A novel uniform magnetic field coil structure with two pairs of saddle coils nested is proposed in order to minish the aspect ratio without losing much field uniformity, which is significant in innovative miniature atomic instruments and sensors. Optimal configuration parameters are obtained from the Taylor expansion of the magnetic field. The remainder terms are used to estimate the field uniformity and optimize the configuration while minishing computation time. Compared with traditional saddle coils, the nested saddle coils have unique advantages in miniature applications where the aspect ratios are strictly limited. The optimized nested saddle coils were manufactured by Flexible Printed Circuit (FPC) technology and the magnetic field uniformities were verified experimentally using a flux-gate magnetometer. Furthermore, the example application of a miniature nuclear magnetic resonance (NMR) gyro demonstrates the practical use of the nested coils.

## I. INTRODUCTION

A transverse magnetic field whose direction is perpendicular to the axis of one thin cylindrical shell is widely used in many atomic systems, e.g. Magnetic Resonance Imaging (MRI), NMR spectrometer, atomic magnetometers and atomic gyros.^{3,4,11,12,14,16} A three-dimensional magnetic field system can be readily produced by two orthogonal transverse magnetic field coils and a solenoid coil by multiplexing the same cylindrical construction. In most atomic sensors, a highly uniform magnetic field is required and high-resolution experiments usually demand field homogeneities of a few parts per million within a (sample) volume of about a cubic centimeter.^{1}

We focus on the design of a miniaturized transverse uniform field coil which is used in atomic gyros, atomic magnetometers or other high precision atomic sensors. Unlike in MRI applications where high power transverse RF magnetic fields are usually needed, some weak magnetic fields are needed in high precision atomic sensors. Meanwhile, due to the use of optical detection methods in high precision atomic sensors, transverse coils needn’t be used as RF detection coils. Therefore, coil constants needn’t be very large to offer large detection sensitivities. In addition, in view of the fact that currents are usually low in high-precision atomic sensors, some electrical characteristics such as resistances and inductances of the coil are ignored temporarily in this paper. For high precision atomic sensors, shortness and field uniformity of the transverse coil are the most important things.

A saddle coil is a kind of widely used traditional transverse magnetic field coil. Geometrical optimization of a saddle uniform magnetic field coil has been widely studied. The central magnetic field of a saddle coil and its second derivatives are given as functions of the coil geometrical dimensions in the paper of D. M. Ginsberg et al. And a saddle coil with an aspect ratio of 2 and circular arcs of 120° with no second derivatives of central field in any direction, in order to ensure the uniformity of the field, has been proposed.^{5} Based on the studies of D. M. Ginsberg, the fourth order Taylor expansions of the magnetic field, which can be used to approximately evaluate the degree of field homogeneity, are derived by F. Bonetto et al.^{1}

In addition to this method for obtaining uniform magnetic fields via the Biot-Savart law, there is also a method based on the vector potential in a closed form with elliptic integrals of the first and second kind, developed by H. Hanssum.^{6} Further optimization and analysis similar to the method of D. M. Ginsberg can be performed analytically.^{7–9} Moreover, spherical harmonic expansion, as a separate method for magnetic field uniformity analysis, is also a very important method with theory and project signality.

Not only has the optimal geometry of the saddle coil (i.e., with the most uniform field) been widely studied, but also the uniformity of the field has been discussed when the optimal geometry cannot be reached. In these cases, it is desirable to improve the uniformity as much as possible. A configuration of saddle coils with an aspect ratio of 1.75 is proposed by Seungmun Jeon et al. with the goal to balance the trade-off between coil height and field uniformity.^{10} Certain practical guidelines are proposed briefly by D. M. Ginsberg and F. Bonetto.^{1,5}

In this paper, a novel structure with two pairs of saddle coils nested and an analytical optimization study with the goal to minish the aspect ratio of the coil without losing field uniformity are presented. Although some advanced numerical methods, such as target field method, whose current distribution can be placed on the surface of the cylinder arbitrarily have been widely used in the design of coils, the traditional analytical method still has its own meaning, especially considering the simplicity of the coil simultaneously. This novel configuration is obtained via Taylor series expansion of the magnetic field and optimization of the remainder terms of the expansion. Lastly, the validity and practicability are illustrated with an actual test and an application demonstration in a miniature NMR gyro.

## II. METHOD

### A. Basics of saddle coils

A basic saddle coil system consists of two equal rectangular coils confined to a cylindrical surface (shown in Fig. 1). For a well-designed geometry of the coil that carry the same current, a uniform magnetic field or a gradient magnetic field is generated, depending on whether the current direction is set in Helmholtz or anti-Helmholtz configuration. Central angle and aspect ratio are the only two important geometric parameters of a saddle coil, since the diameter only determines the coil size. In this paper, a saddle coil with a diameter *D* (radius *R*), central angle *ϕ*, and height *h* is discussed. In order to simplify the calculation, a coordinate system with the origin located in the geometric center of the coil is used.

