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Abstract
Existing fiber optics (FOs)-based sensors, including mechanical deformation ones rely on structures embedded along the length of the FO. In this paper, we introduce and evaluate photonic crystals (PCs) embedded into FO cores acting as mechanical deformation sensors which are departing from this classical approach as the PCs are perpendicular to the length of the FO. Another noteworthy difference from classical FO-PC based sensors is that while classical ones rely on amplitude variations, the ones presented here use the phase variations of the electromagnetic components for assessing mechanical deformations. We start with a straightforward rectangular-lattice PC while also exploring a triangular-lattice PC. Light transmission simulations through the proposed FO-PC mechanical deformation sensors were performed using EM Explorer, and revealed their behaviors under small mechanical deformations. These simulations (of the rectangular-lattice and triangular-lattice PCs) show that these two FO-PC mechanical deformation sensors have roughly the same sensitivities while the triangular-lattice PC triggers at a lower threshold than the rectangular-lattice PC.
© 2017 Optical Society of America
1. Introduction
Fiber optics (FOs)-based systems have a wide range of applications, from sensing to communication and imaging [1,2]. When applied to sensing, FOs have been used to measure various parameters, such as: temperature, pressure, vibration, and even chemical concentration [3]. A particular subclass of FO based sensors is represented by mechanical deformation sensors. Their sensitivities are related to the mechanical strain ε = ΔL/L0, the dimensionless ratio of elongation over reference length, which normally is in the mε to με range. Such FO mechanical deformation sensors are widely used in smart structures (which are embedding optoelectronic devices), from composite materials to prosthetic hands [4–6]. These FO mechanical deformation sensors are based on: (i) fiber Bragg gratings, which couple the forward propagating core mode to the backward propagating core mode [7,8]; (ii) long-period fiber gratings, which couple the forward propagating core mode to one, or to a few, of the forward propagating cladding modes [9,10]; (iii) Fabry-Pérot interferometer settings [11,12]; (iv) U-bend fiber sensors based on the photo-elastic effect [13]; and (v) Rayleigh backscattering [14].
Besides FOs (which are widely in use), photonic crystals (PCs) [15,16] represent another class of optical devices having characteristics that allow for envisioning a large variety of applications, including optical on-chip communications [17], interferometers [18], as well as novel sensor systems [19,20]. PCs are nano-structured on geometrically regular (periodic) lattices and exhibit very strong interaction with light, because of the so-called photonic bandgap (PBG) [21,22]. The PBG formation can be tuned by choosing particular combinations of desig n geometries (lattices) and material parameters, in order to have both the macroscopic and the microscopic resonances at the same frequency. The inhibition of light in PCs is strongly (i.e., non-linearly) related to some of their characteristics, like: dimensions, symmetries, topologies, lattice parameters, filing ratio, and refractive index − to name the most important ones. These characteristics have been used to design sensory devices [23,24], but also modern information transmission and processing systems [25].
An early application of PCs has been to send light over long distances with low losses, and this has led to micro-structured FOs. These are known as photonic crystal fibers (PCFs), see Fig. 1(a), and can be divided into [26]: PCFs based on total internal reflection, called index-guiding PCFs - Fig. 1(b), and PCFs based on PBG formation - Fig. 1(c).
Fig. 1 (a) Photonic crystal fibers (PCFs) showing the incident electromagnetic waves E and H; (b) cross-section of an index-guiding PCF; (c) cross-section of a photonic bandgap (PBG) PCF; (d) schematic of the proposed sensor showing a transversal PC embedded in an FO.
