Center Wavelength Shift Research Articles

Full-range displacement sensor with dual FBGs combined cantilever beam based on magnetic grating

We propose a full-range displacement sensor system based on double fiber Bragg gratings (FBGs) with a cantilever beam. This sensor adopts a magnetic scale as a unique transmission mechanism. By combining a simple and low-cost cantilever beam structure makes the sensing probe very easy to realize. The optimal magnetic gap was explored through multiple sets of numerical simulations. The experiment proved the sine relationship between the center wavelength shifts of FBGs and the linear displacement, proving the feasibility of this method. Results indicate that the amplitude of the tensile compression load of FBGs are 169.76 με and 296.12 με. The fitting linearity based on sinusoidal function at an air gap of 3.5 mm is 0.9663 and 0.9566. The direction of motion can be determined by the phase difference between two FBGs. Moreover, the difference of double FBGs can realize the effect of temperature compensation. Thus, this sensor can achieve non-contact, temperature-independent, and full-range measurements. It can also be exploited to measure other parameters such as angular velocity, acceleration, and magnetic field strength.

In-orbit detection of the spectral smile for the Mars Mineral Spectrometer

As a payload of Tianwen-1 (TW-1), the Mars Mineral Spectrometer (MMS) is tasked with acquiring hyperspectral data of the Martian surface to detect material composition. Microdeformations in optical, mechanical, and thermal components result in the MMS experiencing spectral response distortion in orbit, leading to systematic changes in pixel central wavelengths and full width at half maximum (FWHM). Known as the spectral smile, this distortion compromises the accuracy of reflectance inversion and material composition detection. This study introduces a method for detecting the spectral smile through the Martian atmospheric absorption channel, capitalizing on the distinct characteristics of the atmospheric composition and absorption patterns of Mars. A suitable technical route for in-orbit spectral smile detection was established and tested using simulation experiments and MMS-acquired hyperspectral data. Results suggest that the proposed method can attain central wavelength shifts with a maximum error of 0.32 nm and FWHM variations with a maximum error of 1.95 nm. Employing in-orbit spectral smile detection markedly enhances the correction of Martian atmospheric absorption and provides technical support for Martian surface reflectance inversion. https://github.com/wubingnote/MMS-Spectral-Smile.

Four-wave mixing temperature sensor based on graphene oxide-coated microfiber hybrid waveguide

Utilizing a hybrid waveguide comprising microfiber and graphene oxide (GO), this study introduced an innovative temperature sensor based on four-wave mixing (FWM) effect. The methodology involved a tunable continuous wave (TCW) laser and a self-engineered all-fiber mode-locked laser operating at 1560 nm. The microfiber, featuring a minimum diameter of 10 μm, was fabricated through the flame fusion tapering technique, and the microfiber surface was coated with a GO film through the spontaneous evaporation of a GO solution. To improve the stability and robustness of the sensor, a layer of polydimethylsiloxane (PDMS) was applied to the surface as a protective coating. The GO film compensated for dispersion in conjunction with microfiber, enhanced nonlinearity and facilitated low-power FWM phenomena. Furthermore, it increased temperature sensing sensitivity. By measuring the center wavelength shift of the idler and signal light, we achieved a temperature sensing sensitivity of −2.08 nm/℃ from −15 ℃ to 20 ℃ using the CW laser at 1540 nm. This sensor also had a resolution of 0.024 ℃, a response time of 0.12 s and a maximum temporal resolution of 0.08 s. Our study marks the initial investigation of a temperature sensing mechanism employing the FWM phenomenon within this unique hybrid waveguide. It serves as a proof of concept for using a microfiber combined with the two-dimensional material GO film to generate nonlinear FWM for temperature sensing. Continuous optimization of experimental conditions and waveguide dimensions is expected to enhance sensing performance. This opens up new avenues for future research in novel optical sensing technology, particularly in the exploration of high-performance sensors through the integration of nonlinear phenomena with different two-dimensional materials, showcasing substantial research potential.

