## Abstract

In this paper, a strain-insensitive temperature sensor based on a dual polarization fiber grating laser is demonstrated. The laser is fabricated by inscribing two wavelength-matched Bragg gratings in an Er-doped fiber. It emits single-longitudinal-mode output in wavelength domain and generates a RF-domain signal as a beat note between the two polarization modes. A temperature sensor has been exploited by monitoring the beat frequency. The measured temperature sensitivity is −78.46 kHz/°C. Theoretical analysis suggests that the temperature response is a result of both the differences in thermo-optic coefficient and thermal expansion between the core and cladding. In contrast, the sensor is almost insensitive to applied axial strain. We found that the strain insensitivity is due to the compensation between the strain-induced birefringence change and the effect of the elongation/material index change. The proposed sensor can be applied for reliable and precise measurement of temperature independently, towards the applications in structural integrity, oil-well monitoring, aerospace engineering, and process control.

© 2012 OSA

## 1. Introduction

In the past few decades, fiber Bragg gratings (FBGs) have been exploited as photonic sensors for the measurement of a number of physical and chemical parameters [1]. For the FBG sensors, cross sensitivity between different measurands is a disadvantageous effect which decreases the measurement accuracy for practical applications [2]. In order to overcome this problem, two fiber gratings with different responses have been employed to discriminate the applied strain and temperature change [3, 4]. Individual parameters can be resolved by use of a coefficient matrix. Alternatively, a single grating with complex longitudinal/transverse structure or a specially designed package can be used [5–7]. An additional optical structure, such as an in-line interferometer, can be formed to integrate with a FBG for the discrimination [8, 9]. However, it is more favorable to measure temperature and strain or other parameters independently, to reduce the required number of sensors for practical applications. Based on the coupled mode theory, it is known that an ordinary FBG has positive temperature sensitivity due to the thermal expansion and thermal-optic effect. As to the strain response, the decrease in material refractive index partially compensates the effect of grating elongation but the strain sensitivity cannot reach zero [10]. The typical temperature and strain sensitivities are ~10 pm/°C and ~1 pm/με, respectively. To realize temperature-independent strain measurement, the FBG can be bonded or embedded in a material with a negative temperature response [11]. However, it is quite difficult to implement a strain-insensitive temperature FBG sensor, which is useful towards the applications of structural integrity, oil-well monitoring, aerospace engineering, and process control. Other type of fiber optic sensors for strain-insensitive temperature measurement has been reported, e. g., by monitoring the fluorescence intensity of a Yb-doped fiber, but these sensors cannot meet the multiplexing requirement for many applications [12].

Fiber grating laser sensor is a category of FBG sensor which generates a very narrow signal in spectral width and offers high-resolution measurement. The measurement can be carried out by detecting optical wavelength change or beat frequency between the two orthogonal polarizations. The latter one, i. e., the heterodyning fiber grating laser sensors, which converts external perturbations into the change in radio frequency (RF) domain signal, has received great interest in the recent years because it can measure extreme weak signals and the interrogation module is relatively cheap [13, 14]. In this paper, we demonstrate strain-insensitive temperature sensing based on this kind of sensor. A single-longitudinal-mode dual-polarization fiber Bragg grating laser is fabricated by inscribing two wavelength-matched FBGs into a Er-doped fiber. Temperature sensing can be performed by measuring the beat frequency and the sensitivity is −78.46 KHz/ °C. In contrast, the sensor is almost insensitive to axial strain. Theoretical analysis suggests that the temperature response is a result of the core/cladding difference in thermal expansion and thermal-optic coefficient. The strain sensitivity is almost zero, because the strain-induced birefringence change can fully compensate the effect of grating elongation/material-index change. Different from the wavelength-encoded FBG sensors, the proposed sensor can be intrinsically insensitive to applied strain, by optimizing the geometric structure or selecting thermal/elastic parameters of pure and doped silica glass and can be potentially used in many applications.

