## Abstract

Aiming at realizing a static strain sensor of nano-strain resolution, which is required in most geophysical applications, this paper presents a thorough analysis on the strain resolution of a fiber Bragg grating (FBG) static strain sensor interrogated with a narrow linewidth tunable laser. The main noise sources of the sensor are discussed, and the strain resolution is deduced with a cross-correlation algorithm. The theoretical prediction agrees well with our experimental result, and the analysis is further validated by numerical simulations. Based on the analysis, the paper provides the guidelines for optimizing this type of sensor to realize ultra-high resolution. It is shown that with properly designed FBGs and interrogation systems, nano static strain resolution can be realized, as we recently demonstrated in experiment.

©2011 Optical Society of America

## 1. Introduction

Optical fiber sensors based on fiber Bragg grating (FBG) have been widely used in the measurement of parameters such as strain and temperature, which can be transferred into the shift of the FBG’s Bragg wavelength. These type of sensors have the well-known advantages such as light weight, compact size, immunity to electromagnetic interference and multiplexing capability, and have been widely adopted in scientific and industrial applications [1,2]. For most of these applications, a strain resolution of 1 micro-strain (με) is generally satisfactory, while for certain applications in geophysical research and oil industry, a much higher strain resolution down to nano-strain (nε) over large dynamic range at static to low frequency domain (DC~100 Hz) is required. Although several FBG dynamic strain (acoustic) sensors have been reported realizing even better than nano-strain (nε) sensitivity at high frequency (kHz) region [3,4], it is much more difficult to realize a high resolution for static strain sensors suffering from low frequency drift and environmental disturbance. A dynamic sensor can be self-referenced, but a static strain sensor has to be compared with an extra standard which is usually a frequency-stabilized component or an identical sensor head but free of strain. Meanwhile, a strain resolution of 1.2 nε at 1.5 Hz was reported by using a frequency-locked laser [5], and a strain resolution of sub-nε at the frequency of 1-6 Hz was obtained with the so-called Pound-Drever-Hall technique and a molecular absorption as frequency referencing [6]. However, the dynamic range of above sensors is quite small, limited by the linear range of the slope either in FBG’s reflective spectrum or in the molecular absorption. Recently, we developed a static strain sensor interrogated with a commercially available narrow linewidth tunable laser. High resolution down to 2.6 nε has been demonstrated experimentally [7], and good linearity over large dynamic range is well guaranteed by its principle. The sensor provides enough resolution and dynamic range for the monitoring of crustal deformation induced by oceanic tide.

In this paper, we present a thorough analysis on the static strain resolution of the FBG sensor, which employs a narrow linewidth tunable laser source and a cross-correlation algorithm for interrogating the strain-induced Bragg wavelength shift. The influence of major noise sources (or resolution deterioration factors), such as the wavelength inaccuracy of the laser source, the intensity noise, the shape of FBG’s reflective spectrum, etc., on the strain resolution are analyzed, and the expression of strain resolution is derived. Our analysis gives guidelines for the design and the optimization of the FBG sensor to realize ultra-high static strain resolution, and shows that nano static strain sensing can be realized with an FBG sensor if it is designed properly. The theoretical prediction is verified by numerical simulations, and agrees well with our experimental result. This work opens the way to utilize FBG sensors in applications requiring static nano-strain resolution as well as large dynamic range.

## 2. Typical system configuration

The typical configuration of a high-strain-resolution FBG sensor system is shown in Fig. 1
. The system consists of a pair of identical FBGs; one is for the sensing of strain and the other is strain-free working as a reference. The two FBGs are mounted to experience the same temperature change. As a result, the relative Bragg wavelength change indicates the strain information. A narrow linewidth tunable laser is used to interrogate the FBGs, and the lightwaves reflected from the FBGs are detected by photo-detectors. While the wavelength of the tunable laser sweeps over the principal peaks of the FBGs’ reflective spectra, their reflectivity are sampled at a discrete sequence of wavelengths *λ _{i}* with a step of

*dλ*. The reflectivity of the sensing FBG is labeled as

*R*(

*λ*), while the reflectivity of the reference FBG is labeled as

_{i}*R*(

_{R}*λ*). Assuming the two FBGs have the same spectra shape when the sensing FBG experiences strain variation, which is widely accepted, the reflectivity satisfies:

_{i}*λ*is the wavelength shift caused by strain.

