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Fiber Bragg Gratings (FBGs) are among the most popular optical fiber sensors. FBGs are well suited for direct detection of temperature and strain and can be functionalized for pressure, humidity, and refractive index sensing. Commercial setups for FBG interrogation are based on white-light sources and spectrometer detectors, which are capable of decoding the spectrum of an FBG array. Low-cost spectrometers record the spectrum on a coarse wavelength grid (typically 78–156 pm), whereas wavelength shifts of 1 pm or lower are required by most of the applications. Several algorithms have been presented for detection of small wavelength shift, even with coarse wavelength sampling; most notably, the Karhunen-Loeve Transform (KLT) was demonstrated. In this paper, an improved algorithm based on KLT is proposed, which is capable of further expanding the performances. Simulations show that, reproducing a commercial spectrometer with 156 pm grid, the algorithm estimates wavelength shift with accuracy well below 1 pm. In typical signal-to-noise ratio (SNR) conditions, the root mean square error is 22–220 fm, while the accuracy is 0.22 pm, despite the coarse sampling. Results have been also validated through experimental characterization. The proposed method allows achieving exceptional accuracy in wavelength tracking, beating the picometer level resolution proposed in most commercial and research software, and, due to fast operation (>5 kHz), is compatible also with structural health monitoring and acoustics.

Established through the last decade as a prominent sensing technology, Fiber Bragg Gratings (FBGs) are gaining popularity as a sensing technology [

FBGs are compact and reflective optical filters, inscribed within an optical fiber, which reflect a single wavelength [

The latest research efforts are increasing the potential of FBGs, as a sensing approach. Improving over the traditional phase mask fabrication [

Over the last years, commercial interrogators have significantly improved their affordability, consolidating the white-light principle [

Several detection algorithms have been documented, for tracking FBG wavelength shifts in a white-light setup. The simplest approaches are curve fitting [

Recently, a demodulation technique for FBG sensors based on Karhunen-Loeve Transform (KLT) [

In this paper, the KLT algorithm for detection is revised and improved, through the analysis of the whole eigenvalue set, with the goal of improving the algorithm in two areas: (

The paper is arranged as follows. Section

An FBG is a periodic modulation of the refractive index of an optical fiber [

Equations (

Equation (

Equations (

Most FBG interrogators, based on commercial devices or self-assembled in research laboratories, are based on a white-light setup as in Figure

Schematic of a generic FBG interrogation based on white-light setup.

In order to make FBG sensing attractive and affordable, it is desirable to operate with inexpensive and compact detectors [

Spectrometers sample the wavelength with 9-bit (512 pixel) resolution, in most of the cases. This leads to a typical wavelength resolution of 78 pm or 156 pm, which is significantly inferior to optical spectrum analyzer (OSA) instruments that achieve a resolution bandwidth of 5–20 pm. On the other side, spectrometers have lower cost, over one order of magnitude lower than OSAs, have compact size and low power consumption, as they are usually powered via the USB port of a computer, and have faster response, up to 1–5 kHz using high-speed data transfer protocols such as RS-232 [

In the following simulations, we refer to the worst-case scenario, underlying the following hypotheses.

The wavelength axis is quantized over 512 values, from 1520 nm to 1600 nm, as in [

The amplitude axis is quantized over 32768 values, half of the available range of operation, in order to introduce a “safety margin” that avoids spectral saturation in case of power fluctuations.

We neglect the neighboring effect, between different FBGs composing the array. In other words, we assume that the spectral spacing between adjacent FBGs is sufficient to guarantee negligible interference. Thus, we can reduce the performance analysis to a single FBG and extend the results to all FBGs composing the array. This hypothesis is well realistic when it applies to small thermal/strain variations of an FBG, which is the case of interest for this paper, whereas variations will be limited to ±5 pm.

