Full length articleGlobal overview of the sensitivity of long period gratings to strain
Introduction
In 1978, Hill et al. [1] demonstrated that it was possible to induce a periodic modulation of the refractive index of the core of an optical fiber with light. This gave rise to intense research on optical fiber gratings, from which two principal structures emerged. The most commonly known is the fiber Bragg grating (FBG) [2]. The second is the long period grating (LPG) [3], [4], [5]. The period of modulation of FBGs is of the order of a few hundred nanometers, while the period of LPGs is of the order of a hundred microns.
This difference in modulation period leads to significant differences in their behavior. An FBG inscribed in a single mode fiber couples the incident core mode to the contra-propagative core mode. This means that the light is transferred by the grating from the mode propagating in the forward direction to the mode propagating in the backward direction. The degree of coupling depends on the wavelength. It is maximal for the resonant wavelength, called the “Bragg wavelength”. In consequence, the reflection spectrum of the grating exhibits a peak centered on the Bragg wavelength, while its transmission spectrum exhibits a hole around this wavelength.
An LPG couples the core mode to several co-propagative cladding modes. Again, each coupling is maximum for a resonant wavelength, which differs from one mode to another. Because of the coupling, light is injected in the cladding and finally flows out of the fiber due to inhomogeneities. The transmission spectrum then exhibits holes around all resonant wavelengths.
This property of an LPG makes it useful in sensors: any external influence which modifies the period of modulation or the refractive index of the fiber shifts the resonant wavelengths. The measurement of these shifts allows the strength of the influence to be retrieved. The influence of temperature, strain and the refractive index of the surrounding medium on the LPG has been widely studied (see for example [6], [7], [8]). However, the sensitivity of LPGs to strain remains an open question for two reasons. Firstly, LPGs are complex systems which contain several independent parameters. Most published studies consider a basic configuration and then vary each parameter individually. Results obtained in this way are valid for the original configuration but lack generality. Secondly, in all the studies the radial strain is assumed to be linked to the axial strain via Poisson's ratio. This is true if the fiber is fixed at two points on a monitored structure and free to deform in the transverse plane. But it can be completely wrong if the fiber is embedded in a host material [9], [10], where it is necessary to study the sensitivity of LPGs to axial and radial strain simultaneously. The aim of this paper is to remedy this gap, and produce results useful for the design of real-world strain sensors based on LPGs.
Using a statistical approach, we explored a parameter space using ten thousand random samples. Our methodology is explained in Section 2. Section 3 is devoted to the study of the cross-sensitivities, and presents new findings on the relationship between axial strain sensitivity and radial strain sensitivity. The influence of the various parameters is presented in Section 4. Section 5 presents the influence of the parameters on the coupling coefficient. We conclude with a presentation of some rules to optimize the sensitivity of LPGs to strain.
Section snippets
Parameter space
Each LPG has a characteristic periodic modulation of the effective index of the mode propagating in the core. This modulation can be obtained either by modifying the geometry of the fiber or by inducing a variation of the refractive index of the core with UV illumination. In the following, we will consider the latter case.
The fiber is made of two concentric cylinders: the core and the cladding (see Fig. 1). The radius of the core is a1 and the radius of the cladding is a2. The refractive index
Cross sensitivities
The aim of this section is to determine the influence of the axial strain on the sensitivity of the LPG to radial strain, and reciprocally. As explained above, for each configuration, the sensitivity of the resonant wavelength to radial strain, was calculated for different axial strains. We can then define the minimal sensitivity to radial strain:the maximal sensitivity to radial strain:the mean sensitivity to radial strain:
Influence of the parameters
We will now neglect the cross-sensitivities for the reasons explained in the previous section, and consider the sensitivity of to radial strain to be , and the sensitivity of to axial strain to be .
In the range of parameters used, [−20.6 pm/μϵ; −0.55 pm/μϵ] and [−0.3 pm/μϵ; 10.4 pm/μϵ]. Most of the values of are in the range −0.55 pm/μϵ to −2 pm/μϵ (see Fig. 9). The standard deviation of the series is 1 pm/μϵ. Most of the values of fall between −0.3 pm/μϵ
Coupling coefficient
The coupling coefficient κ is a critical parameter. For any given length of an LPG, the depth of the hole in the spectrum directly depends on the coupling coefficient, and hence determines the precision of measurement that is possible. In order to maintain uniform precision during the measurement process, the depth must remain constant, as must the coupling coefficient. It is then necessary to accurately know the influence of strain on the coupling coefficient.
Let us briefly examine the
Conclusion
In this paper we have presented the results of a statistical study on the influence of radial and axial strain on the cladding modes of an LPG. Physical considerations have allowed us to reduce the number of free parameters to four: the order of the cladding mode, the cladding radius, the grating period, and the refractive index of the surrounding medium. The characteristics of ten thousand gratings were randomly chosen in this parameter space. The gratings were submitted to axial and radial
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