Sub-Terahertz Range Fiber Optic Devices for Sensing Applications

Distributed sensing refers to the solution which enables the real-time, continuous measurement at multiple sensing locations (typically, more than 100 sensing nodes). Due to many of its unique advantages, such as small size, light weight, low cost, electromagnetic immunity, high-temperature survivability, and chemical stability, optical fibers have been well accepted as one of the most promising candidates as the platform for distributed sensing applications. Among different fiber distributed sensing methods, optical frequency domain reflectometry (OFDR) represents a particular promising candidate. Based on the frequency modulated continuous wave (FMCW) method, OFDR method is capable of measuring the spatial-continuous weak Rayleigh scattering patterns along the entire length of the fiber under test, with high spatial resolution (~μm) level and moderate interrogation distance (~ km). To extract the structural information from the unmodified communication grade single mode fiber, large interrogation bandwidth is needed. However, this resource of optical bandwidth is very expensive. The external cavity laser is the state-of-the-art frequency sweep laser source for OFDR system, which costs at least $ 20,000. The cost has rendered the OFDR interrogation technique expensive and limits its applications. This dissertation focused on the improvement of OFDR interrogation system by reducing the total system cost. First, a series of sub-terahertz range fiber sensors, including Fabry-Perot cavity sensors and terahertz fiber grating sensors were symmetrically investigated. Fabricated using femtosecond laser micromachining techniques, these sensors allow the OFDR system for low bandwidth interrogation while maintaining the high accuracy measurement. In addition, a sensor fabrication method without stripping the fiber polymer buffer was developed. This is the first time that an in-line grating structure has been fabricated within the core of an optical fiber with an intact buffer coating, allowing the fiber to retain optimal mechanical properties. Second, a series of low cost sweep laser sources were developed as high-linear coherent sweep laser source, suitable for the sub-terahertz range fiber sensor interrogation. Based on current injection modulation methods, the semiconductor lasers were used as the sweep laser sources and the output wavelength was feedback controlled using optical phase locked loop techniques. The method of using VCSEL laser was also investigated to increase the sweep bandwidth. In addition, an all-digital optical phase locked loop system was implemented using the field programmable gate array, which increases the system design flexibility.


Introduction
Distributed optical fiber sensing technology is a thriving branch of sensing technology, due in large part to its ability to surmount many limitations of traditional single-point sensor, enabling a single system to simultaneously span a large number of equivalently individual sensors [1]. These unique advantages have been successfully demonstrated in many application areas, including oil drilling, structural health monitoring, and perimeter security [2]. The recent development of distributed optical fiber sensors with high spatial resolution has expanded this utility considerably, making it an attractive addition to many emerging applications, such as wearable devices, robotics, and surgical instrument [3,4].
The use of Rayleigh scattering as a sensing method has shown particular promise in distributed sensing with high spatial resolution [5]. The unique Rayleigh indices or via exposing a fiber to intense UV light [9]. Recent technology advances have led to increased popularity in the use of femtosecond (fs) lasers in fabricating fiber optic devices [10,11]. Weak reflectors with a reflectivity of ~ −45 dB were achieved to form IFPIs with this fabrication method [12]. An FFT-based method was used to multiplex 3 IFPIs of differing cavity lengths. Very recently, Huang et al. successfully demonstrated an optical carrier based microwave interferometry method to simultaneously interrogate identical and cascaded IPFIs with a length of 12 cm along an optical fiber [13].
However, the low microwave frequency bandwidth limits its spatial resolution.
This letter reports a fiber-inline ultra-weak (<−60 dB) IFPI array fabricated using fs laser for distributed sensing with high spatial resolution. Interrogation approach, fabrication parameters, sensitivity, operation bandwidth, and distributed sensing ability of the proposed IFPI array were experimentally investigated in this letter.

Operation Mechanism
The Schematic of the interrogation system, based on OFDR, is drawn in Figure 1.1 Light from a tunable laser source (TLS, Newport 6428) is split into two paths -"clock" and "signal". "Clock" is an interferometer used to calibrate the non-linear sweep effect of the TLS by providing a corrected time base for a data acquisition card (DAQ, NI 6251 with the sampling rate of 1MHz) during frequency sweep. Light in the "signal" section is split between the reference and measurement arms of an interferometer via a 50/50 coupler (CPL); in the measurement path, an optical circulator (CIR) further splits the light to interrogate the low-reflection IFPI array and returns the reflected light. A polarization controller (PC) is used to tune the state of polarization in the system.
Another 50/50 CPL then recombines the measurement and reference fields. In this setup, the TLS sweeps from 1535 to 1565 nm at a speed of 16 nm/s, covering a total bandwidth of 3.7 THz. Thus, the AC-coupled voltage received by the DAQ is written as: where is the light-to-voltage conversion coefficient of the photodetector, r is the reflection coefficient of the reflector of the IFPI, Iref is the light intensity of the reference   polarizer, and several neutral density (ND) filters. The laser was switched on or off by electrically gating the internal clock. A single-mode optical fiber (Corning, SMF-28e) with the core and cladding diameters of 8.2 and 125 μm, respectively, was used for these experimental trials. After mechanically stripping its buffer, the fiber was cleaned using acetone and clamped onto two bare fiber holders, which were immersed in distilled water during fabrication. The fiber assembly was mounted on a computer-controlled three-axis translation stage with a resolution of 0.1 μm (Newport, Inc.). The fs laser beam was focused inside the optical fiber through a water immersion objective lens (Olympus UMPlanFL 20x) with a numerical aperture (NA) of 0.4. The velocities of the stages were set at 50 μm/s during fabrication. A cuboid region (10×2×10 μm) was inscribed in the center of the fiber from bottom up to cover the whole cross-section of the fiber core. The center of the inscribed region was aligned with the center of the fiber core. Figure 1.2 shows both the microscopic images and reflection distribution of three IFPIs fabricated with different fs laser powers (0.14, 0.12, and 0.1 W). The reflection values measured using the interrogation system were referenced to an angled polished connector (APC), which was previously measured using off-the-shelf precision instruments. The reflectivity of the weak reflectors decreased as fabrication power was reduced. A reflectivity of around −70 dB was achieved with laser beam at 0.1 W. In this study, 0.14 W was chosen as the fs laser power used for IFPI fabrication.
The sensing mechanism of an IFPI is based on tracking the phase shift of the interferogram in response to ambient change. To extract the phase shift information from the signal received from an individual IFPI, the reflection signal in frequency-domain is squared and filtered using a low-pass filter to obtain the interference signal (S): where L is the physical cavity length of IFPI, neff is the effective refractive index of the fiber, f is the frequency of the laser, and c is the velocity of light in vacuum.
Additionally, neffL is considered the optical length of the IFPI cavity. From Equation (2), as optical length increases, the period of the cosine function with respect to laser frequency decreases. As a result, at laser frequencies (~193 THz), the interference signal shifts to smaller frequencies, proportionally.
To investigate ultra-weak IFPI sensors, two IFPIs with lengths of 1 cm and 1 mm, respectively, were fabricated. The temperature response of both IFPIs was measured using a temperature-controlled water bath. This technique is justified by the fact that steepest slope of a sinusoidal function occurs at its zero-crossing position. with the theory value [12]. Comparing the two IFPIs, the interference period with respect to frequency for the 1 cm IFPI is 1/10 of the 1 mm IFPI. As a result, the 1 cm IFPI is more subject to 2π ambiguity problem, limiting its measurement dynamic range.
However, the slope of the 1 cm IFPI at zero-crossing position is ten times higher than that of the 1 mm IFPI, leading to a much higher accuracy. As a result, an IFPI with a longer length is suitable for precisely measuring smaller physical changes.

