A Universal Phase Error Analysis for Optical Frequency Tuning Lasers Utilized in Fiber Sensing with OFDR

: As optical ﬁber sensing has attracted increasing attention due to its advantages such as high accuracy, low costs, and stability, its optical source judgment has become an attractive issue by which to characterize its performance. Optical frequency-domain reﬂectometry (OFDR) has been demonstrated as a means of the ﬁber identiﬁcation (ID) of optical ﬁbers; however, the linearity of the optical frequency tuning rate determines both the spatial resolution and detection range. In this paper, the results from various simulations and experiments show that the phase error from the initial frequency and tuning rate can affect the performance of the OFDR system, which directs the future improvement direction of ﬁber sensing based on such technology.


Introduction
Benefiting from low costs, a high data rate, and relative immunity to electromagnetic interference, optical fibers have been broadly applied in the telecommunication and sensing fields and are an ideal means for image transmission [1,2]. In earlier designs, since a laser source is utilized to emit the original light, it plays a fundamental role in determining the accuracy of the whole system. Unlike optical time-domain reflectometry (OTDR), optical frequency-domain reflectometry (OFDR) works in the frequency domain using continuous wave (CW) light sources, which enable higher signal-to-noise ratios [3]. Moreover, because of the advantages in the light frequency-domain techniques, including their sensitive measuring ability, high spatial resolution, and low-power consumption, there are some applications for these frequency-swept lasers: OFDR-based distributed sensing [4]; spectroscopy [5]; the optical physical unclonable function (PUF) [6] or fiber ID [7]; fiber sensors [8,9]; and the frequency-modulated continuous-wave (FMCW) LiDAR [10]. To meet the highly linear and coherent light source required in the above applications, various laser control methods and designs [11][12][13] have been proposed. Additionally, there are several principles or theories that have been investigated as means of improving the performance, such as the injection locking [14] design by enhancing the modulation bandwidth, integrated structures [15,16] on a single CMOS-compatible chip with multi-channel sensors, parallel coherent ranging by microcombs [17], and polarization control module analysis to suppress the wave-mixing influence [18]. In addition, there are some models that have been proposed to describe the laser's performances as well, including the noise, bandwidth, and polarization [19,20]. Furthermore, only when the precise laser frequency module has been built can the optical system achieve accurate measurements [21].
Considering that the tuning laser applied in the frequency-domain area can generally be categorized as two mechanisms-wavelength modulation and frequency modulation-both of them complete the control by changing the geometry size of the resonator [13,22]. It is no doubt that the former technique suffers from an inverse relation between the frequency and wavelength. In comparison, the latter is utilized to modulate the optical frequency directly, then such a light source can be toned linearly in the frequency domain without an extra transform function. Distributed feedback semiconductor lasers (DFB-SCLs) have been broadly implemented in the telecommunication and sensing fields due to their lightweight design, solid state, simple control, and low power consumption [23]. They are convenient for generating the required highly linear frequency-sweeping optical source by modulating the junction current [24]. Importantly, both the optical frequency linearity and initial frequency [6] of these lasers determine the sensing resolution and distance, and such properties are from the intrinsic structure and material of the laser itself [25].
In this study, in order to analyze the issues influencing optical frequency tuning progress, an analytical model was established to determine the optical frequency-domain measuring system. By taking both the frequency-swept velocity error and initial frequency variance into account, the module was able to quantify the frequency-domain performance. Furthermore, a simulation and various experiments based on DFB-SCLs were completed; both verified the proposed module accurately, promising further improvements to this technology.

