# Dynamic Responses Measured by Optical Fiber Sensor for Structural Health Monitoring
## 2 Optical Fiber Sensing System  
An FBG sensor is formed when a spatially periodic modulation of the core refractive index along a single-mode fiber is created by exposing the optical fiber to an ultraviolet laser beam. As a broadband optical source impinges upon Bragg gratings, a narrow-band spectral peak is reflected, and all other wavelengths are transmitted without change, resulting in a dip in the transmitted spectrum, as shown in Figure 1.  
The Bragg wavelength reflected from the FBG sensor \(\lambda_{B}\) is related to the grating period \(\Lambda\) and the effective refractive index \(n_{0}\), as follows:  
\[\lambda_{B}=2n_{0}\Lambda \tag{1}\]  
An FBG sensor with a constant grating period and refractive index will reflect light at a single Bragg wavelength. Change in the grating period and index of refraction due to the strain and temperature will lead to a shift of the Bragg wavelength. At a constant temperature, the load-induced Bragg  
Figure 1: Transmission and reflectance spectrum from fiber Bragg grating. wavelength shifts for the light propagating along the two eigenaxes are described by [PERSON] [40] using the following equations:  
\[\Delta\lambda_{B,x} =[\varepsilon_{z}-\frac{n_{0}}{2}(p_{11}\varepsilon_{x}+p_{12}( \varepsilon_{y}+\varepsilon_{z}))]\lambda_{B} \tag{2a}\] \[\Delta\lambda_{B,y} =[\varepsilon_{z}-\frac{n_{0}}{2}(p_{11}\varepsilon_{y}+p_{12}( \varepsilon_{x}+\varepsilon_{z}))]\lambda_{B} \tag{2b}\]  
where \(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\varepsilon_{z}\) are the principal strains in the optical fiber and \(z\) represents the axial direction; \(p_{11}\) and \(p_{12}\) are the strain-optic Pockel's constants. When the transverse strains are equal and related to the longitudinal strains by the Poisson's ratio of the optical fiber, i.e., \(\varepsilon_{x}=\varepsilon_{y}=-\
u\;\varepsilon_{z}\). Equation (2a,b) become equivalent, leading to a single Bragg wavelength shift as follows. \[\Delta\lambda_{B}=[1-\frac{n_{0}}{2}(p_{12}-\
u(p_{11}+p_{12}))]\varepsilon_{ z}\lambda_{B} \tag{3}\]  
This shows that FBG sensors are wavelength-modulated sensors. Using Equation (3), the longitudinal strain in the sensor can be determined by the measured Bragg wavelength shift. Long-period fiber gratings (LPFG) are formed by introducing a periodic modulation of the refractive index with a grating period on the order of hundreds of microns along the core of a single-mode fiber. LPFG can couple the fundamental guided core mode to the forward propagating cladding modes of a fiber. Since the coupling is wavelength-selective, LPFG acts as a wavelength-dependent band stop filter. For the band stop filter, the largest rejection ratio is its resonance peak wavelength. The resonance peak wavelength \(\lambda_{L}\) of an LPFG can be determined using the phase matching condition as follows [41]:  
\[\lambda_{L}=(n_{core}-n_{clad})\Lambda \tag{4}\]  
where \(n_{core}\) and \(n_{clad}\) are the effective indices of the core and cladding, respectively; \(\Lambda\) is the grating period. The transmission spectrum of the LPFG used in this study is the 5 th coupling mode, as shown in Figure 2.  
A wavelength meter is commonly used to detect the wavelength. However, the slow response of the wavelength meter is not suitable for dynamic and transient measurements. In this work, the shift of the FBG wavelength is converted to the change of the light intensity measured by a photodetector. The quick response of the photodetector--within microseconds--can be used to measure the dynamic and transient responses. The interrogation scheme for the Bragg wavelength \(\lambda_{B}\) of the FBG sensor is based on a matched LPFG. The principle of using a matched LPFG to interrogate the signal encoded  
Figure 2: Transmission spectrum of LPFG filter. by the Bragg wavelength from a FBG sensor is based on the measurement of the optical intensity.