Experimental Results and Discussions
To demonstrate the distributed sensing capability of ultra-weak IFPIs, three identical 1 cm IFPIs were fabricated and spliced along a single fiber line.  crosstalk was observed between IFPIs in this experiment, indicating that this technology holds considerable potential for distributed sensing applications.
A key advantage of an IFPI array over the Rayleigh scattering method is that an IFPI does not require a large bandwidth to resolve small changes with high spatial resolution. To demonstrate this feature, a 1 cm IFPI was experimentally tested using different laser sweeping bandwidths.   The interferograms were again taken, and the frequency shifts as a function of IFPI location plotted in Figure 1.6 (c). A Gaussian-like temperature distribution was observed, in which the center IFPI experienced a temperature around 1˚C lower than IFPIs at either edge as expected. This experiment proves that continuously cascaded IFPIs can be used for distributed sensing. The spatial resolution is the cavity length of the individual IFPIs, i.e., 1 cm in this test.  (d) To evaluate system-level accuracy, a stability test was conducted by fixing the ambient condition of a 1 cm IFPI. 100 groups of interferogram were generated using this device. The frequency shift of each interferogram relative to its initial status was calculated. The standard deviation of the frequency shift is less than 100 MHz, determining the system's detection limit. Given the experimentally measured sensitivity of −1.5 GHz/˚C, its temperature detection limit is calculated to be less than 0.067 ˚C.
Previous studies of weak FBG sensors suggest that the multiplexing capacity is mainly limited by two types of crosstalk [6]. The first is spectral shadowing-spectral distortion of the downstream devices caused by the insertion loss of the upstream devices; the second is multiple-reflection crosstalk, or the spectral distortion induced by false signals, which undergo multiple reflections between upstream devices, and experiences the same time delay as the real signal. Both types of crosstalk are exponentially proportional to the reflectivity of the device. Ideally, the reflectivity of an IFPI is calculated by doubling the reflectivity of a single reflector. The smallest reflector achieved in our lab was −72 dB, converted to a −69 dB IFPI, which is more than 30 dB

Introduction
The Bragg grating is a mature sensing technique that has been widely used for strain, stress, pressure, and temperature measurement. Through the integration of these periodic structures into a variety of waveguides, the utility of Bragg grating technology has been successfully demonstrated over a broad set of frequency ranges. In the optics domain, incident frequencies in the hundreds of terahertz are routinely used to interrogate fiber Bragg gratings (FBG) [14]. By resolving shifts in the reflected spectra, subtle changes in the parameters of interest can be precisely measured. Similar utility has been demonstrated in the microwave domain (a few gigahertz) through the successful implementation of coaxial cable Bragg gratings (CCBG) fabricated by introducing discontinuities at the centimeter scale [15,16].
Both FBG and CCBG have demonstrated their utility for large scale, multiplexed sensing applications [6,17]. However, these techniques have distinct limitations; the large frequency ranges necessary for interrogation in the optical domain require swept frequency lasers, or a combination of broadband light source and optical spectrum analyzer, with broad ranges (tens of nm, or a few terahertz, at a wavelength of around 1550 nm) [18], while the long pitch-length of CCBGs in the microwave domain limits its spatial resolution (~tens of cm) for sensing applications.
Terahertz frequency sensing has emerged as a promising method of surmounting the limitations of both the optical and microwave domains. Terahertz frequencies lie between the optical and microwave frequency ranges, which are hundreds of terahertz and tens of gigahertz, respectively. As a consequence of this spectral position, terahertz sensing has the potential to marry the positive qualities of both optical and microwave Bragg grating techniques [19]; as compared to a FBG and CCBG, THz gratings require a narrower interrogation bandwidth (hundreds of gigahertz) and have greater spatial resolution (pitch length < 1 mm), respectively.
This possibility has led to recent promising experimental investigation. Zhou et al., using a KrF laser to modify Topas polymer fiber, interrogated the resulting structure using a high frequency vector network analyzer at hundreds of gigahertz [20]. Similarly, Yan et al. used a CO2 laser to modify a step-index polymer fiber, testing the structure using terahertz time-domain spectroscopy at hundreds of gigahertz [21]. Both methods, however, suffer from significant insertion loss as a result of large perturbations of the waveguides, which significantly limits the multiplexing capability of resulting sensors.
Additionally, the need to interrogate these sensors using direct terahertz frequency modulation requires the use of precision instrumentation at considerable expense due to the high attenuation of the interrogating circuit at THz frequencies.
In this letter, we report a terahertz fiber Bragg grating (THz FBG) sensing modality with the potential to overcome these engineering limitations. By using heterodyne mixing, a mainstay of microwave photonics, this technique has the potential to lead to both simplified sensor interrogation using narrow interrogation bandwidths and greatlyenhanced distributed sensing capacity with high spatial resolution [1,13,22,23]. The interrogation system, fabrication parameters, and sensing utility of THz FBGs were experimentally studied, and the results presented in this letter.

