The Optimization of Multimode Fiber Speckle Sensor for Microvibration

. A vibration sensing system with optical fber speckles is demonstrated and optimized with diferent optical fber diameters and speckle statistical algorithms. Te types of fber diameter and material lead to a diferent performance of fber specklegram sensor (FSS), which has been experimentally explored. Te signal intensity, demodulated from the speckles, is diferent when using multimode fbers with diferent diameters. At the same time, the sensing efect of diferent fbers depends on the speckle statistical algorithms. Accordingly, we use diferent statistical methods in theory and experiment to analyze the infuence of fber diameter and speckle statistical methods on the sensing performance. A vibration sensing system with optimized performance is achieved by the optimized types of optical fber and the corresponding optimized algorithms, which are promising for sensing weak vibration, such as detecting.


Introduction
Fiber specklegram sensors (FSS) are a kind of sensor based on the intermodal interference of multimode optical fbers (MMFs) [1]. When coherent light is incident into a multimode fber, speckles will appear at the end of the multimode fber. Te FSSs are capable of performing similarly to the interferometric techniques [2], which have been used for measuring various physical parameters, such as measuring liquid refractive index [3], weight [4], temperature [5], lateral displacement [6], strain [7], vibration [8],and so on. Compared with interferometric optical fber sensors, FSSs have the advantage of low cost, compared with other interferometric optical fber sensors [9].
Tere are various schemes to retrieve sensing information from the specklegrams. Te statistical methods calculate the correlations of specklegrams with outside disturbance, which include normalized inner-product [4], gray-levelco-occurrence matrix [7], frst moment and second moment [9], the sum of squared diferences [8], ratio image uniformity [10], mutual information [11], and so on. Machine learning is also used to recover information [12], but the calculation cost and sensing time are not satisfactory.
Silica MMFs and polymer optical fbers (POF) are two categories of MMFs. Silica MMFs with diameters of 50 or 62.5 microns are widely used in used for shorter-distance data transmission. POFs are made of polymer materials, typically polymethyl methacrylate (PMMA) polymer, which have been widely applied in data transmission and sensing [13]. Diferent types of MMFs lead to diferent types of specklegrams, which cause diferent statistical properties. Accordingly, the same speckle statistics method may lead to diferent statistics results for diferent types of MMFs, which may lead to diferent performances. It is important to choose a matching statistic method for the types of MMFs, which may optimize the measurement cost and sensing efect.
In this paper, the infuence of optical fber and speckle statistical algorithms with diferent parameters on vibration sensing is studied. By fltering the diameter of the optical fber, we obtained the schemes to match the algorithm to MMFs for sensing vibration.

Fundamentals.
Tere are many guiding modes of light in MMFs, and speckles are generated by interference between diferent modes at the end of fbers. When the fber type, diameter, material, and wavelength of incident light are determined, the maximum number of modes in the fber is also determined. Referring to Reference [13], the number of modes in step-index fber, M, is given by the following equation: where V represents the normalized frequency in the fber, which can be expressed by the following equation: where a is the core radius of fber, λ is the wavelength of light, n 1 is the core index of refractive, and n 2 is the cladding index of refractive. Diferent diameters of optical fbers have diferent numbers of modes, which will lead to the number of speckle granules in the specklegram output at the end of the fber, and further afect the sensing sensitivity [13]. It is assumed that the illuminating light emitted into the optical fber is emitted from a highly coherent light source, and all modes in the optical fber are excited. Referring to [1], the far-feld complex speckle distribution of the fber is a coherent superposition of the complex amplitudes of all modes, such as Te light intensity measured by CCD can be expressed by the following equation: where a m , ϕ m are the m-th mode's amplitude and phase, and are the n-th mode's amplitude and phase. Te statistical information of the specklegram represents the sensing information. In this paper, the normalized innerproduct coefcient, gray-levelco-occurrence matrix, frst moment, and zero mean normalized cross-correlation are utilized to demodulate the vibration information, which is depicted as follows.

Normalized Innerproduct Coefcient (NIPC).
Te specklegrams are containing position information and light intensity information. NIPC is used as the statistical characteristic value of the specklegrams for theoretical analysis, and the relationship between the statistical characteristic value of the speckle and the external disturbance is established. NIPC value is given by the following equation: where I 0 indicates the light intensity at position (x, y) in the reference specklegram, and I indicates the light intensity at position (x, y) in the measuring specklegram.