Assessment of the spectral misalignment effect (SMILE) on EarthCARE's Multi-Spectral Imager aerosol and cloud property retrievals

Abstract. The Multi-Spectral Imager (MSI) on board the Earth Cloud, Aerosol and Radiation Explorer (EarthCARE) will provide horizontal information about aerosols and clouds. These measurements are needed to extend vertical cloud and aerosol property information, which is obtained from EarthCARE's active sensors, in order to obtain a full three-dimensional view of cloud and aerosol conditions. Mesoscale weather systems, in particular, will be characterized. The discovery of a non-compliance of the MSI visible–near-infrared–shortwave infrared (VNS) camera’s visible (VIS) and shortwave infrared (SWIR1) channels regarding a spectral central wavelength (CWVL) shift across-track of up to 14 nm (VIS) and 20 nm (SWIR1) led to the need for an analysis regarding its impact on MSI Level-2A aerosol and cloud products. A significant influence of the spectral misalignment effect (SMILE) on MSI retrievals is identified due to the spectral variation in gas absorption, surface reflectance, and aerosol and cloud properties within the spectral ranges of these MSI bands. For example, the VIS channel is positioned in close proximity to the red edge of green vegetation and is impacted by residual absorption of water vapor and ozone. Small central wavelength variations introduce uncertainties due to the rapid change in surface reflectance for conditions with low optical thickness. The present central wavelength shift in the VIS towards shorter wavelengths than at nadir introduces a relative error in transmission of up to 3.3 % due to the increasing influence of water vapor and ozone absorption. We found relative errors in the top-of-atmosphere (TOA) signal due to the SMILE of up to 30 % for low optical thickness over a land surface in that band. Since the magnitude of the impact strongly depends on the underlying surface and atmospheric conditions, we conclude that accounting for the SMILE in Level-2 retrievals or correcting the Level-1 signal will improve MSI aerosol and cloud product quality.

Low Velocity Impact Monitoring of Composite Tubes Based on FBG Sensors.

Carbon fiber reinforced composites (CFRP) are susceptible to hidden damage from low velocity external impacts during their service life. To ensure the proper monitoring of the state of the composites, it is crucial to predict the location of an impact event. In this paper, fiber Bragg grating (FBG) sensors are affixed to the surface of a carbon fiber composite tube, and an optical sensing interrogator is used to capture the central wavelength shift of the FBG sensors due to low-velocity impacts. A discrete wavelet transform is used for noise reduction in the response signals. Then, the differences in the captured response signals of the FBG sensors at different locations of the impact were analyzed. Moreover, two methods were implemented to predict the location of low-velocity impacts, according to the differences in the captured response signals. The BP neural network-based method utilized three data sets to train the neural network, resulting in an average localization error of 20.68 mm. In contrast, the method based on error outliers selected a specific data set as the reference dataset, achieving an average localization error of 13.98 mm. The comparison of the predicted results shows that the latter approach has a higher predictive accuracy and does not require a significant amount of data.

Polymer Waveguide Sensor Based on Evanescent Bragg Grating for Lab-on-a-Chip Applications.

In this work, an evanescent Bragg grating sensor inscribed in a few-mode planar polymer waveguide was integrated into microchannel structures and characterized by various chemical applications. The planar waveguide and the microchannels consisted of epoxide-based polymers. The Bragg grating structure was postprocessed by using point-by-point direct inscription technology. By monitoring the central wavelength shift of the reflected Bragg signal, the sensor showed a temperature sensitivity of -47.75 pm/K. Moreover, the functionality of the evanescent field-based measurements is demonstrated with two application examples: the refractive index sensing of different aqueous solutions and gas-phase hydrogen concentration detection. For the latter application, the sensor was additionally coated with a functional layer based on palladium nanoparticles. During the refractive index sensing measurement, the sensor achieved a sensitivity of 6.5 nm/RIU from air to 99.9% pure isopropyl alcohol. For the gas-phase hydrogen detection, the coated sensor achieved a reproducible concentration detection up to 4 vol% hydrogen. According to the reported experimental results, the integrated Bragg-grating-based waveguide sensor demonstrates high potential for applications based on the lab-on-a-chip concept.

Error analysis and correction method of multi-core fiber sensing

To meet the application requirements of accurate shape sensing for biomedical robotics and flexible morphing structure of aircraft etc, the error analysis and correction method for multi-core fiber is proposed. First, the relationship between the central wavelength shift and curvature of the multi-core fiber is analyzed which is based on the principle of multi-core fiber shape sensing, and a temperature decoupling method was researched. Considering the influence of the fiber structural parameters on curvature sensing, which is simulated by using the Monte Carlo method, and the simulation results show a strong influence among them. Next, to reduce the sensing errors introduced by the structural parameters, an error correction method based on regularized multilayer perceptron (MLP) networks is investigated, the proposed method does not require precise structural parameters of the multi-core fiber. Finally, some experiments with different curvatures are carried out, the results showed that the maximum root mean square error (RMSE) decreases from 2.03 m−1 to 0.007 m−1, and the minimum RMSE decreases from 0.01 m−1 to 0.001 m−1. The method this paper proposed has potential applications in shape sensing in the fields of biomedical engineering and aerospace.