*ε*= Δ

*λ*/

*dλ*is the corresponding index shift. A certain algorithm is then employed to demodulate the wavelength shift Δ

*λ*by comparing the two spectra.

## 3. Major noise sources

Generally, the measured reflectivity of an FBG suffers from the intensity noise of the laser and the photo-detector and from the wavelength inaccuracy of the tunable laser. These errors are the origin of the strain resolution deterioration.

#### 3.1. Relative intensity noise

The calculated reflectivity is the ratio of the measured optical power reflected from the FBG and the incident optical power. The intensity noise in the measurement of optical power includes the shot noise of the light, the electronic noise of the photo-detector, and the quantization noise in the A/D conversion process. Furthermore, the wavelength sweep processing of the tunable laser is often accompanied with respectable intensity variation. All above noises except the quantization noise are influenced by the integration time. For example, Fig. 2
illustrates the intensity noise level of a photo-detector (Agilent, 81635A) as a function of the integration time. Longer integration time can reduce the intensity noise level, but the sensing speed slows down simultaneously. In this paper, the divergence of the measured reflectivity caused by above intensity noises is labeled as Δ*R*_{elec}.

#### 3.2. The wavelength repeatability of the tunable laser

The definition of the wavelength repeatability is the random uncertainty of the logged wavelength of the tunable laser source in repeated sweeps. The wavelength repeatability is noted as *δ*_{1}*λ*. Since the reflectivity of the FBG is a function of the wavelength, *δ*_{1}*λ* is converted into an error in reflectivity. The converting coefficient is the differential coefficient of the FBG’s spectrum, as shown in Fig. 3
. This error in reflectivity is expressed as *R*'(*λ*)·*δ*_{1}*λ*, where *R*'(*λ*) is the differential coefficient of the FBG’s spectrum. Generally *δ*_{1}*λ* is much smaller than the absolute wavelength inaccuracy of the laser. Because the Bragg wavelength is determined by comparison between logged spectra, the absolute wavelength inaccuracy is of less importance.

#### 3.3. Linewidth of the tunable laser

The linewidth of the tunable laser *δ*_{2}*λ* describes the deviation range of quick frequency jitter of the laser. Similar to the wavelength repeatability, the quick frequency jitter is also converted into the error in the measured reflectivity. The amplitude is not larger than *R*'(*λ*)·*δ*_{2}*λ*, as it is averaged to some extent during the integration time.

#### 3.4. Wavelength stability of the tunable laser

The wavelength stability of the tunable laser describes the slow wavelength drift of the laser over long term. This drift causes large error if it is not appropriately compensated. In the proposed high-resolution static strain sensor, the strain information is calculated from the spectral difference between two identical FBGs. This configuration compensates the temperature induced Bragg wavelength shift and the influence from the slow wavelength drift of the laser source at the same time.

In summary, the total noise in the measured reflectivity of the FBG can be expressed as:

where $\delta \lambda =\sqrt{{\left({\delta}_{1}\lambda \right)}^{2}+{\left({\delta}_{2}\lambda \right)}^{2}}$ is the total wavelength inaccuracy.## 4. Cross-correlation algorithm

In our analysis a cross-correlation algorithm is employed to demodulate the wavelength shift Δ*λ* from the measured spectra, because this algorithm is proved to have very good resolution compared with other algorithms such as the centroid detection algorithm (CDA) and the least square fitting algorithm (LSA) [7–9]. Once the spectrum of the FBG free of strain and that of the FBG under strain are recorded, the cross-correlation product is calculated to determine the Bragg wavelength difference between the two spectra:

In Eq. (3), it is assumed that both *R*(*λ _{i}*) and

*R*

_{R}(

*λ*) are equal to zero if the indices lie outside their ranges. This assumption is acceptable as long as the sampling range covers the whole principal peaks of both FBGs.

_{i}*C*(

*j*) has the maximum at

*j*=

*ε*, where the two spectra overlap completely and

*ε*is demodulated from the index when

*C*(

*j*) is at its maximum. Due to the random errors in the measured reflectivity, the retrieved

*ε*deviates from Δ

*λ*/

*dλ*, and the deviation range is the resolution of the sensor. It should be mentioned that although the index of maximum

*C*(

*j*) falls into an integer which is the nearest to

*ε*,

*ε*can be precisely calculated either by interpolation or by curve-fitting around the maximum.