Theory and Implementation
Taking the DFB-SCL laser as an example, the general optical frequency modulating system is depicted in Figure 1. A DFB-SCL provides the optical source and is generated by a time-varying current curve from the driving circuit (DRI) that is controlled by the control loop (CL), and the Mach-Zehnder interferometer (MZI) is utilized to monitor the optical linearity converted by a balanced photodetector (BPD). Additionally, the optical coupler (CPL) is utilized to divide the optical source into two parts: controlling and measuring. As introduced in previous works [11,26], a small MZI delay, τ d , would result in a flat phase and frequency response. During the controlling period, T C , the SCL optical frequency, f opt (t), can be expressed as where f 0 is the initial optical frequency (IOF); υ 0 is the laser frequency slope over time, which is also termed the frequency-sweep velocity (FSV); e(t) is the corresponding frequency error in the laser frequency sweeping; and t is the time. To determine the relations in the e(t) part, it is treated as the tuning frequency, and the initial frequency-varying errors are e(t) = ∆υ(t) + ∆ f 0 (t) (2) where ∆υ(t) and ∆ f 0 (t) are the laser FSV and IOF errors over time, respectively. For a given system, since the , , and are constant, to analyze the elements that influence the measuring accuracy, the remaining parts require quantitative investigation.
Considering that the phase differential of MZI is resolved and can be described as Moreover, the MZI phase, φ MZI (t), is the phase difference between its two arms [27,28], which can be expressed as where τ d is the MZI time delay and E(t) is the integral of e(t), or E(t) = t 0 e(τ)dτ. For a given system, since the f 0 , τ d , and υ 0 are constant, to analyze the elements that influence the measuring accuracy, the remaining parts require quantitative investigation.
Considering that the phase differential of MZI is resolved and can be described as then, the remaining part, e(t), is the optical frequency error that results in the poor performance. Furthermore, if τ d is relatively small enough or close to 0, the relation can be expressed as where e τ d (t) is defined as the approaching frequency error. For the bandwidth-limited laser, with the control period, T C , its relation between bandwidth, B, and frequency-sweep velocity, υ(t), can be expressed as As for the linear frequency tuning laser, υ(t) = υ 0 , which means B = υ 0 T C . From the above equations, the conclusion that the frequency of the MZI signal is independent of the time delay τ d can be reached. In reality, it is a difficult choice to set the MZI delay for a bandwidth-limited laser because a longer delay results in a lower tuning optical frequency-swept velocity that can easily be captured with greater errors in e(t) than the shorter one.
For practical situations, since it is difficult to generate the ideally linear optical frequency tuning laser source, assume that both the tuning velocity error, ∆υ(t), and initial frequency oscillation, ∆ f 0 (t), follow Gaussian distributions: where σ υ and σ f 0 are the standard deviations for the tuning velocity error and initial frequency oscillations, respectively. If the two components in the laser frequency tuning error, e(t), are independent of each other, they are supposed to also follow a Gaussian distribution [29] and can be expressed as From Equation (5), e τ d (t) is also supposed to follow a Gaussian distribution when τ d is much greater than unity: where f s is the system sampling rate, and τ d f s represents the quantified number of the statistics. Furthermore, for the random distribution data that have a mean of 0 [30], the greater the integrating time, the fewer the absolute errors.

