6. Numerical simulation and discussion

### 6.1. Parameters and results

In Ref. [3], the twin-grating fiber optic sensor was applied on frequency-division multiplexing. The numerical results confirmed that the twin-grating sensor has a high resolution and the resolution is a function of cavity length. In the numerical simulation, the number of samples was N = 1024, the noise was in the interval ffiffiffiffiffiffiffiffiffiffi SNR <sup>p</sup> <sup>¼</sup> <sup>100</sup> to ffiffiffiffiffiffiffiffiffiffi SNR <sup>p</sup> <sup>¼</sup> <sup>10</sup><sup>4</sup> . The quasi-distributed sensor parameters and optical signal parameters can be observed in Table 1.

Being aware of that, our goal is to apply the twin-grating interferometer to wavelengthdivision multiplexing, a quasi-distributed fiber optic sensor is numerically simulated as was done in Ref. [3]. The quasi-distributed sensor consists of three twin-grating sensors. The physical parameters are shown in Table 2. In our numerical simulation, we use some parameters from Ref. [3]: LFP1 = LFP2 = LFP3 = LFP4 = 4 (mm), LBG = 0.5 (mm), n = 1.46, N = 1024; the Bragg gratings have rectangular profile; the noise has Gaussian distribution and its value is in the interval ffiffiffiffiffiffiffiffiffiffi SNR <sup>p</sup> <sup>¼</sup> <sup>100</sup> to ffiffiffiffiffiffiffiffiffiffi SNR <sup>p</sup> <sup>¼</sup> <sup>104</sup> . The same values are possible because our goal is to prove the wavelength-division multiplexing. The measurements are in the intervals S1 ! –0.3 to 0.3 nm, S2 ! –0.5 to 0.5 nm and S3 ! –1 to 1 nm. The signal parameters are shown in Table 2 while Figure 5 shows our numerical results: Demodulation errors vs. SNR<sup>1</sup>=<sup>2</sup> . A Laptop Toshiba 45C was used with a 512 Mb RAM and a velocity of 1.7 GHz.

ΔλBG<sup>3</sup> ¼ 3.246 (nm). Finally, the triangle functions have approximately the same bandwidth: νBG<sup>1</sup> ¼1.244 (cycles/nm), νBG<sup>2</sup> ¼ 1:238 (cycles/nm) and νBG<sup>3</sup> ¼ 1.232 (cycles/nm). Thus, we prove that the quasi-distributed sensor applies the wavelength-division multiplexing and the FDM

Sensor number Sensor parameters Signal values

n = 1.46

n = 1.46

n = 1.46

Table 2. Quasi-distributed sensor parameters used and signal values obtained.

Fabry-Perot sensor 1 (S1) LFP1 = 4 (mm) νFP<sup>1</sup> ¼ 4:979 (cycles/nm) (Eq. (2))

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

λBG<sup>1</sup> ¼ 1531:5 (nm) Fabry-Perot sensor 2 (S2) LFP2 = 4 (mm) νFP<sup>2</sup> ¼4.953(cycles/nm) (Eq. (2))

λBG<sup>2</sup> ¼ 1535:5 (nm) Fabry-Perot sensor 3 (S3) LFP3 = 4 (mm) νFP<sup>3</sup> ¼ 4:928 (ciclos/nm) (Eq. (2))

λBG<sup>3</sup> ¼ 1539:5 (nm)

Figure 7 Shows the behavior of demodulation error vs. signal-to-noise rate SNR1/2, if demodulation error is denominated as the resolution. These results confirm good resolution for the twin-grating sensors. Since the FDPA algorithm makes two evaluations of the Bragg wavelength shift [3, 7], the twin-grating sensors have two resolutions: Low resolution σenv and high

> λ2 BGk 12nLFP

specifies the boundary between low resolution and high resolution. Using Eq. (29) and Table 2, the thresholds are approximately S1 ! σenv < 0:0335 nm, S2 ! σenv < 0:0336 nm, and S3 ! σenv < 0:0338 nm. For each local twin-grating sensor, an SNR1/2 threshold is observable:

The threshold values are very close because the twin-grating sensors have similar cavity length

The quasi-distributed fiber optic sensor (Figure 3) would be built on wavelength-division multiplexing and twin-grating interferometers. Our results optimize the sensor's implementation and also permit its design. Local sensor properties, light source characteristics, noise (Gaussian distribution), signal processing and detection technique are considered in our

(29)

.

