# High-Sensitivity Displacement Sensor Using Few-Mode Optical Fibers and the Optical Vernier Effect
###### Abstract  
This paper presents a displacement sensor designed to achieve the Optical Vernier Effect (OVE) through a simple yet robust configuration, enhancing sensitivity and precision in small displacement measurements. The sensor structure comprises a few-mode fiber (FMF) placed between two single-mode fibers (SMF) in an SMF-FMF-SMF (SFS) configuration. A series of distinct configurations of concatenated Mach-Zehnder fiber interferometers (MZFI) were examined, with the lengths of the reference FMF (_FMF\({}_{Rf}\)_) and sensing FMF (_FMF\({}_{Sen}\)_) adjusted to track the spectral envelope shifts. The results demonstrate that the direction of the spectral shift is governed by the ratio between the _FMF\({}_{Rf}\)_ and _FMF\({}_{Sen}\)_ lengths. The sensor achieved a sensitivity of up to 39.07 nm/mm and a magnification factor (_M_ factor) of up to 50.09, demonstrating exceptional precision and adaptability across a range of applications. The proposed configuration also enhances the overall sensor performance, highlighting its potential for broader use in fields requiring precise displacement monitoring. optical vernier effect; few-mode fiber; displacement sensor; optical fiber sensor; Mach-Zehnder interferometer 20241493  
fiber (PMF) and a fiber modal interferometer (FMI) with an SFS structure. [PERSON] et al. [5] demonstrated a static pressure sensor employing two critical-length SFS structures, which provided temperature compensation. [PERSON] et al. [6] developed a high-sensitivity temperature sensor utilizing tapered two-mode fibers (TTMFs) arranged in series, with each TTMF in an SMF-TTMF-SMF configuration. [PERSON] et al. [7] presented a refractive index sensor using FPIs comprising FMF for the sensor and a hollow capillary tube placed between SMFs for reference. [PERSON] et al. [8] demonstrated a temperature sensor by concatenating two interferometers consisting of a glass capillary tube that connects SMF with FMF, coated at the end with polydimethylsiloxane (PDMS). [PERSON] et al. [9] proposed a temperature sensor based on FMF filled with PDMS, utilizing two FPIs with an SMF-air microcavity-FMF-PDMS-FMF structure. [PERSON] ([PERSON]) et al. [10] introduced a novel approach to improve sensitivity by employing a simulated reference arm (SRA) technique, achieving the OVE with a single Multimodal Interferometer (MMI) comprising FMF and a simulated reference signal. [PERSON] et al. [11] also designed an ultra-sensitive strain sensor with two parallel FPIs, where the sensor FPI was composed of TTMF, and the reference FPI included a quartz capillary between two SMFs, forming air cavities. [PERSON] ([PERSON]) et al. [12] reported a highly sensitive salinity sensor based on the virtual Vernier effect in a TTMF fiber with an SMF-TTMF-SMF structure. While these approaches offer significant advances, certain limitations persist. The use of cascaded interferometers to achieve the OVE introduces complexities in system calibration and measurement [4]. Additionally, multimode fiber interferometers are susceptible to noise from environmental variations [5; 10], and the fabrication of tapered fibers remains a challenging process, often compromising sensor durability over time [6; 11]. The use of FPIs with cavities demands complex configurations, requiring greater calibration effort [7]. Recent OVE research has incorporated computational tools for signal prediction and generation, leading to more refined envelope definitions, as demonstrated in the works of [PERSON] and [PERSON] et al. Nonetheless, challenges remain in the precise and reliable extraction of useful signal information, as well as in managing environmental interference on physical reference arms, the high costs and complexity of conventional demodulation, and the stringent requirements for sensor fabrication and interrogation systems [10; 12]. In this work, we propose the concatenation of two MZIs based on SFS sections to achieve the OVE and its harmonics.