# The Effects of Adhesive and Bonding Length on the Strain Transfer of Optical Fiber Sensors
## 3 Mach-Zehnder Interferometric Sensor  
Interferometric sensors have received much attention due to their high sensitivity and high spatial resolution. The Mach-Zehnder optical fiber sensor is perhaps the best known because it was developed first. This interferometer acts in the classic sense by optically interfering the light propagating in the reference and sensing fibers. A schematic diagram of a Mach-Zehnder interferometer is shown in Figure 2. It consists of two \(2\times 2\) couplers at the input and output. The excitation is applied to the sensing fiber, resulting in optical path difference between the reference and sensing fibers. Figure 1: Analytical model of surface bonded optical fiber [16]. Figure 2: Mach-Zehnder interferometer. The light intensity of the output of the Mach-Zehnder interferometer can be expressed as [19]  
\[ \begin{split} I=2A^{2}\left(1+\cos\Delta\phi\right)\\ \Delta\phi=\frac{2\pi n_{0}}{\lambda}\left\{1-\frac{n_{0}}{2}\left[ \left(1-\
u\right)p_{12}-\
u p_{11}\right]\right\}\int_{\Gamma}\epsilon_{1}^{f }dx\end{split}  \tag{2}\]  
where \(\Delta\phi\) is the optical phase shift, \(n_{0}\) is the refractive index of the optical fiber, \(\lambda\) is optical wavelength, \(\
u\) is the Poisson's ratio, \(P_{11}\) and \(P_{12}\) are the Pockel's constants, \(\epsilon_{1}^{f}\) is the strain of the optical fiber. Since the terms in front of the integral sign of \(\Delta\phi\) are constants for any given optical fiber system, the total optical phase shift \(\Delta\phi\) is proportional to the integral of the optical fiber strain. Thus, by measuring the total optical phase shift, the integral of the optical fiber strain can be easily obtained as follows:  
\[\int_{\Gamma_{s}}\epsilon_{1}^{f}dx=\frac{\Delta\phi}{\frac{2\pi n_{0}}{ \lambda}\left\{1-\frac{1}{2}n_{0}^{2}\left[\left(1-\
u_{f}\right)p_{12}-\
u_{ f}p_{11}\right]\right\}} \tag{3}\]  
The light intensity shown in Equation (2) is a cosine function of the optical phase shift. As the optical phase shift \(\Delta\phi=2\pi\), the light intensity goes through a complete cycle. The corresponding integral of the strain is  
\[ \begin{split}\Delta s=&\int_{\Gamma_{s}}\epsilon_{1 }^{f}dx=\frac{2\pi}{\frac{2\pi n_{0}}{\lambda}\left\{1-\frac{1}{2}n_{0}^{2} \left[\left(1-\
u_{f}\right)p_{12}-\
u_{f}p_{11}\right]\right\}}\\ =&\frac{\lambda}{n_{0}\left\{1-\frac{1}{2}n_{0}^{2} \left[\left(1-\
u_{f}\right)p_{12}-\
u_{f}p_{11}\right]\right\}}\end{split}  \tag{4}\]  
The integral of the strain in Equation (4) represents the total change of the length of the sensing fiber, which is surface bonded on the host material. Thus, the average strain transferred to the surface bonded optical fiber for one complete cycle of light intensity, _i.e._, per fringe can be obtained  
\[\Delta\epsilon_{avg}=\frac{\Delta s}{s}=\frac{\int_{\Gamma_{s}}\epsilon_{1}^{ f}dx}{2L_{f}}=\frac{\lambda}{2L_{f}n_{0}\left\{1-\frac{1}{2}n_{0}^{2}\left[ \left(1-\
u_{f}\right)p_{12}-\
u_{f}p_{11}\right]\right\}} \tag{5}\]  
The total average strain in the surface bonded optical fiber can be calculated by counting the number of fringes as follows. \[\left(\epsilon_{1}^{f}\right)_{\text{exp}}=m\Delta\epsilon_{avg}=\frac{m \lambda}{2L_{f}n_{0}\left\{1-\frac{1}{2}n_{0}^{2}\left[\left(1-\
u_{f}\right) p_{12}-\
u_{f}p_{11}\right]\right\}} \tag{6}\]  
where \(m\) is an integer denoted as the number of fringes. Thus, the experimental result of the average strain of the surface bonded optical fiber using the Mach-Zehnder interferometric technique can be determined from Equation (6). The maximum experimental error is equal to the strain transferred to the optical fiber per fringe as shown in Equation (5). Theoretical prediction of the average strain of the surface bonded optical fiber can be deduced from Equation (1) as  
\[ \begin{split}\left(\epsilon_{1}^{f}\right)_{the}=\frac{\int_{ \Gamma_{s}}\epsilon_{1}^{f}(x)dx}{s}=\frac{\int_{\Gamma_{s}} \frac{\epsilon_{0}}{\pi r_{f}^{2}}\frac{\epsilon_{0}}{\pi r_{f}^{2}}\frac{1}{ \epsilon\left(2\pi r_{f}E_{h}+\frac{1}{E_{f}}\right)}}{2L_{f}}\end{split}