Estimation of absolute distance and high-frequency vibration from the modulated SM-OFI signal using compound mutated genetic algorithm

Highlights

• The paper presents a method to estimate the absolute distance and the vibration frequency from the modulated SM-OFI signal using a compound mutated genetic algorithm (CM-GA).

• The algorithm is designed to process the SM-OFI signal experimentally obtained during the five carrier signal cycles (blocks).

• The comprehensive statistical analysis depicted a linear agreement between the actual and estimated sensing values.

• The method provide a simple and compact solution to the absolute distance and high-frequency vibration measurements under the weak feedback conditions.

• The optical arrangement is suitable for the industrial and scientific environments, where the dimension of the sensor is a decisive parameter.

Abstract

Contactless measurement of absolute distance or high-frequency vibration is a prominent factor for many industrial & scientific applications like predictive maintenance, non-destructive testing, and reverse engineering, etc. In recent years, self-mixing optical feedback interferometry (SM-OFI) has proven its superiority over other conventional optical methods because of its simple, compact and less expensive structure. Typically, the extraction of the sensing parameters from the modulated SM-OFI signal relies on the analysis of interferometric fringes that often leads to a non-optimal solution for weak feedback conditions. In this paper, the authors introduce a novel method to process the SM-OFI signal that employs a multi-objective compound mutated genetic algorithm to search for optimum solutions. To the author's knowledge, this is the first attempt to measure the vibrating frequency and the absolute distance between the laser-diode front facet and the vibrating target, simultaneously using a multi-objective global optimization method instead of frequency analysis. The validity of the proposed method was experimentally tested on a vibrating object located 2 to 20 cm away from the laser diode and having a vibration frequency between 2 and 10 kHz. The proposed method exhibits an excellent accuracy of 1% & 0.98% with an SNR of 42 dB & 19 dB for absolute distance & vibration-frequency measurement, respectively. The method also possesses a resolution of 0.6 mm during absolute distance measurement and a percentage of error ranging between 2.3 to 8% during high-frequency vibration measurement, both under the weak feedback conditions. The comparison with other methods also supports the potential of the proposed method for the applications where absolute distance and frequency of the vibrations are required to measure simultaneously.

Introduction

Absolute distance and frequency of vibration are the two crucial metrological parameters that often require in various scientific and industrial applications. To measure these parameters, to date various optical sensors have been developed based on different optical methods such as light intensity variations, triangulation, time-of-flight, and confocal. Although these methods exhibit higher accuracy but comprise a complex and bulky optical arrangement. Thus, it becomes expensive, inconvenient and sometimes impossible to place at an inaccessible location of measurement. Therefore, to provide a low-cost, simple and compact [1] solution self-mixing (SM) optical feedback interferometry (OFI) is being investigated by the researchers.

Self-mixing is a coherent-interferometric process, transpires between the resonant laser light and the partially reflected light from the target surface. The self-mixing process modulates the threshold gain, junction voltage and lasing spectrum of the laser according to the strength of the interference. Mathematically, the self-mixing effect is a well-defined process explained with a set of stochastic delay differential equations [2], known as the Lang-Kobayashi equations [3]. The self-mixing effect is being widely explored as a contactless sensing tool to detect displacement, distance, velocity, vibration amplitude, lens thickness, etc. [4], [5], [6], [7], [8]. Apart from their metrological applications, SM-OFI is also being investigated in different directions like particle-sizing [9], flow profiling [10] and even in the biomedical sector [11].

The extraction of the desired parameter from the SM-OFI signal typically relies on the fringes produced during the interferometric process [12]. During the last decade, several methods have been developed to find a conclusive value of the sensed parameter, such as fringes counting [13], Fourier transformation [14], interpolated fast Fourier transformation [15], Hilbert transformation [16], multiple signal classification [17]. The core concept of these methods is to process the beat frequency of the fringes. Thus, any damage in the signal spectrum during the processing leads to an inaccurate estimation of the parameter value. To suppress such uncertainties, a genetic algorithm (GA) is being explored by various research groups in recent years and has successfully implemented to find linewidth enhancement factor [18] and absolute distance of non-vibrating target [19].

Principally, GA is a global optimization, and search technique based on the natural mechanism of selection. It does not require any information about the derivatives, rather it provides a list of optimum solutions rather than a single solution [20]. The main limitation of the GA is its premature convergence in the presence of many local minima's. Fig. 1 illustrates several local minima observed during the measurement of the absolute distance. A common strategy to avoid the premature convergence of the algorithm is to produce more fit individuals (solution) in every generation through improvising the mutation process.

There are different mutation methods available in the literature such as schema mutation [21], dynamic mutation [22], compound mutation [23], cluster mutation [24] and hyper-mutation [25]. Among these methods, the best-fit method for the present scenario is the compound mutation method because

• It seeks an optimum solution as compared to the previous solution in the current population only, thus reduces the time complexity.

• It always maintains the diversity in the population, thus assuring avoidance of the local convergence problem.

Thus, generates a more appropriate solution even for the weak feedback conditions [26].

The main aim of this paper is to test and evaluate the performance CM-GA for the estimation of the absolute distance and the high-frequency vibration of the target. The rest of the paper is organized as follows: Section 2 discusses the basic mathematical formations and concepts of the SM-OFI. Section 3 explains the key methodologies used in the paper. Section 4 clarifies the experimental setup and data preparation steps. Section 5 collects all the results and discusses the issues that leftover in the previous sections. Section 6 concludes the paper and lays the foundation for future directions.