## Abstract

A new method based on fractal theory is proposed to analyze velocity sensing. The waveform of a self-mixing speckle signal is processed as a pattern of a fractal. Fractal boxes are defined as a set of grids used to divide the fractal pattern, and box-counting (BC) is introduced to characterize the statistical property of a speckle signal. A group of simulated speckle signals are analyzed by calculating the BCs corresponding to different velocities of the object. A linear dependence between the BCs of speckle signals and velocities is obtained, the result of which is validated by the analysis of a group of signals obtained from experiments. The performance of the fractal analysis is compared with those of the previous analysis methods. Better linearity and higher measurement sensitivity of the fractal analysis are indicated. The experimental result shows that the fractal method can be used as a valid analysis tool for the self-mixing speckle signal in velocity sensing.

© 2008 Optical Society of America

## 1. Introduction

Recently, the research of laser self-mixing speckle interference (SMSI) has attracted some attention for potential applications in optical sensors. The typical application of SMSI is in velocity sensing. With the aid of self-mixing speckle interference, one can obtain self-mixing speckle signals. The speckle signal is the random intensity fluctuation carrying velocity information of the dynamic random rough surface of an object, which is illuminated by a laser. To retrieve the velocity information of an object, some analysis methods have been used in previous work, such as pulse counting, the autocorrelation function, and fast Fourier transform (FFT) [1-6]. Each of those analyses has obtained an approximate linear relationship between the speckle signal and the velocity of an object. The analyzed results only reflect dependence between the mean frequency of the signal and the velocity of the object. In fact, we find that in the experiments, the average amplitude of the signal also changes with the velocity. The previous analyses only considered the fluctuant frequency of the signal but did not consider the amplitude of the signal to different velocities. Thus, a new analysis method that can simultaneously consider the frequency and amplitude of the signal is necessary. Fractal analysis is proposed as the requirement.

In self-mixing speckle experiments, signals that represent light intensity changes along a time axis are generated in a laser cavity by illuminating a random rough surface. These are temporal sequences that have fractal properties for each pixel of the speckle signals. The purpose of this paper is to propose a new approach with which to analyze self-mixing speckle signals. The proposition is owing to two factors: fractal analysis is an effective approach to study the temporal or spatial-ordered non-linear events that exist in nature, and it has rarely been used to study the dynamic speckle signal, though it was applied to a wide variety of scientific problems, including spatial speckle patterns [7-10]. In this paper, the analysis is carried out by calculating the BCs of a speckle waveform. The validity of this method is demonstrated by the analysis of speckle signals obtained from numerical simulation and experiments. The performance of fractal analysis is compared with those of pulse counting, the autocorrelation function, and FFT.

## 2. Principle

In Fig. 1, the curve is a section of the waveform of a self-mixing speckle signal. From the method proposed by Mandelbrot [11], the signal waveform obtained from SMSI is considered as a fractal pattern F in the plane of intensity versus time(*f*(*t*)~*t*). In this plane, a set of grids with scale *δ* is used to divide the plane. As a result, a mesh is created. The grid that is crossed by the waveform is called a fractal box, and the BC in scale *δ* is defined as *N _{δ}*(

*F*).

Equation (1) mathematically defines the dimension of BC (BCD) as a limit:

where *Dim* is the BCD. The BC *N _{δ}*(

*F*) indicates the complexity and irregularity of the waveform, which is related to the frequency spectrum and the amplitude of the signal. The BC

*N*(

_{δ}*F*) characterizes the statistical property of a signal, and it varies with the change of scale

*δ*. The BCD indicates that the complexity of the waveform increases as scale

*δ*decreases.

Because the speckle signal is discrete, scale *δ* must be an integer and greater than or equal to 1. In fact, the dimension of BC exists only in a special scale range (*δ*
_{1},*δ*
_{2}), in which the slopes of log*N _{δ}*(

*F*) versus log

*δ*approximately maintain a constant, so that the range (

*δ*

_{1},

*δ*

_{2}) is called a scaling range. In the scaling range, the figure

*F*is a fractal and the BCD is unique; thus, the definition of the BCD in Eq. (1) is modified:

where Δ represents the change of value.

