2. A quasi-distributed fiber optic sensor

1. Introduction

180 Numerical Simulations in Engineering and Science

wavelength.

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:

method is known as direct spectrometric detection technique.

sensors´ resolution is high and the resolution depends on the cavity length.

• 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

The frequency-division multiplexing scheme [3] is illustrated in Figure 1b for a quasidistributed 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 wavelengthdivision multiplexing (WDM). The cross-talk noise is eliminated because all Bragg grating had low reflectivity, r < 1% [5]. In this sensor, the reflection spectrum is the superposition of all frequency components which are produced by all local interferometers. The detection

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) [6–9]. 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

Figure 2 illustrates the optical system under study. The optic system consists of a quasidistributed 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

λBG<sup>1</sup> 6¼ λBG<sup>2</sup> 6¼ … 6¼ λBGk 6¼ … 6¼ λBGK (3)

λ2 BG1

≈ …≈ νFPK (4)

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183

<sup>4</sup>nΔ<sup>λ</sup> (5)

�i2πλνdλ (6)

4πnLFPð Þ λ � λBGk λ2 BGk " # ! <sup>e</sup>

�iωt dt

(7)

Interference patterns have approximately the same frequency,

size can be found in the interval of [3]

LFPmax <sup>¼</sup> <sup>λ</sup><sup>2</sup>

Rð Þ¼ ν

ð ∞

X K

k¼1 2ak

�∞

BG1

<sup>ν</sup>FP<sup>1</sup> <sup>≈</sup> <sup>ν</sup>FP<sup>2</sup> <sup>≈</sup> …<sup>≈</sup> <sup>ν</sup>FPk <sup>¼</sup> <sup>2</sup>nLFP

λ2 BGk

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

From Eqs. (1) and (4), the cavity length defines the frequency of all interference patterns. Its

where Δλ is the spectrometer resolution, LFPmin ¼ 2LBG is the minimum cavity length, and

the Fourier domain phase analysis (FDPA) algorithm does not accept additional information or loss of information. The maximum cavity length is delimited because the OSA spectrometer has a limit of full-width half-maximum (FWHM). Figure 3 illustrates the optical spectrum.

<sup>4</sup>nΔλ is the maximum cavity length. The minimum cavity length is delimited because

LFPmin ≤ LFP ≤ LFPmax ! 2LBG ≤ LFP ≤

To know the frequency spectrum Rð Þν , the Fourier transform is applied to Eq. (1),

ð ∞

RTð Þ λ e

�∞

sinc<sup>2</sup> <sup>2</sup>n1LBGð Þ <sup>λ</sup> � <sup>λ</sup>BGk λ2 BGk " # ! <sup>1</sup> <sup>þ</sup> cos

RTð Þ¼ ν

Substituting Eq. (1) into Eq. (6), the spectra RTð Þν is

Figure 3. Optical signal detected by the optical analyzer spectrometer.

πn1LBG <sup>λ</sup>BGk � �<sup>2</sup>

Figure 2. A quasi-distributed sensor based on the WDM technique and twin-grating sensors.

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 LFP but each twin-grating sensor has its own Bragg wavelength. Thus, the WDM technique is generated and the FDM technique is eliminated. All Bragg gratings have the same length LBG and a typical reflectivity of 1%. The low reflectivity eliminates cross-talk noise.

### 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

$$R\_T(\lambda) = \sum\_{k=1}^{K} 2a\_k \left[ \left( \frac{\pi n\_1 L\_{BG}}{\lambda\_{BGk}} \right)^2 \text{sinc}^2 \left( \frac{2n\_1 L\_{BG} (\lambda - \lambda\_{BGk})}{\lambda\_{BGk}^2} \right) \right] \left[ 1 + \cos \left( \frac{4 \pi n L\_{FP} (\lambda - \lambda\_{BGk})}{\lambda\_{BGk}^2} \right) \right] \tag{1}$$

