In recent years, the field of fiber optic sensing technology has experienced significant advancements^[1]. Fiber Bragg gratings (FBGs), occupying a pivotal role in this research area, have found applications across a diverse range of disciplines including chemistry^[2], biology^[3], and medicine^[4]. Accompanying the continuous expansion of sensing monitoring capabilities, the monitoring range has evolved from single-point to distributed multi-point sensing^[5], triggering a transformation in FBGs from single inscription to array multiplexing^[6]. Traditional FBGs are typically manufactured by meticulously fusing individual gratings into a small-scale array, which demands meticulous control over the grating’s uniformity and fusion loss, thus potentially capping the number of multiplexing. Jiang et al.^[7] successfully mastered the continuous inscription technique of ultra-weak fiber Bragg grating (uwFBG) arrays in the fiber preparation process through a single exposure using an excimer laser, its key advantages being large-scale multiplexing capacities and minimal crosstalk interference, aspects that have fostered vigorous development in multiple engineering fields^[8].

In the uwFBG system, uwCFBGs hold significant potential for engineering applications^[9]. Owing to its wider bandwidth, coherent vibration detection based on narrow linewidth lasers presents unique advantages^[10]. However, as bare uwCFBG arrays bear a resemblance to single-mode fibers in terms of shape and exhibit poor mechanical strength and environmental adaptability, it necessitates a secondary coating to prepare uwCFBG cables of diverse structures to cater to the requirements of various engineering applications^[11]. Given that the uwCFBG array has no weld points, it can achieve a swift and integrated coating process in the cable preparation^[12]. This process is fast, stable, and consistent, but under tension and stress, the center wavelength of the uwCFBG array is susceptible to shifts.

The challenge lies in swiftly detecting and accurately calculating the shift in the center wavelength and utilizing this feedback to fine-tune the parameters of the coating process. This has emerged as a key issue for the practical application of uwCFBG array multiplexing technology. Currently, two primary methods exist for detecting the center wavelength of FBGs. The first is the spectral maximum method, including the adaptive threshold method^[13] and the Steger image correction method^[14]. However, for the relatively plateaued peak of the chirped spectrum, there is a significant discrepancy between the maximum value and the center wavelength. The second is the Gaussian function fitting method, encompassing genetic algorithms^[15], ant colony algorithms^[16], neural network algorithms^[17], and particle swarm algorithms^[18]. Yet this method overlooks the broad pulse shape of the uwCFBG spectrum, making the accurate determination of the center wavelength position challenging when the waveform is asymmetrical^[19]. In the field of fiber optic inertial navigation technology, the center wavelength of the spectrum is typically identified as the centroid of the broadband optical power spectral density^[20]. Drawing inspiration from this, researchers employ the spectral odd-even function method to ascertain the maximum value, thereby identifying the center wavelength of the spectrum^[21,22]. While this algorithm performs effectively on fiber Bragg grating (FBG) with a single peak, its application to uwCFBG with a relatively flat peak necessitates the computation of multiple sets of maximum values to determine the optimal value. Moreover, it is significantly susceptible to noise interference.

To achieve an accurate center wavelength position amid the conditions of wide bandwidth and spectral jitter in the uwCFBG, we propose a novel center wavelength detection algorithm for a chirped pulse, which is based on the resolution of the correlation coefficient. This is accomplished by shifting the original signal and its symmetric function to derive the correlation coefficient. Subsequently, the moving step that corresponds to the maximum correlation coefficient is determined, which in turn allows for the identification of the center wavelength position that corresponds to the broadband reflection spectrum. This approach is impervious to spectral jitter and offers a simpler calculation, which holds significant engineering implications for the calibration of uwCFBG parameters and other related applications.

When the change rate of the chirp increases or the excimer laser spot is not uniform, the reflection spectrum formed by the weakly reflected chirp pulse incident on the broadband light source will cause the jitter reflection spectrum to appear, as shown in Fig. 1^[23].

