# In Situ Study of the Magnetic Field Gradient Produced by a Miniature Bi-Planar Coil for Chip-Scale Atomic Devices

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Coil Design

#### 2.2. Theory of the Gradient Measurement: Relaxation of Nuclear Spins under Inhomogeneity Magnetic Field

## 3. Experimental Setup

#### 3.1. The Measurement Platform

#### 3.2. Coil Fabrication

#### 3.3. Sensor Head Fabrication

## 4. Results

#### 4.1. Coil Constants Measurement

#### 4.2. Magnetic Field Gradient-Induced Relaxation Measurement

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The figure shows the design of the bi-planar coil for the x and y directions, where the red and blue lines represent different directions of current flow.

**Figure 2.**The figure displays the design of the bi-planar coil for the z direction, where the red lines represent the same direction of current flow.

**Figure 4.**The experimental setup for testing the bi-planar coil. PD: photo diode, NDF: neutral density filter, PBS: polarization beam splitter, GT: Glan–Taylor prism, PEEK: polyether ether ketone, PM: polarization maintaining. The bi-planar coils are set in the $xy$ plane. The two planes of the bi-planar coils are in parallel. The three pair bi-planar coils can produce three dimension magnetic fields. The $cos\left(\theta \right)$ coils are in a cylinder shape. The cylinder axial direction is the same as the heating laser beam direction. The $cos\left(\theta \right)$ coils [39] are arranged in such a way that they can produce z and y direction homogeneous magnetic fields. The cylinder also configures a Lee-Whiting coil, which could produce an x direction homogeneous magnetic field [40].

**Figure 5.**Typical Free Induction Decay (FID) signal of ${}^{131}$Xe nuclear spins under a holding magnetic field produced by the bi-planar coil ${B}_{y}$.

**Figure 6.**The relationship between the current in the bi-planar coil and the magnetic field strength measured by the precession frequency of ${}^{131}$Xe nuclear spins for the x or y direction.

**Figure 7.**The FID signal of the ${}^{131}$Xe nuclear spins under a y direction bigger coil ($cos\left(\theta \right)$ coil) system with more uniform magnetic field gradients. The nuclear quadrupolar interaction induced beating signals can be resolved. The relaxation of the nuclear spins is smaller.

**Figure 8.**The FID signal of the ${}^{131}$Xe nuclear spins under a bigger coil system with more uniform magnetic field gradients. The magnetic field was in the z direction. The relaxation rate was 0.30 ${\mathrm{s}}^{-1}$.

**Figure 9.**The FID signal of the ${}^{131}$Xe nuclear spins under the z bi-planar coil. The relaxation rate was 0.50 ${\mathrm{s}}^{-1}$.

**Figure 10.**The magnetic field gradient produced by the y direction bi-planar coils. Due to symmetry, the x direction field gradient is similar. The color represents the error of the magnetic field, which is defined in Equation (2).

**Figure 12.**The more detailed plotting of the magnetic field gradient produced by the y direction bi-planar coil. ‘y gradient’ represents $\partial {B}_{y}/\partial y$ and ‘z or x gradient’ represents ‘$\partial {B}_{z}/\partial z$ or $\partial {B}_{x}/\partial x$ in the figure. Due to symmetry, the x direction bi-planar coil is similar.

**Figure 13.**The more detailed plotting of the magnetic field gradient produced by the z direction bi-planar coil. ‘z gradient’ represents $\partial {B}_{z}/\partial z$ and ‘x or y gradient’ represents $\partial {B}_{x}/\partial x$ or $\partial {B}_{y}/\partial y$ in the figure.

**Figure 14.**The relation between the magnetic field and the relaxation rate for the y bi-planar coil. Here, we did not directly measure the relationship between the magnetic field gradient and the relaxation rates. However, we directly measured the relationship between the magnetic field and relaxation rate because the magnetic field gradient can be calculated to be ${\u03f5}_{avg}B/0.2$ cm in our design. The x bi-planar coil is similar.

Direction | Coil ${}^{\mathit{a}}$ | Precession ${}^{\mathit{b}}$ | Theoretical |
---|---|---|---|

${B}_{y}/{B}_{x}$ | 49 nT/mA | 51 nT/mA | 42 nT/mA |

${B}_{z}$ | 535 nT/mA | 574 nT/mA | 476 nT/mA |

Item | ${\mathit{B}}_{\mathit{x}}$ (Lee-Whiting) | ${\mathit{B}}_{\mathit{x}}$ (bi) | ${\mathit{B}}_{\mathit{z}}$ (cos) | ${\mathit{B}}_{\mathit{z}}$ (bi) |
---|---|---|---|---|

Slope of curve | 5979 mV/mA | 1226 mV/mA | 2479 mV/mA | 16,600 mV/mA |

Coil constant | 240 nT/mA | 49 nT/mA | 80 nT/mA | 535 nT/mA |

Coil Type ${}^{\mathit{a}}$ | ${\mathit{B}}_{\mathit{y}}$ and ${\mathit{B}}_{\mathit{x}}$ Relaxation | ${\mathit{B}}_{\mathit{z}}$ Relaxation |
---|---|---|

Big coil | 0.30 ${\mathrm{s}}^{-1}$ | 0.30 ${\mathrm{s}}^{-1}$ |

Bi-planar coil | 0.50 ${\mathrm{s}}^{-1}$ | 0.50 ${\mathrm{s}}^{-1}$ |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Wang, J.; Zhang, N.; Wang, J.; Ma, Y.; Yu, M.; Wang, Y.; Zhao, L.; Jiang, Z.
In Situ Study of the Magnetic Field Gradient Produced by a Miniature Bi-Planar Coil for Chip-Scale Atomic Devices. *Micromachines* **2023**, *14*, 1985.
https://doi.org/10.3390/mi14111985

**AMA Style**

Chen Y, Wang J, Zhang N, Wang J, Ma Y, Yu M, Wang Y, Zhao L, Jiang Z.
In Situ Study of the Magnetic Field Gradient Produced by a Miniature Bi-Planar Coil for Chip-Scale Atomic Devices. *Micromachines*. 2023; 14(11):1985.
https://doi.org/10.3390/mi14111985

**Chicago/Turabian Style**

Chen, Yao, Jiyang Wang, Ning Zhang, Jing Wang, Yintao Ma, Mingzhi Yu, Yanbin Wang, Libo Zhao, and Zhuangde Jiang.
2023. "In Situ Study of the Magnetic Field Gradient Produced by a Miniature Bi-Planar Coil for Chip-Scale Atomic Devices" *Micromachines* 14, no. 11: 1985.
https://doi.org/10.3390/mi14111985