A Theoretical Study and Numerical Simulation of a Quasi-Distributed Sensor Based on the Low-Finesse Fabry-Perot Interferometer: Frequency-Division Multiplexing

The application of the sensor optical fibers in the areas of scientific instrumentation and industrial instrumentation is very attractive due to its numerous advantages. In the industry of civil engineering for example, quasi-distributed sensors made with optical fiber are used for reliable strain and temperature measurements. Here, a quasi-distributed sensor in the frequency domain is discussed. The sensor consists of a series of low-finesse Fabry-Perot interferometers where each Fabry-Perot interferometer acts as a local sensor. Fabry-Perot interferometers are formed by pairs of identical low reflective Bragg gratings imprinted in a single mode fiber. All interferometer sensors have different cavity length, provoking frequency-domain multiplexing. The optical signal represents the superposition of all interference patterns which can be decomposed using the Fourier transform. The frequency spectrum was analyzed and sensor’s properties were defined. Following that, a quasi-distributed sensor was numerically simulated. Our sensor simulation considers sensor properties, signal processing, noise system, and instrumentation. The numerical results show the behavior of resolution vs. signal-to-noise ratio. From our results, the Fabry-Perot sensor has high resolution and low resolution. Both resolutions are conceivable because the Fourier Domain Phase Analysis (FDPA) algorithm elaborates two evaluations of Bragg wavelength shift.


Introduction
Bragg grating has a very particular peak in its reflection spectrum; the peak is centered at the Bragg wavelength λ BG = 2nΛ [1], where Λ is the grating pitch and n is the effective fiber refraction index. The operational principle of a fiber Brag grating sensor is based on the spectral We analyzed the optical signal and then the quasi-distributed sensor's properties were defined, for example, minimum and maximum cavities, number of samples, spatial resolution, and multiplexing capability of a twin-grating fiber sensor. All parameters are expressed in terms of physical parameters and instrumentation characteristics. Then, the quasi-distributed sensor was numerically simulated (in operation) and we obtained the graph of demodulation errors vs. signal-to-noise ratio. From our numerical results, the cavity length augments the resolution and all Fabry-Perot sensors have two resolutions: a high resolution and low resolution. The cavity length, low resolution, and noise system define the transition between both resolutions. In general, our theoretical analysis and numerical simulation permit its optimal implementation and its design. Figure 1 shows our optical sensing system schematically. The optical system consists of a broadband source, an optical circulator 50/50, an optical spectrometer analyzer (OSA spectrometer), a personal computer and a quasi-distributed sensor. The quasi-distributed sensor can be implemented by using a serial array of low-finesse Fabry-Perot interferometers [29,30]. The local sensors are formed by pairs of identical low reflective Bragg gratings imprinted in a single mode fiber. Each Fabry-Perot interferometer has a unique optical path length which obtains the frequency-division multiplexing (FDM). The Bragg gratings have approximately the same length and typical reflectivity of 0.1%. Thus, wavelength-division multiplexing was eliminated for our optical sensor. represented mathematically. We analyzed the optical signal and then the quasi-distributed sensor's properties were defined, for example, minimum and maximum cavities, number of samples, spatial resolution, and multiplexing capability of a twin-grating fiber sensor. All parameters are expressed in terms of physical parameters and instrumentation characteristics. Then, the quasi-distributed sensor was numerically simulated (in operation) and we obtained the graph of demodulation errors vs. signal-to-noise ratio. From our numerical results, the cavity length augments the resolution and all Fabry-Perot sensors have two resolutions: a high resolution and low resolution. The cavity length, low resolution, and noise system define the transition between both resolutions. In general, our theoretical analysis and numerical simulation permit its optimal implementation and its design. Figure 1 shows our optical sensing system schematically. The optical system consists of a broadband source, an optical circulator 50/50, an optical spectrometer analyzer (OSA spectrometer), a personal computer and a quasi-distributed sensor. The quasi-distributed sensor can be implemented by using a serial array of low-finesse Fabry-Perot interferometers [29,30]. The local sensors are formed by pairs of identical low reflective Bragg gratings imprinted in a single mode fiber. Each Fabry-Perot interferometer has a unique optical path length which obtains the frequency-division multiplexing (FDM). The Bragg gratings have approximately the same length and typical reflectivity of 0.1%. Thus, wavelength-division multiplexing was eliminated for our optical sensor. is the length of gratings, 1 is the minimum cavity length, is the spatial resolution, is the m-th cavity length, is the maximum cavity length and OSA is the Optical Spectrometer Analyzer.