Smartly designed PCs can be used to control light traveling through [27], and the fact that PBG PCFs are highly sensitive implicitly advocated for using them as sensors (e.g., for refractive index sensing [28], or in biomedical applications [29,30]). To the best of our knowledge, PBG PCF sensors developed so far have been limited to bi-dimensional (2D) PCs aligned along the length of the FOs, on the axis in Fig. 1(a). This paper is proposing and studying sensor structures using bi-dimensional (2D) PCs lattices embedded inside an FO which are perpendicular (i.e., in the plane) to the length of the FO, on the axis in Fig. 1(d). Additionally, as opposed to classical approaches (which rely on amplitude variations [31–34]), the FO-PC sensors we propose work differently as relying on phase variations of the electromagnetic components for assessing mechanical deformations.
The remainder of the paper is structured as follows. In Section 2 the proposed FO-PC mechanical deformation sensor is presented conceptually, and embodied on two different PC lattices: a rectangular lattice (RL) and a triangular lattice (TL). Results of simulations performed with the two PC lattices (RL and TL) are reported in Section 3, with discussions following in Section 4, our aim being to see if and how the two different PC lattices affect the FO-PC sensors’ behavior in general, and the sensitivity thresholds in particular. Conclusions and future directions for research are ending the paper.
2. Concept and embodiments of the proposed FO-PC sensor
The index-guiding PCFs and the PBG PCFs were built as microstructures parallel along the length of the FO () in Fig. 1(a). Departing from this approach, we have advocated (preliminary studies [35,36]) for a different FO-PC sensor concept, namely to embed a 2D PC structure perpendicular to the length of the FO. Fabricating such structures could rely on lithographic techniques, while nano-imprint and even directional photo-fluidization imprint [37] might be considered. Additionally, a very fresh and thorough review of micro-hole drilling [38] shows precision down to 500 nm, which is expected to improve. Obviously, fabrication techniques for integrated circuits can and have been used [39,40], and currently are the ones to envisage. Light still travels along the length of the FO (i.e., on the Ox axis), but it is perpendicular to both (Oy) and (Oz) axes. The outcome of such a design is that a force applied along the length of the FO (Ox) will induce mechanical deformations which will intimately affect the PC by modifying its dimensions. These tiny geometric modifications will alter the PC dielectric properties and affect non-linearly the light traveling through the PC. Therefore, information regarding the force applied to the FO, and the associated mechanical deformations (elongations), can be inferred from particular changes of the light traveling through the FO-PC mechanical deformation sensor.
As already mentioned, here we investigate and compare two possible PCs lattices [41,42]. We start with a RL (suggested in [35,36]) while continuing with a TL (see Fig. 2), as a TL is smaller and tends to have a larger PBG. For both of them we have used cylindrical holes of diameter ϕ, together with mode-matching cylindrical holes of diameter φ < ϕ. The positions of these cylindrical holes are determined by the type of lattice used (RL or TL), by the initial offset, c, and by the periodicity of the lattice a. Additionally, b > a defines the size of a cavity, like a defect in the lattice. The periodicity a is the elementary step of the lattice, i.e., the distance between two adjacent cylindrical holes.
Fig. 2 3D view (at scale) of the two PCs: (a) RL; (b) TL. Cross section (not at scale, for enhanced visibility and ease of understanding) along the length of the FO (α) showing the positioning of the cylindrical holes: (c) RL (i = 1…7); (d) TL (i = 1…10).
The size of the cavity b is larger than a and is used in particular places of the PC, the distance between some of the adjacent cylindrical holes becoming larger than the periodicity of the lattice. In particular, in the RL case we have used two cavities, as shown in Fig. 2(c), while the TL has only one cavity, as shown in Fig. 2(d). Another noticeable difference is that the RL we have used has only one input and one output row of mode matching holes (thinner cylindrical holes of diameter φ), as opposed to TL which has two input and two output such mode matching rows, as presented in Figs. 2(c) and 2(d). The mode matching holes in the RL case are on the lattice grid, which is not the case for the TL. The mode matching holes in the TL case are positioned symmetrically to the nearby two rows which form the TL, i.e., like mirror reflections. In the end, the RL and TL structures have 7 and respectively 10 rows of cylindrical holes.
