Fiber Curvature Research Articles

Highly sensitive curvature fiber sensor based on an enhanced core diameter mismatch.

In this study, a curvature fiber sensor based on an enhanced core diameter mismatch is experimentally proposed and theoretically investigated. The structure is fabricated by splicing two types of step multimode and coreless fibers to excite the high-order cladding modes to improve the curvature sensitivity. Experimental results show that the highest curvature sensitivities of the structure reach -114.74 nm/m-1 in the Dip 1272 nm, -91.08 nm/m-1 in Dip 1408 nm, and -61.10 nm/m-1 in Dip 1644 nm in the measuring range of 0-0.49778 m-1. Meanwhile, the sensor's temperature and strain responses were also tested, which shows little influence on the curvature measurement. Additionally, the proposed fiber sensor exhibits features of easy fabrication, simple structure, and high mechanical strength. This study proposes a device for curvature measurement with potential use in material mechanics and optical fiber sensor design.

Distributed Temperature and Curvature Sensing Based on Raman Scattering in Few-Mode Fiber

We propose and experimentally demonstrate a distributed temperature and curvature sensing method based on spontaneous Raman backscattered lights in a few-mode optical fiber. A loop structure is used to demodulate the distributed temperature information using the Raman anti-Stokes (AS) scattering lights. And the fiber curvature is estimated by the loss of Raman Stokes (S) light. This system can measure temperature and curvature at the same position of the fiber simultaneously. The experimental results show that the temperature uncertainty and spatial resolution can reach up to 0.24 °C and 1.8 m on 6 km sensing fiber. Besides, the maximum measurement error is 0.11 cm within the curvature measurement range of 0.5–3 cm.

Study on distinct quality properties of cashmere from local goat breeds of Mongolia

The research was carried out in 2021 to determine the specific quality properties of local Mongolian breeds of cashmere goats. The research covered 122 herder households from 13 soums in 8 aimags. A total of 4,560 samples were collected from 1,522 goats representing the main factors determining cashmere quality: ecological habitat, various herd management practices, local breeds, age and sex of goats, and cashmere colour. Cashmere fibre diameter (FD), fibre curvature (FC), and coefficient of variation of fibre diameter (CVFD) were tested in 1,522 samples, and cashmere length was tested in 1,387 samples. The overall mean fibre diameter (MFD) was 16.1 μm, mean length (FL) was 44.9 mm, and fibre curvature (FC) was 61.9°/mm. Cashmere from one-year-old goats was significantly finer than cashmere from older goats (about 1.0 μm, P<0.05). Cashmere from does in the samples tested was significantly finer than cashmere from the bucks. Over the range in mean fibre diameter, from 13.5 to 19.0 µm, fibre curvature declined from 87.8 to 44.2°/mm. Of all cashmere samples, 40.6% belonged to super fine (≤15.5µm), 43.7% and 12.1% had a fibre diameter between 15.51-16.80 and 16.81-17.50 μm, respectively, and were suitable for knitwear. Only 3.6% of samples tested were between 17.51 to 19.00 μm and may only be suitable for weaving. The average CVFD values of cashmere of Mongolian breeds of cashmere goats was 22.5 ± 0.04%, with an individual range of 17.0-26.9%. This shows that it has lesser variation than Alashan white cashmere goats of Inner Mongolia Autonomous Region, China, where CVFD ranged from 27.09% to 41.39%. Compared to cashmere from China, Tajikistan, and Kyrgyzstan, with mean fibre curvature of 46, 46, and 58mm, respectively, the cashmere of local Mongolian breeds of goats would be considered curvier and longer, which makes stronger yarns. Short, higher crimped, softer cashmere may be preferred for woollen-spun yarns, but longer, lower crimped, softer cashmere may be preferred for woven yarn destined for knitwear. Fibre curvature of Mongolian cashmere goats declined an average of 6.1°/mm for a 1µm increase in MFD, which is similar to the 5.8°/mm observed in Chinese Liaoning goats, but less than the 13.6°/mm measured in controlled experiments with Australian goats. Significant differences were found between the cashmere of local Mongolian breeds of goats kept in open, natural rangelands and herded in a nomadic way in distinct ecological conditions of harsh continental climate compared to those breeds of goats kept in intensive farming. This shows the potential to offer distinct quality cashmere from local Mongolian breeds while maintaining proper management and selective breeding.

Multi-channel curvature sensor based on fiber bending loss wavelength and SPR.