Because of the random noise described in section 3, the measured spectrum is the sum of real spectra and the random noise. The correlation product is then modified as:

*ε*= 0 is assumed to simplify the analysis without any influence on the resolution.

The first term *C _{R}*(

*j*) in Eq. (4) is the auto-correlation curve of the real spectrum without noise, as shown in Fig. 4 (the black dashed line). It has a peak (labeled P) at

*j*= 0 and declines when

*j*diverges from 0. The second term

*C*(

_{N}*j*) is the cross-correlation of the real spectrum and the random noise. It makes the actual cross-correlation curve

*C’*(

*j*) fluctuate, as shown as the red line in Fig. 4. Because only the relative amplitude is concerned with the position of the maximum, we shift

*C’*(

*j*) to pass through point P, the peak of

*C*(

_{R}*j*), as shown as the dashed red line in Fig. 4. The random noise makes the shifted curve fluctuate around the auto-correlation curve. The deviation is a function of the index

*j*:

*Δ*and

_{N,elec}*Δ*are the noises caused by the relative intensity noise and the wavelength inaccuracy, respectively.

_{N,λ}*Δ*has a mean of zero and a standard deviation as:

_{N}The gray zone in Fig. 4 shows the fluctuation range of the shifted cross-correlation curve. The peak of *C’*(*j*) appears in the region satisfying

The factor of 1/2 in Eq. (7) is because only the positive Δ_{N} (*j*) causes the peak deviating from *j* = 0 while the negative value does not. Once the expressions of *C _{R}*(

*j*) and

*σ*(

*Δ*(

_{N}*j*)) are obtained, the wavelength resolution can be calculated from Eq. (7).

## 5. Model of FBG’s spectrum

*N*-order Gaussian curve is employed as the model of the FBG’s spectrum for quantitative analysis:

*λ*=

*λ*

_{0}+

*i*·

*dλ*,

*dλ*is the wavelength sweep step and

*λ*

_{width}=

*w*·

*dλ*is the full width at 1/

*e*maximum of the FBG’s spectrum.

*n*is the order of Gaussian function, working as a shape parameter in the model. For FBGs with smooth spectra (maximum reflectivity less than 80%),

*n*is 1; for a high reflectivity FBG with a spectrum of a flat peak and sharp slopes,

*n*is a large integer. With this expression, we obtain following expressions around

*j*= 0:

As long as the sampling range covers the whole principal peak of the FBG, which is usually satisfied in experiments, above approximations describe the expression with good precision. While *w* varies from 16 to 1024, and *n* varies from 1 to 5, the approximation of Eqs. (9) and (10) has an error less than 1%. For Eq. (11), the error is less than 5% for *n* = 1 and a little larger when *n* ≥ 2. With above approximations we calculate the resolution with Eq. (7) as

*λ*is the wavelength resolution of the sensor,

_{R}*dλ*the wavelength sweep step of the laser,

*σ*(Δ

*R*) the standard deviation of the relative intensity noise,

*σ*(

*δλ*) the standard deviation of wavelength inaccuracy. From the wavelength resolution, the strain resolution can be simply deduced by the strain-wavelength coefficient of about 1.2 pm/με.

In our previous experiment, a static strain resolution of 2.6 nε was demonstrated using the same configuration described in this paper [7]. The experimental parameters used in the previous experiments are listed in Table 1 . With these parameters, the strain resolution of the sensor is theoretically calculated to be 2.3 nε using Eq. (12). This value is very close to the experimental result, showing the effectiveness of the theoretical analysis presented in this paper.

The analysis is further verified by numerical simulation using a LabVIEW program. Two sets of noises levels are used as simulation conditions as shown in Fig. 5 . Under each condition, the Bragg wavelength is calculated for 1000 times, and then the standard deviation (i.e., the resolution) is obtained and compared with that calculated from Eq. (12). The resolution by simulation is well described by the analytical result, proving that Eq. (12) is proper in describing the resolution of the sensor.

## 6. Guidelines for sensor design and optimization

Based on the above analytical result of the resolution given in Eq. (12), we discuss the guidelines for the sensor design and optimization, that is, how to design the system and choose the FBG, the tunable laser, the photo-detector, and the system configuration to realize nano-strain resolution.