Table 1 presents in a compact form the four parameters a, b, ϕ, and φ mentioned above for both RL and TL under no mechanical stress, i.e., when there is no force applied (). These values have been optimized in [41,42] both for 1D and 2D PCs structures for microcavity filtering. When considering both the two different layouts in Fig. 2, as well as the different dimensions presented in Table 1, it becomes clear why the two PCs end up having different overall sizes, namely L0 = 3000 nm for the RL, and L0 = 2380 nm for the TL one (smaller as expected, although having 3 more rows of cylindrical holes).
Table 1. Parameters for RL and TL when [41,42].
As a remark, the parameter c was introduced only for explaining the fact that the drawings do not start exactly on the first row of cylindrical holes (i.e., tangent to the cylinders), but this parameter was not used further in simulations.
If a force is applied along the length of the FO (and perpendicular to the PCs), a mechanical deformation (elongation) ΔL will result - Fig. 3. ΔL is obtained from Hook’s law:
where L0 and A0 are the initial length of the PC structure and the cross section area of the FO. The initial length L0 was estimated before, while the cross section of the FO can be determined from the fact that the diameter is 100 µm. The last element missing is the Young’s modulus Y. The core and the cladding of the FO are made of GaAs and, Al0.6Ga0.4As respectively [43]. For the two different wavelengths λ = 1.5 μm and λ = 0.8 μm where RL and TL exhibit PBGs, these two materials have an absorption coefficient k in the range , and the Young’s modulus for AlyGa1-yAs is (see [43]):When (the core of the FO is ) we obtain Y = 8.53 × 1010 N/m2. Only this value of Y will be used further in the simulations. The PC structures are entirely embedded in the core of the FO (i.e., cladding is not used). All these parameters are presented in a compact form in Table 2, which includes also the elongations ΔL for RL and TL when F = 1N.Table 2. Young modulus (), length (), cross section (), and elongation () for RL and TL when .
The external force applied will intimately affect the positions of each and every row of cylindrical holes, as shown in Fig. 3. All the cylindrical holes in a row will be shifted along the length of the FO with a displacement (elongation) Δxi, where i = 1…7 for RL and i = 1…10 for TL. These displacements depend on the initial position xi of the row and on the external force F applied along the length of the FO. That is why our next step was to determine the positions of the centers of the rows of cylindrical holes forming the PC, namely xi + Δxi, when F = 1 N. Table 3 presents these values for both RL and TL.
Table 3. Positions of the centers of the rows along the length of the FO when F = 1 N.
The column labeled xi gives the positions of the centers of the rows of cylindrical holes when no force is applied (i.e., when F = 0 N). When F > 0 each position is displaced due to an elongation Δxi which can be readily calculated using Hooke’s law, as we have done before, i.e., Δxi = (Fxi)/(YA0). The positions of the centers of the rows shift by Δxi to become xi + Δxi. For F = 1 N all of these are detailed in Table 3. As the PC sensors are embedded within an FO, the origins of the reference systems for the PC sensors are relative, and only the shifts (between cylinders) count. For the EM Explorer simulations (to be detailed in Section 3), reducing the volumes of the PCs translates into significant reductions of simulation times (that is why c was ignored). Obviously, when applying a force to the PC sensors all the cylinders are going to shift. As we want the tightest volumes (for the shortest simulation times) we have considered that the origins of the reference systems (for EM Explorer) are exactly in the center of the first raw of cylinders (i.e., at x1) and that the origin moves with this first row of cylinders. This explains why Δx1 = 0 (the origin shifting accordingly) and, taking RL as an example, why x1 + Δx1 = 238 nm, i.e., the same as x1 (in Table 3): x1 is in one system of reference, while x1 + Δx1 is in another (shifted) system of reference.
