2.1.

Optical Fiber Characteristics

The nonlinear fiber used in this experiment is a bismuth oxide $\left({\mathrm{Bi}}_2{\mathrm{O}}_3\right)$ based optical fiber $\left(\mathrm{Bi}\text{-}\mathrm{NLF}\right)$ kindly supplied by Asahi Glass Co. Ltd. The nonlinear coefficient of this fiber is $\gamma =1360{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$ , its length $L=2.6\mathrm{m}$ , the attenuation $\alpha =2.2\mathrm{dB}∕\mathrm{m}$ , and the chromatic dispersion $D_c=-270\mathrm{ps}{\mathrm{nm}}^{-1}{\mathrm{km}}^{-1}$ .

A traditional experimental setup was used for measuring the FWM efficiency of the $\mathrm{Bi}\text{-}\mathrm{NLF}$ that corresponds to the ratio between the optical power of the conversion signal and the optical power of the pump signal. At fiber input, pump power was $20.23\mathrm{dBm}$ and signal power was $2.76\mathrm{dBm}$ . Polarizations were aligned to maximize FWM interaction and the signal wavelength was varied in $0.1\mathrm{nm}$ steps. The resulting power of the FWM converted signal is represented in Fig. 1. The FWHM bandwidth is approximately $2.5\mathrm{nm}$ . This value is acceptable for a sensing head, corresponding to a range of $250°\mathrm{C}$ and $2500\mathrm{\mu }\mathrm{\epsilon }$ for temperature and strain measurement. Nevertheless, Lee ⁹ demonstrated the FWM effect using $40\mathrm{cm}$ of the same fiber and the reported bandwidth was approximately $6\mathrm{nm}$ . Due to high nonlinear coefficient and low dispersion slope, this FWM bandwidth is approximately constant within the entire C band $(1530–1565\mathrm{nm})$ .

Fig. 1

Four-wave mixing efficiency using $\mathrm{Bi}\text{-}\mathrm{NLF}$ .

2.2.

Experimental Results

Figure 2 shows the proposed fiber ring resonator that incorporates a $3\mathrm{dB}$ optical coupler, the $\mathrm{Bi}\text{-}\mathrm{NLF}$ , and an optical amplifier with up to $30\mathrm{dBm}$ of output power. The Bragg grating sensor is connected to one of the coupler output ports. The sensor was fabricated using the following fiber: Corning SMF28 standard single-mode optical fiber, germanosilicate core with $3\mathrm{mol}$ % of ${\mathrm{GeO}}_2$ , cold-hydrogenated at $100\mathrm{atm}$ . The Bragg wavelength of the FBG sensor is $1548.7\mathrm{nm}$ with $\sim 100\%$ reflectivity. In the other coupler output port a pump laser diode with $10\mathrm{mW}$ of optical power at $1548.2\mathrm{nm}$ was connected through an optical circulator. All measurements were recorded using an optical spectrum analyser with $0.05\mathrm{nm}$ of resolution. The sensing head was bonded to a translation stage (TS) with a displacement resolution of $1\mathrm{\mu }\mathrm{m}$ and placed in a tubular furnace, with a resolution of $0.1°\mathrm{C}$ .

Fig. 2

Fiber ring laser sensor.

The response to temperature and strain of the grating sensor was determined separately. Figure 3 shows the experimental results obtained when temperature was fixed to $20°\mathrm{C}$ and strain varied up to $1000\mathrm{\mu }\mathrm{\epsilon }$ . The strain coefficients of the wavelength peak and efficiency conversion are shown in Table 1. Figure 4 presents the temperature results when the strain was fixed and the temperature varied between $20°\mathrm{C}$ to $100°\mathrm{C}$ . Table 2 shows the temperature coefficients.

Fig. 3

Response of the fiber laser sensor when temperature is fixed and strain is varied.

Fig. 4

Response of the fiber laser sensor when strain is fixed and temperature is varied.

Table 1

Strain coefficients.

Table 2

Temperature coefficients.

The signal wavelength $\left({\lambda }_{signal}\right)$ and the efficiency conversion $\left(\eta \right)$ were used and applied in the matrix method.⁴

The matrix method can be used to determine the $N$ parameters to which the sensor head is sensitive. However, this is only possible if $N$ different characteristics of the sensing head structure can be independently determined, and such characteristics change differently under the action of the $N$ measurands. In a particular case where the sensor head has linear sensitivities for all parameters, it is possible to write $N$ independent equations allowing explicit solutions for the actual value of each measurand, even in the situation where all of them are changing. This concept is better illustrated in the case where only two measurands are considered, for instance, the strain-temperature discrimination, which will be addressed hereafter.

Strain and temperature can be obtained simultaneously using the following matrix equation:

In such conditions, Eq. 1 can be used in order to have an explicit matrix equation that allows simultaneous measurement of $\mathrm{\Delta }T$ and $\mathrm{\Delta }\epsilon$ for this sensing head:

Table 3

Condition numbers of different dual grating sensors.10

The performance of the matrix was evaluated when the sensing head undertook strain variations in a range of $1000\mathrm{\mu }\mathrm{\epsilon }$ at a fixed temperature $(T=50°\mathrm{C})$ and the other way around, i.e., temperature variations in a range of $100°\mathrm{C}$ for a specific applied strain $(\epsilon =500\mathrm{\mu }\mathrm{\epsilon })$ . The results have maximum errors of $\pm 1.8°\mathrm{C}$ and $\pm 12\mathrm{\mu }\mathrm{\epsilon }$ , for temperature and strain, respectively. The maximum errors include the errors that come from the use of the linear regression fit, and the errors from the use of the matrix method.