Operation Mechanism
The schematic of the interrogation system, based on optical frequency domain reflectometry (OFDR), is shown in Figure 2.1 [24,25]. The light generated by the tunable laser source (TLS) is split by a 90/10 optical coupler (CPL) into two paths, "clock" and "signal". The "clock" path with two 50/50 CPLs is an interferometer that provides the sample clock for the data acquisition card (DAQ), compensating for the non-linearity of the tunable laser. The sampling rate of DAQ is 14 MSa/sec. The light in the "signal" path is split using a 50/50 coupler into a reference arm and a detection arm; a circulator (CIR) guides the reflected light from the THz FBG structure, which is   with each FBG containing equally N reflection points with a period of Δz. The total AC coupled voltage received by the DAQ can then be expressed as: where η is the light-to-voltage conversion coefficient of the photodiode, r is the reflection coefficient of the FBG reflectors, Iref is the light intensity of the reference arm, β is the propagation constant, zref is the length from the reference arm to the PD, and zm is the length from the start of the m th FBG to the PD. The intensity of the reflected light can be obtained using a Fourier transform. In this study, Rayleigh scattering is defined as the noise floor, resulting in a signal-to-noise ratio (SNR) of ~ 23 dB. Signal from target FBGs from the fiber under test can be extracted via a Butterworth band-pass filter.
The sensing mechanism is based on tracking the frequency shift of the THz FBG signal related with the ambient change. To extract the THz FBG signal shift, a self-mixing technique was applied to the extracted individual THz FBG signal. The reflected signal in frequency domain is squared and filtered using a low-pass filter to obtain the FBG reflection spectrum: FBG results in enhanced signal quality factor (Q-factor). However, the trade-off is the mitigated spatial resolution due to the increased grating length.
To investigate the potential utility of a THz FBG as a temperature sensor, a THz FBG was fabricated using a fs laser power of 0.11 W, a period length of 1 mm, and 20 reflection points. The THz FBG was placed in a temperature-controlled water bath and the sensor's temperature response measured. GHz. Figure 2.4(c) plots the temperature response from 50 ºC to 65 ºC. Using this configuration, the temperature sensitivity for the THz FBG was observed to be approximately -1.32 GHz/ºC. It is worth noting that the sensitivity of THz FBG is much larger than conventional microwave grating due to the fact that the interrogation window in the proposed setup is in optical range, and the grating resonant peak under test is at a much higher order in comparison with 1st order in a microwave grating. For 1 mm grating, the resonant peaks range from 1923th to 1967th order, given that the laser tuning bandwidth is from 1525 to 1555 nm.
In order to investigate the distributed sensing capability of the system, a 40reflection point THz FBG was fabricated with a period of 1 mm, shown in Figure 5(a).
The system was calibrated using the reflectivity of an APC connector, which was previously measured to be -60 dB using precision instrument.   10 10  addition, the ultraweak reflection nature of so fabricated THz FBGs promises a huge multiplexing capacity [27].
A key feature of THz FBGs is that they require a much narrower detection bandwidth than FBGs in the optical frequency range while maintaining good spatial resolution. To demonstrate this feature, a THz FBG with 20 reflection points was tested using differing sweeping bandwidths from a tunable laser. Figure 2.6 (a-c) shows the spectra of the sensor under test using these differing laser sweep bandwidths. Figure 2.6 (d) shows the temperature response for each different bandwidth, which are observed to agree well with each other. These results demonstrate that, when compared with a conventional optical FBG, use of a THz FBG can effectively reduce detection bandwidth.
To evaluate system-level accuracy, a stability test was conducted by fixing the temperature of a 1 mm, 20 reflection point THz FBG. 100 spectra were recorded using

Introduction
Femtosecond laser has been attracting more interests in micromachining [28,29].
Due to the non-linear absorption, the laser induced breakdown could precisely deliver the high-intensity energy to the focal point causing minimum damage to the surrounding material [30]. Recently, terahertz fiber Bragg grating (THz FBG) was successfully fabricated by the femtosecond laser [16]. The pitch length was setup in sub-centimeter scale to achieve the THz domain detection. Ultra-weak (<-80 dB) reflectors were periodically seeded along the single mode fiber and no crosstalk was found. The THz FBG holds the great potential for high resolution distributed sensing. Before the fabrication of the THz FBG, it is necessary to remove the buffer from femtosecond laser processing region due to the different material absorption between the buffer coating and silica glass.
However, practically the bare fiber is easy to break with poor mechanical property. It is hard to handle the bare fiber without the coating material during the fabrication process.
The THz FBG sample was also fragile during the delivery and testing. This is especially true for long distributed THz FBG arrays. Although it is possible to recoat the bare fiber sensor with UV curable material after the fabrication, the recoating length was limited.
Here stresses the engineering challenge: during the femtosecond laser fabrication, can we process the sample without stripping the buffer.
This letter reports a new femtosecond (fs) laser fabrication method to modify the ultra-weak waveguide structure on the THz FBG sensor. The buffer was able to remain on the senor during the fs laser fabrication process, which significantly enhance the mechanical prosperity of the sensor. Experiments were conducted between the THz FBG with previous fs laser fabrication method and the one we proposed with intact polymer buffer coating. Highly similar results were observed.