For the particular task of sensing small mechanical deformations the useful domain corresponds to forces up to a few , as pointed out in the literature [30], but also justified by applications [34] and by the resilience of PCFs [44]. That is why the process presented in Table 3 for F = 1 N was repeated for F = 2 … 10 N (i.e., forces up to 10 N with a step of 1 N. The positions of the centers of the rows of cylindrical holes forming the RL and the TL PCs were determined for each of these forces. The resulting PC structures (11 for RL and 11 for TL) where used to define input geometries for EM Explorer [45].
3. Simulation results
We have used EM Explorer [45], which relies on the Finite Difference Time Domain (FDTD) method to solve Maxwell equations. It is a CAD tool for electromagnetic (EM) waves solver for scattering problems of periodic structures illuminated by an arbitrary incident field. Detailed simulations were performed both for the RL and the TL structures shown in Figs. 2 and 3. The RL and TL geometries have been modified as detailed in the previous section (for external forces ranging from 1 N to 10 N, with a 1 N step). We have accounted for the shifts in positions due to the applied external forces (as explained in the previous section), but we have neglected both the modifications of the holes (i.e., the fact that circles are becoming ellipses) and the tightening of some distances (i.e., the cylindrical holes in a row are going to be slightly closer to one another). Obviously, both of these are happening, but we have estimated (and tested) that they are well below thresholds which could affect the results of the simulations.
Before starting the simulation process the wavelength where each of the two lattices exhibits the PBG were studied (see [41,42]). Consequently, the wavelength has been set to λ = 1.5 μm for RL and respectively to λ = 0.8 μm for TL. All the simulations have been performed at these PBG wavelengths (λ), having air as background, and using a mesh size (Δx, Δy, Δz). The mesh size for obtaining accurate approximations of Maxwell’s equations when light is traveling through periodic nanometric structures is recommended to be [46]:
Here nmax was estimated for the two different λ by first of all calculating the photon energy (0.826 eV and respectively 1.549 eV) and afterwards mapping energies into nmax using the tables and graphs presented in [47].For all the simulations performed on the geometrically different RL and TL we have used as boundary conditions for the (Δx, Δy, Δz) volume the perfectly matched layers (PMLs) for the absorption of EM waves [48]. The PML boundary conditions provide a reflectionless interface (absorbing without reflection the EM waves) between the region of interest and the PML at all incident angles.
Table 4 details the settings we have used for all the simulations performed on the 11 different RL and on the 11 different TL, and also shows that Eq. (3) is satisfied.
Table 4. Simulations settings for RL and TL.
One last issue with respect to simulation settings is that we have bounded the number of iterations to a maximum of 10,000 for limiting the execution time (but in all the cases reported here convergence was achieved after less than two thousand iterations).
The simulations under varying elongation forces have revealed both the electric and the magnetic components of the EM field, showing both the amplitude and phase of each of these two vectors and on each of the three axes (Ox, Oy, and Oz). Some but not all of these simulations - of the way light propagates through the two lattices presented in Fig. 2 - are detailed in Figs. 4 and 5. The EM components were obtained from EM Explorer and imported in Matlab to plot their variations with respect to F, as presented in Figs. 4(a) and 5(a). These show how the phase and amplitude of E and H on each axis vary when light propagates through the RL and TL, respectively, for elongation forces from 0 to 10 N. From Fig. 4(a) the plot of the phase of Hx (the magnetic component on Ox) reveals a 2π abrupt change when the force F changes from 2 N to 3 N (or vice versa).
Fig. 4 RL simulations: (a) phase and amplitude of the transmitted EM components for external forces from 0 to 10 N; (b) colored coded images of the phase and amplitude of the EM components.
Fig. 5 TL structure: (a) phase and amplitude of the transmitted EM components for external forces from 0 to 10 N; (b) colored coded images of the phase and amplitude of the EM components.
This steep variation will be used to estimate the sensitivity of the RL sensor. Similarly, from Fig. 5(a) one can see that the phase of Ey (the electrical component on Oy) for the TL has a 2π variation when the force changes from 1 N to 2 N (or vice versa). This steep variation will be used to estimate the sensitivity of the TL sensor.
