Abstract : This paper presents a detailed theoretical derivation and analysis of the temperature and strain sensing sensitivity of Bragg fiber gratings (FBG) and long-period fiber gratings (LPFG), deriving some important conclusions that provide guidance for the application of fiber gratings in the sensor field. Keywords : Bragg fiber grating; long-period fiber grating; fiber grating sensor; temperature sensitivity; strain sensitivity. In recent years, the application of fiber gratings in sensing has attracted increasing attention. Fiber grating sensors (FGS) have many advantages that other sensors cannot match: anti-electromagnetic interference, small size, light weight, good temperature resistance, strong multiplexing capability, long transmission distance, corrosion resistance, and high sensitivity. FGS has broad application prospects. As early as 1988, it was successfully used in the aerospace field as an effective non-destructive testing technology. It can also be used in chemical, industrial, power, hydropower, shipbuilding, and coal mining fields. Currently, the application of FGS in "smart and smart buildings" has become a research hotspot for experts and scholars. FGS is used for sensing because its resonant coupling wavelength shifts with changes in external parameters. Based on this, changes in external conditions such as temperature or stress applied to the fiber optic cable will cause variations in the effective refractive index and grating period of the fiber, resulting in changes in the grating coupling wavelength (i.e., wavelength drift). By measuring the magnitude of the wavelength drift, the changes in temperature or stress can be measured. This paper derives expressions for the temperature and strain sensitivity of long-period fiber gratings and short-period fiber gratings, and concludes that by selecting appropriate parameters, the problem of cross-sensitivity between temperature and strain in fiber grating sensing can be solved using long-period fiber gratings, and that temperature and strain can be measured simultaneously using a single long-period fiber grating. 1 Working principle of long and short period fiber grating sensing According to the coupled mode theory, the following phase matching conditions must be met for two transmission modes to couple in a fiber grating [1]: Δβ1－β2＝Δβ＝2π/Λ, where β1 and β2 are the propagation constants of the two modes to be coupled; Δβ is the difference between the two propagation constants; Λ is the grating period. In a Bragg fiber grating, the energy of the forward-propagating guided mode is coupled into the reverse guided mode to form a reflection peak. The grating equation of the Bragg fiber grating is [2] λB＝2neffΛ, where neff is the effective refractive index of the fiber grating; λB is the resonant coupling wavelength of the fiber grating. When the external temperature or stress acts on the grating and causes its effective refractive index and grating period to change, the Bragg wavelength of the grating is easily changed (i.e., moved). By measuring the magnitude of ΔλB, the magnitude of the change in external temperature or stress can be determined. In long-period fiber gratings, energy is coupled from the forward-propagating guided mode to the cladding mode, forming multiple absorption peaks in its transmission spectrum. The grating equation of a long-period fiber grating is [2, where n[sub]c0[/sub] is the effective refractive index of the guided mode; is the effective refractive index of the nth cladding mode; λn is the coupling wavelength from the guided mode to the nth cladding mode (for simplicity, the order n is omitted in the following formulas). Similarly, when external conditions such as temperature or stress act on the grating, causing changes in the effective refractive index and grating period of its core and cladding, the wavelength of the grating will change (i.e., shift). The magnitude of Δλn can also determine the change in external temperature or stress. 2 Temperature and Axial Strain Sensitivity of Bragg Fiber Grating When the external temperature, strain and other parameters change, the Bragg wavelength drift is known from the Bragg fiber grating equation as [3]  2.1 Temperature Sensitivity When the temperature of the fiber grating changes, the period and effective refractive index of the fiber grating change due to thermal expansion effect and thermo-optic effect (assuming that its stress is constant). The change of the grating period caused by thermal expansion effect is [3] Where ξ is the thermo-optic coefficient of the fiber, representing the rate of change of refractive index with temperature. Substituting equations (2) and (3) into equation (1), and according to the Bragg grating equation, the temperature sensitivity of the fiber Bragg grating is 2.2 Strain Sensitivity Strain causes the stretching and elastic-optic effect of the grating period, while the elastic-optic effect causes the change of the refractive index of the fiber, thereby causing the Bragg wavelength to drift. Assuming that the grating is only subjected to axial strain (temperature is constant). The change of the grating pitch caused by axial strain is the change of the effective