The signal parameters are: RTð Þ λ is a set of interference patterns, λ is the wavelength, λBGk is the kth Bragg wavelength, ak are amplitude factors, n<sup>1</sup> is the amplitude of the effective refractive index modulation of the gratings, LBG is the length of gratings, n is the effective index of the core, LFP is the cavity length, and k is the number of twin-grating sensors. The optical signal has the next characteristics: the enveloped function is a sinc one and its width ΔBGk is defined as the spectral distance between its +1 and –1 zeros,

$$
\Lambda\_{\rm BG1} \neq \Lambda\_{\rm BG2} \neq \Delta\_{\rm BG3} \neq \dots \neq \Delta\_{\rm BGk} = \frac{\lambda\_{\rm BGk}^2}{n\_1 L\_{\rm BG}} \neq \dots \neq \Delta\_{\rm BGK} \tag{2}
$$

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

Twin-Grating Fiber Optic Sensors Applied on Wavelength-Division Multiplexing and Its Numerical Resolution http://dx.doi.org/10.5772/intechopen.75586 183

$$
\lambda\_{\text{BG1}} \neq \lambda\_{\text{BG2}} \neq \dots \neq \lambda\_{\text{BGk}} \neq \dots \neq \lambda\_{\text{BGK}} \tag{3}
$$

Interference patterns have approximately the same frequency,

$$\nu\_{\rm FP1} \approx \nu\_{\rm FP2} \approx \dots \approx \nu\_{\rm FPk} = \frac{2nL\_{\rm FP}}{\lambda\_{\rm BGk}^2} \approx \dots \approx \nu\_{\rm FPK} \tag{4}$$

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]

$$L\_{FP\text{min}} \le L\_{FP} \le L\_{FP\text{max}} \to 2L\_{BG} \le L\_{FP} \le \frac{\lambda\_{BG1}^2}{4n\Delta\lambda} \tag{5}$$

where Δλ is the spectrometer resolution, LFPmin ¼ 2LBG is the minimum cavity length, and LFPmax <sup>¼</sup> <sup>λ</sup><sup>2</sup> BG1 <sup>4</sup>nΔλ is the maximum cavity length. The minimum cavity length is delimited because the Fourier domain phase analysis (FDPA) algorithm does not accept additional information or loss of information. The maximum cavity length is delimited because the OSA spectrometer has a limit of full-width half-maximum (FWHM). Figure 3 illustrates the optical spectrum.

To know the frequency spectrum Rð Þν , the Fourier transform is applied to Eq. (1),

$$R\_T(\nu) = \int\_{-\infty}^{\infty} R\_T(\lambda) e^{-i2\pi\lambda\nu} d\lambda \tag{6}$$

Substituting Eq. (1) into Eq. (6), the spectra RTð Þν is

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 LFP but each twin-grating sensor has its own Bragg wavelength. Thus, the WDM technique is generated and the FDM technique is eliminated. All Bragg gratings have the same length LBG and a typical

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

> sinc<sup>2</sup> <sup>2</sup>n1LBGð Þ <sup>λ</sup> � <sup>λ</sup>BGk λ2 BGk

The signal parameters are: RTð Þ λ is a set of interference patterns, λ is the wavelength, λBGk is the kth Bragg wavelength, ak are amplitude factors, n<sup>1</sup> is the amplitude of the effective refractive index modulation of the gratings, LBG is the length of gratings, n is the effective index of the core, LFP is the cavity length, and k is the number of twin-grating sensors. The optical signal has the next characteristics: the enveloped function is a sinc one and its width ΔBGk is defined

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

1 þ cos

BGk n1LBG 4πnLFPð Þ λ � λBGk λ2 BGk

6¼ … 6¼ ΔBGK (2)

(1)

" # !

" # !

<sup>Δ</sup>BG<sup>1</sup> 6¼ <sup>Δ</sup>BG<sup>2</sup> 6¼ <sup>Δ</sup>BG<sup>3</sup> 6¼ … 6¼ <sup>Δ</sup>BGk <sup>¼</sup> <sup>λ</sup><sup>2</sup>

reflectivity of 1%. The low reflectivity eliminates cross-talk noise.

Figure 2. A quasi-distributed sensor based on the WDM technique and twin-grating sensors.