Spectral analysis of beam-combining-systems based on laser array with smile effect

The smile effect in semiconductor laser arrays can cause spectral deterioration in spectral beam combining (SBC) systems. We analyze this effect using position error and pointing deviation and propose a coupled-cavity resonance model based on the external-cavity optical feedback and the multibeam interference theory to study locked spectra. Our results indicate that the central emitter exhibits excellent spectral purity, and the edge emitters show side peaks and a decrease in peak intensity. With the increase in the position error and pointing deviation, the emitters experience an obvious spectral shift and an expansion of the spectral width. Pointing deviation has a greater impact on wavelength locking and array spectral width than position errors. Increasing the focal length and incident angle can reduce shifts and spectral width. Furthermore, adopting a grating with a large line density can also decrease its center wavelength shift and compress the spectral width of the combined beam.

Assessment of the Influence of Instrument Parameters on the Detection Accuracy of Greenhouse-Gases Absorption Spectrometer-2 (GAS-2)

Satellite-based monitoring of atmospheric greenhouse gas (GHG) concentrations has emerged as a prominent and globally recognized field of research. With the imminent launch of the Greenhouse-Gases Absorption Spectrometer-2 (GAS-2) on the FengYun3-H (FY3-H) satellite in 2024, there is a promising prospect for substantial advancements in GHG detection capabilities. Crucially, the accurate acquisition of spectral information by GAS-2 is heavily reliant on its instrument parameters. However, the existing body of research predominantly emphasizes the examination of atmospheric parameters and their impact on GHG detection accuracy, thereby leaving a discernible gap in the comprehensive evaluation of instrument parameters specifically concerning the acquisition of atmospheric greenhouse gas concentration data by GAS-2. To address this knowledge gap, our study employs a radiation transfer model grounded in radiation transfer theory. This comprehensive investigation aims to quantitatively analyze the effects of various instrument parameters, encompassing crucial aspects such as spectral resolution, spectral sampling rate, signal-to-noise ratio, radiometric resolution, and spectral calibration accuracy (including instrument line shape function, central wavelength shift, and spectral resolution broadening). Based on our preliminary findings, it is evident that GAS-2 has the necessary spectral resolution, spectral sampling rate, and signal-to-noise ratio, slightly surpassing existing international instruments and enabling a significant detection accuracy level of 1 part per million (ppm). Moreover, it is essential to recognize the critical impact of instrument spectral calibration accuracy on overall detection precision. Among the five commonly used instrument line shape functions, the sinc function has the least impact on detection accuracy. Additionally, GAS-2’s radiance quantization depth is 14 bits, which is comparable to similar international payloads and maintains a root mean squared error below 0.1 ppm, thus ensuring a high level of precision. This study provides a comprehensive evaluation of the influence of GAS-2’s instrument parameters on detection accuracy, offering valuable insights for the future development of spectral calibration, the optimization of similar payload instrument parameters, and the overall improvement of instrument quantification capabilities.

Optical Fiber Sensor with Novel Structure and Its Applications in Oil and Gas Exploration

Compared to electronic sensors, Optical Fiber Sensors (OFSs) have received increasing attention due to their advantages, such as small size, light weight, anti-electromagnetic interference, easy reuse, and remote sensing. This paper proposes a novel all-fiber mode interferometer. This interferometer excites higher-order cladding modes through mode mismatch between standard single-mode fibers and thin cores. There is interference between higher-order cladding mode and core layer mode, resulting in interference fringes used as sensing signals. Sensitivity testing is conducted on the designed OFS to demonstrate whether it meets the application requirements for petroleum exploration. Transmission-type Thin-Core Fiber Modal Interferometers (TCFMIs) with different lengths of Thin-Core Fibers (TCFs) (20 mm, 40 mm, and 60 mm) are produced in the experiment. Among them, TCFMIs at 60 mm TCF length can obtain ideal sensing signals. The TCFMI (with a length of 20 mm TCF) is encapsulated in a self-made aluminum groove in rosin to test its Refractive Index (RI). The results show that as the RI increases, the central wavelength shifts towards the long wavelength direction. Its sensitivity reaches 146 nm/R.I.U. When the temperature is increased, the central wavelength shifts towards the long wavelength direction, resulting in lower temperature sensitivity. In the sensitivity test, the phase change obtained by designing OFSs is proportional to the vibration acceleration. It is fixed on the vibration table to keep the acceleration of the vibration table constant and adjust the vibration frequency of the vibration table. The results show that the vibration spectral line of the sensor is relatively flat within 100 Hz, and resonance occurs within the range of 200 Hz to 350 Hz. Through phase demodulation, sensors loaded with different oscillator masses increase linearly in the low-frequency range. When the vibration frequency approaches the resonance frequency of the sensor, the phase sensitivity of the sensor increases nonlinearly.