Fiber Bragg gratings and interferometric curvature sensors are easily disturbed by axial strain and temperature, and cascaded multi-channel curvature sensing is difficult. In this letter, a curvature sensor based on fiber bending loss wavelength and the surface plasmon resonance (SPR) mechanism is proposed, which is insensitive to axial strain and temperature. In addition, fiber bending loss valley wavelength demodulation curvature improves the accuracy of bending loss intensity sensing. Experiments show that the bending loss valley of single-mode fiber with different cut-off wavelengths has different working bands which is combined with a plastic-clad multi-mode fiber SPR curvature sensor to realize a wavelength division multiplexing multi-channel curvature sensor. The bending loss valley wavelength sensitivity of single-mode fiber is 0.8474 nm/m-1, and the intensity sensitivity is 0.0036 a.u./m-1. The resonance valley wavelength sensitivity of the multi-mode fiber SPR curvature sensor is 0.3348 nm/m-1, and the intensity sensitivity is 0.0026 a.u./m-1. The proposed sensor is insensitive to temperature and strain, and the working band is controllable, which provides a new, to the best of our knowledge, solution for wavelength division multiplexing multi-channel fiber curvature sensing.

Modeling the dynamics of partially wetting droplets on fibers

We study the motion of liquid droplets moving along partially wettable fibers. Employing lattice Boltzmann simulations, we observe three dynamic regimes: compact droplet, droplet breakup, and droplet oscillation. The transitions between these regimes depend not only on the droplet Bond number but also on the fiber curvature. In particular, droplet breakup is promoted with increasing fiber curvature. Furthermore, we can identify different scaling laws for the droplet velocity when the droplet size is large or small compared to the fiber radius.

A fiber sensor for different curvature measurement based on nano-erbium-ytterbium co-doped fiber cascade structure as both source and sensor

A fiber sensing system based on Nano-Erbium-Ytterbium co-doped fiber (Nano-EYDF) is proposed and demonstrated, where it works as both source and sensor. Especially, a fiber curvature sensor based on Mach-Zehnder interferometer is fabricated in a Nano-EYDF, which is a cascaded structure composed of single-mode fiber (SMF), Nano-EYDF and SMF. Based on the principle of evanescent wave coupling, the light transmitted in the SMF core can be coupled into the Nano-EYDF cladding. Therefore, the Nano-EYDF core mode and cladding mode that satisfy the phase matching conditions can couple evanescent waves to form a double-beam interference. When the external environment changes, the interference spectrum will change, so the curvature sensing can be realized by observing the wavelength shift. The experimental results show when the length of the Nano-EYDF is 20 mm and 25 mm, the transmission spectrum of the sensor appears blue shift in the downforce curvature range of 0–1.76 m−1, and the curvature sensitivity is −9.763 nm/m−1 and −12.787 nm/m−1 respectively. In the propulsion curvature range of 0–0.94 m−1, the transmission spectrum exhibits blue shift, and the curvature sensitivity can be up to −9.261 nm/m−1 and −9.411 nm/m−1, respectively. The sensor is easily fabricated and has high sensitivity, and demonstrates good application prospects in curvature measurement fields such as 3D shape reconstruction and geological monitoring.

A novel plasmonic sensor using solid core D-shaped negative curvature optical fiber with AU–TiO2 layer

Recently, unique structure fiber enacts a significant role in sensing applications. A novel D-shaped negative curvature fiber based surface plasmon resonance (SPR) sensor is proposed. The design consists of negative curvature solid core fiber surrounded by a cladding composed of double-layer adjoined airholes. An outside layer of the cladding contains semi-circular airholes, and the interior layer has small circular air holes to overwhelm the higher-order modes. The external sensing approach detects changes in the refractive index between 1.33 and 1.37. Employing gold as a plasmonic material, SPR is developed on top of the photonic crystal fiber (PCF). Between the gold layer and the PCF, a small layer of titanium dioxide is placed for binding the plasmonic material to the fiber. And it acts as an adhesive layer. The finite element method is used to perform numerical analyses. After improving the structural parameters, an enhanced sensitivity of 26 000 nm RIU−1 is achieved. Because of its enhanced sensitivity, this sensor is used in a bio-medical application to detect and prevent diseases in the early stage.

