# Simultaneous Measurement of Refractive Index and Flow Rate Using a Co\({}^{2+}\)-Doped Microfiber
## 2 Materials and Methods
### Principle of the Proposed Sensor  
It is well known that the FBG center wavelength \(\lambda_{B}\) and the resonant wavelength \(\lambda_{cl,k}\) of the \(k\)th cladding mode can be shown as [18; 26]  
\[\lambda_{B}=2n_{eff,co}\cdot\Lambda \tag{1}\]  
\[\lambda_{cl,k}=(n_{eff,co}+n_{eff,cl,k})\cdot\Lambda \tag{2}\]  
where \(n_{eff,co}\) and \(n_{eff,cl,k}\) are the effective index of the core mode and the \(k\)th cladding mode in the Co-MFBGS, respectively; \(\Lambda\) is the Bragg grating period. As the core mode was confined in the core of the Co-MFBGS, it was insensitive to the surrounding LRI. Meanwhile, the cladding modes of the Co-MFBGS due to the evanescent field are very sensitive to the LRI. So, the Co-MFBGS can be used for LRI sensing, and from Equation (2) the wavelength drift of the cladding mode caused by LRI change is rewritten as  
\[\Delta_{\lambda cl,k}(RI)=\Delta n_{eff,cl,k}(RI)\cdot\Lambda \tag{3}\]  
where \(\Delta n_{eff,cl,k}(RI)\) is the \(k\)th cladding mode LRI change. The wavelength interval \(\lambda_{\text{int}}\) is defined as  
\[\lambda_{\text{int}}=\lambda_{B}-\lambda_{cl,k} \tag{4}\]  
It is the wavelength difference between core mode and \(k\)th cladding mode. Combining Equations (3) and (4), the wavelength interval shift \(\Delta\lambda_{\text{int}}(RI)\) caused by the change of LRI can be written as  
\[\Delta\lambda_{\text{int}}(RI)=\Delta\lambda_{cl,k}(RI)=\Delta n_{eff,cl,k}(RI )\cdot\Lambda \tag{5}\]  
The wavelength drift of FBG caused by temperature change is well known as [27]  
\[ \begin{array}{c}\Delta\lambda_{B}(T)=2n_{eff,co}\cdot\Lambda(\frac{1}{n_{ eff,co}}\frac{d_{inf,co}}{dI}+\frac{1}{\Lambda}\frac{dA}{dI})\Delta T\\ =\lambda_{B}(\alpha+\beta)\Delta T\end{array}  \tag{6}\]  
\[ \begin{array}{c}\Delta\lambda_{cl,k}(T)=\lambda_{cl,k}(\frac{1}{n_{eff,co} +n_{eff,cl,k}}\frac{d(n_{eff,co}+n_{eff,cl,k})}{dI}+\frac{1}{\Lambda}\frac{dA} {dI})\Delta T\\ =\lambda_{cl,k}(\gamma+\beta)\Delta T\end{array}  \tag{7}\]  
where \(\Delta\lambda_{B}(T)\) and \(\Delta\lambda_{cl,k}(T)\) are the wavelength changes of the core mode and \(k\)th cladding mode of FBG caused by temperature change, \(\Delta T\) is the temperature change, \(\alpha=\frac{1}{n_{eff,co}}\frac{dn_{eff,co}}{dI}\) is the thermo-optic coefficient of the optical fiber core, \(\beta=\frac{1}{\Lambda}\frac{dA}{dI}\) is the grating coefficient of thermal expansion, and \(\gamma=\frac{1}{n_{eff,co}+n_{eff,cl,k}}\frac{d(n_{eff,co}+n_{eff,cl,k})}{dI}\) is the thermo-optic coefficient of the optical fiber cladding. Cobalt is a kind of light absorption material and is easy to get, and the CDOF can be heated by the pump laser as a result of the non-radiation effect; meanwhile, the temperature of the Co-MFBGS increases and the wavelengths of the core mode and cladding modes have a redshift. When the Co-MFBGS was used as an LFR sensor, parts of the sensor heat can be carried away by the fluidic sample [28], which would lead to a decrease in the sensor's temperature and a blue shift in the wavelengths of the core mode and cladding modes. So, the LFR can be measured by detecting the wavelength shift of the Co-MFBGS. Depending on the theory of hot-wire, the relationship between the LFR of the fluidic sample and heat loss is shown as [29]  
\[Q=\Delta T(A+B\
u^{i}) \tag{8}\]  
where \(Q\) is the power absorbed by the CDOF, \(\
u\) is the LFR, and \(A\), \(B\) and \(i\) are empirical coefficients. So, Equations (6) and (7) can be rewritten as  
\[\Delta\lambda_{B}(T)=\lambda_{B}(\alpha+\beta)\frac{Q}{A+B\
u^{i}} \tag{9}\]  
\[\Delta\lambda_{cl,k}(T)=\lambda_{cl,k}(\gamma+\beta)\frac{Q}{A+B\
u^{i}} \tag{10}\]  
Additionally, the wavelength interval shift \(\Delta\lambda_{\text{int}}(T)\) caused by the change of temperature can be deduced as  
\[ \begin{array}{l}\Delta\lambda_{\text{int}}(T)=\Delta\lambda_{B}(T)-\Delta \lambda_{cl,k}(T)\\ =(\lambda_{B}\alpha+\lambda_{B}\beta-\lambda_{cl,k}\gamma-\lambda_{cl,k}\beta) \frac{Q}{A+B\
u^{i}}\end{array}