( ) and ( ) Spectrums
When the quasi-distributed sensor does not have external perturbations, the optical signal ( ) will be the superposition of all interference patterns, ( ) is the optical signal detected by the OSA spectrometer and ( ), ( ), ( ) … ( ) are interference patterns generated by all interferometer sensors. Considering the physical parameters, the optical signal can be re-written as [27] ( ) = 2 sin 2 ( − ) 1 + cos 4 ( − ) (2) Figure 1. Sensing system: L BG is the length of gratings, L FP1 is the minimum cavity length, L SR is the spatial resolution, L FPm is the m-th cavity length, L FPM is the maximum cavity length and OSA is the Optical Spectrometer Analyzer.

R T (λ) and R T (ν) Spectrums
When the quasi-distributed sensor does not have external perturbations, the optical signal R T (λ) will be the superposition of all interference patterns, R T (λ) is the optical signal detected by the OSA spectrometer and R 1 (λ), R 2 (λ), R 3 (λ) . . . R M (λ) are interference patterns generated by all interferometer sensors. Considering the physical parameters, the optical signal can be re-written as [27] where λ is the wavelength, a m is amplitude factor, n 1 is the amplitude of the effective refractive index modulation of the gratings, L BG is the length of gratings, λ BG is the Bragg wavelength, n is the effective index of the core, L FPm is the mth cavity length, and M is the number of low-finesse Fabry-Perot interferometers (local sensors). Analyzing the optical signal (2), all interference patterns have a similar enveloped function (sinc function), the sinc function is the reflection spectrum of the gratings, the width ∆ BG is defined as the spectral distance between its +1 and −1 zeros, Each interference pattern has its own frequency component. There are M modulate functions where the frequency component ν FPm will be To know the frequency components, we apply the Fourier transform to the optical signal R T (ν) is the frequency spectrum, F { } is the Fourier operator, and ν is the frequency. Substituting Equations (2)-(4) into Equation (5), the frequency spectrum is (6) Invoking the convolution properties and Fourier operator, we have the symbol ⊗ indicates the convolution. Using the identities cos 2 (ϕ) = 1 2 (1 + cos(2ϕ)), , c m are amplitude factors, and ν BG is the bandwidth In addition, ν FPm is the center position of each triangle function. Here, all frequency components were separated as Figure 2 illustrates.
In addition, is the center position of each triangle function. Here, all frequency components were separated as Figure 2 illustrates.

R T (λ, δλ) and R T (ν, δλ) Spectrums
When the quasi-distributed sensor has external perturbations, the measured temperature or string affects the gating period Λ, the refraction index n, the length of gratings L BG , and cavity length L FPm [27]. In turn, interference patterns have a small shift in response to a measured variation, and the optical signal detected by the OSA spectrometer is The optical spectrum R T (λ, δλ) can be expressed as where R T (λ, δλ) is the optical signal due to external perturbations and δλ BGm is the Bragg wavelength shift due to measured change. Now, we estimate their frequency components through Invoking the shift property, the Fourier transform is Observing the Equation (13), the frequency spectrum R T (ν, δλ) is the multiplication between R T (ν) (Equation (8)) and a set of phases. Those phases contain the information about the perturbations.

Cavity Length
For all quasi-distributed sensors based on interferometers (optical fiber), the cavity length is a very important parameter since it defines the sensor characteristics. Their limits depend of instrumentation, local sensor characteristics, and signal demodulation. In the following sections, we determine minimum and maximum cavities where the low-finesse Fabry-Perot interferometer can be applied.

Minimum Cavity Length
The Fourier Domain Phase Analysis (FDPA) algorithm was developed for the twin-grating fiber optic sensor [27]. This algorithm does not accept additional information and does not lose information, therefore, good signal detection and good frequency component identification are necessary. From Figure 2, first frequency components ν FP1 can be defined by The condition (14) eliminates the overlapping between components, ν FP1 and ν FP0 . Using the Equations (4) and (9), we have 2nL FP1 As n 1 ≈ n, the minimum cavity length will be It is not possible to have smaller cavities because the FDPA algorithm cannot demodulate the optical signal.

Maximum Cavity Length
The optical sensing system applies the direct spectroscopic detection [4]. This technique uses an optical spectrometer analyzer which defines the maximum cavity length L FPM . The OSA spectrometer has a limit for the optical signal detection. The limit is the Full-With Half-Maximum (FWHM). Considering the sampling theorem, the OSA spectrometer can detect the signal if and only if the next condition is true, where ∆λ FPmin is the minimum period detectable (FWHM) and ∆λ is its spectrometer resolution. Then, the maximum frequency component can be expressed as From Figure 2 and Equation (4), last frequency component ν FPM can be determined by Combining Equations (17)- (19), the maximum cavity length is Equation (20) indicates the maximum cavity length where OSA spectrometer can detect the optical signal. It is not possible to have bigger cavities because the instrumentation cannot detect the optical signal. Using Equations (16) and (20), the cavity length can be within the interval of

