1. Introduction

Microstructured Optical Fibers (MOF) have been a subject of extensive research for over two decades [1] and applications in various fields of photonics have been found (i.e. supercontinuum sources [2], novel fiber lasers [3], fiber-optic communication [4] and optical fiber sensors [5]). One of the main features of MOFs, differentiating them from traditional optical fibers, is that their optical properties can be adjusted by modifying their geometry. For example dedicated geometries allow an ultrahigh birefringence to be reached, two orders of magnitude higher than in traditional highly birefringent fibers [6,7]. Also, the sensitivity in MOFs can be significantly improved in comparison to traditional optical fibers. The highly asymmetrical designs reported in [8] showed polarimetric sensitivity to hydrostatic pressure at a level of −43 rad/MPaˣm at 1.55 μm. Similar pressure sensitivities, but with a positive sign, have been also reached in microstructured polymer fibers [9]. It is noteworthy that even small variations of the fiber’s structural parameters (i.e. air hole size, lattice constant, filling factor, etc.) may lead to a significant change in its parameters (e.g. supercontinuum outcome [10], macrobend performance [11] or pressure sensitivity [12]).

While the influence of MOF geometry on sensitivity to pressure [8,12], temperature [13,14] or transversal strain [15] has already been reported, there is a limited number of reports on the influence of the microstructure on longitudinal strain sensitivity. In [16], Pang et al. investigated the fundamental mode’s phase sensitivity to longitudinal strain in commercially available hollow core and highly nonlinear photonic crystal fibers (PCFs). The reported results show that the normalized phase sensitivity to longitudinal strain in a hollow core fiber can be increased by approx. 20% in comparison to traditional all silica single mode fibers (SMF). On the other hand, the strain sensitivity in the investigated highly nonlinear PCF is comparable to standard SMF strain sensitivity, which is in agreement with what was reported in [17], where the strain sensitivities of three different types of hexagonal lattice PCFs were investigated.

As far as we are aware, our report is the first paper presenting a thorough analysis of the influence of an MOF’s air hole diameter on phase sensitivity of the fundamental mode to longitudinal strain. We show that while the axial stress component remains constant with air hole diameter change, the stress components along the axes perpendicular to the fiber are strongly affected by air hole size. What is more, we verify the numerical data with experimental results obtained for four dedicated fiber samples drawn from the same preform and differing in air hole sizes. Based on an all fiber Mach-Zehnder interferometer set-up we show that the fibers with larger air holes have significantly higher effective refractive index sensitivity to longitudinal strain, which is in agreement with the developed theoretical model.

2. Modelling of longitudinal strain in microstructured fibers

The normalized phase sensitivity (S) of a mode propagating in an optical fiber exposed to longitudinal strain (ε) can be given by Eq. (1) [16]:

To analyze the influence of the fiber’s microstructure geometry (i.e. air hole size and distribution) on n_eff sensitivity we designed a dedicated three air hole fiber presented in Fig. 1, which is a simplified version of the fiber previously reported for fiber Bragg grating inscription [18]. The lattice constant, (Λ) defined as the distance between the centers of the core and the air holes, was set to 9.3 µm. The fiber’s cladding had a standard diameter of 125 µm and the 8 µm diameter core was doped with 4 mol% of germanium dioxide (GeO₂). The GeO₂ doping level, comparable to this used in standard SMF, provides sufficient refractive index contrast to efficiently confine light in the core. The air holes shape the mode distribution and are responsible for introducing accessory stress to the fiber core region under externally applied longitudinal strain. The air hole diameter (β) was set in simulations as a variable ranging from 4 µm to 9.6 µm.

 

Fig. 1 3D model of the investigated fiber used in numerical simulations.

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To model the effective refractive indices’ sensitivity to longitudinal strain we used commercially available fully vectorial Finite Element Method (FEM) software (COMSOL). In the calculations, we applied at least 900 000 finite elements on a full three-dimensional fiber model with Perfect Electric Conductor (PEC) boundary conditions. The axial stretch induced stress causes the initially isotropic silica to become birefringent with the principal refractive indices (n_x, n_y and n_z) given in the most general case by the following set of equations:

In Fig. 2 we show the results of the simulations with the calculated (according to Eq. (1)) normalized effective refractive index sensitivity under a longitudinal strain of 0.5 mε. It is clearly seen that an increase of air hole diameter results in an increase of S_n (blue line). This effect is caused by the fact that while the z component (along the fiber axis) of stress remains constant despite the variable air hole diameter, the x and y components change. Larger air holes introduce additional compressive stress in the x-y plane around the core area (as seen on the cross sections on the right side of Fig. 2). This stress can be quantified by τ, which is defined as an integral of x and y stress components in the core area (depicted on Fig. 2 as a red line) and is a power function of the air-hole diameter.