Operation Mechanism
Both grating structures were interrogated using optical frequency domain reflectometry (OFDR) [17,18] and fabricated using a Ti: Sapphire fs laser (Coherent, Inc.) micromachining system. Single mode optical fiber (Corning, SMF-28) with core and cladding diameters of 8.2 and 125 µm, respectively, and a dual acrylate buffer was used in the fabrication of both gratings.
In the case of the stripped-buffer grating, the laser power used during fabrication was 0.11 W, resulting in the creation of a weak core mode reflection. This laser power was strong enough to be absorbed by the buffer material of the optical fiber to melt and deform the polymer layer, defocus the fs laser beam and fail the grating fabrication. This justifies the necessity to remove the buffer prior to fabrication of the grating structure.
Reflection signals recorded using this method were measured to be 22 dB above the Rayleigh level, which was considered the noise floor in this study, offering good signal quality for detection.  In order to fabricate a similar grating structure in which the buffer layer of the fiber under process was preserved, fs laser power was adjusted to 0.085 W; at this reduced level, the laser energy was not absorbed by the dual acrylate coating at a level sufficient to melt the fiber buffer. It is worth noting that the melting power of buffer materials under this setup is 0.05 W when laser beam is focused on polymer coating. The power 0.085 W used in grating fabrication here is focused on the fiber core, explaining that it is 70% higher than the threshold power to melt polymer. While core reflectors were still inscribed along the length of the fiber under process, these perturbations within the optical fiber core were reduced to the point of being invisible using conventional light microscopy. Signals recorded from these reflectors were measured to be 10 dB above the noise floor, again sufficient for good signal quality.  Similarly, to the strain testing set-up, temperature sensing using the two gratings was conducted by simultaneously interrogating the two sensors in series with both A liner relationship was found between the temperature drop and the sensor frequency shift, with R2 values for both gratings greater than 0.9994 and trend line slopes of -1.2791 GHz/°C and -1.3041 GHz/°C for the uncoated and buffer-coated gratings, respectively. As the heat capacity of water is larger than the buffer coating material, the sensor with the intact buffer coating cooled slightly faster than the grating without the buffer coating, which is surmised to account for the slight difference between the twotemperature testing result trend line slopes.

Experimental Results and Discussions
The results of both strain and temperature testing demonstrate that the fiber grating structure with an intact buffer coating has analogous sensing capabilities to the uncoated grating structure fabricated using conventional methods.

Conclusions
To conclude, this letter reports a new femtosecond laser fabrication method to modify the ultra-weak waveguide structures. To our best of knowledge, this is the first time to process the fiber sensing device without stripping the buffer. The fs laser output power was carefully adjusted to avoid the buffer coating melt and deformation, which enables grating fabrication with intact buffer, and largely enhances the sensor mechanical property. Strain and temperature test were conducted on both the uncoated buffer THz FBG sensor and the coated one. Highly identical results were observed, indicating that the proposed method holds great potential for distributed sensing measurement.

Introduction
Optical fiber sensors have a unique set of characteristics that make them particularly useful as strain and temperature sensors [23,26,[34][35][36][39][40][41]. Their chemical stability and immunity to electromagnetic interference make them ideally suited to sensing in harsh environments, while the ease with which multiple sensors can be fabricated in series and simultaneously interrogated allows for multiplexed sensing across considerable distances (~km), making optical fiber sensors a viable solution in structural health monitoring, energy, and aerospace applications [1,13,42,43].
Fiber Bragg gratings in particular have demonstrated their utility in these areas, and are a fundamental element of many optical fiber sensing techniques. Recently, fiber Bragg gratings fabricated with grating structures corresponding to the terahertz range (THz FBGs) have demonstrated additional characteristics that make them particularly well-suited to applications requiring large-scale, multiplexed sensing techniques [24,44]. Chief among these advantages are their narrow interrogation bandwidth (hundreds of gigahertz) and their high spatial resolution (<1 mm).
Despite these advantages, however, current THz FBG sensing techniques remain limited by their relatively narrow dynamic range. This limitation is chiefly due to the use of higher-order resonant peaks (1923 rd to 1967 th order with a pitch length of 1 mm) for parameter measurement and signal processing, which reduces dynamic range [44].
This trade-off is particularly limiting in the case of structural health monitoring, an area in which the need for structural data gathered during episodes of high strain and  It is known that the quality factor of the resonance peak can be significantly improved by introducing a defect into periodic structures, such as photonic crystals or FBGs [20,45,46]. This letter reports a grating structure interrogated in the terahertz range that surmounts this limitation. By interrogating the structure using lower-order (9 th order) resonance peaks and introducing a π-phase-shift into a THz FBG structure, the dynamic range of the resulting sensor was substantially increased and the corresponding loss of accuracy minimized, respectively. Experimental investigation of the π-phase-shifted THz FBG was conducted and compared against a THz FBG that was not phase-shifted, but was otherwise identically fabricated and interrogated in series along the same optical fiber. The results of this investigation, as well as a theoretical simulation modeling device physics, are presented in this letter.

Operation Mechanism
Optical frequency domain reflectometry (OFDR) was used as the interrogating system in this study [47]. The theoretical framework used to model the π-phase-shifted THz FBG is as follows: M π-phase-shifted THz FBGs are embedded along the detection arm of the system, with each π-phase-shifted FBG containing N reflectors (N is even) and a pitch length Δz. The total AC coupled voltage received by the DAQ can then be expressed as: where η is the light-to-voltage conversion coefficient of the photodiode, r is the reflection coefficient of the FBG reflectors, Iref is the light intensity of the reference arm, Data from the π-phase-shifted THz FBG is extracted from the total M FBGs using a Butterworth band-pass filter; data from the unmodified THz FBG is similarly

