3. Number of samples

In Ref. [11] the twin-grating sensor was applied for the temperature measurement. The wavelength shift sensitivity to a temperature change was estimated to be 0.00985 nm/o C. The demodulation signal was done using the Fourier domain phase analysis algorithm. The optical signal was acquired applying direct spectrometric detection. This detection technique uses an optical spectrometer analyzer; then, the acquired optical signal becomes discrete. The signal samples RT λ<sup>j</sup> are taken as wavelengths <sup>λ</sup><sup>j</sup> <sup>¼</sup> <sup>λ</sup>min <sup>þ</sup> <sup>j</sup>δλs, where <sup>j</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …, N � 1, <sup>N</sup> is the number of samples. The interval working is λ<sup>w</sup> ¼ λmax � λmin: λmax is the maximum wavelength, λmin is the minimum wavelength and δλ<sup>s</sup> is the wavelength step.

From Figure 4, the maximum frequency νmax is

$$\nu\_{\text{max}} = \nu\_{\text{FPK}} + \frac{1}{2}\nu\_{\text{BGmax}}\tag{17}$$

4. Capacity of wavelength-division multiplexing

Figure 5.

ters can be observed in Figure 3.

Figure 5. λw, λmin, λmax, and Δλop representation.

In Refs. [4, 5] two experimental sensing systems where twin-grating fiber optic sensors were applied on wavelength-division multiplexing were reported. The first optical system consisted of two wavelength channels. Both channels were centered around 815 and 839 nm. The second optical system consisted of three wavelength channels. The channels were around 1542, 1548, and 1554 nm. Therefore, based on the Bragg grating characteristics, the twin-grating interferometer can be applied in wavelength-division multiplexing if and only if each interferometer sensor has its own Bragg wavelength: λBGk ¼ 2n1Λk, where Λ<sup>k</sup> is the period [1]. In this case, each interference pattern has its own bandwidth in the wavelength domain. The patterns are in the interval of λmin until λmax (interval working λw); it is not possible in other positions, see

Twin-Grating Fiber Optic Sensors Applied on Wavelength-Division Multiplexing and Its Numerical Resolution

http://dx.doi.org/10.5772/intechopen.75586

Let us introduce the operation range Δλop; the operation range defines the interval in which an interference pattern can move into the wavelength domain. Each interferometer sensor has its own operation range and overlapping is not acceptable. To calculate the capacity of wavelength-

K is the number of local sensors and wavelength channels. λw, λmin, λmax, and Δλop parame-

<sup>¼</sup> <sup>λ</sup>max � <sup>λ</sup>min Δλop

(23)

187

division multiplexing K, we use the interval working λ<sup>w</sup> and the operation range Δλop

<sup>K</sup> <sup>¼</sup> <sup>λ</sup><sup>w</sup> Δλop

Substituting Eqs. (4)–(11) into Eq. (17), the maximum frequency is

$$\nu\_{\text{max}} = \frac{2nL\_{\text{FPmax}}}{\lambda\_{\text{BG1}}^2} + \frac{2n\_1L\_{\text{BG}}}{\lambda\_{\text{BG1}}^2} \tag{18}$$

When we substitute the maximum cavity length (Eq. (5)) into Eq. (18), the parameter νmax takes the form

$$\nu\_{\text{max}} = \frac{1}{2\Delta\lambda} + \frac{2n\_1 L\_{\text{BG}}}{\lambda\_{\text{BG1}}^2} \tag{19}$$

Applying the sampling theorem, the sampling frequency ν<sup>s</sup> is

$$\nu\_s \ge 2\nu\_{\text{max}} = \frac{1}{\Delta\lambda} + \frac{4n\_1 L\_{\text{BG}}}{\lambda\_{\text{BG1}}^2} \tag{20}$$

Since <sup>ν</sup><sup>s</sup> <sup>¼</sup> <sup>1</sup> δλ<sup>s</sup> , we have

$$
\delta\lambda\_s \le \frac{\Delta\lambda\lambda\_{\rm BG1}^2}{\lambda\_{\rm BG1}^2 + 4n\_1L\_{\rm BG}\Delta\lambda} \tag{21}
$$

Finally, the number of samples N is given by

$$N = \frac{\lambda\_w}{\delta \lambda\_s} = \frac{\lambda\_w \left(\lambda\_{\rm BG1}^2 + 4n\_1 L\_{\rm BG} \Delta \lambda\right)}{\Delta \lambda \lambda\_{\rm BG1}^2} \tag{22}$$

Samples N depend on twin-grating sensor properties, the spectrometer resolution, and the interval working. The number of samples is a very important parameter for the twin-grating sensor demodulation because it affects the sensor's resolution.