Design and analysis of tunable optical filter based on silicon-on-insulator photonic crystal for visible light communication

In this paper, tunable narrow bandwidth band-stop and band-pass optical filters for visible light communication (VLC) are designed and numerically investigated using a photonic crystal (PC) structure. Designing a tunable filter for visible light communication is crucial for enhancing spectral utilization and maximizing communication efficiency. A rectangular defect with optimized sides length utilized in the center of the PC structure based on the desired filtering characteristics. To obtain a favorable narrow bandwidth filter for visible light wavelengths, this defect is filled with a specific liquid crystal. In addition, an electric field is applied to the proposed structure in order to change the refractive index of the liquid crystal and achieve tunability in the desired range. The functional wavelength ranges of band-pass and band-stop filters are 550–570 nm and 565–585 nm, respectively. Applying an electric field causes a shift in the central wavelengths of the filters. It is shown that a 1.5 nm central wavelength shift for a band-stop filter and a 0.9 nm central wavelength shift for a band-pass filter can be achieved by changing the voltage source up to 1.5 V. In addition, the impact that the variation in the length of the rectangular defects’ sides has on the filtering characteristics of the proposed device is studied.

A Novel Catheter Shape-Sensing Method Based on Deep Learning with a Multi-Core Optical Fiber.

In this paper, we propose a novel shape-sensing method based on deep learning with a multi-core optical fiber for the accurate shape-sensing of catheters and guidewires. Firstly, we designed a catheter with embedded multi-core fiber containing three sensing outer cores and one temperature compensation middle core. Then, we analyzed the relationship between the central wavelength shift, the curvature of the multi-core Fiber Bragg Grating (FBG), and temperature compensation methods to establish a Particle Swarm Optimization (PSO) BP neural network-based catheter shape sensing method. Finally, experiments were conducted in both constant and variable temperature environments to validate the method. The average and maximum distance errors of the PSO-BP neural network were 0.57 and 1.33 mm, respectively, under constant temperature conditions, and 0.36 and 0.96 mm, respectively, under variable temperature conditions. This well-sensed catheter shape demonstrates the effectiveness of the shape-sensing method proposed in this paper and its potential applications in real surgical catheters and guidewire.

Beam smoothing and polarization randomizing using quasi-broadband laser

We propose a quasi-broadband laser scheme to achieve comparable beam smoothing and polarization randomizing performance with the broadband laser in high-power lasers. The quasi-broadband laser can be multi-central frequency and multi-color laser in which each beamlet is a superposition of frequency-modulated lasers with frequency shift, i.e., multi-frequency, and the central wavelengths of the beamlets are different from each other, i.e., multi-color. The quasi-broadband laser can also be multi-color spectrally incoherent laser whose beamlets are from broadband sources with central wavelength shift. In addition, the half-aperture polarization control plate array is applied to randomize the polarization and to enhance the smoothing effect. Results indicate that, the two quasi-broadband laser schemes can generate a rapidly temporally-varying light field with broadened spectrum and degenerated coherence, benefiting the randomization of polarization state and improvement of the irradiation uniformity.

Tunable-wavelength broadband spectrum dissipative soliton mode-locked fiber laser

The tunable-wavelength broadband spectrum dissipative soliton (DS) is demonstrated in Yb-doped mode-locked laser using an all-fiber structure saturable absorber (SA), that is, graded index multimode fiber (GIMF)-step index multimode fiber (SIMF)-GIMF with 4.70% modulation depth and 4.55 mW saturation power. Increasing pump power alters intra-cavity balance between loss and gain and nonlinear and dispersion, causing wavelength adjustment and spectrum broadening. The center wavelength shifts towards shorter wavelength with rang of 1037.77–1024.83 nm, corresponding 3-dB bandwidth widens from 6.10 to 9.23 nm and pulse width narrows from 23.80 to 13.38 ps. The signal-to-noise ratio (SNR) is 71.2 dB at the repetition rate of 14.63 MHz. The GIMF-SIMF-GIMF structure has a huge potential to serve as both SA and filter component for the tunable-wavelength broadband spectrum Yb-doped mode-locked fiber laser (MLFL), which is favorable for developing the field of nonlinear microscopy and material processing.

