Extracting vibrational parameters from the time-frequency map of a self mixing signal: An approach based on Wavelet analysis Ajit Jha∗ , Santiago Royo ∗ , Member, IEEE, Francisco Azcona∗ , Student Member, IEEE, Carlos Yanez∗ ∗ Centre
for Sensors, Instruments and and Systems Development (CD6) UPC-BarcelonaTech Rambla Sant Nebridi 10, Terrassa E08222, Barcelona, Spain Email:
[email protected]
Abstract—The characterization of the time-frequency map that is obtained from wavelet analysis allows us to view the temporal and spectral components of non-stationary signals simultaneously. Wavelets thus provide a complete platform from where the parameters of interest in the signal in time and/or frequency domain can be extracted. In this paper, we analyse the conventional self mixing interferometry signal (SMS) in the timefrequency domain by using wavelet analysis, in order to extract vibrational features of the target directly from the SMS, to detect the moment in time when the direction of the target changes, the velocity of the target or its displacement.
I.
I NTRODUCTION
Self mixing interferometry (SMI) is a laser interferometry technique which needs just a single laser diode to perform nanometer scale measurements in an extremely compact way. The laser acts as coherent source, detector and interfering medium all by itself. From the theoretical point of view, SMI is a technique described by the principle of optical self injection (OSI) in which a certain fraction of light emitted by the laser is injected back inside its cavity upon striking the target, allowing it to interfere with the existent stationary wave. This gives rise to an interference pattern causing periodic fluctuations of intensity and frequency of the emitted light, due to changes induced in the gain of the lasing medium. This fluctuation in intensity is then picked up by the already existent monitor photodiode placed at the rear part of the laser diode, or can be measured as a voltage fluctuation at the junction of the diode. Mathematically, the phase and power of the field subject to optical feedback may be described as in [1] [2] and the set of equations that govern OFI summarized as below φf (t) = φo (t) − Csin(φf (t) + tan−1 α) Pf (t) = Po [1 + mF (φf (t))]
(1) (2)
where φf (t) = 2πff (t)τext (t), and φo (t) = 2πfo τext (t) are the round trip phase of the OFI signal and the roundtrip phase of the standalone laser signal, respectively; Pf (t) and Po (t) are the output power of the laser under feedback and of the standalone laser respectively; α is the linewidth enhancement factor, m is the modulation index and C the is feedback parameter that determines the number of active laser modes [3] [4]. More specifically, C determines the shape of the OFI signal and the importance of the hysteresis effects present in it. For instance, for C≈1, the shape is sawtooth, while for C
