# Long-Period Grating with Asymmetrical Modulation for Curvature Sensing
## 1 Introduction  
In recent years, the monitoring of the health conditions of magnificent structures such as bridges and power grid towers has attracted great attention. The measurement of curvature is of special importance in structural health monitoring for civil engineering and aerospace industries. Curvature monitoring based on the optical method could make entire sensors compact, simple, accurate, and reliable [1]. Compared with traditional electrical sensors, optic fiber sensors offer a number of advantages such as fast response, remote monitoring, resistance to harsh environments, and immunity to electromagnetic interference. Furthermore, their compact size, lightweight, and flexibility make it easy for sensors to be embedded into structures in order to monitor certain parameters like strain, temperature, and curvature, making the use of optical fiber sensors an ideal solution in several industrial applications like civil engineering or robot arms. Based on the difference in device configurations, fiber curvature sensors could be roughly divided into categories based on interferometers and fiber gratings [2]. Curvature sensors based on interferometers usually adopt a splicing single-mode fiber (SMF) with a special optical fiber, such as photonic crystal fiber (PCF), and multicore fiber [3, 4]. And to improve the sensing characteristics, some of them usually use the method of lateral-offset splicing. For example, [PERSON] et al. proposed a curvature sensor consisting of a lateral-offset spliced single-mode four-core single-mode fiber Mach-Zehnder interferometer and achieved a curvature sensitivity of \(-18.75\) nm/m\({}^{-1}\)[4]. The lateral splicing could effectively improve the sensitivity, but the sensor structures are usually fragile [5]. The other types of curvaturesensors are those based on fiber gratings, including the fiber Bragg grating (FBG) [6; 7; 8] and long-period fiber grating (LPFG) [9; 10]. These types of sensors usually just use single-mode fibers to fabricate the curvature sensor. Compared with FBGs, LPFGs are usually more sensitive to fiber bending since cladding modes are excited in the fiber sensor. LPFG usually has periodic refractive index (RI) change or a periodic geometry structure along the fiber length, which can couple light from the guided core mode to the cladding mode at phase-matched wavelengths. The periodic change in RI or geometry along the fiber length can be realized by various means such as UV light illumination, CO2 laser irradiation, electric-arc discharge, and femtosecond (fs) laser pulse irradiation [9; 10]. Several methods were utilized to fabricate LPFGs, such as laser-introduced refractive index modulation, or periodical splicing of different types of optical fibers. [PERSON] et al. proposed a bending sensor formed by periodical splicing of SMF and a multimode fiber (MMF) [11]. In other cases, a CO2 laser or femtosecond laser are used to induce LPFGs [12] in optical fibers. Other methods of mechanically introducing LPFGs have also been proposed, e.g., using laminated plates [13] or 3D-printed periodic grooved plates [14]. For curvature measurement, the determination of the bending direction is of importance in many practical applications, and sensors that can measure both curvature magnitude and direction are preferred. Introducing an asymmetric structure to a cylindrical fiber is one of the feasible methods to recognize bending directions [15]. For example, extra asymmetric grooves could be carved to fabricate an LPFG curvature sensor that is direction-dependent. Asymmetric microhole-structured LPFGs in SMF and PCF are possible through the use of femtosecond laser drilling. The fiber core is not penetrated by the bottom but by the side wall of the microholes. These asymmetric microhole-structured LPFGs can be readily fabricated in SMF and PCF [16]. [PERSON] et al. reported a highly sensitive vector bending sensor that uses a CO2 laser to fabricate a D-shaped LPFG and achieves high sensitivity in four directions [9].