Gray-LevelCo-Occurrence Matrix (GLCM)
. GLCM is a square matrix related to the texture of the image. Te elements of the matrix are equal to the number of gray levels in the image. Te GLCM is calculated to show how often a pixel with gray-level (or grayscale intensity) value i occurs to adjacent pixels with the value j. If the size of the image is M × N, the gray level is G, and its G × G size GLCM matrix. Te elements can be expressed as Among them, P (x, y) represents the selected pixel and d(d x , d y ) represents the ofset value.
Te eigenvalues of the gray-levelco-occurrence matrix are contrast, autocorrelation, entropy, and homogeneity. In sensing, the eigenvalues of the contrast and homogeneity of the gray-levelco-occurrence matrix are used as the statistical eigenvalues of speckles. Homogeneity, which measures the closeness of the distribution of elements in the GLCM to the GLCM diagonal, is given by the following: Te contrast, which represents the clarity of the image and the depth of the texture groove, is given by the following equation:

First Moment/Second Moment.
Te frst moment refers to associating the position information with the intensity on each pixel in the category of the matrix, which is an analogoue of the moment of a fat plate with uneven mass distribution in mechanics. Te mean values in the x and y directions, μ x and μ y are given as follows: .

(9)
Te general formula of the p-th radial moment is given by the following equation: Although the expressions of the frst moment and the second moment are diferent in value, the overall trend is the same. Terefore, it can be regarded as achieving similar statistical efects with the frst moment and the second moment.

Zero Mean Normalized Cross-Correlation (ZNCC).
In ZNCC, the intensity of each pixel in the specklegram is subtracted from its own mean value, to calculate the correlation. Accordingly, ZNCC is more robust to linear and uniform brightness changes, which is given by the following equation: Similar to the NIPC algorithm, when the reference specklegram is the same as the measured specklegram, the value is 1. When the specklegram deviates from the reference specklegram, ZNCC will decrease accordingly. I 0 can be set as a calibration point, which can improve the dynamic range of the measurement. It is diferent from the NIPC algorithm.

Experiment Setup.
Te experimental setup for the speckle vibration sensing system is shown in Figure 1. Te light source is a semiconductor laser of 650 nm. A CCD (Hikvision MV-CA013-21UM, USB3.0) acquires the specklegram from the MMFs, which is set at 30FPS. According to the Nyquist sampling law, we can measure vibration signals below 15 Hz.
In this paper, we choose 4 types of step-type silica MMFs and 3 types of POFs made of PMMA with diferent diameters and numerical aperture (NA), provided commercially, which are shown in Table 1.
Vibration on fber is excited by the motor, which is set at the speed of 26.5 rpm (round per minute, RPM). Te corresponding frequency of vibration is 0.4417 Hz. Te vibration can be represented by the change of speckle statistics of the fber specklegram. Here we choose seven commercial optical fbers, as shown in Table 1. Under the disturbance of the vibration signal of fxed frequency, the specklegrams were acquired continuously for two minutes with CCD at the acquisition speed of 30 frames per second (FPS), which forms an independent data set. Te corresponding sensing curve can be obtained by processing the specklegrams with diferent speckle statistical methods, and the sensing curve takes time as the horizontal axis. Repeating the experiment 5 times to take the average value.
As the core diameter of the same fber increases, the size of the speckle granules gradually decreases, and the number of speckle granules gradually increases, as shown in Figure 2. Based on the known fber parameters and the wavelength information of the light source, we can calculate the maximum number of modes that can be transmitted in diferent fbers. Te maximum number of modes M that can be transmitted by stepped silica fbers has been given in Table 1. Teoretically, a larger diameter of the fber supports more number of modes transmitted in the fber, which afects the sensitivity of FSSs.

Sensing Results of Vibration.
Ideally, when the fber is not disturbed, the speckle statistics value should remain constant. However, due to temperature drift and other NOSIE, the statistical value is not constant. Here, we test the stability of sensing systems by getting specklegrams without vibration. Te standard deviation of the sensing value describes the stability of the sensing process. Under the same set of data, the coefcients of variation are 1.5837 × 10 − 6 (NIPC), 5.3349 × 10 − 3 (GLCM-contrast), 1.7229 × 10 − 3 (GLCMhomogeneity), 0.9414 × 10 − 4 (frst-moment), and 1.6269 × 10 − 5 (ZNCC). NIPC and ZNCC have good stability. Figure 3 shows the speckle vibration frequency sensing curve of the silica step-index fber with diferent core diameters and polymer fbers based on the NIPC method and its fast Fourier transform image. Te baseline is ftted by the least square and is removed.
In Figure 3, the vibration signals based on fbers with diferent diameters are demodulated by NIPC. Te peaks of the curve refect the sensitivity of sensing. With the same vibration, the curves of NIPC values of diferent fbers are diferent. As the fber core diameter increases, the sensing value gradually increases. But when the diameter becomes larger to a certain value, the amplitude of the sensing curve shows a downward trend. When the linearity of the vibration sensing curve becomes very poor, the tiny vibration signal will vanish in the sensing system.
Te main frequency of the vibration sensing curve after the Fourier transform can be used as the recovered vibration frequency. It can be seen that the specklegrams from various optical fbers have demodulated the vibration frequency information. Except for the slightly diferent recovery results of 50 μm silica fber, the recovery results of other diameters are very accurate. In addition, it can also be combined with the linearity of the NIPC vibration sensing curve to evaluate the sensing quality. For example, in Figure 3      For the results of the frst moment in Figure 5, for the silica fbers, the periodic pulse peaks are distinguished, and the main frequency after demodulation is also accurate, with 200 μm being the best. For the POFs, the pulse peak is not obvious, and the frequency cannot be calculated. Only 250 μm polymer optical fber can demodulate the valid vibration signal frequency information. Some of the others can also demodulate information, but its amplitude is equivalent to or smaller than the amplitude of the interference frequency.
In Figure 6, the ZNCC sensing curves obtained by the sensing system, give obvious periodic pulse peaks. However, the sensing curve after preprocessing still fuctuates. As the fber core diameter increases, the peak value of the pulse peak in the sensing curve shows an upward trend. However, as the fber core diameter increases and exceeds a certain value, the linearity of the system sensing curve shows a downward trend.