Experimental Results and Discussions
In order to experimentally investigate the sensing utility of the π-phase-shifted fiber Bragg grating, a strain test was conducted. A Ti:Sapphire femtosecond laser (Coherent, Inc.) was used to fabricate both gratings within single-mode optical fiber (Corning, SMF-28) at a laser power of 0.11 W [37,38]. Each grating contains 18 reflection cavities with a pitch-length of 1 mm. A π-phase-shift (1.5 times the pitch length) was added halfway along the length of the phase-shifted THz FBG.
Both the π-phase-shifted and unmodified THz FBGs were tested in series and interrogated simultaneously. The optical fiber with both sensors was secured to an optical bench on one end, with the other end free to hang. Weights were sequentially added to the free end of the optical fiber, and the resulting output from each sensor was recorded using the method described above. In all, 20 weights, with a total mass of 124 g, were added to the free end of the fiber at 6.2 g intervals. In order to quantify the strain change along the optical fiber generated by the addition of each set of weights, the unmodified THz FBG was interrogated using higher order modes, as described in previous work [44]. Due to the fact that the sequential addition of each weight to the fiber under test resulted in an incremental phase shift of less than 2π radians, this method provided a direct measure of strain along the fiber. Importantly, however, the total change after the addition of all the weights resulted in a combined shift greater than 2π radians using higher-order modes, which is beyond the detection limit of the earlier approach, but within the limit of the method described in this letter. Using this technique, total strain change of 1.0 mε in intervals of approximately 71.5 µε was measured over the course of the experimental trial.
As both sensors were placed along the same optical fiber and tested simultaneously, crosstalk between the two sensors was considered. By using ultraweak reflectors in the  FBGs was effectively minimized [24] The results of this experiment are shown in Fig. 3. The comparison between the spectra of π-phase-shifted THz FBG and unmodified THz FBG clearly showed that the quality factor of the resonance peak/dip improves by introducing a π-phase-shift into the normal THz FBG. As a result, the π-phase-shifted THz FBG demonstrated much improved linearity between strain change and frequency shift, with a R 2 value of 0.9962 and a trend-line slope of -0.0011 GHz/µε. In contrast, data collected from the unmodified THz FBG demonstrated less linearity, with an R 2 value of 0.6699. These results demonstrate that the π-phase-shifted THz FBG performed substantially better than the unmodified THz FBG in the detection of strain changes along the length of the optical fiber under test. In addition, as observed in the experiments, the intensity of reflection spectrum decreases as the loaded strain increases. This is attributed to the fact that the refractive index contrast, or reflectivity, of the fs modified reflectors decreases under strain. Importantly, the experimental results also showed that the signal of the proposed THz FBG was sufficiently strong for strain sensing within the testing range up to 1 mε.

Conclusions
To conclude, this letter demonstrates that the application of a π-phase-shift to THz FBGs provides the enhanced accuracy necessary to use lower-order resonance peaks for sensor interrogation, allowing for expanded dynamic range relative to the previously reported method using higher-order resonance peaks. This sensing strategy was theoretically-modeled and experimentally investigated using both a π-phase-shifted THz FBG and an unmodified THz FBG in series along the same optical fiber under test.
The resulting linearity of the peak shift of the π-phase-shifted THz FBG (R 2 = 0.9962) demonstrates that this technique has the potential to lead to the application of THz FBGs to domains in which large dynamic range is a critical engineering requirement.
Structural health monitoring provides an example of one such area; by allowing for enhanced real-time monitoring of critical infrastructure, THz FBGs with enhanced dynamic range have considerable potential societal benefit.

Introduction
Optical fibers have seen increasing use as sensing elements for strain, stress, temperature and refractive index measurements in a wide variety of engineering applications. A more recent application of fiber sensors is to determine angle and position information used for three-dimensional shape sensing. Due to its compact size and ability to conform to relatively complex shapes, optical fiber can be easily fixed along an object of interest to mirror its orientation. Multiple strain sensors can then be integrated together to form a single sensing probe aligned to the changing contour along the object. By using a multi-core fiber or a multi-fiber bundle packaging method, three or more strain sensors can be packaged in parallel together as one sensing element to measure strain information in three dimensions, data that can ultimately be used to find the angle and position information necessary for three-dimensional shape sensing.

Different fiber distributed strain sensors have been proposed and commercialized
to reach this engineering goal. The two main research areas undergirding this technology are fiber Bragg gratings (FBG) and coherent optical frequency domain reflectometry (C-OFDR). Both methods use differing fundamental physics to reach the same goal; distributed strain sensing along an optical fiber.
A fiber Bragg grating (FBG) is a wavelength-based fiber sensor [14,48]. Periodic reflectors are inscribed along a single mode fiber by UV laser to form a wavelengthspecific dielectric mirror that reflects a certain wavelength and transmits the rest.
Multiple FBGs can be integrated along one sensing probe using wavelength multiplexing to achieve distributed strain measurements. Miller et al. have proposed several FBG-based structures used to resolve the spatial angle information using multiple FBGs, which include multi-core fibers [49,50] and three-fiber bundles [51] as two-axis fiber inclinometers. More recently, researchers have demonstrated using a femtosecond laser to directly inscribe FBG waveguide structures into a single coreless fiber [52] and standard single mode fiber [53] for three-dimensional shape sensing applications.
Coherent optical frequency domain reflectometry (C-OFDR) [2,5,[54][55][56][57][58][59] is an alternative, state-of-the-art distributed sensing technology. By sweeping the optical frequency with a tunable laser, the Rayleigh backscatter profile along an optical fiber can be measured, which correlates to strain along the fiber under test. This interferometric measurement is capable of maintaining a terahertz-level detection bandwidth with high spatial resolution. Using three or more probes, C-OFDR has successfully demonstrated its ability to deliver three-dimensional shape sensing using both the multicore fiber [60][61][62] and multiple fiber packing methods [63].
Recently, the use of terahertz fiber Bragg grating (THz FBG), an extension of C-OFDR interrogation, has been demonstrated [44,64]. A high-power femtosecond laser is employed to inscribe ultra-weak periodic reflectors (< -70 dB) within the core of a standard single mode optical fiber, providing distributed interferometric strain measurements along the fiber under test and allowing the system to achieve enhanced sensitivity compared with traditional Rayleigh scattering-based (< -80 dB) C-OFDR techniques. By combining three or more of these THz FBG sensors into a single fiber sensor bundle, the system is able to determine the spatial angle information necessary to act as a two-axis optical fiber inclinometer, an important step towards threedimensional shape sensing.
This manuscript reports a terahertz fiber grating-based two-axis optical fiber inclinometer fabricated using ultra-weak reflection arrays (-70 dB). Three identical THz FBGs were aligned and packaged as a single fiber bundle. The differing strain distributions across the three THz FBGs within the sensing probe were measured and used to determine the spatial angle information along the fiber bundle. The inclinometer was tested at eight azimuthal angles (from 0° to 315°). The standard deviation of the greatest inclination angle error was 0.048° and the inclination angle stability was 0.030°.
No cross-talk was found between the ultra-weak reflection arrays.
To bind the modified fibers into a three-fiber bundle, the three SMF fibers with THz FBG arrays were secured equidistant to one another, as shown in Figure 5.2 (a). The three modified fibers were aligned using reference marks ~5 cm beyond the length of the sensing arrays. Two 40 mm heat melt tubes (EVA) and a 90 mm heat shrink tube were placed outside the aligned sensing arrays, as illustrated in Figure 5.1 (a). Hot air (~300 F°) was blown from ~1 cm away and swept across the length of the bundle at a speed of ~1 cm/s. After subsequent cooling, the three THz FBG sensing arrays were firmly fixed in place, shown in Figure 5.1 (b).
To assemble the inclinometer, one end of the packaged bundle was positioned using a screwed ferrule into the center of a 360° rotational mount (Newport Inc.) along the Z axis to give azimuthal rotation control, as illustrated in Figure 5.1 (b). The other end formulae (16). The azimuthal angle φ was defined as the sensor rotation along the Z axis. As the sensor was deflected towards the Y+ direction, the inclination angle θ at a certain azimuthal angle φ, shown in Figure 5.3 (b), was defined as the acute angle between the Z axis and line passing through the 6th and 7th sensing elements.
In order to account for packaging inconsistencies such as sensing array misalignment or heat shrink tube nonuniformity, calibration must be performed. First the cross-plane of each sensing element, the relative positions of the three strain sensing points were calibrated. As illustrated in Figure 5.4 (d), the bundle was bent towards three different azimuthal angles φ1, φ2, and φ3, where sensing arrays #1, #2, and #3 were each positioned on top, respectively. The differing strain distributions for the seven sensing points were then measured along each sensing array. For each sensing element, a two-dimensional cross-plane was extracted to calculate the relative position of the The initial condition is defined where the relative distance between the sensing points along sensing array #1 to the X-axis is normalized to 1. The relative positions of the three sensing points for each sensing element cross-plane were then solved, shown in Figure 5.5 These coefficients were converted to polar form and applied to solve the Frenet-Serret equations.
Second, the absolute radius can be calibrated based on the assumption that all seven strain sensing points along the sensing array #1 share the same distance to the center of the inclinometer bundle. Based on the first calibration results, another radius coefficient is set to match the calculated position and the absolute deflection by the Y-directional stage at each azimuthal angle. The stability test of the proposed inclinometer was conducted with a fixed 0˚ azimuthal angle and 0˚ inclination angle. In total 250 sets of tests were repeated to measure this initial inclination angle. The standard deviation of the measured angle was 0.030°, indicating a good repeatability of this device.