refractive index, which is expressed by the elastic-optic coefficient matrix Pij and the strain tensor matrix εj as [5] In the formula, i=1, 2, 3 represent the x, y, z directions respectively. Assuming the shear strain is zero, the strain tensor matrix εj is expressed as [3]  The elastic-optical matrix is ​​[5] In the formula, P11, P12, P44 are elastic-optical coefficients; ν is the Poisson coefficient of the fiber material; P44＝（P11－P12）/2. From equations (6), (7) and (8), we get: Substituting equations (5), (9), and (10) into equation (1), we get the axial strain sensitivity as 3 Long-period fiber grating temperature and strain sensitivity � 3.1 Temperature sensitivity When the external temperature changes, the temperature sensitivity can be obtained by differentiating the long-period grating equation with respect to temperature T [4] The change in grating period caused by thermal expansion effect is given by the formula, where αc0 is the thermal expansion coefficient of the fiber; ξc0 and ξc1 are the thermo-optical coefficients of the fiber core and cladding respectively. � 3.2 Axial strain sensitivity (assuming the temperature is constant, only affected by axial strain) When the strain changes, the axial strain sensitivity is obtained by differentiating the long-period grating equation with respect to ε[4]. The grating period change caused by strain is given by the formula, where pc0 and pcl are the effective elastic-optic coefficients of the fiber core and cladding, respectively. 3.3 Effect of cladding parameters on wavelength when temperature or stress changes. It can be seen from equations (15) and (19) that the temperature sensitivity coefficient K_T and strain sensitivity coefficient K_ε of the long-period fiber grating are not only related to the fiber core parameters, but also to the cladding parameters. Taking the commonly used germanium-silicon fiber as an example, let Λ = 200 μm, nc0 = 1.45, pc0 = 0.22, ξc0 = 6.7 × 10-6, αc0 = 0.5 × 10-6, and the relative refractive index difference Δ = 5 × 10-3. The wavelength change of the grating when the temperature changes can be expressed as 6.9 × 10-6 and 7.1 × 10-6. The relationship between temperature change and wavelength change is shown in Figure 1. The wavelength change of the grating under varying axial stress can be expressed as shown in Figure 2, where the strain and wavelength change are obtained when pc1 takes values ​​of 0.20, 0.21, 0.22, 0.23, and 0.24 respectively. From Figures 1 and 2, it can be seen that when the effective elastic-optical coefficient pc1 of the cladding is the same as the effective elastic-optical coefficient pc0 of the fiber core, the strain sensitivity of the long-period grating is almost zero, and the thermo-optical coefficient is also the same. Therefore, by selecting appropriate cladding parameters, a long-period grating insensitive to temperature or strain can be fabricated, thus solving the problem of cross-sensitivity between temperature and strain in fiber optic grating sensors. Furthermore, different cladding modes have different effective refractive indices, resulting in different temperature and strain sensitivities. Utilizing this, by monitoring the shift in absorption wavelength between two different cladding modes, a single long-period fiber optic grating sensor can simultaneously measure temperature and strain. For example, assuming that the temperature and strain sensitivity coefficients KT1, Kε1 and KT2, Kε2 of the first and second-order cladding modes have been experimentally obtained, a system of two equations can be established: If the offsets Δλ1 and Δλ2 of the two wavelengths are measured, the temperature T and strain ε can be obtained by solving the system of equations (22). 4 Conclusion This paper introduces the working principle of FGS and derives the expressions for the temperature and strain sensitivity coefficients of long-period fiber gratings and Bragg fiber gratings based on coupled-mode theory. It can be seen that: (1) The temperature and strain sensitivity coefficients of long-period fiber gratings are not only related to the core parameters, but also to the cladding parameters; (2) For long-period fiber gratings, by selecting appropriate cladding parameters, the temperature or strain sensitivity of the fiber grating can be made almost zero, so as to make a temperature or strain insensitive fiber grating to solve the problem of its temperature and strain cross-sensitivity; (3) A long-period fiber grating sensor can be used to measure temperature and strain simultaneously. These conclusions have certain reference value for the design, fabrication and application of fiber grating sensors. References [1] Turan Erdogan. Fiber grating spectra [J]. J. Lightwave Technol., 1997, 15(8): 1277-1295. [2] Alan D, Kersey, Michael A, et al. Fiber grating sensors [J]. J. Lightwave Technol., 1997, 15(8): 1443-1463. [3] Liao Yanbiao. Fiber Optics [M]. Beijing: Tsinghua University Press, 2001. [4] Ye CC, James S W. Simultaneous temperature and bend sensing with long-period fiber gratings [J]. Opt. Lett., 2000, 25(14): 1007-1009. [5] Lan Xinju. Laser Technology [M]. Wuhan: Huazhong University of Science and Technology Press, 1995.