Capacity of Frequency-Division Multiplexing
In the quasi-distributed sensor, each low-finesse Fabry-Perot interferometer generates an interference pattern and then each pattern produces a channel in the frequency domain. The enveloped function produces the bandwidth ν BG and the modulate function provokes the frequency components −ν FPm , ν FP0 , and ν FPm . The term ν FP0 contains information from all Fabry-Perot interferometers while −ν FPm and ν FPm contain similar information from the mth Fabry-Perot sensor. From Figure 2, we have the next condition In other words, the capacity of frequency-division multiplexing M is given by the relation between last and first frequency components. Substituting the Equations (14), (15), and (17) into (22), the capacity M can be re-written as Finally, substituting the Equations (16) and (20) into Equation (22), we have This expression gives the limit for the multiplexing capacity within one wavelength channel.

Number of Samples
When the optical spectrometer analyzer instrument acquires the optical signal, the reflection spectrum is recorded as a series of digital samples. If a minimum and maximum wavelength within a working interval λ w = λ max − λ min , then λ max is the maximum wavelength, λ min is the minimum wavelength and δλ is the wavelength step. The signal samples R T (λ k ) are taken as wavelengths λ k = λ min + kδλ where k = 0, 1, 2, . . . , N − 1, N is the number of samples. The representation of such a signal in the Fourier domain is also discrete. Therefore, we obtain the next condition from Figure 2 ν where ν max is the maximum frequency, ν s is the sampling frequency, and the Nyquist theorem was considered. Substituting Equations (9) and (19) into Equation (25), we have Since ν s = 1 δλ , we have Finally, the number of samples is The number of samples depends of optical system parameters.

Digital Demodulation
The demodulation is the complete signal processing algorithm developed for a quasi-distributed sensor based on the low-finesse Fabry-Perot interferometers. The complete processing algorithm combines the Fourier Domain Phase Analysis (FDPA) algorithm and a bank of M filters. The FDPA algorithm was described in [27] while the bank of filters is where the symbol ⊗ indicates the convolution operation, the rect function is definition as where δ is the Dirac delta. Invoking the Dirac delta properties, the bank of M filters is The bank filter of M filters is a series of rect functions where ν FPm is the central position and ν BG is its bandwidth.
The digital demodulation consists of two phases: calibration and measurements. In the calibration, there are four steps: (4) we calculate its complex conjugate R * m (ν) where * indicates a complex conjugate. In the measurement, there are seven steps: (4) the relative phase ϕ rel is calculated, (5) the ambiguity 2πP is eliminated and then absolute phase ϕ abs is calculated, (6), the Bragg wavelength shift is computed, and (7) a digital adaptive filter is applied [37].
Due to the presence of the noise in the original signal, the calculated phase will be fluctuating. To minimize the noise influence and provide the best estimate, the absolute phase is multiplied with a set of coefficients. Those coefficients act as an adaptive filter. Figure 3 illustrates the digital demodulation schematically.

Calibration Meassurements
Calibration Measurements

Parameters and Results
To test and compare our theoretical analysis, we performed a numerical simulation of a quasi-distributed sensor based on low-finesse Fabry-Perot interferometers. Three Fabry-Perot

Parameters and Results
To test and compare our theoretical analysis, we performed a numerical simulation of a quasi-distributed sensor based on low-finesse Fabry-Perot interferometers. Three Fabry-Perot sensors were simulated. Their physical parameters can be observed in Table 1. Discrete spectrums were simulated using the physical parameters. Noise was simulated by adding to those samples pseudorandom numbers with Gaussian distribution; the interval was from √ SNR = 10 0 to √ SNR = 10 4 . Typical of Bragg gratings with rectangular profiles, a refractive index modulation was used. In most of our numerical experiments, the number of samples was equal to 1024 (Fast Fourier transform algorithm was considered). For each local sensor, the reference spectrum and 50 measurements were simulated. The measurements were in the intervals of S1 → 0 to 0.2 nm, S2 → 0 to 0.4 nm, and S3 → 0 to 0.7nm. Figure 4 shows the spectrum R T (λ), Figure 5 shows the spectrum R T (ν), and Figure 6 presents our numerical results: Demodulation errors vs SNR 1/2 . A Laptop Toshiba 45C was used, with 512 Mb of RAM memory and a velocity of 1.7 GHz. Table 1. Quasi-distributed sensor parameters.

