [ ] t t D F m P t P F λ π 4 . 1 0 + = (1) where P 0 is the emitted optical power in the free running state, m is the modulation index, D(t) is the displacement of the Usman Zabit, Francis Bony, and Thierry Bosch are with the Université de Toulouse, LOSE, INPT, F-31071 Toulouse Cedex 7 (phone: 33(0)561588237; e-mail: usman.zabit@ enseeiht.fr). Aleksandar Rakic is with School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia. target and λ F (t) is the emission wavelength subject to optical feedback. F is the familiar cosinus function, that becomes progressively distorted at higher feedback levels. This new approach allows reducing the number of external optical components as compared to [1] and is faster than a previously reported approach based on a phase unwrapping method (PU) [2]. Moreover, it successfully compensates for the fringe-loss under moderate feedback for harmonic vibrations and so improves the sensor accuracy. II. DISPLACEMENT RECONSTRUCTION Figure 1 presents a typical SM set-up where the built-in photodiode of the LD package is used to retrieve the SM signal caused by a target displacement D(t)=Nλ 0 /2 + ε, where N is the number of fringes, λ 0 is the LD wavelength under free running conditions, and ε is the excess fringe ‘ε’ (< λ 0 /2). A piezoelectric transducer (PZT) from Physik Instrumente (P753.2CD), equipped with a capacitive feedback sensor for direct-motion metrology with a resolution of 2 nm is used as a target. The LD, driven by a constant injection current, is a Hitachi HL7851G emitting at λ 0 =785 nm. Fig.1. Schematic diagram of SM vibrometer set-up. The proposed method (Fig. 2) starts by reconstructing the phase from the SM signal. The SM fringes, under moderate feedback are characterised by the sharp edges (termed here transitions) of the saw-tooth shape signal and the hysteresis [Fig. 3. (b)]. The transitions are detected as tr(n), where n = 0, 1, 2, … is the number of fringes detected. Each transition is stored in the Transition Memory block including its respective sign ‘sgn’ and the number of samples (representing the time) counted since the previous transition. Based on the sign of the previous two detected transitions, the algorithm either selects a “slope of sgn×2π” block or a “slope of sgn×(m + ½)π” block DOI: 10.1109/LPT.2010.2040825 (c) 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. A Self-mixing Displacement Sensor with Fringe-loss Compensation for Harmonic Vibrations U. Zabit, T. Bosch Senior Member, IEEE, F. Bony and A. D. Rakic Senior Member IEEE T