2.1. Optical signal

RTð Þ¼ <sup>λ</sup> <sup>X</sup>

K

k¼1 2ak

small variation, the optical signal will be

182 Numerical Simulations in Engineering and Science

πn1LBG λBGk � �<sup>2</sup>

as the spectral distance between its +1 and –1 zeros,

$$R(\nu) = \int\_{-\infty}^{\infty} \sum\_{k=1}^{K} 2a\_k \left[ \left( \frac{\pi n\_1 L\_{\rm BG}}{\lambda\_{\rm BG}} \right)^2 \text{sinc}^2 \left( \frac{2n\_1 L\_{\rm BG} (\lambda - \lambda\_{\rm BG})}{\lambda\_{\rm BG}^2} \right) \right] \left[ 1 + \cos \left( \frac{4 \pi n L\_{\rm FP} (\lambda - \lambda\_{\rm BG})}{\lambda\_{\rm BG}^2} \right) \right] e^{-i\omega t} dt \tag{7}$$

Figure 3. Optical signal detected by the optical analyzer spectrometer.

Solving the transformation, the frequency spectrum is defined by

$$R\_T(\nu) = \sum\_{k=-K}^{K} R\_k(\nu) = \sum\_{k=-K}^{K} c\_k \text{tri}\left(\frac{\nu - \nu\_{\text{FPk}}}{\nu\_{\text{BGk}}}\right) \tag{8}$$

This frequency spectrum is the superposition of a set of triangle functions, where a triangle function is defined as trið Þ¼ x 1 � j j x j j x ≤ 1 0 otherwise � , Rkð Þν is the Fourier transform of the k-th interference pattern, ck are amplitude factors, νFPk is the center position of each peak and νBGk is the bandwidth of each peak.

$$
\omega\_{\rm BGk} = \frac{4n\_1 L\_{\rm BG}}{\lambda\_{\rm BGk}^2} \tag{9}
$$

In the frequency spectrum, the component νFP<sup>0</sup> contains information from all twin-grating sensors, the positive components νFP1, …, νFPK; the negative components �νFP1, …, � νFPK contain the same information. The minimum bandwidth νBGmin is

$$\nu\_{\text{BGmin}} = \frac{4n\_1 L\_{\text{BG}}}{\lambda\_{\text{BGK}}^2} \tag{10}$$

and the maximum bandwidth νBGmax is

$$
\omega\_{\text{BGmax}} = \frac{4n\_1 L\_{\text{BG}}}{\lambda\_{\text{BG1}}^2} \tag{11}
$$

RTð Þ¼ <sup>λ</sup>; δλ <sup>X</sup>

form

lation signal.

K

k¼1

Rkð Þ¼ λ � δλ<sup>k</sup> R1ð Þþ λ � δλ<sup>1</sup> … þ Rkð Þþ λ � δλ<sup>k</sup> … þ RKð Þ λ � δλ<sup>K</sup> (13)

�i2πλνdλ (14)

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185

�i2πλνdλ (15)

�i2πνδλkRkð Þ<sup>ν</sup> (16)

where RTð Þ λ; δλ is the optical signal due to external perturbations and δλ<sup>k</sup> is the Bragg

RTð Þ λ; δλ e

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

Rkð Þ λ � δλ<sup>k</sup> e

wavelength shift due to measured change [11, 12]. Its frequency spectrum RTð Þ ν; δλ is

∞

�∞

X K

k¼�K

Using the Fourier transform properties and solving, the frequency spectra RTð Þ ν; δλ takes the

K

k¼�K e

RTð Þ ν; δλ is the multiplication between Rð Þν and a set of phases. Each phase contains information of each twin-grating sensor and then the FDPA algorithm can be applied in the demodu-

∞

�∞

RTð Þ¼ <sup>ν</sup>; δλ <sup>X</sup>

RTð Þ¼ <sup>ν</sup>; δλ <sup>ð</sup>

RTð Þ¼ <sup>ν</sup>; δλ <sup>ð</sup>

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

Figure 4. Frequency spectrum determined from the optical signal.

Figure 4 shows the frequency spectrum Rð Þν . Based on Figure 4, all twin-grating interferometers produce approximately the same frequency components. This is possible because all interferometers have the same cavity length and all enveloped functions are approximately similar.