Taylor series expansion is among the best tools for the analysis of field uniformity. From the Taylor series expansion, it follows that the derivatives of each order of the central magnetic field, at least the lowest orders, should be zero to ensure the field uniformity. This can be easily done by adjusting the geometric parameters of the coil.

where *i* = *x*, *y*, *z* denotes each component of the magnetic field, *n* denotes the total order of the derivatives, and $\u2211m1+m2+m3=mo(xm1,ym2,zm3)$ denotes *m*-th order infinitesimal of the spatial variables.

In order to obtain the coefficient of each order of the Taylor series expansion, an exact analytical expression of the magnetic field is needed. The exact analytical expression can be derived by applying the Biot-Savart Law to the four arc and straight sections separately.

The magnetic field at the origin of the coordinate system and its lowest derivatives reflect the coefficient of the Taylor series are calculated in order to analyze the magnetic field uniformity.

where $i\u2192$ is the unit vector along the *x* direction, *μ*_{0} = 4*π* × 10^{−7} H/m is the permeability of free space, *N* is the number of turns in the coil, *I* is the current flowing in the coil, *D* is the coil diameter, *h* and *ϕ* are the height and central angle, respectively, and *s* = 1 + (*h*/*D*)^{2} is a parameter describing the aspect ratio.

Due to the axisymmetric nature of the coil, the three-direction component of the magnetic field will have some terms of zero during the Taylor expansion process. *B*_{x} odd-order derivatives with respect to *x*, *y*, or *z*; *B*_{y} odd-order derivatives with respect to *z* or even-order derivatives with respect to *x* or *y*; and *B*_{z} odd-order derivatives with respect to *y* or even-order derivatives with respect to *x* or *z* are all null because of symmetry reasons.^{1} Therefore, only a few derivatives survive. The surviveing second and fourth order derivatives agreeing with D. M. Ginsberg^{5} and F. Bonetto^{1} are calculated as follows:

where *A*, *B*, *C* are introduced to simplify the expression without physical meaning.

The central angle *ϕ* and the aspect ratio *h*/*D*, which is equivalent to the parameter *s*, are the only two important geometric parameters, since the diameter *D* only determines the size of the coil. Therefore, the problem of the field uniformity can be solved by finding certain combinations of *ϕ* and *s* for which the first few derivatives (usually the first two) of the magnetic field at the origin are equal to zero. The first-order derivative of the magnetic field at the origin is zero due to the symmetry naturally. The three components of the second-order derivative should be set to zero by designing an appropriate coil configuration. In particular, it follows from the laws of electromagnetism that the sum of the three components of second derivatives is zero. Therefore, we obtain a system of two independent equations that determine the two geometric parameters *ϕ* and *s*.

A solution with *s* = 5 and *ϕ* = 120° is obtained from this set of equations. With this ideal configuration, field uniformity in the center of the coil can be optimized theoretically. It is worth mentioning that the fabrication tolerance usually causes a loss to field uniformity in a practical application. However, it also follows that the aspect ratio *h*/*D* must be 2, which is a burden for miniature atomic sensors.

### B. Nested saddle coils

#### 1. Design method of nested saddle coils

In this section, the field uniformity is considered in cases where the aspect ratio of the coil is limited by spatial considerations. Therefore, only one variable parameter *ϕ* remains, and the three components of the second derivatives cannot be zero simultaneously.

Certain practical guidelines have also been established in previous research. One configuration of saddle coils with a aspect ratio of 1.75 and *ϕ* = 120° is proposed by Seungmun Jeon et al. in order to miniaturize the aspect ratio of coils while preserving most of the uniformity.^{10} However, the field uniformity rapidly falls off if the aspect ratio is further reduced. D. M. Ginsberg and F. Bonetto showed that the parameter *ϕ* can be set to a value which ensures the second-order derivatives with respect to *x* be zero if the optimal aspect ratio cannot be reached.^{1,5} While field uniformity along the *x* axis can be reached with this optimal design, the field uniformity along the *y* and *z* axes is usually not satisfactory.

Mathematically, it is an overconstrained problem since there are two equations but only one free parameter when the aspect ratio is limited to a fixed value. In order to introduce one or more free parameters, a novel configuration with two saddle coils nested (shown in Fig. 2) is proposed. Although the nested configuration is similar to Birdcage coils or cosine theta coils in shape to some extent, their interests are different. The cosine theta coil replaces the continuous current densities with discrete wires, and also the infinitely long cylinder with a finite aspect ratio. However, the nested configuration tries to make the field uniform using Taylor series expansion analytically. For the nested coils, length and field uniformity are the most important things.