## 2. Experiment

Figure 1
shows the experimental setup for strain-insensitive temperature sensing. The employed active fiber is an Er-doped fiber. The core and cladding diameters are 3.05 μm and 125.6μm, respectively. The numerical aperture (N. A.) is 0.23. The core ellipticity is 97.5%, measured by the manufacturer. The Er-doped fiber has a peak absorption of 11 dB/m at 980 nm. The fiber grating laser is fabricated by directly inscribing two wavelength-matched Bragg gratings in the Er-doped fiber with a 193 nm excimer laser and a phase mask. Hydrogen loading or other photosensitization treatment is not needed, because the 193 nm ultraviolet exposure induces a two-photon excitation process over the fiber core, which enables highly efficient grating inscription. The beam-scanning method is used to reach a good reflective-wavelength match, and also to narrow the reflection band for the purpose of suppressing possible lasing at unwanted frequency. The laser cavity has a length of 2 cm, containing a 7.2-mm-long high-reflectivity grating, a 6.8-mm-long grating with lower reflectivity, and a 6 mm grating spacing. A 980 nm pump is launched into the Er-doped fiber laser through a wavelength division multiplexer (WDM). A photodetector (PD) and a RF spectrum analyzer are used to monitor the RF spectrum of the beat note. Figure 2
shows the measured RF spectrum of the beat signal. The beat frequency is 368.2 MHz and the signal-to-noise ratio is higher than 50 dB. The beat frequency Δ*ν* is determined by [13, 14]

*c*is the speed of light in vacuum,

*λ*

_{0}denotes the laser wavelength,

*n*

_{0}= (

*n*

_{s}+

*n*

_{f})/2 represents the average effective index and

*B*=

*n*

_{s}-

*n*

_{f}is modal birefringence,

*n*

_{s}and

*n*

_{f}are the effective indexes of the Er-doped fiber along the slow and fast axis, respectively (

*n*

_{s}>

*n*

_{f}). The intra-cavity birefringence

*B*is estimated to be 2.7 × 10

^{−6}based on Eq. (1). The intra-cavity birefringence is mainly determined by the intrinsic birefringence of the active fiber, and can be somewhat changed by the ultraviolet illumination. In the inset of Fig. 2, the output spectrum of the fiber grating laser is shown when the pump power is 115 mW. The lasing wavelength is 1529.87 nm.

The temperature and strain responses of the sensor have been tested. Figure 3 shows the measured temperature response of the proposed sensor, within the range from 20 °C to 100 °C with a step of 10 °C. It was tested when the applied axial strain is 0 and 400 με, respectively. The result indicates that the temperature response can be hardly affected by the applied strain. The average temperature sensitivity is −78.46 kHz/ °C. Considering the resolution of the RF spectrum analyzer of 1 kHz, the ideal resolution of the temperature sensor is 0.012 °C.

Figure 4 shows the measured strain responses of the sensor at 20 °C and 50 °C, respectively. The strain response is measured by elongating the sensor by use of two linear stages. The beat frequency is almost insensitive to the applied strain. The beat frequency fluctuations are ± 0.13 MHz and ± 0.12 MHz at 20 °C and 50 °C, respectively.

## 3. Theoretical analysis

To better understand the temperature and strain responses, it is necessary to clarify how the intrinsic birefringence in an optical fiber is formed. According to [15], in the fiber drawing process, a certain ellipticity of fiber core could be introduced due to the technical limitation. As a consequence, the geometric imperfection induces a modal birefringence *B*_{G}, which is defined as “geometric birefringence” in the text. The amplitude of *B*_{G} apparently relies on the fiber core ellipticity. In addition, when the fiber is cooled down from the melting temperature, a thermal stress is induced at the core/cladding interface, due to the mismatch in thermal expansion. As a result, an additional birefringence *B*_{S} is introduced owing to the structural imperfection, defined as “thermal-stress birefringence”. The amplitude of *B*_{S} is relevant with multiple factors, including the fiber core ellipticity, the melting temperature, and the thermal and elastic parameters of the fiber glass. We have set up a model and carried out a mode solving process by use of a finite-element-method (FEM) software. In the calculation, the fiber core diameter is 3.05 and 3.126 μm along two orthogonal polarizations. The refractive indexes of the core and cladding are 1.462 and 1.444, respectively. The calculated geometric birefringence *B*_{G} is 1.88 × 10^{−6}. The result suggests that the intrinsic birefringence 2.7 × 10^{−6} of the Er-doped fiber is mainly determined by the geometric imperfection, and the thermal stress also makes a considerable contribution.

Considering that the lasing wavelength equals the reflective wavelength of the fiber gratings, substituting the phase matching condition ${\lambda}_{0}=2{n}_{0}\Lambda $ into Eq. (1), the beat frequency can be expressed by

where Λ is the grating pitch. Equation (2) indicates that when the fiber laser is subjected to external perturbations, the beat frequency shifts as a result of the changes of the modal birefringence*B*, the effective index

*n*

_{0}and grating pitch Λ. Changes of

*n*

_{0}and Λ can be simply characterized by thermo-optic/elastic-optic coefficient and fiber elongation, respectively. The analysis of birefringence change, however, is much more complicated, because both the geometric and thermal-stress birefringence could change and the overall effect cannot be simply considered as the sum of the two contributions. However, we can independently investigate the two effects to figure out which factors affect the sensitivities. The elastic and thermal parameters are not given in full detail by the manufacturer but reasonable parameters used in previous reports can be employed in our analysis. Note that the core/cladding difference in elastic and thermal properties can make considerable contribution to the sensitivities, and the fiber cannot be considered as a homogeneous material as described in [16]. In the following text, the temperature and strain responses are theoretically investigated, respectively.