### 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

$$\begin{aligned} R\_T(\lambda, \delta \lambda) &= \sum\_{k=1}^K 2a\_k \left[ \left( \frac{\pi n\_1 L\_{\rm BG}}{\lambda\_{\rm BG}} \right)^2 \text{sinc}^2 \left( \frac{2n\_1 L\_{\rm BG} (\lambda - \lambda\_{\rm BG} - \delta \lambda\_k)}{\lambda\_{\rm BG}^2} \right) \right] \\ &\left[ 1 + \cos \left( \frac{4 \pi n L\_{\rm FP} (\lambda - \lambda\_{\rm BG} - \delta \lambda\_k)}{\lambda\_{\rm BG}^2} \right) \right] \end{aligned} \tag{12}$$

It can also be expressed as

Twin-Grating Fiber Optic Sensors Applied on Wavelength-Division Multiplexing and Its Numerical Resolution http://dx.doi.org/10.5772/intechopen.75586 185

Figure 4. Frequency spectrum determined from the optical signal.

Solving the transformation, the frequency spectrum is defined by

RTð Þ¼ <sup>ν</sup> <sup>X</sup>

�

contain the same information. The minimum bandwidth νBGmin is

function is defined as trið Þ¼ x

184 Numerical Simulations in Engineering and Science

the bandwidth of each peak.

and the maximum bandwidth νBGmax is

2.2. Optical signal produced by external perturbation

K

k¼1 2ak

1 þ cos

RTð Þ¼ <sup>λ</sup>; δλ <sup>X</sup>

similar.

spectrometer is

It can also be expressed as

K

Rkð Þ¼ <sup>ν</sup> <sup>X</sup>

This frequency spectrum is the superposition of a set of triangle functions, where a triangle

interference pattern, ck are amplitude factors, νFPk is the center position of each peak and νBGk is

<sup>ν</sup>BGk <sup>¼</sup> <sup>4</sup>n1LBG λ2 BGk

In the frequency spectrum, the component νFP<sup>0</sup> contains information from all twin-grating sensors, the positive components νFP1, …, νFPK; the negative components �νFP1, …, � νFPK

> <sup>ν</sup>BGmin <sup>¼</sup> <sup>4</sup>n1LBG λ2 BGK

> <sup>ν</sup>BGmax <sup>¼</sup> <sup>4</sup>n1LBG λ2 BG1

Figure 4 shows the frequency spectrum Rð Þν . Based on Figure 4, all twin-grating interferometers produce approximately the same frequency components. This is possible because all interferometers have the same cavity length and all enveloped functions are approximately

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

> 4πnLFPð Þ λ � λBGk � δλ<sup>k</sup> λ2 BGk

sinc<sup>2</sup> <sup>2</sup>n1LBGð Þ <sup>λ</sup> � <sup>λ</sup>BGk � δλ<sup>k</sup> λ2 BGk

" # ! (12)

" # !

πn1LBG λBGk � �<sup>2</sup> K

cktri <sup>ν</sup> � <sup>ν</sup>FPk νBGk � �

, Rkð Þν is the Fourier transform of the k-th

(8)

(9)

(10)

(11)

k¼�K

k¼�K

1 � j j x j j x ≤ 1 0 otherwise

$$R\_T(\lambda, \delta\lambda) = \sum\_{k=1}^{K} R\_k(\lambda - \delta\lambda\_k) = R\_1(\lambda - \delta\lambda\_1) + \dots + R\_k(\lambda - \delta\lambda\_k) + \dots + R\_K(\lambda - \delta\lambda\_K) \tag{13}$$

where RTð Þ λ; δλ is the optical signal due to external perturbations and δλ<sup>k</sup> is the Bragg wavelength shift due to measured change [11, 12]. Its frequency spectrum RTð Þ ν; δλ is

$$R\_T(\nu, \delta \lambda) = \int\_{-\infty}^{\infty} R\_T(\lambda, \delta \lambda) e^{-i2\pi\lambda\nu} d\lambda \tag{14}$$

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

$$R\_T(\nu, \delta\lambda) = \int\_{-\infty}^{\infty} \sum\_{k=-K}^{K} R\_k(\lambda - \delta\lambda\_k) e^{-i2\pi\lambda\nu} d\lambda \tag{15}$$

Using the Fourier transform properties and solving, the frequency spectra RTð Þ ν; δλ takes the form

$$R\_T(\nu, \delta \lambda) = \sum\_{k=-K}^{K} e^{-i2\pi \nu \delta \lambda\_k} R\_k(\nu) \tag{16}$$

RTð Þ ν; δλ is the multiplication between Rð Þν and a set of phases. Each phase contains information of each twin-grating sensor and then the FDPA algorithm can be applied in the demodulation signal.