Simulations and Experiments
To verify the influence caused by the τ d , the following parameters shown in Table 1 are set to complete the simulations. From Equation (5), it can be concluded that only when the time delay, τ d , is tiny enough can the precise mathematic model, e(t), be captured. However, as introduced above, it is difficult to choose between the time delay and accuracy of the MZI frequency; even though a lower time delay results in the more accurate tuning rate, a greater time delay leads to fewer standard deviations, termed σ sim . To quantify the performance of the results, the simulation error, err σ (τ d ), rate is defined as Figure 2 illustrates the simulations with various time delays, τ d . To express the difference in the frequency domain, Figure 2(a1-c1) shows the fast Fourier transform (FFT) results, in which the frequency of the maximum value is supposed to be 125 GHz/ms. Then, there are the two periods of the MZI signal in the time domains shown in Figure 2(a2-c2), in which the amplitude is set as a united value to simplify the phase part of the tuning frequency laser, and it is clear that the time accuracy decreases when the time delay increases, which is caused by the limited system sampling rate. Figure 2(a3-c3) depicts the short-time FFT (STFFT) results for the various simulations with noise, in which the Hamming window is selected to complete the FFT calculations. As for the statistics of the simulations, the results are shown in Figure 2(a4-c4), and the detailed values for the simulation of the approaching frequency error are illustrated in Table 2.
As mentioned in the theory section, the error rate between the theory and simulation standard deviations increases, but the absolute value of the simulation mean decreases with increases in the time delay.
To empirically evaluate the accuracy of the proposed phase error model of the SCL and the optimized algorithm for verifying the relations of the fiber subsets, experimental testing with a semiconductor laser (Eblana EP1550-NLW-B, 1550 nm) was carried out using the parameters shown in Table 3, and with a 125 Msamples/second system sampling rate.
2(a2-c2), in which the amplitude is set as a united value to simplify the phase part of the tuning frequency laser, and it is clear that the time accuracy decreases when the time delay increases, which is caused by the limited system sampling rate. Figure 2(a3-c3) depicts the short-time FFT (STFFT) results for the various simulations with noise, in which the Hamming window is selected to complete the FFT calculations. As for the statistics of the simulations, the results are shown in Figure 2(a4-c4), and the detailed values for the simulation of the approaching frequency error are illustrated in Table 2.    A comparison between the various time delays, including 10 ns, 100 ns, and 250 ns, in which the MZI frequency was set to 6.25 MHz and 12.5 MHz, is depicted in Figure 3. Similar to the simulation results, as the time delay increased, the distortions in the tuning laser increased simultaneously; the FFT results from the MZI signals in the region of interest (ROI) are shown in Figure 3(a1-c1). As mentioned in the theory section, the tuning rate was limited by both the time delay and the laser tuning bandwidth. Compared with Figure 3(a3-c3), there was a roughly constant value that is the product between the tuning rate, υ 0 , and the control period, T C , which is supposed to be the optical bandwidth of the laser.  Therefore, it can be safely concluded that only when a short time delay and lowoptical-frequency tuning velocity are utilized in the controlling system can an a SCL with limited optical bandwidth generate the highly linear optical source to capture the ideal measurements. Furthermore, to analyze the influence of the tuning rate at the same time delay, we set up experiments with the parameters shown in Table 4. Since the time delays utilized in the experiments were the same, the phase error from the τ d could be treated equally. The results are shown in Figure 4.
From the experiments mentioned above, it was clear that the phase error of the optical tuning source increased with increases in the frequency tuning velocity, as the sideband noise shown in Figure 4(a1-c1) and Figure 3(b3) illustrate. However, both the system sampling rate and the optical bandwidth limited the accuracy of the sensing system that Photonics 2022, 9, 922 7 of 9 was based on the optical continuous wave, even though there were some extra approaches utilized to minimize such influence to improve the stability and application areas.

Conclusions
In summary, this work demonstrates and experimentally evaluates an OFDR fiber sensing system that can directly simulate and judge the stability of semiconductor lasers for sensing applications, and such a module has been proved by various parameters. Generally, since the core of the OFDR system is the constant tuning rate of the SCL, the optical error plays a significant role when precise results are required. Thus, a detailed error module would inspire the research direction of prior OFDR systems used for sensing and further broaden their application in other areas, including optical sensors, laser spectroscopy, and SCL performance characterization.  Therefore, it can be safely concluded that only when a short time delay and lowoptical-frequency tuning velocity are utilized in the controlling system can an a SCL with limited optical bandwidth generate the highly linear optical source to capture the ideal measurements.

Conclusions
In summary, this work demonstrates and experimentally evaluates an OFDR fiber sensing system that can directly simulate and judge the stability of semiconductor lasers for sensing applications, and such a module has been proved by various parameters. Generally, since the core of the OFDR system is the constant tuning rate of the SCL, the optical error plays a significant role when precise results are required. Thus, a detailed error module would inspire the research direction of prior OFDR systems used for sensing and further broaden their application in other areas, including optical sensors, laser spectroscopy, and SCL performance characterization.

Conflicts of Interest:
The authors declare no conflict of interest.