#### 6.1. FBG spectrum

The first important issue is to select the FBG with suitable spectrum, including the shape of profile and the bandwidth. According to Eq. (12), the resolution of FBG sensors with different shape parameter *n* is shown in Fig. 6
. The 1st-order Gaussian shape FBG has better resolution with narrower bandwidth, which is more preferable in practice because it requires smaller wavelength tunable range, and then can achieve higher measurement speed. We can see from Fig. 6 that, with the given *dλ*, *δλ*, and Δ*R*, which are practical values for commercially available devices, nano-strain resolution is achievable with a proper FBG.

From Fig. 6 we also find that an optimized bandwidth exists to have best resolution. From Eq. (12) the optimized bandwidth is calculated when *n* = 1:

And the corresponding wavelength resolution of the sensor is:

Equations (13) and (14) show the achievable resolution of the sensor at given laser source and photo-detectors.

In practice, the optimized bandwidth of FBG may not be usable due to the limitation of the tunable range of the laser source or due to the limitation on measurement speed. We compared the resolution with FBGs of different bandwidths at fixed intensity noise level and wavelength inaccuracy, respectively. As shown in Fig. 7 , narrower bandwidth FBG is preferred to achieve higher resolution when using a laser source with good wavelength inaccuracy. On the other hand, a broad bandwidth FBG is suitable to make full use of the low-intensity-noise photo- detectors as shown in Fig. 8 .

#### 6.2. Sweep step of the laser source

From Eq. (12) we know that the sweep step *dλ* plays an important role in the resolution. The resolution is proportional to the square root of *dλ* at a fixed noise level. Using a shorter sweep step (a smaller *dλ* value) while keeping the integration time of the photo-detector constant produces the best resolution. It consumes, however, longer time to complete one sweep. If the sweep speed remains constant, on the other hand, using a shorter sweep step deduces the integration time at the photo-detector, which enhances the intensity noise level. As a result, the resolution is not necessarily improved. Therefore, the most suitable sets of parameters depend on a specific application’s requirements on the measurement speed and the resolution.

#### 6.3. Wavelength repeatability and linewidth of the tunable laser source

The total wavelength inaccuracy of the sensor is the sum of the linewidth and the wavelength repeatability of the tunable laser source. For most tunable lasers, the wavelength repeatability is much larger than their linewidth, and thus plays the dominant role in this type of sensors. As a result, the linewidth of the laser source is of less importance here compared with in wavelength-fixed laser based dynamic FBG strain sensors, because the Bragg wavelength is obtained from all of the sampled points of the FBG reflection spectrum integrally in this type of sensors so that the quick frequency jitter is averaged. In choosing a tunable laser source, more attention should be paid to higher wavelength repeatability instead of narrower linewidth, as long as using a single mode laser source normally with a linewidth of ~MHz.

## 7. Conclusion

It is well know that FBG sensors can be used for static strain sensing at a resolution of about 1 micro-strain as a bench mark. In this paper, we proved that a nano-order static strain resolution can be realized with FBG sensors if the sensor systems are designed properly.

We thoroughly analyzed the performance of FBG static strain sensors interrogated with narrow linewidth tunable laser sources, discussed the noise sources, and deduced an expression of the strain resolution of the sensors with the cross-correlation demodulation algorithm. The resolution calculated by using the expression is in good agreement with our experimental result, and is further verified by numerical simulation. Based on the analysis, we provide the guidelines for designing and optimizing this type of sensors to realize nano-strain resolution for static strain sensing, including the design and selection of the system configuration incorporating wavelength/temperature compensation, the cross-correlation demodulation algorithm, the spectrum profile of the FBG, the wavelength inaccuracy of the tunable laser sources, and the intensity noise level of the photo-detectors.

Based on the theoretical analysis and guidelines, we developed ultra-high resolution static strain sensors for geophysical applications, and currently we are employing them in monitoring oceanic tide induced crustal deformation.

## Acknowledgments

The authors thank Prof. Kazuo Hotate at the University of Tokyo, Japan, for helpful discussions and comments. This work is supported by the Strategic International Cooperation Program from Japan Science and Technology Agency (JST), the Grant-in-Aid for Scientific Research (A) from Japan Society for the Promotion of Science (JSPS), and the Global Center of Excellence (G-COE) Program from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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