 

Fig. 2 Left - normalized n_eff sensitivity to longitudinal strain (blue line) and integral of x and y stress components in the core area (red line) in the function of MOF air hole diameter. Right – normalized sum of x and y stress components in two fiber cross sections (with air hole diameter equal to 6.4 µm and 10 µm for top and bottom image respectively) under strain (below zero values represent compressive stress).

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As a result of the compressive stress which appears during elongation, the effective refractive index of the propagated mode increases. On the other side, the fiber elongation results in tensile stress in the z axis, which leads to a decrease in the effective refractive index of a propagated mode. As the z component stress has larger values than the x-y components, the net value of S_n is negative and increases with air hole diameter.

3. Microstructured fiber characterization and phase sensitivity measurements

Four fiber series (A-D) differing in air hole diameter were fabricated from a single preform with the stack and draw technique. Scanning electron microscope (SEM) images of the cross sections of drawn fibers are presented in Fig. 3 while the structural parameters of all the fiber series are listed in Table 1. Fiber A, with all air holes collapsed, can be used as a reference fiber, with expected properties equivalent to a standard SMF. Fibers B-D have air holes with diameters ranging from 5.3 µm (fiber B) to 18 µm (fiber D). The initially round core (fiber C) becomes triangular when the air hole size decreases (fibers A and B). This is due to the fact that the cladding material is drawn towards the shrinking air voids. On the other hand air holes with diameter exceeding the lattice pitch (fiber D) compress the core into a three armed star shape. The fibers were simulated numerically (with geometry data taken from real, SEM captured structures) using the finite difference method for effective refractive indices of the fundamental mode (Table 1). As the core shape is not ideally circular, the fibers were also investigated numerically for phase modal birefringence (defined as the difference of effective refractive index of the two orthogonally polarized modes B = n_effx - n_effy). The results proved low birefringence in the range of 10⁻⁵ for fiber D and 10⁻⁶ for fibers A-C.

 

Fig. 3 SEM images of the drawn fibers.

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Table 1. Geometrical and propagation parameters of the drawn microstructured fibers.

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The microstructured fiber samples (approx. 0.5 m length) were spliced into SMF-MOF-SMF patchcords using standard telecom fusion splicers. The patchcord’s insertion loss measured at 1550 nm varied between 0.1 dB and 1.3 dB for fiber A and fiber D respectively. The samples were then spliced as sensing arms of a standard fiber Mach-Zehnder interferometer shown in Fig. 4 and characterized with respect of phase sensitivity to longitudinal strain. We used a 1550 nm DFB laser source and a simple photodetector connected to an oscilloscope. Elongation of approx. 0.1 mm (resulting in 0.2 mε) was applied gradually using a linear translation stage with a micrometer screw. The phase change introduced by the sample elongation was monitored as the change of signal’s intensity on the oscilloscope. The interference pattern showed a clear sinusoidal trend, which proves single mode operation in all patchcords. The normalized phase sensitivity was calculated from the recorded number of fringes (N_f) for a given elongation (ΔL) according to Eq. (5) [16].

 

Fig. 4 Schematic of the Mach-Zehnder interferometer measurement setup.

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The measurement was repeated 10 times for each fiber’s sample and the observed maximum inaccuracy in the fringe count was ± 1, which results in an inaccuracy in the S_n measurement at a level of ± 0.01 ε⁻¹. Furthermore, we measured the S_n of a standard single mode fiber (SMF-28) using the same measurement setup for reference. The results are presented in Table 2. A clear relationship between the air hole size and the effective refractive index sensitivity is observed, which is in good agreement with the theoretical analysis presented above. The experimental dependency is stronger than the numerical one. This effect may result from the fact, that fibers fabricated using the stack and draw method are characterized by certain initial stress which is “pre-frozen” in the structure during the manufacturing process, hence they may show higher strain sensitivities (this is why fiber A, with no air holes, has a higher S_n than an SMF).

Table 2. Measured effective refractive index sensitivities to axial strain.

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4. Conclusions

According to our best knowledge, this is the first report showing a clear dependency between a microstructured fiber’s geometry (namely air hole sizes) and effective refractive index sensitivity to longitudinal strain. The theory presented, confirmed by experimental data, enables tailoring the phase sensitivity of a mode by proper design of MOF’s air hole arrangement. For a single mode fiber with air hole diameter of 18 µm we have shown an increase (in comparison to a standard SMF) of the normalized refractive index sensitivity by over 33% (with S_n = −0.14 and S_n = −0.21 for developed MOF and standard SMF, respectively). Such an increase results in a normalized phase sensitivity to longitudinal strain of 0.86 ε⁻¹, which is over 10% higher than values reported for a typical nonlinear MOF [16] and proves (contrary to [17]) the usability of MOFs in axial strain sensing. Potential applications for the reported effect may be found in differentiating the strain response of fiber optic sensors as well as in highly sensitive longitudinal strain transducers. Moreover, as temperature phase sensitivity of optical fibers is mainly dependent on material properties and the doping level [13], the reported methodology may be used to design enhanced strain sensitivity pure silica microstructured fibers with decreased temperature cross sensitivity.