A combinatorial approach to the elastic response of electrospun mats: Architectural framework and single fiber properties

Electrospun is a unique class of porous and heterogeneous materials with multi-length-scales constituents that offer a rich variety of surface functionalities to serve a host of applications. Upscaling the electrospun materials from the laboratory to the industry is often limited by the lack of understanding of their mechanical properties. Herein, we developed a theoretical framework to predict the elastic constants of the electrospun mats that hinges on the concept of elastic properties of constituent fibers, three-dimensional (3D) alignment of fibers, and local fiber curvature. Enabled by continuum-based micromechanical approaches, this framework successfully predicted the elastic moduli regardless of bead-string morphology and local architectural heterogeneities present within the electrospun mats. The 3D fiber orientation distribution obtained using X-ray nano-computed tomography (nanoCT) analysis served as a key input for the validation of the analytical model. In general, the predicted elastic moduli are in reasonably good agreement with the experimental data of randomly oriented and preferentially aligned polylactic acid (PLA)-based electrospun mats. To demonstrate our analytical model's versatility and reliability, another set of PA6(3)-based electrospun mats has been chosen from the literature for validation purposes. The parametric analysis has been performed to provide a roadmap to improve the elastic moduli of electrospun mats and justify the assumed values of some of the key attributes.

582. REPORT ON A 42 YEAR EXPERIENCE WITH THE LAPAROTOMIC/LAPAROSCOPIC HELLER-DOR OPERATION PERFORMED WITH INTRAOPERATIVE MANOMETRY FOR ESOPHAGEAL ACHALASIA

Abstract In 1979 our surgical group proposed the Heller-Dor operation (HD) to accomplish new concepts of surgical pathophysiology: to abolish the lower esophageal sphincter with the division of the U-and sling fibers of the lesser gastric curvature, to prevent GER with an effective but not too effective partial fundoplication which protects the myotomy surface. Aim is to report very long-term results of the treatment of esophageal achalasia with the Heller-Dor operation performed with intra operative manometry. In the period 1979-2021 HD was in first instance adopted in 202 patients (97 men; median age 45 years; IQR 31–59) in laparotomy (L-HD) and in 162 (79 men; age 48 years; IQR 35–63) in laparoscopy (V-HD). At intraoperative manometry, the complete abolition of the high-pressure zone was always obtained and the Dor length and pressure were trimmed according to reference values. Follow-up consisted of interview, endoscopy and barium swallow performed every 3 years. The overall outcome was graded from excellent to poor according to the severity of symptoms and esophagitis. Median follow-up was 14.8 years in HD, 7.5 years in V-HD (p<0.001). Esophageal diameter and residual barium column decreased substantially during the first 3 years after the operation, but not successively. In the L-HD group poor results were 22 (10.9%): due to esophagitis in 17 (8.4%) and to recurrent dysphagia in 5 (2.5%). In V-HD, 4 (2.5%) had esophagitis and 2 (1.2%) severe dysphagia. Outcome was satisfactory in 89% of L-HD and in 96% of V-HD (p<0.001). Postoperatively, according to the Kaplan Mayer curve, dysphagia, reflux symptoms and esophagitis occurred within 60-65 months after surgery. Timing of dysphagia relapse and post-myotomy reflux occurrence indicate that after five years, surgical results can be considered definitive. The 96% of satisfactory results we achieved with VL-HD, at a median follow-up time of 8 years, shows the potentiality of this surgical technique. In our experience, the learning process, not the follow-up length or the laparotomy/laparoscopy method is the cause of improved results of the laparoscopy HD.

Community-based vicuña (Vicugna vicugna) shearing in the arid Puna of Argentina: Body weight and fiber traits obtained during the chakus

