# Enhancing Precision in SQUID Sensors: Analyzing Washer Geometry Dependence at the Microscale
## 5 Discussion  
In this study, we have endeavored to establish a methodology that integrates numerical computations with the SuperScreen library to simulate and optimize washer coils within SQUIDs. Our analysis encompasses simulations of various washer coil shapes, with a specific focus on their magnetic field response, screening current behavior, and self-inductance. In traditional SQUID design practices [36], the exploration of washer geometric  
Figure 12: Comparison of the signal-to-noise ratio (SNR) as a function of the washer diameter for different shapes in SQUID. Compares the signal-to-noise ratio (SNR) for square (blue solid line), octagonal (green solid line), and circular (red solid line) washers across diameters ranging from 150 \(\upmu\)m to 250 \(\upmu\)m.  
Figure 13: Correlation matrices for washers of different shapes: **(a)** Heat map for a square washer. **(b)** Heat map for a circular washer. **(c)** Heat map for an octagonal washer. These matrices display correlations among the parameters. The correlation coefficients range from \(-1\) (complete negative correlation) to \(+1\) (complete positive correlation), with the color intensity indicating the strength of the correlation. parameters has often been limited to rudimentary numerical calculations of inductance, relying on rough empirical formulas. However, this approach lacks the depth required to comprehensively understand how various physical effects influence device performance. Such limitations hinder the precise determination of subsequent design parameters, as inaccuracies in washer inductance can significantly impact the selection of shunt resistors and the fabrication process of Josephson junctions, ultimately affecting the sensitivity and noise levels of the final device. To address these shortcomings, our study introduces an innovative approach that utilizes a Python package for the first time to model, simulate, and visualize the electromagnetic responses, screening currents, inductance coefficients, and other critical parameters of different washer geometries. Using the research methods outlined above, we have drawn conclusions regarding the geometric dependence of washers, highlighting the advantages of octagonal and circular washers over traditional square washers in device applications. The simulation results reveal that the magnetic field at the boundary of the square washer is 0.23 mT, compared to 0.20 mT for the circular washer and 0.18 mT for the octagonal washer, indicating that the optimized octagonal washer significantly reduces boundary effects. These findings demonstrate that octagonal washers not only mitigate boundary effects more effectively but also exhibit greater stability throughout the manufacturing process. In contrast, square washers, under magnetic dipole field conditions simulating real-world applications, exhibit vortex formation at their edges, diminishing screening currents and impacting the conversion efficiency from captured magnetic flux to output voltage. Additionally, our analysis reveals that the significantly higher slope of the inner diameter versus inductance curve for the square washer at 2.8391, compared to octagonal and circular washers, is detrimental to device design. Our analysis highlights significant deviations between the simulated inductance values and those predicted by empirical formulas, most notably in square washers with an 80.8% error, followed by circular and octagonal washers. The differences we noted stem primarily from the pronounced edge effects introduced by the acute angles in polygonal washers, markedly influencing their inductance. Furthermore, our research indicates that excessively enlarging the washer's inner diameter not only increases device noise but also leads to problematic coupling with subsequent superconducting electronic circuits. This effectively undermines the advantages of an increased effective area, resulting in diminished magnetic flux-to-voltage conversion coefficients. These issues manifest as low sensitivity and high device noise within the SQUID system.