Discussion.
For the same specklegrams, the four speckle statistical methods exhibit diferent performances. Ideally, for the same vibration signal, the larger the diameter of the  fber is, the larger the peak amplitude of the sensing curve pulse is and the more accurate the FFT frequency is. However, due to the increase of modes, the infuence of interference will grow greater in the experiment.
For all the speckle statistical methods, a thin optical fber always has less speckle information. Te statistical value fuctuates greatly, which leads to a nonhorizontal baseline, as shown in Figure 3(a). A large diameter eliminates this problem of baseline. Each speckle statistical method demodulates the vibration signal by calculating the changes in speckle images. When the number of speckles is too low, the statistical values cannot refect the vibration signal well.
On the contrary, if the speckle granules are too small, the CCD may not be able to capture them. A high-resolution camera is needed, which increases the cost of the sensor. Furthermore, the infuence of disturbance on speckles will also increase with the increase in the number of modes. Te infuence of interference on speckles will also increase with the increase in the number of modes. Tese infuences gradually overlap. Finally, the pulse amplitude of the sensing curve becomes uneven with gradually decreasing, shown in Figure 3(k) and Figure 6(g).
In the experiments to test the stability of the algorithm, the sensing values of ZNCC and NIPC are basically constant. NIPC and ZNCC methods are cross-correlations between the intensity information and spatial information. For fbers of diferent diameters, the sensing curve has obvious repetitive pulse peaks, and the number of times matches the set frequency. ZNCC can also adjust the I 0 point by itself to improve the dynamic range, which improves the linear range and improve the sensitivity of the sensor by manually zeroing. In addition, a timely update of reference pictures can also reduce drift caused by environmental interference, and avoid linearity degradation caused by excessive baseline foating.
In contrast, the GLCM algorithm still has strong fuctuations in sensing data even without vibration. When the experimental fber becomes larger, the number of modes will increase, and the infuence of disturbance will become larger, accordingly, the oscillation of the sensing curve will become more obvious. When performing low-frequency vibration sensing, the distinction between the vibration signal and the interference signal is low. At the experimental frequency, even if the data has been preprocessed, the sensing curve still cannot form an obvious periodic pulse peak. As shown in Figure 5, only two sensing curves can demodulate the external vibration signal.
Te stability of FM is between GLCM and ZNCC, under appropriate diameters, which demodulate obvious periodic pulse peaks. However, there are many burrs in the pulse curve, and the main frequency of FFT is not clearly distinguished from other frequencies.
Optimized statistical methods should be selected for fbers of diferent diameters, based on criteria such as a smooth sensing curve, a uniform periodic pulse peak, and distinct main frequency amplitude after FFT. Based on the above selection criteria, the matching scheme of the vibration sensing system can be given, as shown in Table 2.

Conclusions
Several statistical methods are introduced, and the optimization analysis of vibration sensing is performed with the addition of low-frequency vibration signals. Both NIPC and ZNCC show good stability and can recover more accurate sensing information for fbers of diferent diameters. ZNCC can adjust the dynamic range by zeroing, making the application scenarios of vibration sensing wider. GLCM can very fnely refect the texture of the image and accurately refect the weak change of the fber state, but in the lowfrequency vibration sensing scene, its demodulation and stability are not good. FM also has stability problems. Te trend of the corresponding P-th moment algorithm is the same, but the amplitude is diferent. Tis feature can be used to adjust application scenarios.
When there is less speckle information for a 50 μm silica fber, the sensing curve of each statistical method still fuctuates after baseline ftting, which afects the frequency recovery. Te larger diameter of the fber, the greater the peak amplitude of the sensing curve pulse, and the higher the demodulated main frequency amplitude. However, afected by stability, the more modes, the more afected the amplitude of the pulse peak, and the higher the requirements for sampling equipment.
Finally, for diferent speckle statistical methods, suggested fber diameters for vibration sensing systems are given.

Data Availability
Te data used in this study are generated from the measures in the paper.