4. Capacity of wavelength-division multiplexing

3. Number of samples

186 Numerical Simulations in Engineering and Science

RT λ<sup>j</sup>

the form

Since <sup>ν</sup><sup>s</sup> <sup>¼</sup> <sup>1</sup>

δλ<sup>s</sup>

, we have

Finally, the number of samples N is given by

In Ref. [11] the twin-grating sensor was applied for the temperature measurement. The wave-

ulation 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

 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

> 1 2

> > þ

2n1LBG λ2 BG1

> 4n1LBG λ2 BG1

BG<sup>1</sup> <sup>þ</sup> <sup>4</sup>n1LBGΔ<sup>λ</sup> Δλλ<sup>2</sup> BG1

BG<sup>1</sup> <sup>þ</sup> <sup>4</sup>n1LBGΔ<sup>λ</sup> (21)

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

2Δλ þ

Δλ þ

Δλλ<sup>2</sup> BG1

2n1LBG λ2 BG1

νmax ¼ νFPK þ

<sup>ν</sup>max <sup>¼</sup> <sup>2</sup>nLFPmax λ2 BG1

<sup>ν</sup>max <sup>¼</sup> <sup>1</sup>

<sup>ν</sup><sup>s</sup> <sup>≥</sup> <sup>2</sup>νmax <sup>¼</sup> <sup>1</sup>

λ2

<sup>¼</sup> <sup>λ</sup><sup>w</sup> <sup>λ</sup><sup>2</sup>

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

δλ<sup>s</sup> ≤

<sup>N</sup> <sup>¼</sup> <sup>λ</sup><sup>w</sup> δλ<sup>s</sup>

sensor demodulation because it affects the sensor's resolution.

C. The demod-

(18)

(19)

(20)

(22)

νBGmax (17)

length shift sensitivity to a temperature change was estimated to be 0.00985 nm/o

minimum wavelength and δλ<sup>s</sup> is the wavelength step.

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

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

From Figure 4, the maximum frequency νmax is

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 Figure 5.

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

$$K = \frac{\lambda\_w}{\Delta \lambda\_{\rm op}} = \frac{\lambda\_{\rm max} - \lambda\_{\rm min}}{\Delta \lambda\_{\rm op}} \tag{23}$$

K is the number of local sensors and wavelength channels. λw, λmin, λmax, and Δλop parameters can be observed in Figure 3.

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

To illustrate, the next numerical example is presented.

Example 1: A broadband light source has the interval from λmin ¼ 1470 nm to λmax ¼ 1620 nm. If the operation range is selected to <sup>Δ</sup>λop <sup>¼</sup> 6 nm, the number of local sensors is <sup>K</sup> <sup>¼</sup> <sup>1620</sup>�<sup>1470</sup> 6 ¼ 25. The quasi-distributed sensor would have 25 twin-grating sensors and the signal will have 25 wavelength channels. Here, two important points can be mentioned: 1) The parameter Δλop permits the selection of the number of local sensors; 2) the cost per sensing point can be reduced combining the wavelength-and-frequency division multiplexing.