In the Argentine Puna, innovation proposals for the production of vicuña fiber represent a strategic opportunity for local communities. However, the development of these proposals, essential for the addition of local value, is affected by the lack of information on the relationship between fiber quality and processing. Thus, this study aimed to characterize the variability in both the quantity and quality of vicuña fiber obtained by shearing in the Argentine Puna, in the context of sustainable management plans presented by different indigenous communities of the Yavi district, Jujuy province. Samples (n = 150) were taken during shearing in locations at low (<3814 masl, n = 5) and high altitude (>4222 masl, n = 4). The following were recorded: sex, age, and body weight (BW, kg) of the animals, fleece weight (FW, g), staple length (SL, mm), mean fiber diameter (MFD, µm), mean diameter of down fibers (DDO, µm), mean diameter of hair (DHA, µm), coarse hair content (CHC, %), proportion of down fibers (PWB, %) and mean fiber curvature (MFC, °/mm). In adult vicuñas BW and SL were significantly affected by altitude, FW was affected by sex, and all the other quality variables were affected by the sampling location. Adult vicuñas BW from high locations were lighter than those from low locations (38.6 vs. 41.6 kg, P < 0.01). SL was greater in animals from low locations than in those from high locations (42.4 vs. 38.8 mm, P < 0.05). Average FW was 228 g, with animals from low locations showing more variable FW than those from high locations. Females produced lighter fleece than adult males (216 vs. 244 g, P < 0.01). MFD, DDO and DHA, CHC, PWB and MFC presented averages of 13 µm, 12.6 µm, 53 µm, 0.9 %, 87.0 % and 87°/mm, respectively. For young vicuñas, the BW and FW were 30.9 kg and 213 g, respectively. Differences among sampling locations were significant for all quality variables studied and could present implications for the textile processing of vicuña fiber.

Femtosecond inscription of large-area fiber Bragg gratings for high-power cladding pump reflection.

A new, to the best of our knowledge, method for inscribing fiber Bragg gratings inside a fiber's cladding based on the motorized rotation of the fiber is reported. By minimizing the aberrations induced by the fiber curvature on the femtosecond writing beam, this technique based on a phase mask allows to cover large transverse areas of a standard high-power fiber's cladding. With this approach, a first-order Bragg grating was inscribed in the pure-silica inner cladding of a 20/400-µm fiber. It was then implemented as a pump reflector at the end of a 36-m-long Yb-doped fiber laser reaching 600 W of output power, confirming the power handling capabilities of such a component. Comparison of the laser performances with and without the pump reflector showcases its great potential for increasing pump absorption inside cladding-pumped fiber lasers, which paves the way for significantly reducing their active fiber length.

Large measurement-range and low temperature cross-sensitivity optical fiber curvature sensor based on Michelson interferometer

A large-measurement-range fiber curvature sensor based on Michelson interferometer is proposed and investigated. The sensor is fabricated by sandwiching a bitaper between a single-mode fiber (SMF) and a piece of multimode fiber (MMF), and the MMF end face is multiply discharged to form a fiber sphere. The mode interference is accumulated in the MMF for the bitaper could excite high-order modes and couple them with core mode. The sensor can be defined as a cantilever beam structure, in which the fiber sphere serves as a mass block, while the bitaper and MMF act as cantilever beam. The reflection interference spectrum is sensitive to ambient curvature, so a large-range curvature detection is realized. Experimental results show that the sensor has a sensitivity of −0.6888 nm/m−1 and −1.812 dB/m−1 in the curvature range of 0–12.2894 m−1. The maximum temperature sensitivity is 0.0078 nm/°C and 0.007 dB /°C in the temperature range of 30–150 °C, and the maximum temperature cross-sensitivity is −0.0113 m−1/°C, which could be neglected in the application. The sensor has advantages of low cost, easy fabrication, large curvature detection range, and low temperature cross-sensitivity, making it has broad application prospects in the curvature sensing field.

A High-FOM surface plasmon resonance sensor based on MMF-TUMMF-MMF structure of optical fiber

The figure of merit (FOM) of conventional surface plasmon resonance (SPR) sensors is low due to the low sensitivity and wide full width at half maxima (FWHM) caused by many low-intensity high-order cladding modes. To address this problem, this paper proposes a SPR sensor based on U-shaped tapered arm cascade of multimode fiber (TUMMF). The U-shaped macro-bend increases the intensity of the light field leaking into the cladding, and the tapered area makes the number of higher-order modes smaller to achieve high sensitivity and narrow FWMH sensing performance. Through simulation and experimental verification, the sensitivity of the sensor is proportional to the fiber curvature and taper ratio, and the FWMH is inversely proportional. Therefore, the parameters of the sensor are optimized by integrating the FOM and the mechanical strength of the sensor. The optimized sensor parameters are 3 mm bending diameter, 0.85 taper ratio, and 152 nm gold film thickness. The sensor was tested in the refractive index (RI) sensing range of 1.3324–1.3731 and showed a sensitivity of 4816.092 nm/RIU and a FWMH of 97.54 nm. The FOM of the proposed sensor is 49.37 /RIU, which is 3.18 times higher than that of the U-shaped multimode fiber SPR sensor. The probe has the advantages of simple fabrication, high environmental stability and easy modification, and has a wide range of applications in bioenvironmental and other fields.