Vector fiber Bragg gratings accelerometer based on silicone compliant cylinder for low frequency vibration monitoring

Vector accelerometer has attracted much attention for its great application potential in underground seismic signal measurement. We propose and demonstrate a novel vector accelerometer based on the three fiber Bragg gratings (FBGs) embedded in a silicone rubber compliant cylinder at 120° distributed uniformly. The accelerometer is capable of detecting the orientation of vibration with a range of 0°–360° and the acceleration through monitoring the central wavelength shifts of three FBGs simultaneously. The experimental results show that the natural frequency of the accelerometer is about 85 Hz, and the sensitivity is 84.21 pm/g in the flat range of 20 Hz–60 Hz. Through experimental calibration, the designed accelerometer can accurately obtain vibration vector information, including vibration orientation and acceleration. In addition, the range of resonant frequency and sensitivity can be expanded by adjusting the hardness of the silicone rubber materials. Due to the characteristics of small size and orientation recognition, the accelerometer can be applied to low-frequency vibration acceleration vector measurement in narrow spaces.

Investigation of Hybrid Remote Fiber Optic Sensing Solutions for Railway Applications

Fiber optic sensing (FOS) has become a well-known technology in response to the rising demands of the railway transportation field despite the abundance of electronic sensing systems in the market. FOS application boasts an all-in-one solution that is both efficient and versatile. In order to enhance the understanding of the capabilities of FOS, this paper presents a hybrid fiber optic sensing system with an improved sensing ability to facilitate transportation applications for primary or secondary security interfaces. The hybrid sensing scheme incorporates two different sensing systems designed for long-distance applications. The first system employs a coding technique for the transmitted pulses, which provide information on train location through cross-correlation with the reflected pulses from fiber Bragg grating (FBG) sensors located along the railway. The proposed system can accurately predict the train’s location up to a precision of one cm. The second system examines the wavelength drift of the reflected signal from the FBG sensor affected by the train using a tunable optical filter and photodetector. It determines essential parameters such as the train’s location, speed, and direction by measuring the Bragg wavelength shift and its direction. The effect of the train movement and speed on the applied strain on the FBG sensor is calculated in this work and applied to the simulation to determine the train’s location, speed, and direction. A calibration table facilitates the correlation between the train speed and the shift in the FBG center wavelength, which helps ensure accurate results. The hybrid fiber optic sensing system is designed to facilitate railway transportation applications’ sustainability and security.

A PhC-SOA based cancerous cell detection biosensor

In this paper, we present a novel method to design an ultra-small photonic integrated biosensor to detect cancerous cells. The proposed biosensor is based on the self-phase modulation in PhC-SOA, inducing a frequency shift on a pulse traveling through the device. The amount of the frequency chirp depends on the group velocity of the active medium waveguide being determined by the refractive index of the microfluidic infiltrating the holes around the waveguide. The refractive index of the microfluidic is also determined by the cell type that can be normal or cancerous. Since the refractive index of a cancerous cell is higher than that of a normal one, the group index of the waveguide and the output chirp will decrease. By measuring the amount of the output chirp, we can detect the cell type. The Simulation results showed that for a 0.02 change in the refractive index of the cell, a 3.71 nm central wavelength shift occurred for a 10-ps 7-mW gaussian pulse input with a central wavelength of 1533.53 nm. In terms of the wavelength shift, the sensitivity and figure of merit are 185.5 and 530, respectively. To detect the cell type, we integrated a PhC channel drop filter to drop the chirped signal due to the cancerous cell infiltration. Designing an appropriate PhC-CDF leads to achieving an ultra-small cancerous detection cell biosensor with more than 97% precision.