Lu et al.^[21] used a piecewise Gaussian function to represent asymmetric FBG, but uwCFBGs have larger wavelength bandwidths of 3 dB and flatter wavelength regions near the center. We assume that the envelope of the complex amplitude of the wavelength pulses could be described by a Gaussian function, in this case, the spectral waveform function for the uwCFBG can be expressed as f(k)=A1e−(k−x1)2λ2+A2e−(k−x2)2λ2+n,x0<k≤x3,(1)where n is the noise on the uwCFBG spectrum and the noise magnitude is related to the chirp change rate and the non-uniformity degree of the uwCFBG spot, λ is the pulse duration, and A1 and A2 are the amplitudes of two Gaussian waveforms. After symmetric inversion along the y-axis, the new reflection spectrum expression is f(−k)=A1e−(−k−x1)2λ2+A2e−(−k−x2)2λ2+n,−x3<k≤−x0.(2)

Taking a function of f(k) symmetric with respect to the vertical axis yields f(−k+x), where x is the step, and the function moves to the horizontal axis. At this time, the correlation coefficient is constructed and can be expressed as fcor(x)=∑k=1N[f(k)−Ef(k)][f(−k+x)−Ef(−k+x)]∑k=1N[f(k)−Ef(k)]2∑k=1N[f(−k+x)−Ef(−k+x)]2,x=[1,2,3,…,2N],(3)where x is the moving step of the relative position of the original function f(k) and the symmetric function f(−k+x), N is the sampling number of the original function, Ef(k) and Ef(−k+x) are the average values of the original function and the symmetric function within the sampling number, and dynamic correlation coefficient results can be obtained by moving x.

Figure 2 shows the flow chart of the proposed method in this paper. The original spectral function f(k) is symmetrically transformed to get f(−k), and the transformed function needs to be translated in the horizontal coordinate by stepping the x point to get f(−k+x). The correlation coefficient between f(k) and f(−k+x) is calculated in real time during the step process until the statistics of the two functions are completed from the beginning of contact to the complete separation. Currently, the maximum point of the correlation coefficient curve is calculated, and the step size is transformed according to the index of the extreme point, and then the central wavelength of the uwCFBG is indexed. When the correlation coefficient is the maximum, the original function and the center of gravity of the symmetric function coincide. At this time, the horizontal and vertical coordinates of the original function corresponding to the center of gravity can be indexed through x, that is, the central wavelength position.

To verify the effectiveness of the proposed algorithm, we use segmenting function superimposed Gaussian noise to simulate the uwCFBG spectra under different jitter conditions. By adjusting the noise amplitude to 0, 0.01, 0.05, and 0.10 V, the accuracy of the barycenter positioning results of the chirped wide pulse reflection spectra under different conditions is compared.

Figure 3 shows the original function and even function obtained by spectral map transformation under different noise amplitudes, where the collapse at the top of the spectrum is to simulate the difference in reflectivity caused by the non-uniform spot in the writing process of the uwCFBG. As can be seen from Fig. 2, with the increase of the spectral jitter, the overall image distortion of the uwCFBG spectrum is relatively serious, and there are multiple peaks. It can be predicted that it is easy to obtain a large error using the spectral maximum method to directly solve the central wavelength.

To verify the validity of our proposed algorithm, we analyze the accuracy of the spectral barycenter results using the parity function decomposition algorithm and the correlation coefficient solving method, respectively. Reference [21] has proved that, as the relative position of the original function and the odd function changes, the maximum value of the difference between them also changes constantly, so when the maximum value of the floating state is at the minimum, it corresponds to the position of the center of gravity of the spectrum. When there are multiple peaks, they need to be analyzed one by one. Therefore, in the simulation results, we need to compare the results of fpeak1(x) and fpeak2(x) so that the maximum value of the output corresponds to fodd(x), and then find the step size corresponding to the smallest fodd(x) to retrieve the corresponding position of the spectral center of gravity. In the process of solving the correlation coefficient, there is no need to compare the peak maximum many times, and the calculation result is relatively simple.

Figure 4 shows the decomposition of the parity function and the solving results of the correlation coefficient under different noises. In Fig. 3(a), n=0V, the algorithm based on parity function decomposition can obtain min[fodd(x)] with x=54, and the algorithm based on correlation coefficient decomposition can also obtain max[fcor(x)] with x=54, indicating that the center wavelength of the original spectral function f(k) and f(−k+x) coincide in the case of x=54. In Figs. 3(b) and 3(c), when n=0.01V and n=0.05V, the result of the parity function decomposition algorithm fluctuates obviously, but min[fodd(x)] can still be obtained with x=53, and the calculation accuracy becomes worse, while the algorithm proposed by us is not affected by noise. When the spectral jitter noise continues to increase, the parity function decomposition algorithm in Fig. 3(d) can no longer obtain accurate min[fodd(x)] when n=0.10V, and the method is invalid. However, the proposed algorithm can still obtain accurate positioning results, which proves the high reliability of the central wavelength-dependent correlation coefficient localization algorithm.