<sup>2</sup> <sup>≈</sup> 101:<sup>19</sup>, S3 1539 ð Þ! :5 nm SNR<sup>1</sup>=<sup>2</sup> <sup>≈</sup> 101:<sup>2</sup>

ΔλBG<sup>1</sup> ¼ 3:213 (nm) (Eq. (3))

191

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

ΔλBG<sup>2</sup> ¼ 3:229 (nm) (Eq. (3))

ΔλBG<sup>3</sup> ¼ 3:246 ðnm) (Eq. (3))

<sup>ν</sup>BG<sup>1</sup> <sup>¼</sup>1.244 (cycles/nm) (Eq. (9)) <sup>L</sup>BG = 0.5 (mm)

<sup>ν</sup>BG<sup>2</sup> <sup>¼</sup> <sup>1</sup>:238 (cycles/nm) (Eq. (9)) <sup>L</sup>BG = 0.5 (mm)

<sup>ν</sup>BG<sup>3</sup> <sup>¼</sup>1.232 (cycles/nm) (Eq. (9)) <sup>L</sup>BG = 0.5 (mm)

σenv <

<sup>2</sup> <sup>≈</sup> 101:<sup>15</sup>, S2 1355 ð Þ! :5 nm SNR<sup>1</sup>

and the enveloped functions have approximately the same bandwidth ΔλBGk.

technique was eliminated.

resolution σint. the threshold [3].

<sup>S</sup>1 1531 ð Þ! :5 nm SNR<sup>1</sup>

6.2. Discussion

Analyzing Tables 1 and 2, the sensing system presented in Figure 3 is based on wavelengthdivision multiplexing where their wavelength-channels are 1531.5, 1535.5, and 1539.5 nm. The frequency-division multiplexing was eliminated since the frequency components are νFP<sup>1</sup> ¼ 4:979 (cycles/nm), νFP<sup>2</sup> ¼4.953 (cycles/nm), and νFP<sup>3</sup> ¼ 4:928 (cycles/nm). All interference patterns have approximately the same bandwidth: ΔλBG<sup>1</sup> ¼ 3:213 (nm), ΔλBG<sup>2</sup> ¼ 3:229 (nm), and


Table 1. Sensor parameters and signal parameters used in the frequency-division multiplexing [3].

Twin-Grating Fiber Optic Sensors Applied on Wavelength-Division Multiplexing and Its Numerical Resolution http://dx.doi.org/10.5772/intechopen.75586 191


Table 2. Quasi-distributed sensor parameters used and signal values obtained.

ΔλBG<sup>3</sup> ¼ 3.246 (nm). Finally, the triangle functions have approximately the same bandwidth: νBG<sup>1</sup> ¼1.244 (cycles/nm), νBG<sup>2</sup> ¼ 1:238 (cycles/nm) and νBG<sup>3</sup> ¼ 1.232 (cycles/nm). Thus, we prove that the quasi-distributed sensor applies the wavelength-division multiplexing and the FDM technique was eliminated.

Figure 7 Shows the behavior of demodulation error vs. signal-to-noise rate SNR1/2, if demodulation error is denominated as the resolution. These results confirm good resolution for the twin-grating sensors. Since the FDPA algorithm makes two evaluations of the Bragg wavelength shift [3, 7], the twin-grating sensors have two resolutions: Low resolution σenv and high resolution σint. the threshold [3].

$$
\sigma\_{\rm env} < \frac{\lambda\_{\rm BG}^2}{12nL\_{\rm FP}} \tag{29}
$$

specifies the boundary between low resolution and high resolution. Using Eq. (29) and Table 2, the thresholds are approximately S1 ! σenv < 0:0335 nm, S2 ! σenv < 0:0336 nm, and S3 ! σenv < 0:0338 nm. For each local twin-grating sensor, an SNR1/2 threshold is observable:

<sup>S</sup>1 1531 ð Þ! :5 nm SNR<sup>1</sup> <sup>2</sup> <sup>≈</sup> 101:<sup>15</sup>, S2 1355 ð Þ! :5 nm SNR<sup>1</sup> <sup>2</sup> <sup>≈</sup> 101:<sup>19</sup>, S3 1539 ð Þ! :5 nm SNR<sup>1</sup>=<sup>2</sup> <sup>≈</sup> 101:<sup>2</sup> . The threshold values are very close because the twin-grating sensors have similar cavity length and the enveloped functions have approximately the same bandwidth ΔλBGk.