Application of Neural Network Algorithms for Central Wavelength Determination of Fiber Optic Sensors

Fiber Bragg gratings are sensitive elements in fiber optic sensor networks, and this paper discusses the practicalities of using neural network algorithms to determine their central wavelengths. The problem is to determine the central wavelength of a single sensor, the parameters of which are obtained using a low-resolution spectrum analyzer. The configuration of the neural network and the algorithm for producing the training and control datasets are specified. The training results for the selected neural network configuration demonstrated that the proposed method could determine the position of the central wavelength with a resolution two and a half orders of magnitude higher than the resolution of the input data sampling. The obtained results demonstrate that the approach makes it possible to determine the FBG central wavelength shift with an error not exceeding ~0.5 pm at a spectrum analyzer resolution of 167 pm.

Relaxed-tolerance subwavelength grating coupler

Short-wavelength mid-infrared (MIR) silicon photonics at 2–2.5 μm wavelengths has a wide range of applications in optical communications, chemical analysis, and environmental monitoring. In this spectral region, subwavelength grating (SWG) couplers can be fabricated by using commercial multi-project wafer (MPW) services, due to their larger device feature sizes compared with those in the telecommunication band. However, conventional SWG couplers are still sensitive to variations in dimensions produced by errors in fabrication processes, restricting the fabrication reproducibility of the devices. Here, we demonstrated a relaxed-tolerance SWG coupler that has relaxed fabrication tolerance to overcome this limitation. The approach is based on the design of the SWG coupler with dual-hole structures to reduce the influence of random fabrication variations on the grating effective refractive index. Theoretically, the slope of the center wavelength shift of the relaxed-tolerance SWG coupler to the variation of the subwavelength hole size is 0.58, while that is 0.63 for the conventional SWG coupler. We demonstrated the relaxed-tolerance SWG coupler with a peak coupling efficiency of −6.2 dB at a wavelength of 2.038 μm with a 1-dB bandwidth of about 30 nm. Additionally, fabrication reproducibility and optical fiber coupling tolerance are also presented. The study paves a promising way toward developing short-wavelength MIR grating couplers with high performance, excellent repeatability, and cost-efficiency.

Ultrasonic sensing directivity of π-phase-shifted fiber Bragg grating hydrophone

<inline-formula><tex-math id="M1">\begin{document}${\text{π }}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M1.png"/></alternatives></inline-formula>-phase-shifted fiber Bragg grating with a short effective sensing length becomes one of research hotspots in ultrasonic sensing, because light undergoes strong localization centered at its phase shift position. To investigate the directional sensing characteristics of <inline-formula><tex-math id="M2">\begin{document}${\text{π }}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M2.png"/></alternatives></inline-formula>-phase-shifted fiber Bragg grating as hydrophone, the theory of sound propagation in layered media is used to calculate the strain of fiber core, then the transfer matrix method based on the coupled-mode theory in optics is used to calculate the shift of central wavelength in optical reflection spectrum. Results of strain and wavelength shift under obliquely incident ultrasonic from 1－10 MHz are divided into A area, B area, and C area, and analyzed by numerical calculation and simulation calculation. Axial strain and elasto-optical strain change the grating period and effective refractive index by the mechanical effect and elasto-optical effect, respectively, thereby resulting in wavelength shift. In A area (frequency below 5 MHz, incident angle below <inline-formula><tex-math id="M3">\begin{document}$15^\circ $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M3.png"/></alternatives></inline-formula>), the axial strain nearly equals zero, thus elasto-optical effect plays a predominant role in wavelength shift. The maximal response occurs at vertical incidence, and then obviously declines with angle increasing. The maximum is essentially unchanged with grating length. In B area and C area (angle above <inline-formula><tex-math id="M4">\begin{document}$15^\circ $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M4.png"/></alternatives></inline-formula>), both mechanical effect and elasto-optical effect contribute to wavelength shift. In B area (frequency below 5 MHz), the amplitude of strain is the largest in three areas. A peak of wavelength shift appears at the same angle of the peak of strain, where exists the interference of the guided wave in fiber with the direct ultrasonic wave form water. The peak amplitude of wavelength shift decreases with grating length increasing. In C area (frequency below 5 MHz), the amplitude of strain is larger than in A area, but the wavelength shift is smaller, which is correlated to its higher axial wave number. Comparing the results in three areas, it is clear that the wavelength shift is larger at lower frequency and at vertical incidence. Experiments on 3 MHz and 5 MHz are then performed with a π-phase-shifted fiber Bragg grating. The experimental result accords well with the theoretical result. The research is important in practically using the <inline-formula><tex-math id="M5">\begin{document}${\text{π }}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222154_M5.png"/></alternatives></inline-formula>-phase-shifted fiber Bragg grating in ultrasonic sensing.