Dual hollow-core negative curvature fiber polarization beam splitter covering the O + E + S + C + L communication band

In this paper, a dual hollow-core negative curvature fiber is proposed for the polarization beam splitter. The effects of the structure parameters of the dual hollow-core negative curvature fiber on the coupling length and coupling length ratio of the x -polarized and y -polarized core modes and the higher-order mode extinction ratio are analyzed by the finite element method. Moreover, the normalized output powers of the x -polarized and y -polarized modes in the cores A and B and the corresponding extinction ratio are also investigated by the mode coupling theory. The simulation results show that a dual hollow-core negative curvature fiber polarization beam splitter with the length of 6.45 cm can achieve a broad bandwidth of 400 nm (1.23–1.63 µm), covering the O + E + S + C + L communication band. In addition, the higher-order mode extinction ratio is greater than 100 in the considered wavelength range, which means that it has good single-mode characteristics. It is believed that the proposed dual hollow-core negative curvature fiber polarization beam splitter will have significant applications in the optical communication system.

New Method for a SEM-Based Characterization of Helical-Fiber Nonwovens.

The lack of tools particularly designed for the quantification of the fiber morphology in nonwovens, especially the multi-level structured fibers, is the main reason for the limited research studies on the establishment of realistic nonwoven structure. In this study, two polymers, cellulose acetate (CA) and thermoplastic polyurethane (TPU), which have different molecular flexibility, were chosen to produce nonwovens with helical nanofibers. Focusing on the nonwovens with helical fibers, a soft package was developed to characterize fiber morphologies, including fiber orientation, helix diameter, and curvature of helix. The novelty of this study is the proposal of a method for the characterization of nanofibrous nonwovens with special fiber shape (helical fibers) which can be used for curve fibers. The characterization results for the helical-fiber nonwoven sample and the nonwoven sample with straight fibers were compared and analyzed.

Feature Extraction Enabled Deep Learning From Specklegram for Optical Fiber Curvature Sensing

A novel approach for multimode fiber (MMF) curvature sensing based on the texture feature of the specklegram from the fiber facet is proposed, developed and demonstrated. Different from previously reported methods, the texture feature vector of the fiber specklegram is used as the descriptor of the curvature of the fiber. After the specklegrams from two kinds of MMF under different curvatures are recorded, the texture feature vector of each specklegram is first extracted using the uniform local binary pattern algorithm. Then, the texture feature vector is mapped into the target curvature in a nonlinear manner using a one-dimensional convolutional neural network (1D CNN). In the experiment, the model trained by the texture features of fiber specklegrams with a core diameter of 50 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\mu }\text{m}$ </tex-math></inline-formula> exhibits high prediction accuracy and good generalization ability in the curvature range of 1.55-6.93 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{m}^{-1}$ </tex-math></inline-formula> . In the first experiment, good recognition results are provided by the texture feature of the MMF specklegram, for which the average curvature recognition accuracy is 100%. In the second experiment, the prediction error for 91.4% of the samples in the test set is within a range of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\pm }{\mathbf {{0}.{1}\,\,m}}^{\mathbf {-{1}}}$ </tex-math></inline-formula> . In addition, this approach has significantly better prediction ability and robustness than the traditional approach, which processes the specklegram image directly using a two-dimensional convolutional neural network (2D CNN) for sensing. The experimental results demonstrate the fiber curvature sensing capability of the proposed method based on the analysis of texture feature using a 1D CNN.

Machine Learning Assisted High-Sensitivity and Large-Dynamic-Range Curvature Sensor Based on No-Core Fiber and Hollow-Core Fiber

Simultaneously increasing the sensitivity and dynamic range of an optical fiber sensor is desired and yet challenging. In this article, we demonstrate an optical fiber curvature sensor based on a no-core fiber (NCF) cascaded with a hollow-core fiber (HCF), realizing simultaneously high sensitivity and a large dynamic range with the assistance of machine learning analysis. The sensor is fabricated by simply fusion splicing a section of NCF and HCF to two single-mode fibers (SMFs), forming the SMF-NCF-HCF-SMF hybrid structure. It is shown that the multimode interference in the NCF can increase the sensitivity of the device for curvature measurements, compared to the conventional SMF-HCF-SMF structure. However, the enhanced sensitivity poses a limitation on the dynamic range of the proposed curvature sensor. We propose the use of machine learning to analyze the measured spectra of the sensor device, achieving one-to-one mapping between the measured raw spectrum and the exerted curvature on the sensor, and thereby the issue of the limited dynamic range is resolved. The proposed strategy, enabling the co-existence of high sensitivity and a large dynamic range, is a highly generalizable technique and can be extended to other optical fiber sensors for measuring other physical, chemical, and biological quantities in different applications.