### 6.2. Discussion

6. Numerical simulation and discussion

was N = 1024, the noise was in the interval ffiffiffiffiffiffiffiffiffiffi

<sup>p</sup> <sup>¼</sup> <sup>100</sup> to ffiffiffiffiffiffiffiffiffiffi

SNR <sup>p</sup> <sup>¼</sup> <sup>104</sup>

In Ref. [3], the twin-grating fiber optic sensor was applied on frequency-division multiplexing. The numerical results confirmed that the twin-grating sensor has a high resolution and the resolution is a function of cavity length. In the numerical simulation, the number of samples

SNR

Being aware of that, our goal is to apply the twin-grating interferometer to wavelengthdivision multiplexing, a quasi-distributed fiber optic sensor is numerically simulated as was done in Ref. [3]. The quasi-distributed sensor consists of three twin-grating sensors. The physical parameters are shown in Table 2. In our numerical simulation, we use some parameters from Ref. [3]: LFP1 = LFP2 = LFP3 = LFP4 = 4 (mm), LBG = 0.5 (mm), n = 1.46, N = 1024; the Bragg gratings have rectangular profile; the noise has Gaussian distribution and its value is in

prove the wavelength-division multiplexing. The measurements are in the intervals S1 ! –0.3 to 0.3 nm, S2 ! –0.5 to 0.5 nm and S3 ! –1 to 1 nm. The signal parameters are shown in Table 2 while Figure 5 shows our numerical results: Demodulation errors vs. SNR<sup>1</sup>=<sup>2</sup>

Analyzing Tables 1 and 2, the sensing system presented in Figure 3 is based on wavelengthdivision multiplexing where their wavelength-channels are 1531.5, 1535.5, and 1539.5 nm. The frequency-division multiplexing was eliminated since the frequency components are νFP<sup>1</sup> ¼ 4:979 (cycles/nm), νFP<sup>2</sup> ¼4.953 (cycles/nm), and νFP<sup>3</sup> ¼ 4:928 (cycles/nm). All interference patterns have approximately the same bandwidth: ΔλBG<sup>1</sup> ¼ 3:213 (nm), ΔλBG<sup>2</sup> ¼ 3:229 (nm), and

sensor parameters and optical signal parameters can be observed in Table 1.

Laptop Toshiba 45C was used with a 512 Mb RAM and a velocity of 1.7 GHz.

Sensor number Sensor parameters Signal values Twin-grating sensor 1 (S1) LFP1 = 4 (mm) ΔλBG ¼ 3:22 (nm)

n = 1.46

Twin-grating sensor 2 (S2) LFP2 = 8 (mm) ΔλBG ¼ 3:22 ðnm)

n = 1.46

Twin-grating sensor 3 (S3) LFP3 = 16 (mm) ΔλBG ¼ 3:22 (nm)

n = 1.46

Table 1. Sensor parameters and signal parameters used in the frequency-division multiplexing [3].

λBG ¼ 1532:5 (nm)

λBG ¼ 1532:5 (nm)

λBG ¼ 1532:5 (nm)

<sup>p</sup> <sup>¼</sup> <sup>100</sup> to ffiffiffiffiffiffiffiffiffiffi

SNR <sup>p</sup> <sup>¼</sup> <sup>10</sup><sup>4</sup>

. The same values are possible because our goal is to

<sup>ν</sup>BG <sup>¼</sup> <sup>1</sup>:23 (ciclos/nm) <sup>L</sup>BG = 0.5 (mm)

<sup>ν</sup>BG <sup>¼</sup> <sup>1</sup>:23 (ciclos/nm) <sup>L</sup>BG = 0.5 (mm)

<sup>ν</sup>BG <sup>¼</sup> <sup>1</sup>:23 (ciclos/nm) <sup>L</sup>BG = 0.5 (mm)

νFP<sup>1</sup> ¼ 4:95 (ciclos/nm)

νFP<sup>2</sup> ¼ 9:91 (ciclos/nm)

νFP<sup>3</sup> ¼ 19:82 (ciclos/nm)