Experimental Results and Discussions
The maximum inclination measurement range was 2.5° with steps of 0.085° and the error standard deviation was 0.036°. In these tests, the inclination measurement range was limited by the maximum frequency shift within 1 period of the implemented THz FBG, which is 100 GHz for 1 mm pitch length.

Conclusions
To conclude, this manuscript reports a new two-axis optical fiber inclinometer based on terahertz fiber Bragg grating (THz FBG) structures. Inclination angles from 0° to 1.7° were tested at eight azimuthal angles (from 0° to 315°), covering one full rotation. The standard deviation of the largest inclination angle error was 0.048° and the observed inclination angle stability was 0.030°. In addition, the 3-dimensional spatial positions for all sensing elements were solved, indicating this sensor holds substantial potential for three-dimensional distributed sensing.

Introduction
Terahertz fiber Bragg gratings (THz FBGs) have been investigated as sensor elements for a number of distributed strain and temperature sensing applications [44].
Femtosecond laser micromachining techniques have been used to fabricate these millimeter-range weak periodic structures within the cores of otherwise unmodified communications-grade optical fibers [26]. GHz within 1 ms was achieved using the system. In order to experimentally investigate the system, a strain test was performed; highly linear results (R 2 = 0.996) were observed, with a sensitivity of −0.142 GHz/µɛ and a standard deviation of 1.167 µɛ. A multiplexing test was also conducted, and no cross-talk observed between the sensor elements. These results demonstrate that this system holds substantial potential as a method of dynamic distributed strain sensing.

Operation Mechanism
A schematic of the interrogation system, based on coherent optical frequency domain reflectometry (C-OFDR) [25,64,71], is shown in Figure 6.1. An arbitrary wave  where ( ) is the output frequency of the DFB laser, 0 is the initial start sweep optical frequency, ( ) is the LDC current dependent gain, 0 is the LDC external modulation voltage to current coefficient, and ( ) is the output modulation voltage waveform from the AWG. To simplify the equation, a sweep coefficient ( ) can be defined as: An extra MZI interferometer with two 3-dB couplers and a delay of 28.4 ns was constructed for this iteration process. The laser sweep speed ( ), related with laser frequency ( ), can then be measured using this MZI with a delay length of : where ( ) is the radio frequency output of the BPD after the MZI.
The iteration process is as follows: For the n th round of sweep ( ≥ 0) using the n th AWG modulation waveform ( ), the n th laser sweep speed ( ) can be measured based on eq. (3). Using the n th sweep coefficient ( ), the next round (n+1) th sweep coefficient +1 ( ) can be generated: where ̅ is the target sweep speed and initial sweep coefficient 0 ( ) = ̅ . The (n+1) th AWG modulation voltage waveform +1 ( ) can then be determined based on eq.(3): For the first sweeping round (n = 0), the initial AWG output waveform 0 ( ) is set as a ramp function from 0 to 1 Volt over 1 ms. The resulting laser optical sweep speed 0 ( ) is measured using a short-time FFT algorithm and plotted in Figure 6.2 (a). The target frequency sweep speed was set to 140 GHz/ms. After 6 rounds of feedback predistortion iterations, the resulting measured laser sweep speed is plotted in Figure 6.2 (b). The DFB laser sweep from 0.1 ms to 1 ms is used to interrogate the THz FBG sensors, where the sweep speed is between 130 and 150 GHz/ms. After re-sampling the data using the reference clock arm, 6k data points are generated, covering an interrogation bandwidth of 110 GHz. A 1 ms DC voltage output follows the 1 ms precalibrated waveform to allow the laser sweep speed to return to zero.
A Ti:sapphire femtosecond laser micromachining system (Coherent, Inc.) was employed to fabricate the THz FBG sensing units [24,73]. The pitch length was set to 1 mm, resulting in a 100 GHz resonation period, which can be fully covered by the