#### A. Temperature response

Deduced from Eq. (2), the temperature sensitivity can be expressed by

*α*and

*β*represent the thermal expansion coefficient and thermo-optic coefficient of silica glass, respectively. The item (

*α +*2

*β*) depicts the effects of fiber elongation and the refractive-index change of silica glass. The item $\frac{1}{B}\frac{\delta B}{\delta T}$ denotes the normalized birefringence change induced by temperature variation.

The effects of changes in *B*_{G} and *B*_{S} on temperature sensitivity are investigated, respectively. The former one is first calculated, regardless of the thermal stress. In the calculation, the thermal expansion coefficient is 5 × 10^{−7} /°C, which is uniform over the whole fiber to avoid inducing any thermal stress. The thermo-optic coefficient of silica glass, i. e., the fiber cladding, is 7.5 × 10^{−6} /°C. The geometric-birefringence change mainly comes from the difference in thermo-optic coefficient between the core and cladding, considering that the thermal expansion is rather small. As described in [17], the doped core has a smaller thermo-optic coefficient than the cladding. We thereby define a normalized difference in thermo-optic coefficient *η* = (*β*_{co}-*β*_{cl})/*β*_{cl}. Then mode solving is performed by used of the FEM software with the composed model. The fundamental-mode indexes and the modal birefringence are calculated at the original temperature *T*_{1} and *T*_{2} = *T*_{1} + Δ*T*, respectively. The index distribution at *T*_{2} over the fiber cross section is determined by *β*_{co}, *β*_{cl}, and Δ*T*. With the calculated change in birefringence, the temperature sensitivity can be obtained based on Eq. (3). The sensitivity is then calculated for different *η,* by repeating the above process. Figure 5(a)
shows the calculated temperature sensitivity as a function of *η*. The temperature sensitivity is negative because the geometric birefringence decreases with temperature as a result of the difference in thermo-optic coefficient. A larger difference results in a higher temperature sensitivity. The calculated sensitivity reaches ~-70 kHz/°C for *η* = −30%, according to the calculated result.

The above calculation suggests that the intrinsic birefringence is partially induced by the thermal stress, which indicates that the fiber core is in compression at room temperature and therefore the cladding has a larger thermal-expansion coefficient than the core. When the environmental temperature rises, the thermal stress partially releases because the temperature is closer to the melting temperature. Therefore, the decrease in birefringence leads to a negative temperature sensitivity. Now the effect of change in thermal-stress birefringence is calculated in detail. Similarly, a normalized difference is defined as *κ* = (*α*_{co}-*α*_{cl})/ *α*_{cl}. By use of FEM computing, the temperature sensitivity is calculated with the following steps: First, calculate the effective indexes for the fundamental mode along the two orthogonal polarizations and thus the mode birefringence at the original temperature *T*_{1} with the mode solver. Second, calculate the thermal stress change over the fiber cross section at temperature *T*_{2} = *T*_{1} + Δ*T*, taking the fiber core ellipticity and the thermal-expansion coefficients into account. A new index distribution over the fiber cross section is determined, as a consequence of the elastic-optic effect. Third, calculate the corresponding birefringence for the “heated” fiber. Substituting the calculated birefringence change into Eq. (3), the temperature sensitivity can be obtained. In the calculation, we assume the birefringence changes linearly with temperature, which is reasonable for temperature under 350 degrees Celsius [15]. The thermo-optic coefficients in the core and cladding are equally 7.5 × 10^{−6} /°C. The difference in elastic properties between the core and cladding, including the Young’s modulus and Poisson ratio, is not taken into consideration. Figure 5(b) shows the calculated temperature sensitivity independently induced by the thermal stress as a function of *κ*. The calculated sensitivity is negative, in accordance with our predication. A higher sensitivity is resulted with higher difference *κ*. The calculated sensitivity induced by thermal-stress relaxation can be over −80 kHz/°C for *κ* = −30%.