Sensor Number Sensor Parameters Signal Values
Low-finesse Fabry-Perot interferometer 1 (S1)        From Tables 1 and 2, the simulated quasi-distributed sensor satisfies the instrumentation and signal requirements. Observing Table 1 and Figures 4 and 5, numerical results are in concordance with the theory. Thus, we confirm our theoretical analysis. Our numerical results can be observed in Figure 6. The theoretical analysis and our numerical results are in concordance with experimental results presented by Shlyagin et al. [30]; frequency-division multiplexing can be implemented based on a twin grating sensor. The presented study optimizes significantly the quasi-distributed sensor implementation and the sensibility of local sensors. To develop the sensing system based on the frequency-division multiplexing (Figure 1), the broadband light source can have the If the OSA spectrometer has ∆λ = 10 pm (typical value), the quasi-distributed sensor will have its limits as Table 2 illustrates. From Tables 1 and 2, the simulated quasi-distributed sensor satisfies the instrumentation and signal requirements. Observing Table 1 and Figures 4 and 5, numerical results are in concordance with the theory. Thus, we confirm our theoretical analysis. Our numerical results can be observed in Figure 6. The theoretical analysis and our numerical results are in concordance with experimental results presented by Shlyagin et al. [30]; frequency-division multiplexing can be implemented based on a twin grating sensor. The presented study optimizes significantly the quasi-distributed sensor implementation and the sensibility of local sensors. To develop the sensing system based on the frequency-division multiplexing (Figure 1), the broadband light source can have the following parameters: a central wavelength of λ c = 1532.5 nm, λ min = 1520 nm, and λ max = 1570 nm. The low reflectivity eliminates the cross-talk noise and its value can be selected from the references [29,38]. Figure 6 shows the behavior of Demodulation errors vs signal-to-noise rate SNR 1/2 . If the demodulation error is denominated resolution, then low-finesse Fabry-Perot has two resolutions: low resolution and high resolution. Two resolutions are possible because the FDPA algorithm does two evaluations of the Bragg wavelength shift [27,37]. All Fabry-Perot sensors have similar low resolution, however, each local sensor has its own high resolution. The high resolution depends of cavity length. If the cavity length is bigger than the Fabry-Perot sensor, it will have better resolution.

Discussion
Based on our theoretical analysis and numerical simulation, the quasi-distributed sensor would be built on the low-finesse Fabry-Perot interferometer. Our theoretical analysis optimizes its implementation. Instrumentation, local sensor properties, noise (Gaussian distribution), and signal processing were considered. The quasi-distributed sensor has good sensitivity and excellent resolution. All Fabry-Perot sensors have two resolutions: low resolution and high resolution (See Figure 6). Low resolution was obtained when the Bragg wavelength shift was evaluated with an enveloped function. High resolution was obtained when the Bragg wavelength shift was evaluated combining the enveloped and modulated functions [27,37].
When the noise is big, signal-to-noise ratio (SNR) is small. In this case, the FDPA algorithm cannot evaluate the Bragg wavelength shift, causing the transition from high resolution to low resolution. This can be observed in Figure 6. As the (necessary) signal is within the interval of -π to π, and based on the signal detection theory, the thresholding value is where σ env is the low resolution (resolution by enveloped function) and ∆λ FPm = 1 ν FPm is the period of our frequency component. The threshold divides between low and high resolutions. Substituting Equation (4) into Equation (32), we have From Equation (33), each low-finesse Fabry-Perot interferometer has its own thresholding value. This one depends on the cavity length, Bragg wavelength, and refraction index. For example: our Fabry-Perot sensors have next thresholding values, S1 → 0.033 nm, S2 → 0.016 nm, and S3 → 0.008 nm. The thresholding value is smaller if the cavity length is bigger.
In the quasi-distributed sensor, ghost interferometers are eliminated if the separation between any two interferometers satisfies the expression L sp > L FPM , where L sp is the spatial resolution. If Fabry-Perot interferometers are formed by uniform unapodized gratings with equal length L BG , the bandwidth of each peak is given by Equation (9). To be separated in the frequency domain, two peaks should not overlap. This condition imposes the following constraints: the minimum distance between centers of gratings for the shortest interferometers is 2L BG , and the difference in the cavity lengths of any two Fabry-Perot interferometers must exceed 2L BG .
Our future research work is in the following direction: wavelength-division multiplexing (WDM) can be implemented based on the low-finesse Fabry-Perot interferometers. The theoretical resolution is another direction. Technical applications are possible, for example: temperature, strain, humidity, force measurement, and oil detection.

Conclusions
The quasi-distributed optical fiber sensor based on the low-finesse Fabry-Perot interferometer was studied theoretically and simulated numerically. Theory and simulation are in concordance. Our study considers quasi-distributed sensor properties, local sensor properties, signal processing, noise source, frequency-division multiplexing, and instrumentation. Our numerical results showed that all Fabry-Perot sensors have two resolutions: low resolution and high resolution. Low resolution is similar for all sensors, however, each Fabry-Perot sensor has its own high resolution. The thresholding value (from high resolution to low resolution) was defined in terms of low resolution and physical parameters.
The quasi-distributed sensor has potential industrial applications, for example: structure monitoring, security system, humidity sensing, and level sensing.