For a more nuanced view, Figs. 4(b) and 5(b) present colored coded images of the phase and amplitude variations of the EM components for F = 0, 5, and 10 N.
4. Discussion
As mentioned in the Introduction, classical elongation sensors tend to rely on Bragg peak wavelength variations, which in many cases are reported with respect to strain variations (ε = ΔL/L0). Here are a few recent examples from the literature: 1.3 pm/με [31], 11.22 dB/mε [32], 7.96 dB/mε [33], and in 13.01 pm/με [34].
It can be seen that even these are reporting either pm or dB per strain. Still, there are many other cases, especially for mechanical sensors relying on other principles, which report sensitivity as the ratio of the output with respect to the measured property. As an example Liang et al. [49] reported a sensitivity of −14.595 nm/N, while this is linked to wavelength shifts. All of these make a direct comparison far from straightforward, as besides having different sizes (i.e., L0) the outputs of the sensors might end up being reasonably different. As the sensors presented in this paper rely on phase variations (determined by elongation forces), even the accuracy of the interferometer used for measuring the phase variations would affect such estimates. That is why we have decided to report sensitivities as pm/°.
The RL has phase variations which can be used for detecting mechanical deformations having a sensitivity of:
For the TL, the phase of varies from 3.089 to –3.058 radians. This means that the FO-PC mechanical deformation sensor embedding the TL exhibits an phase variation of Δθ = 3.05 - (- 3.089) = 6.14 radians = 351.8° for a mechanical deformation ΔL = 3.55 nm. Hence, the sensitivity of the TL used as a mechanical deformation sensor is:
The sensors we have presented have 4.48 nm/N (RL) and 3.55 nm/N (TL).
Both phase variations for RL (Hx) and TL (Ey) are presented in Fig. 6. We notice the similarity of their profiles, both exhibiting sharp variations. The only discrepancy is that the forces where they exhibit phase variations are slightly different: RL triggers between 2 N and 3 N, while TL triggers between 1 N and 2 N.
5. Conclusions
This paper has presented and analyzed two FO mechanical deformation sensors based on PCs embedded into FO cores perpendicular to the length of the FO. The two PCs are using a RL [35,36] and respectively a TL structure. These two FO-PC mechanical deformation sensors have been presented and evaluated through simulations.
The simulations performed have shown that measuring the phase variations of the EM components can provide accurate information on small mechanical deformations. On one hand, by comparing the slopes of phase vs. force/elongation it was inferred that both sensors behave similarly, i.e., have roughly the same sensitivities, with a small advantage for the TL (which can be mainly attributed to the smaller size L0 of the TL-based sensor). On the other hand, each sensor has a different triggering threshold: the RL-based sensor triggers between 2 N to 3 N, while the TL-based sensor has the advantage that it triggers at lower forces, between 1 N to 2 N. The two sensors also have different outputs: the RL exhibits sharp phase variations on Hx, while the TL shows them on Ey (Fig. 6).
Further research will consider a complex Young’s modulus (as the structure is a mixture of cladding, core and air holes), could look at other lattice designs, and should make provisions for faster simulations. These would be needed for multi-parametric analyses optimizing the sensing performance. Finally, incorporating new materials (e.g., fluids filling the air holes) might expand the versatility of these mechanical sensors to other parameters (e.g., temperature), and expand their use, for example beyond axon swellings that are in the few Angstroms to 1 nm range [50].
Funding
European Union through the European Regional Development Fund under the Competitiveness Operational Program (BioCell-NanoART = Novel Bio-inspired Cellular Nano-architectures, POC-A1-A1.1.4-E 30/2016); Romanian Authority for Scientific Research (CNDI–UEFISCDI) (PN-III-P2-2.1-BG-2016-0297, http://3om-group-optomechatronics.ro/).
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