The reflection spectrum, prior to be quantized, is perturbed by additive white noise (AWN). The AWN provides in this case a worst-case scenario, because unlike

The SNR figure is defined as the ratio between the variance of the useful signal (i.e., the reflection spectrum of the FBG) and the noise variance.

The spectrum of the optical source is normalized.

The parameters chosen for simulation are

In these conditions, Figure

Spectrum of an FBG centered at 1550 nm, in absence of noise, generated by means of coupled mode theory [

The wavelength detection procedure is an algorithm that applies to the decoded spectrum, after the inner portion correspondent to the FBG has been isolated. We assume that the wavelength axis

As in [

Subsequently, the array

Then, we apply the KLT to the matrix

Calling

In the original implementation of the KLT, the highest rank eigenvalue

High-rank eigenvalue

In the small signal analysis, we are interested in detecting small wavelength shifts around the FBG central wavelength

In first place, we focus on the FFT operation in (

The Capon estimator, described in [

The PSD is computed for each

The second modification lies in the choice of the high-rank eigenvalue as a metric to estimate the wavelength shift. This choice, initially drawn in [

Figure

Comparison of different eigenvalue tracking approaches, in small FBG wavelength shift: after applying (

The choice of

To recap, Figure

Proposed algorithm for small signal decoding of FBGs based on KLT and Capon estimator.

In this section, the performance analysis of the FBG decoding method is outlined, using the simulation framework as in Section

Figure ^{−1}.

Variation of the second-rank eigenvalue

In this small signal analysis, the root mean square error (RMSE) between the wavelength shift and its estimate obtained with KLT can be evaluated. For

This concept is expanded for the evaluation of the accuracy. In Figure

Wavelength detection accuracy (99% interval of confidence) as a function of SNR. Data are reported for different values of

Considering

Response of the KLT detection algorithm to a 100-Hz sine wave.

An important factor for consideration is the computation time and how it relates to the length of the size

Accuracy and computation time as a function of

| Accuracy (pm) | Computation time ( |
---|---|---|

3 | 0.45 | 108 |

5 | 0.22 | 126 |

8 | 0.21 | 145 |

12 | 0.21 | 186 |

15 | 0.21 | 209 |

25 | 0.22 | 334 |

A brief experimental characterization has been set up, with the purpose of validating the prior performance analysis and showing that the KLT is effective in real-case scenarios. The setup in Figure

Figure

Wavelength shift, estimated in experimental measurements carried out with a commercial spectrometer, operating at 5 kHz. (a) shows the response to a 25 Hz exponentially damped tone, while (b) shows the response in absence of stimuli.

The results, in line with the previous simulative analysis, confirm that the KLT algorithm is successful in decoding small variations of wavelength shift, well below the picometer level, despite the coarse wavelength sampling. The algorithm for detection is also computationally fast, compatible with structural health monitoring and low-frequency acoustics, as well as static sensing.

FBGs are strongly increasing their penetration into sensing technologies, and among fiber-optic sensors they are one of the most popular approaches. The FBG market is consolidating and detection hardware is becoming more affordable. Furthermore, recent innovations such as draw-tower grating [

On the other side, in order to become attractive and competitive with mainstream sensing technologies, it is essential to reduce the cost of sensing equipment. Low-cost spectrometers are becoming more attractive, but they sample the FBG spectra on a coarse wavelength grid, which is unpractical to detect small shifts with precision. Most datasheets mark the wavelength detection accuracy to ~5 pm, when spectral sampling is 156 pm (512 pixels over 80 nm).

In this paper, an algorithm based on KLT is proposed with the aim of improving the detection of small wavelength shift. The FBG spectrum, as acquired by the spectrometer, is processed using a Capon estimator and then a KLT. As a result, sub-pm wavelength shift can be easily detected, with typical RMSE ranging from 0.02 to 0.2 pm, and subpicometer accuracy, depending on the SNR. The algorithm is fast, compatible with 5 kHz operation [

The author declares that there is no conflict of interests regarding the publication of this paper.