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

## Abstract

In this work, the twin-grating fiber optic sensor has been applied on wavelength-division multiplexing. A quasi-distributed sensor formed by three local twin-grating sensors, is numerically simulated. The wavelength channels were 1531.5, 1535.5, and 1539.5 nm. The numerical simulation shows the resolution vs. signal-to-noise rate. Three local twin-grating sensors have approximately the same resolution because all local sensors have the same cavity length and the wavelength channels are very close. All local sensors have two numerical resolutions because the Fourier domain phase analysis algorithm makes two evaluations of the Bragg wavelength shift. The transition between both resolutions can be calculated with the parameters: cavity length, Bragg wavelength channel, refraction index, and enveloped resolution. This transition depends on the noise system, demodulation algorithm, instrumentation, and local sensor properties. A very important point is, a theoretical analysis will permit to know the exact resolution for each local twin-grating sensor.

### Keywords

- twin-grating fiber optic sensor
- wavelength-division multiplexing
- numerical resolution
- quasi-distributed sensor
- numerical simulation

## 1. Introduction

Optic fiber sensors (OFSs) exhibit small dimensions; they are light weight and made of a dielectric material, vitreous silica. Some measurable parameters are temperature, strain, humidity, pressure, salinity, current, voltage, and concentration. Fiber sensors have as good resolution and accuracy as electronic and mechanical sensors. For this reason, OFSs are very active worldwide. An optic fiber sensor can be extrinsic or intrinsic. In an extrinsic sensor, the fiber acts as a means of getting the light to the sensing localization. In an intrinsic sensor, perturbations act on the fiber and the fiber in turn changes some characteristics of the light inside the fiber [1]. Both sensors types find potential industrial applications. On the other hand, a fiber sensor can also be spatially classified as a distributed sensor, a quasi-distributed sensor, or a point sensor. A distributed sensor is sensitive along its entire length. A quasi-distributed fiber optic sensor is not sensitive along its entire length, but is locally sensitized at various points. A point sensor is sensitive at a specific point along its entire length. In particular, a quasi-distributed sensor uses multiplexing techniques and their combinations. Two fundamental techniques are wavelength-division multiplexing (WDM) and frequency-division multiplexing (FDM). In Ref. [2], Grattan and Sun described the WDM technique:

The WDM technique received little attention due to the initial high cost of components such as wavelength selective couplers and filters. However, the widespread use of Bragg grating systems has opened up a range of possibilities for the use of wavelength-division multiplexing. Figure 1a illustrates a scheme of a quasi-distributed sensor based on the Bragg gratings; its configuration is serial and each Bragg grating has its own Bragg wavelength.

The frequency-division multiplexing scheme [3] is illustrated in Figure 1b for a quasi-distributed sensor based on the twin-grating fiber optic sensor. Each twin-grating sensor [4, 5] consists of two identical Bragg gratings and acts as a local sensor. In this configuration, there are * m*-twin-grating sensors in serial connection. Each interferometer has its own cavity length. However, all Bragg gratings have the same Bragg wavelength to eliminate wavelength-division multiplexing (WDM). The cross-talk noise is eliminated because all Bragg grating had low reflectivity,

Nowadays, the quasi-distributed sensor finds potential application in civil engineering (strain and temperature measurements), industrial process (temperature, strain, level, and pressure measurements), military application (vibration detection), sport science (vibration and strain), and aircraft (strain, vibration, and pressure measurements) [6, 7, 8, 9]. This sensor type reduces the cost by sensing point. In this work, a quasi-distributed sensor based on wavelength-division multiplexing and twin-grating sensor is discussed and simulated. The results show the numerical resolution in terms of Bragg wavelength shift. The results demonstrate that twin-grating sensors´ resolution is high and the resolution depends on the cavity length.

