High-precision sensing requires a balance between accuracy and measurement range, a challenge for interferometric sensors. While interferometry offers exceptional accuracy due to the wave properties of coherent electromagnetic radiation (achieving precision levels better than 10⁻²¹ in gravitational wave detectors and 10⁻¹⁷ in ultrastable lasers), the periodic nature of the interference signal limits the range. This limitation hinders applications in precision manufacturing, biomedical sensing, and structural health monitoring, where both high accuracy and wide range are crucial. This research proposes moderate-coherence sensing, utilizing a laser with limited coherence to overcome the range restriction while maintaining high accuracy. The approach combines the sensitive interference signal with measurements of the interferometric visibility, enabling a significant expansion of the measurement range without sacrificing precision. The experimental validation focuses on length sensing to demonstrate the effectiveness of this novel approach.

The paper references prior work demonstrating the high precision of interferometric sensors in various applications, including gravitational wave detection and ultrastable lasers. It acknowledges the limitations of interferometric techniques regarding measurement range, citing existing methods like active control with movable mirrors to extend the range but highlighting the complexity and bandwidth limitations of these approaches. The authors also cite literature on the use of interferometry in precision manufacturing, biomedical sensing, and structural health monitoring, emphasizing the need for a sensor that combines high accuracy with a wide range and high bandwidth. Finally, the concept of complex lasers with controllable coherence is mentioned, laying the groundwork for the proposed moderate-coherence sensing method.

The core of the methodology lies in using a low-finesse Fabry-Pérot cavity and a Fabry-Pérot laser diode with multiple excited modes. The limited coherence of this laser, determined by the Gaussian medium gain spectrum, is key. The reflected power from the cavity is calculated by superimposing the individual quasi-coherent emission lines. The calculation involves expressing the laser gain as a Gaussian function, calculating the roundtrip single-pass phase for each emitted mode as a function of cavity length, determining the power transmitted through the cavity for each mode, and finally obtaining the total reflected power. The interferometric visibility, defined as the ratio of maximum to minimum power values of the fringes, is a crucial parameter. Simulations explored the dependence of visibility on laser gain width and laser line spacing, revealing that wider laser gains (hence, lower coherence) lead to improved visibility over a larger range. The experimental setup included a temperature and current-stabilized Fabry-Pérot laser diode, an isolator and circulator to prevent back reflections, a fiber beam splitter to provide a reference signal, a balanced detector, and a low-finesse cavity constructed using two wedged silicon wafers. One mirror was fixed, while the other was mounted on a linear stage with piezoelectric actuation for precise length control. Measurements were taken semi-automatically, combining manual offset adjustments with automated piezo-controlled scanning. Data acquisition used a measurement card to record the photodetector signal and optical reference signal at each step. Real-time measurements involved scanning the cavity length over half a wavelength using a ring piezo, capturing fringe maxima and minima to calculate the cavity length using software algorithms. The visibility was computed, and a mid-fringe section was extracted and filtered, allowing for precise calculation of the mid-fringe position using linear regression. Combining this position and the measured visibility via a lookup table enabled determination of the actual cavity length. Accuracy and reproducibility were assessed by performing 50 nanometer-step experiments and 50 repetitive measurements at different cavity lengths.

The experimental results strongly validated the theoretical predictions. The measured visibility closely matched the theoretical calculations, confirming the dominant role of coherence modulation. The system achieved an accuracy of 1 nm over a measurement range of 120 µm, corresponding to a relative uncertainty of less than 0.00083%. Real-time measurements demonstrated a measurement bandwidth exceeding 20 kHz. The nanometer-step experiments and repetitive measurements confirmed the high accuracy and reproducibility of the system, even at longer cavity lengths where visibility is reduced. Standard deviations in repetitive measurements were less than ±0.3 nm for shorter distances and less than ±0.6 nm for the longest distance (102.782 µm). The slight increase in error at longer distances is attributed to decreased visibility and smaller mid-fringe increments.

The results demonstrate the success of the moderate-coherence sensing technique in achieving a previously unattainable combination of high accuracy, wide measurement range, and high bandwidth. The close agreement between experimental and theoretical results validates the theoretical model and demonstrates the dominant influence of coherence modulation on the system's performance. The real-time capability opens the door for applications requiring dynamic length monitoring. The technique's versatility extends beyond length sensing to any application leveraging changes in cavity length, such as temperature, acceleration, and pressure sensing. The high accuracy and wide range achieved overcome limitations of traditional interferometric techniques, offering significant advantages in diverse fields.

This study introduces and validates moderate-coherence sensing, a technique combining high accuracy, wide range, and high bandwidth in optical cavity-based measurements. Experimental results confirm sub-nanometer accuracy across a 120 µm range with a bandwidth over 20 kHz. Future work could explore applications beyond length sensing and investigate potential improvements in accuracy and range by optimizing laser parameters and cavity design.

While the study achieved remarkable results, a limitation is the reliance on a lookup table for the mid-fringe position calculation, which could be refined with more extensive simulations and data acquisition. The increased noise at longer distances could be addressed through further optimization of the experimental setup and signal processing techniques. The current study focuses on length sensing; exploring broader applications will be necessary to fully assess the versatility and limitations of the technique in different contexts.