. The quasi-distributed

. A

6.1. Parameters and results

190 Numerical Simulations in Engineering and Science

the interval ffiffiffiffiffiffiffiffiffiffi

SNR

The quasi-distributed fiber optic sensor (Figure 3) would be built on wavelength-division multiplexing and twin-grating interferometers. Our results optimize the sensor's implementation and also permit its design. Local sensor properties, light source characteristics, noise (Gaussian distribution), signal processing and detection technique are considered in our numerical simulation. Our experimental results (Figure 7) corroborate well functionally. Two resolutions are also confirmed σBG for each local sensor: low resolution σenv and high resolution σint. Both resolutions depend on noise system, cavity length, instrumentation, sensor properties, and the digital demodulation algorithm.

applications, the quasi-distributed sensor can be applied for temperature monitoring, gasoline detection (security), strain measurement, and level liquid measurement. Our analysis makes an excellent contribution to quasi-distributed sensor implementation because all local sensors will have high resolution (see Figure 7), high sensibility, low cost by sensing point, and the

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

In this work, a quasi-distributed fiber optic sensor was numerically simulated. The sensor was based on twin-grating sensors and wavelength-division multiplexing. The numerical results show the resolution for each local twin-grating sensor. Local sensors have approximately the same resolution because all twin-grating sensors have the same cavity length and the wavelength channels are close. Two resolutions were obtained for each local sensor. Our numerical results show that the quasi-distributed sensor has potential industrial application: temperature measurement, strain measurement, pressure measurement, humidity monitoring, and security

Authors thank PRODEP 2017 No. F-PROMEP-39/Rev-04 SEP-23-005 (number DSA/103.5/16/ 10313), PRODEP 2017 Project No. 236110 of found 1.1.9.25 (Agreement RG/003/2017) and

, Nancy Elizabeth Franco Rodríguez<sup>4</sup>

1 Electronic Department, CUCEI, University of Guadalajara, Guadalajara, Jalisco, México

3 Department of Engineering Projects, CUCEI, University of Guadalajara, Guadalajara,

4 Computer Department, CUCEI, University of Guadalajara, Guadalajara, Jalisco, México 5 Department of Computer Science and Engineering, CUValles, University of Guadalajara,

6 Faculty of Chemical Science, Colima University, Coquimatlán, Colima, Mexico

2 Mathematics Department, CUCEI, University of Guadalajara, Guadalajara, Jalisco, México

, Antonio Casillas Zamora1

,

, Alex Guillen Bonilla<sup>5</sup> and

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

193

quasi-distributed sensor can be designed without other requirements.

7. Conclusion

system.

Acknowledgements

Author details

Juan Reyes Gómez<sup>6</sup>

Jalisco, México

Ameca, Jalisco, México

Gustavo Adolfo Vega Gómez<sup>1</sup>

PRODEP 2017 Project No 238635 (511-6/17-8091).

José Trinidad Guillen Bonilla1,2\*, Héctor Guillen Bonilla3

\*Address all correspondence to: trinidad.guillen@academicos.udg.mx

A twin-grating fiber optic sensor and an optical fiber sensor based on a single Bragg grating will have the same resolution if and only if FDPA algorithm cannot eliminate the 2πP ambiguity. In this case, the twin-grating sensor has only low resolution because the FDPA algorithm evaluates the Bragg wavelength shift with the enveloped function [11]. However, if the optical sensing system has small noise, then the twin-grating sensor has high resolution, since the Bragg wavelength shift was evaluated combining the enveloped and modulated functions [3, 11]. Additionally, three twin-grating sensors have approximately the same resolution because three wavelength channels are very close and all interferometer sensors have the same cavity length. In conclusion, the sensor´s resolution is high while the FDPA algorithm can acceptably demodulate the optical signal.

The presented study optimizes the quasi-distributed sensor which was shown in Figure 3. Combining our study (this work) and the analysis presented in Ref. [3], the experimental sensing system described by Shlyagin et al. [4, 5] can be optimized. The optimization will be on signal processing, local sensor properties, sensitivity, resolution and instrumentation parameters. Additionally, the cost per sensing point is considerably reduced.

Our future work has the following direction: a theoretical analysis and practical application. In the theoretical analysis, frequency-and-wavelength division multiplexing can be implemented based on the twin-grating interferometer; resolution is another direction. In the practical

Figure 7. Numerical results obtained from the numerical experiments.

applications, the quasi-distributed sensor can be applied for temperature monitoring, gasoline detection (security), strain measurement, and level liquid measurement. Our analysis makes an excellent contribution to quasi-distributed sensor implementation because all local sensors will have high resolution (see Figure 7), high sensibility, low cost by sensing point, and the quasi-distributed sensor can be designed without other requirements.