Experimental Results and Discussions
In order to experimentally investigate the chirp-DFB laser-based interrogator system, a strain test was conducted. One THz FBG with a pitch length of 1 mm and 20 reflector points was fabricated using 0.12 W femtosecond laser power and connected to the interrogation system. One end of the sensor under test was secured to an optical bench while the other end was left free to hang, shown in measurements were gathered; the frequency domain interferogram shifts were determined, shown in Figure 4, and resulted in a standard deviation of 1.167 µɛ.
To demonstrate multiplexing capability, another 1 mm pitch length, 20-point THz FBG was fabricated using 0.14 W femtosecond laser power and cascaded in-line with the first THz FBG, as shown in the schematic setup in Figure 5  fabrication. The precise mechanism of this lack of a clear peak for this particular 0.14 W THz FBG will be the subject of future research; however, this lack did not prevent the authors from observing frequency shifts during testing, as evidenced in Figure 6.5 (d).
Although the chirp-DFB lasers are a suitable method of interrogating THz FBGs, they present several limitations. The maximum interrogation range in the frequency domain is limited by the sweep bandwidth of the chirp-DFB laser, which is approximately 100 GHz; consequently, large strain changes with frequency shifts greater than 100 GHz cannot be resolved. Additionally, the precise starting optical frequency for each frequency sweep may vary; a calibration method that precisely tunes the starting frequency will be investigated in subsequent work. Although the intensity output of laser does vary due to the current injection modulation, the fact that changing output intesnity did not prevent the interrogation to measure multiplexed strain changes using Thz FBGs. Optical or electrical automatic gain control module can be added to potentially solve this problem for particular applications.

Conclusions
To conclude, this letter reports a chirped DFB laser-based interrogator for THz  [24,44,64,71]. By definition, a sub-THz-FS is an optical fiber-inline structure with characteristic geometries in the millimeter or sub-millimeter range that can be interrogated using sub-THz bandwidths in the optical frequency band [74]. Uniquely, sub-THz-FSs allow systems to simultaneously achieve distributed strain and temperature measurements with high-accuracy and high spatial resolution using a narrow interrogation bandwidth. Previously, the interrogation system of sub-  [66,78]. Although effective, these methods add complexity, cost, and increased device footprint to ECLs.
In contrast, a distributed feedback laser (DBF) is also capable of mode-hop-free wavelength tuning via modulating its injection current, without the need for moving geometric-optic components. However, there are two critical fundamental challenges associated with using this frequency sweep technique for C-OFDR-based applications: (1) a limited tuning bandwidth (~100 GHz), and (2) a nonlinear relationship between injection current and laser frequency, leading to inconsistent sweep velocities. The challenge of limited bandwidth, which restricts the spatial resolution of many C-OFDR applications, is readily overcome using sub-THz-FSs due to their unique, proven ability to facilitate narrow interrogation bandwidth operation [24,44,74]. Thus, inconsistent sweep velocity represents the key remaining challenge precluding the use of tunable DFB lasers for sub-THz-FS sensor interrogation. Efforts have been made to overcome this remaining limitation by implementing an auxiliary sampling clock; however, due to the Nyquist criterion, the delay line for the interferometer used in the sampling clock must be at least four times longer than the total length of sensing arm. This long delay line makes the interrogation system more susceptible to ambient noise, and, given the same sweep velocity, necessitates the use of high frequency electronics, resulting in increased design complexity and system cost.
This letter described an alternative approach that actively linearizes the frequency sweep in order to overcome the remaining challenge of inconsistent sweep velocity directly, allowing for purely-electronically modulated DFB lasers to be used for sub-THz- previously reported results obtained using an ECL. Additionally, a soldering iron was employed as a heat source to form a temperature distribution along a continuously cascaded sub-THz-FS array to demonstrate its high spatial resolution distributed sensing capability.

Operation Mechanism
A schematic of the described interrogation system is shown in Figure 7 condition; over that span, a signal-to-noise ratio (SNR) above 50 dB was achieved. During testing, a resting period of 5 ms followed each 9 ms sweep in order to discharge the capacitors in the loop filter, resulting in a total 14 ms for each complete pulse cycle and a reputation rate of 71 Hz. To determine the noise of the system, 1 second of data with 71 chirped laser pulses was recorded. The Fourier transform of this data is plotted in Figure   7.3 (b). A 71 Hz frequency period was observed due to the reputation rate. The full width at half maximum (FWHM) of the peak envelope using a Gaussian curve fit was measured to be 116 Hz.
A homodyne configuration was constructed using two 2×2 3-dB couplers, depicted in the sensing module of Figure 7.1. The input light is split into two paths via the first coupler, with one path serving as the reference arm and the other path directed into the sensing arm including a sub-THz-FS array. The sensing arm was terminated using an anti-reflection cut. The reflected light from sub-THz-FSs was then combined with light from the reference arm through the second coupler. A photodetector and a single channel AC-coupled 12-bit ADC was used to record the resulting data. The sampling rate of the ADC was set to 8 MSa/s with a matched anti-aliasing filter. The digitized raw data was then fed into a DSP module. In order to investigate the sensing capability of the described concept, a 20-pt periodic weak reflection sub-THz-FS array with a 1 mm pitch length was fabricated along a single mode fiber (SMF-28, Corning, Inc.) using a Ti: Sapphire femtosecond laser (Coherent, Inc.) [26,38]. During interrogation and signal processing, the sub-THz-FS array was considered as 9 cascaded sub-THz-grating sensor units using a 4-mm wide moving