Cellular electron tomography of the apical complex in the apicomplexan parasite Eimeria tenella shows a highly organised gateway for regulated secretion.

The apical complex of apicomplexan parasites is essential for host cell invasion and intracellular survival and as the site of regulated exocytosis from specialised secretory organelles called rhoptries and micronemes. Despite its importance, there are few data on the three-dimensional organisation and quantification of these organelles within the apical complex or how they are trafficked to this specialised region of plasma membrane for exocytosis. In coccidian apicomplexans there is an additional tubulin-containing hollow barrel structure, the conoid, which provides a structural gateway for this specialised apical secretion. Using a combination of cellular electron tomography and serial block face-scanning electron microscopy (SBF-SEM) we have reconstructed the entire apical end of Eimeria tenella sporozoites; we report a detailed dissection of the three- dimensional organisation of the conoid and show there is high curvature of the tubulin-containing fibres that might be linked to the unusual comma-shaped arrangement of protofilaments. We quantified the number and location of rhoptries and micronemes within cells and show a highly organised gateway for trafficking and docking of rhoptries, micronemes and microtubule-associated vesicles within the conoid around a set of intra-conoidal microtubules. Finally, we provide ultrastructural evidence for fusion of rhoptries directly through the parasite plasma membrane early in infection and the presence of a pore in the parasitophorous vacuole membrane, providing a structural explanation for how rhoptry proteins may be trafficked between the parasite and the host cytoplasm.

A novel method of curvature demodulation for Mach-Zehnder interferometer optical fiber curvature sensors

This paper constructs a novel curvature demodulation method for Mach-Zehnder interferometer (MZI) optical fiber sensors. The method is composed of two parts: spectral curve decomposition and support vector machine regression (SVR) of curvature. The authors fabricated the MZI fiber optic curvature sensor and carried out curvature loading experiments. The curvature demodulation method proposed in this paper includes the following steps: spectral data acquisition, data normalization, coefficient vectors calculation, and curvature calculation. The minimum root mean square errors (RMSE) of the curvature demodulation method is 0.02133 m−1. This method has two characteristics: wide curvature demodulation range and unitive curvature regression expression. The curvature demodulation method is not only limited to the MZI sensor curvature demodulation but also applicable to the demodulation of other physical quantities with the spectral data as input. This paper provides a new idea for solving similar demodulation problems.

Theory of phase transmission fibre-optic deformation sensing

SUMMARYWe present a theory and conceptual examples for fibre-optic deformation sensing based on phase changes of transmitted light. As a first result, we establish an exact relation between observable phase changes and the deformation tensor along the fibre. This relation is nonlinear and includes effects related to both local changes in fibre length and deformation-induced changes of the local refractive index. In cases where the norm of the deformation tensor is much smaller than 1, a useful first-order relation can be derived. It connects phase changes to an integral over in-line strain along the fibre times the local refractive index. When spatial variations of the refractive index are negligible, this permits the calculation of phase change measurements from distributed strain measurements, for instance, from distributed acoustic sensing (DAS). An alternative form of the first-order relation reveals that a directional sensitivity determines the ability of a point along the fibre to measure deformation. This directional sensitivity is proportional to fibre curvature and spatial variability of the refractive index. In a series of simple conceptual examples, we illustrate how a seismic wavefield is represented in a phase change time-series and what the role of higher-order effects may be. Specifically, we demonstrate that variable curvature along the fibre may lead to a multiplication of seismic waves, meaning that a single seismic wave appears multiple times in a recording of optical phase changes. Furthermore, we show that higher-order effects may be observable in specific scenarios, including deformation exactly perpendicular to the fibre orientation. Though higher-order effects may be realized in controlled laboratory settings, they are unlikely to occur in seismic experiments where fibre geometries are irregular and waves asymptotically propagate in all directions with all possible polarizations as a consequence of 3-D heterogeneity. Our results provide the mathematical foundation for the analysis of emerging transmission-based fibre-optic sensing data, and their later use in seismic event characterization and studies of Earth structure.