## 2. A quasi-distributed fiber optic sensor

Figure 2 illustrates the optical system under study. The optic system consists of a quasi-distributed fiber optic sensor which is based on wavelength-division multiplexing (WDM) and twin-grating sensors. The sensing system has five fundamental components: an optical broadband source, an optical circulator 50/50, an optical spectrometer analyzer (OSA spectrometer), a personal computer, and a quasi-distributed sensor. In particular, the quasi-distributed sensor consists of a serial array of twin-grating sensors. Each twin-grating sensor acts as a low-finesse Fabry-Perot interferometer [3, 10]. All interferometers have the same cavity length

### 2.1. Optical signal

In the sensing system presented in Figure 2, the signal from each local sensor is returned by reflection from each twin-grating sensor, where each twin-grating interferometer has its own wavelength. The signal returned to the detector is monitored with the OSA spectrometer; the intensity at each wavelength corresponds to the measurement each local sensor. When the quasi-distributed sensor does not have external perturbations and interference patterns have small variation, the optical signal will be

The signal parameters are: * k*th Bragg wavelength,

Each interference pattern has its own central Bragg wavelength and the next condition is true

Interference patterns have approximately the same frequency,

From Eqs. (1) and (4), the cavity length defines the frequency of all interference patterns. Its size can be found in the interval of [3]

where

To know the frequency spectrum

Substituting Eq. (1) into Eq. (6), the spectra

Solving the transformation, the frequency spectrum is defined by

This frequency spectrum is the superposition of a set of triangle functions, where a triangle function is defined as * k*-th interference pattern,

In the frequency spectrum, the component

and the maximum bandwidth

Figure 4 shows the frequency spectrum

### 2.2. Optical signal produced by external perturbation

When the quasi-distributed sensor has external perturbations due to the temperature or strain, Bragg gratings and cavity length have an elongation. In turn, interference patterns have a small shift in response to a measured variation. The optical signal detected by the OSA spectrometer is

It can also be expressed as

where

Substituting Eq. (13) into Eq. (14), the spectra is now

Using the Fourier transform properties and solving, the frequency spectra

## 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

From Figure 4, the maximum frequency

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

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

Applying the sampling theorem, the sampling frequency

Since

Finally, the number of samples

Samples

## 4. Capacity of wavelength-division multiplexing

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:

Let us introduce the operation range * K*, we use the interval working

To illustrate, the next numerical example is presented.

** Example 1:** A broadband light source has the interval from

## 5. Demodulation signal

In this section, we present the demodulation signal for the quasi-distributed sensor based on wavelength-division multiplexing. The signal processing combines the Fourier domain phase analysis (FDPA) algorithm, a bank of * K* filters and a band-pass filter. The FDPA algorithm was described and also applied in Refs. [3, 11]. The bank of

*filters can be defined as*K

where the symbol

and * K* filters is

The signal * k*th interference pattern. The spectrum

where the rect function (Eq. (27)) has the next definition

The filter

Basically, the digital demodulation consists of two stages: calibration and measurement. The calibration stage is developed once; five steps are necessary and the references are generated. The calibration considers the signal acquisition

## 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

Sensor number | Sensor parameters | Signal values |
---|---|---|

Twin-grating sensor 1 (S1) | _{FP1} = 4 (mm) | |

_{BG} = 0.5 (mm) | ||

= 1.46 | ||

Twin-grating sensor 2 (S2) | _{FP2} = 8 (mm) | |

_{BG} = 0.5 (mm) | ||

= 1.46 | ||

Twin-grating sensor 3 (S3) | _{FP3} = 16 (mm) | |

_{BG} = 0.5 (mm) | ||

= 1.46 | ||

Being aware of that, our goal is to apply the twin-grating interferometer to wavelength-division 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]: _{FP1} = _{FP2} = _{FP3} = _{FP4} = 4 (mm), _{BG} = 0.5 (mm), * n* = 1.46,

*= 1024; the Bragg gratings have rectangular profile; the noise has Gaussian distribution and its value is in the interval*N

Sensor number | Sensor parameters | Signal values |
---|---|---|

Fabry-Perot sensor 1 (S1) | _{FP1} = 4 (mm) | |

_{BG} = 0.5 (mm) | ||

= 1.46 | ||

Fabry-Perot sensor 2 (S2) | _{FP2} = 4 (mm) | |

_{BG} = 0.5 (mm) | ||

= 1.46 | ||

Fabry-Perot sensor 3 (S3) | _{FP3} = 4 (mm) | |

_{BG} = 0.5 (mm) | ||

= 1.46 | ||

Analyzing Tables 1 and 2, the sensing system presented in Figure 3 is based on wavelength-division 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

Figure 7 Shows the behavior of demodulation error vs. signal-to-noise rate SNR^{1/2}, 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

specifies the boundary between low resolution and high resolution. Using Eq. (29) and Table 2, the thresholds are approximately S1 → ^{1/2} threshold is observable:

### 6.2. Discussion

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

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

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 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.

## 7. Conclusion

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 system.

## Acknowledgments

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

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