Experimental Results and Discussions
Butterworth bandpass filter with a step size of 2-mm. In this case, each sensor unit contains 4 reflection peaks. This signal processing method has been systematically investigated in the previous publications [24,44,74]. The interferograms of the target sensor units were extracted using a self-mixing method and a low-pass filter. Changes in strain or   groups of measurements were collected; the maximum standard deviation among these sensor units was calculated to be 0. 16 GHz, corresponding to a detection limit of 1.11 µɛ.
The starting sweep frequency was evaluated by measuring the starting frequency of the entire system over 1000 captures, and the standard deviation of start frequency was 109 MHz.
In order to demonstrate the distributed sensing capability of the system, a dynamic temperature test was conducted. A schematic illustration of the testing setup is shown in  can also be applied to resample the data and correct for non-linear sweep speeds [82].
However, this sampling clock method increases both system sampling and signal processing complexity.
A closed loop control system based on an optical phase-locked loop (OPLL) that modifies laser output frequency in real-time offers an alternative approach to precisely controlling optical sweep speed [72,83]. Recently, a digital-controlled chirped pulse laser based on a digital phase-locked loop (DPLL) design was reported based on modular electronic design [84]. A digital phase comparator (XOR gate) was utilized to extract phase errors between the output of a Mach-Zehnder interferometer, which converted laser sweep speed to a radio-frequency signal, and a reference oscillator. corresponding to a strain sensing instability of 0.79 µɛ. In order to evaluate the potential of the ADPLL-based SV-LLPG as an element of an OFDR system, the system was used to interrogate a sub-THz-FS array. Highly linear results and a sensitivity of -0.1346 GHz/µɛ were observed, which agrees well with previously reported sensing results obtained using an ECL [24,44].

Operation Mechanism
A schematic of the described interrogation system is shown in Figure 8.1. In the SV-LLPG module, a DFB laser is employed as the frequency sweep source, which is injection current-modulated using a time-varying voltage via a laser control circuit. An isolator is placed at the laser output to eliminate reflection. Using a 90/10 coupler, 10% of the output power is directed in to the MZI and 90% of the power into the sensing module to interrogate the sub-THz-FS array. The MZI is constructed with two 3-dB couplers with a constant delay, , of 11.334 ns. Under the assumption that the DFB laser is operated at a constant sweep velocity, the AC-coupled current output i(t) at the photo diode after the MZI as a function of time can be expressed as: ( ) = ( ) 2 8 cos[2 ( 0 + ) ] where A(t) is the electrical-field amplitude directed into the MZI as a function of time, η is the responsivity of the photo diode, f_0 is the initial frequency of the DFB laser during sweeping, ν is the optical sweep velocity, and t is the time. A beat frequency in the radio frequency (RF) range less than 250 kHz, which is linearly proportional to laser sweep velocity, is generated through this fixed delay MZI. Due to the current injection modulation, the intensity of the DFB laser output varies as a function of time. To account for this effect, an automatic gain control (AGC) transimpedance amplifier is used to adjust the amplitude of AC-coupled photodiode output signal. This photodiode has a bandwidth of 1MHz. A high-speed voltage comparator with a bandwidth of 50 MHz is used to convert the analog beat signals into digital signals, which is sent into a digital input port of a FPGA evaluation board with a 100 MHz system clock. In order to improve the initial laser sweep linearity before phase locking, a pre-distortion voltage waveform is pre-calibrated using a feedback iteration method [82], resulting in an output frequency sweep velocity at about 12.5 GHz/ms. This open loop pre-distortion voltage waveform is then stored within the FPGA memory. A type-II phase comparator, constructed with two D flip-flops and an AND gate [87], is used to extract the phase difference between the input digital signal and the on-board reference frequency clock fR at 140 kHz. The resulting phase error signals are then fed into a loop controller constructed using an integrator to further modify the pre-distortion current modulation waveform. The digital output of the FGPA is converted to an analog signal using a digital-to-analog converter (DAC) module with a refresh cycle of 1 µs and sent to the laser driver circuit. Hz frequency period was observed due to the repetition rate described above. The full width at half maximum (FWHM) of the peak envelope using a Gaussian curve fit was measured to be 90 Hz.
Along the sensing module, a homodyne interferometry structure is constructed using two 2×2 3-dB couplers as shown in Figure 8.1. The input light is split into two paths via the first coupler, with one serving as reference arm and the other path directed into the sensing arm, which includes a sub-THz-FS array. The sensing arm is terminated using an anti-reflection cut. The reflected light from the sub-THz-FS is then combined with light from reference arm via the second coupler. A photodetector and a single channel AC-coupled 8-bit analog-to-digital converter (ADC) is used to record the resulting data. The sampling rate of the ADC is set to 8 MSa/s with a match anti-aliasing filter. The digitized raw data is then fed into a DSP module. In order to investigate the sensing capability of the described SV-LLPG system, a 20-pt periodic weak reflection sub-THz-FS array with a 1 mm pitch length was fabricated along a single mode fiber (SMF-28, Corning, Inc.) using a Ti: Sapphire femtosecond laser micromachining system (Coherent, Inc.) [24,44,64,73]. During interrogation and signal processing, this sub-THz-FS array was considered to be 9

Experimental Results and Discussions
cascaded sub-THz-grating sensor units using a 4-mm wide moving Butterworth bandpass filter with a step size of 2-mm. Each sensor unit contains 4 reflection peaks.
This signal processing method has been systematically investigated in the previous publications (7,22). A self-mixing method and a low pass filter is applied to extract the resulting interferograms. Changes in strain along the optical fiber result in optical path length (OPL) changes between the weak reflectors, which generate a phase-shift in the interferograms that are used to measure strain changes along the sensor probe.
To evaluate the strain sensing capability of the system, a series of static strain tests were conducted. One end of the fiber under test (FUT) was secured to an optical bench while the other end was left free to hang. Weights were sequentially added to the free

Conclusions
To conclude, this manuscript reports a FPGA-controlled sweep velocity-locked laser pulse generator (SV-LLPG) design. A DFB laser is employed as the sweep source and an ADPLL control system is used to lock the laser sweep velocity to an on-board reference clock. Highly linear chirped laser pulses with a bandwidth of 111.16 GHz were demonstrated. A sweep velocity of 12.35 GHz/ms was achieved for 9 ms within each chirped pulse at a 50 Hz pulse repetition rate. To investigate system sensing utility, the SV-LLPG prototype was used as an element of an OFDR system to interrogate a sub-THz-FS array. A static strain test was conducted and highly linear results were observed.
The proposed device holds the promise to deliver a low size, weight and power (SWaP) and affordable interrogator for distributed fiber sensing applications. In addition, the FPGA based design makes it easier to be integrated and adopted for various applications in the future.