In order to calculate the BCD of a speckle signal, first the BC corresponding to each scale *δ* should be calculated, and then the scaling range will be found. Finally, the BCD of the signal will be calculated. The steps of the calculation are listed as follows:

(1) Select scale *δ*, *viz.*, the length of the grid side. Using scale *δ* to divide *M*, the total length of the horizontal data, the horizontal axis is divided into *M*/*δ* at uniform intervals:

When *M*/*δ* is not an integer, the intervals will be:

$$[\left(\mathit{floor}\left(\frac{M}{\delta}\right)\xb7\delta +1\right),M]\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\mathit{the}\phantom{\rule{0.5em}{0ex}}\mathit{last}\phantom{\rule{0.5em}{0ex}}\mathit{interval}$$

The operation of *floor* is to round down to the next allowable integer value.

(2) In the interval of *m*, use scale *δ* to divide the maximum value of the signal data and then obtain the integer part of the result. The same operation is done with the minimum value in this interval, and another integer is also obtained. The absolute value of the subtraction operation between the two integers gets the BC in the interval *m*. If the remainder of the value of scale*δ* dividing the maximum is not equal to zero, the BC will be increased by one.

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\mathrm{min}\frac{\left[f\left(\left(m-1\right)\delta +1\right),f\left(m\delta \right)\right]}{\delta +1}$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\left(\mathrm{max}\frac{\left[f\left(\left(m-1\right)\delta +1\right),f\left(m\delta \right)\right]}{\delta \ne 0}\right).$$

(3) Add the BC of all the intervals, and the total BC of the signal waveform will be obtained.

(4) Processes (1)–(3) are repeated for different scales of *δ*, and then the BC N_{δ}(*F*) corresponding to *δ* is obtained.

The logarithm operation is done with scale *δ* and the corresponding BC *N _{δ}*(

*F*). The relationship between log

*N*(

_{δ}*F*) and log

*δ*is plotted in Fig. 2. The shape of this plot is related to the frequency and amplitude of the signal waveform and contains massive information about the object. In Fig. 2, a section of straight line (or nearly a straight line) locates in the scaling range, and the mean slope of the line represents the BCD of the signal. To accurately determine the scaling range, the slope versus scale

*δ*is further plotted in Fig. 3. In this figure, the range (

*δ*

_{1},

*δ*

_{2}), in which a horizontal line is located, is the scaling range. In the scaling range, the mean value of the data point located in the horizontal line is the BCD.

The BCD can be used as one measure to determine whether a signal is a fractal or not. When a speckle signal is a fractal, we can introduce the BC as a parameter to characterize the statistical property of the signal. For a fixed signal, the BC is determined by scale *δ*. The smaller the value of *δ* is, the bigger the BC is. However, since the speckle signal we obtained is discrete, *δ* must be an integer and greater than or equal to 1. When *δ*=1, the biggest BC will be obtained, which best characterizes the speckle signal in detail. Therefore, in calculating the BC of the speckle signal, we select *δ*=1 and the calculation is carried out as follows:

(1) Multiply a common number to make all the speckle waveform data become integers. This operation does not change the waveform of the speckle signal but enlarges the amplitude. (2) Provided that the enlarged speckle waveform is expressed with the function *f* (*t*),*t*=1,2,…,*n*., then the BC is calculated by

The BC of a speckle signal relates to the amplitude and frequency of the waveform containing the velocity information of the object being tested. Calculating the BCs of the speckle signals corresponding to different velocities, one can find a relationship between the BC and the velocity. The relationship can be use in velocity sensing.