The analysis in this subsection suggests that the temperature sensitivity can be a result of changes in both geometric and thermal-stress birefringence, which are induced by the differences in thermal-optic coefficient and thermal expansion coefficient between the core and cladding, respectively. Both the two kinds of birefringence decreases with temperature and therefore the temperature sensitivity is negative and cannot reach zero.

#### B. Strain response

The strain response is also analyzed by investigating the effects of geometric and thermal-stress birefringence changes. The strain sensitivity can be derived from Eq. (2) as

*p*

_{e}= 0.22 is the effective elastic-optic coefficient to characterize the index change of silica glass and “1” represents the elongation. The item $\frac{1}{B}\frac{\delta B}{\delta \epsilon}$ is the normalized birefringence change induced by the axial strain.

To calculate the geometric-birefringence change, the normalized difference of *p*_{e} between the core and cladding is defined as *γ* = (*p*_{e,co}-*p*_{e,cl})/*p*_{e,cl}. The parameter *p*_{e} measures the index change rate with applied strain. This parameter relates to several factors including material stress-optic coefficient tensor and Poisson ratio, and we cannot determine whether the core has a larger or smaller *p*_{e} than the cladding. The geometric birefringence of the strained fiber changes as a result of the difference in elastic-optic coefficient. The modal birefringence is calculated for unstrained fiber and strained fiber with 1000 με, respectively, by use of the FEM software. The cross-sectional index distribution of the strained fiber is determined by *p*_{e,co}, *p*_{e,cl} and applied strain. The strain sensitivity for different *γ* can be calculated, based on Eq. (4). Figure 6(a)
shows the calculated strain sensitivity as a function of *γ*. For γ<0, i. e., the core has a smaller elastic-optic coefficient than the cladding, the index step increases under axial strain and causes an increment in geometric birefringence. As a result, it is possible to compensate the effect depicted as (1-2*p*_{e}) and obtain zero strain sensitivity. According to the calculated result, the strain sensitivity is zero for *γ* = −1.55%, i. e. *p*_{e,co} = 0.2166, as a result of the compensation.

When a material is stretched in one direction, it usually tends to shrink in the other two directions perpendicular to the direction of stretch. The ratio of transverse strain and axial strain is defined as Poisson ratio *μ*. For silica glass *μ* = 0.17. This parameter relies on material composition and can be different for the core and cladding. That is to say, for a strained fiber, the core and cladding shrink with different rates over the cross section and thereby an additional stress is formed at the core-cladding interface. The induced cross-sectional stress distribution can be first calculated, and the corresponding index distribution is thereby determined by the elastic-optic effect. The birefringence can be calculated through the mode solving process for unstrained and strain fiber, respectively, and the strain sensitivity can be obtained based on Eq. (4). Figure 6(b) shows the calculated strain sensitivity induced by the additional stress change as a function of τ = (*μ*_{co}-*μ*_{cl})/ *μ*_{cl}. The result suggests that a zero strain sensitivity can be achieved when *μ*_{co} is a slightly larger than *μ*_{cl}.

In this subsection, the calculated result shows that the effect of fiber elongation and material-index change expressed as (1-2*p*_{e}) in Eq. (4) can be compensated by the strain-induced birefringence change. As a result, a zero strain sensitivity can be obtained in the experiment.

In addition, positive strain sensitivities have been observed for Er/Yb co-doped or other types of fibers in the previous reports [18] and [19]. Based on the calculated result in Fig. 6, we can determine that the strain sensitivity mainly comes from the difference in Poisson ratio. The core has a smaller Poisson ratio than the cladding, which induces an additional compression subjected to the core and thereby leads to an increment of birefringence.

## 4. Conclusion

In this paper, a strain-insensitive temperature sensor has been demonstrated by use of a dual-polarization fiber grating laser. Temperature sensing is carried out by measuring the beat frequency between the two orthogonal polarization states generated by the laser. The sensor is found insensitive to axial strain. Detailed theoretical investigation has been performed to explain the temperature and strain responses. Both the geometric birefringence and the thermal-stress one can change by temperature and axial strain and make contribution to the sensitivities. We found that the birefringence decreases with temperature and results in a negative sensitivity, which is in accordance with the measured result. In contrast, the strain-induced birefringence change can compensate the effect of the fiber elongation and absolute material-index change. As a result, zero strain sensitivity can be obtained. Compared with the conventional FBG sensors, the proposed sensor can be intrinsically insensitive to axial strain and therefore can find potential applications in smart structure and aerospace engineering.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (60736039 61177074, and 11104117) and the Fundamental Research Funds for the Central Universities (21609102).

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