The demodulation of the uwCFBG is an important way to observe spectrum, reflectivity, and other parameters. The uwFBGs combined with optical frequency domain reflection technology (OFDR) can realize the demodulation of the fiber-optic hydrophone array^[24]. By separating and optimizing the beat signal, the position information of the uwFBG with high spatial resolution is extracted using the spectrum information of the beat signal, and the reflection spectral information of the uwFBG is restored by combining the Hilbert transform. The wavelength demodulation of the grating is realized. As the spectrum of the uwCFBG is relatively wide^[25], the light grating requires a certain wavelength scanning bandwidth. In this paper, we use the light grating interrogator (LGI-100B) produced by Sentek Instrument to realize the wavelength calculation for the uwCFBGs.

The demodulation system is shown in Fig. 5. The light wave emitted by the amplified spontaneous emission (ASE) source passes through the intensity modulator and enters the chirped fiber Bragg grating array to be tested through circulator 1 (CIR1). The light signal reflected by the array is amplified by the erbium-doped fiber amplifiers (EDFA) and enters circulator 2 (CIR2). The light wave reflected by the uwCFBG is sent to the photodetector (PD). The signal is converted into an electrical signal by the photodetector, and the phase comparator is used to detect the phase of the signal and the reference signal.

Figure 6 shows the reflection spectrum of the 284-element optical fiber hydrophone array coated with acrylic resin, and it shows a good overall consistency. Because the measured uwCFBG is written by a monopulse laser at one time, the reflectivity is extremely low, about −30dB. It can be found from the spectral pattern diagram that the entire array is 1.5 km, and the spectral patterns of all primitive grating are in good agreement. Secondly, it can be seen from Fig. 6 that the central wavelength of the uwCFBG array is between 1551 and 1555 nm, preliminarily. Meanwhile, the 3 dB bandwidth of the spectrum is greater than 3 nm, which significantly increases the bandwidth compared with ordinary weak reflection FBGs. The chirp rate of the phase mask is 2 nm/cm, and the overall length of the uwCFBG can be deduced to be about 7.5 mm.

To verify the effectiveness of our proposed algorithm, 10 gratings are selected to calculate the central wavelength comparison, among which wavelength 1 is obtained using the parity function decomposition algorithm, and wavelength 2 is obtained based on the correlation coefficient solving algorithm.

Figure 7 shows the calculation results of 30 central wavelengths of the uwCFBG under different algorithms. Due to the large number of array multiplexes, we analyzed the first 10, central 10, and final 10 uwCFBGs in the array, respectively, which were composed of uwCFBG001-010, uwCFBG101-110, and uwCFBG201-210. It can be clearly seen that the central wavelength calculated by our proposed algorithm is more consistent. Then, all the uwCFBG spectra of the array are statistically analyzed according to the calculation results of the central wavelengths of the two algorithms, and the central wavelength of the grating is about 1553.3 nm. The center wavelength calculation based on traditional parity function decomposition results in a range of 0.73 nm and a variance of 0.29, while the algorithm based on correlation coefficient decomposition results in a range of 0.19 nm and a variance of 0.08. At the same time, due to the slight difference in the single writing process of the array itself, it is normal that the center wavelength is offset from each other, so the calculation results of the center wavelength are also affected to a certain extent. However, the comparative analysis shows that the results of our proposed algorithm are more stable and more suitable for the jitter wideband chirped grating spectral detection scene.

In this paper, we propose a method for calculating the central wavelength of weakly reflective chirped grating arrays based on correlation coefficients. First, the spectral function of the dithered grating is constructed by the combination of step function and Gaussian noise, and the steps of solving the correlation coefficients are derived. Second, the calculation results of the chirped grating center wavelength under different noise conditions are designed based on the theory. The qualitative analysis proves that the calculation results of our proposed algorithm are simpler and are not affected by noise. We select 30 uwCFBGs from a 284-element grating array to calculate the central wavelength, and use the range and variance results to quantitatively prove the robustness of our proposed algorithm. This algorithm is of great significance to the parameter calibration of wideband jitter uwCFBGs.