## 3. Experiments

To demonstrate the validity of the fractal method, we first used a group of speckle signals produced by numerical simulation to do the fractal analysis experimentally. The signals are shown in Fig. 4 representing a group of velocities that are 98mm/s, 167mm/s, 242mm/s, 340mm/s, 431mm/s, and 580mm/s. Using Eq. (7), the BC of each speckle signal is calculated. We plot each BC versus the corresponding velocity in Fig. 5; a linear relationship between the BC and the velocity is indicated. The linear relation can be expressed as Y=1710+41.73X, and the linearity reaches to 100%. The fractal analysis of the simulated speckle signals obtained a numerical result that is a linear relationship between the BC and the velocity.

We also used a group of speckle signals produced by self-mixing speckle experiments to do the same analysis. With the aid of the experimental setup presented in the previous study, SMSI in DFB lasers [6], a group of speckle signals are experimentally obtained and illustrated in Fig. 6. Each signal corresponds to a velocity of the rough surface of the object. The BCs of the speckle signals corresponding to different velocities are calculated in Eq. (7). A plot representing the experimental result is shown in Fig. 7. It is seen that the linear relationship between the BC and the velocity is retained. The linear relationship can be expressed as Y=1774+61.34X. The related linearity coefficient is 98.9%. The fractal analysis of the experimental signals achieves an experimental result that is a linear relationship between the BC and the velocity.

The numerical result is in agreement with that of the experiment, which indicates that fractal analysis is valid for determining the self-mixing speckle signal in velocity sensing.

## 4. Comparison

A comparison is made between fractal analysis and other approaches. In previous studies, pulse counting, the autocorrelation function, and FFT were usually used to analyze the SMSIs. Here, we also use these approaches to analyze the same group of experimental signals that were analyzed in fractal analysis. The results are given in Fig. 8. As a brief introduction, the main point of pulse counting is to calculate the number of pulses in the measuring time by making a logical product of the speckle pulse train and a gate pulse [1]. The autocorrelation method is used to calculate the autocorrelation time τ* _{c}*, which is defined as the time delay at which the normalized autocorrelation function drops to

*1*/

*e*[2], and the FFT method is used to calculate the mean frequency of the spectrum to assign the bandwidth variation of the FFT to the speckle signal [4]. Figures 8(a), 8(b), and 8(c) show that the mean pulse frequency, autocorrelation time, and mean spectrum frequency are approximately linearly changed with the velocity, and the linearity coefficient of each is 97.982%, 97.487%, and 85.227%, respectively.

A further comparison of these processing results is made. The calculated values in Figs. 7 and 8 are normalized by dividing each series by their maximum value, and the normalized results can be compared in the same plot as shown in Fig. 9. The black squares, red circles, green up-triangles, and blue down-triangles represent the normalized values corresponding to the results of fractal, pulse counting, autocorrelation, and FFT analyses, respectively. The lines that connect a series of normalized values represent the four analysis methods used in speckle signal processing. The slope of the measuring line shows the measurement sensitivity. It is seen that in the range of velocities from 100mm/s to 600mm/s, the normalized parameter of the fractal analysis is more sensitive to the variation of the velocity than the others. This indicates that fractal analysis has a higher measure sensitivity compared with pulse counting, the autocorrelation function, and FFT. Why did fractal analysis have better performance? A reasonable explanation is that the conventional measures that rely on extracting characteristic scales (mean pulse frequency, correlation time, and mean spectral frequency) are not adequate for characterizing such speckle signals. Fractal analysis is based on structure, which can simultaneously consider the frequency and amplitude of the signal. Fractal BC is a more suitable parameter for describing existing differences in speckle signals.

## 5. Conclusion

This paper proposed the fractal method to analyze self-mixing speckle signals in velocity sensing. The algorithm of box counting was introduced theoretically. The validity of this method is demonstrated experimentally by processing speckle signals obtained from numerical simulation and experiments. The processing result shows a better linearity and higher measurement sensitivity compared with those of pulse counting, autocorrelation, and FFT. The results indicate that the fractal method can be used as a valid analysis tool for self-mixing speckle signals in velocity sensing.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education grant 20050319007.

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