With a nested configuration such as this one, there are five adjustable parameters, the aspect ratios *α*_{1}(*s*_{1}), *α*_{2}(*s*_{2}), the central angles *ϕ*_{1}, *ϕ*_{2} and the ampere-turn ratio of two coils *t* = (*NI*)_{2}/(*NI*)_{1} . As there are only two equations, three of the five parameters should be fixed in advance. In this case, *α*_{1}, *α*_{2} and *t* are chosen in advance to satisfy the aspect ratio and structural simplicity goals. After that, *ϕ*_{1} and *ϕ*_{2} can be solved from the two equations as follows:

It is notable that the value of $si\u221272(si\u22121)12(3si\u221215)\u2009sin\varphi i2(i=1,2)$ is negative, with the limitation of 1 < *s*_{1}, *s*_{2} < 5. Therefore, the ampere-turn ratio of the two coils (*NI*)_{2}/(*NI*)_{1} is negative. In other words, the directions of the current flow in the two coils has to be opposite. To some extent, this is similar to the use of inverted currents in electromagnets to achieve small aspect ratio.

On the one hand, the aspect ratio should be decreased as far as possible to minish the volume of atomic sensors. On the other hand, the coil also should be big enough to hold some necessary devices such as a gas cell and photodetectors, what’s more, the aspect ratio also should be large enough to ensure the field uniformity. The case in which the aspect ratio of the coil is less than or equal to 1 is considered in this paper. Assuming that *s*_{1} = 5/4(*α*_{1} = 1/2), *s*_{2} = 2(*α*_{2} = 1) and (*NI*)_{2}/(*NI*)_{1} = −3, the solution of *ϕ*_{1} = 102.920° and *ϕ*_{2} = 115.187° can be obtained with the nested saddle coils configuration. Moreover, assuming that *s*_{1} = 25/16(*α*_{1} = 3/4), *s*_{2} = 2(*α*_{1} = 1) and (*NI*)_{2}/(*NI*)_{1} = −2, the solution of *ϕ*_{1} = 113.237° and *ϕ*_{2} = 116.709° can be obtained. Significantly, the ampere-turn ratio (*NI*)_{2}/(*NI*)_{1} is set to -3 or -2 to keep coil simplicity, however, each number of turns should be large enough to avoid perturbations in field homogeneity induced by the wires especially when a single turn is used.

The validity of the nested coils is demonstrated by numerical calculation examples. The coil diameter is set to 50 mm and the current is set to 1A. A cube with 10 mm side length located in the center of the coil is considered. In order to give a visual representation that removes the effect of different coil diameters, all lengths are expressed in form of the relative magnitude with respect to the diameter. In addition, the uniformity of each coil is evaluated according to the relative deviation with respect to the center field.

The two nested coils described above and three other existing configurations of single saddle coils with aspect ratios of 2, 1.75 and 1 are analyzed. The single coil with an aspect ratio of 2 is considered to be the ideal configuration, while the coil with an aspect ratio of 1.75 from Ref. 12 is considered as a compromise between the aspect ratio and the field uniformity. The single coil with an aspect ratio of 1 modified with the method proposed in Refs. 6 and 7, with the second derivatives of *B*_{x} with respect to *x* equal to zero is also calculated for comparison. Specific configurations of these five coils are shown in Table I.

Magnetic field uniformities produced by these five coils are shown visually by the following graphics. Fig. 3 shows the magnetic field relative errors of different coils along three coordinate axes. Fig. 4 shows the magnetic field relative errors of different coils on three coordinate planes. Both figures show that although the aspect ratios of the two nested saddle coils are reduced by 50% compared with ideal saddle coil, the field uniformities are still satisfactory. In order to evaluate the magnetic field uniformity of the coil in the entire target region, the probability distribution curve of the magnetic field uniformities in the target region is introduced. The relative errors of all points in the entire target region are considered as probability samples. Fig. 5 shows the probability distribution curves of the magnetic field uniformities of all the five coils. It shows that although the uniform performances of the magnetic field of the nested coils are somewhat reduced compared to the ideal coil, the nested coils are still good choices for small applications considering their smaller aspect ratios.

#### 2. Optimization of nested saddle coils

Nested saddle coils have advantages of balancing the trade-off between the aspect ratio and the field uniformity. However, the number of solutions is infinite since the three parameters *s*_{1}(*α*_{1}), *s*_{2}(*α*_{2}) and (*NI*)_{2}/(*NI*)_{1} are arbitrarily given in advance. Parameters will be optimized in this section to find appropriate configurations. Since the first three derivatives are equal to zero, the fourth becomes the main term of the remainder and can be used to estimate the uniformity of the magnetic field. Furthermore, considering all points in the region, the sum absolute values of fourth-order derivatives at the coil center can be minimized in order to obtain the most uniform field.

Some restrictive conditions must be applied in order to ensure the simple and compact structure. Firstly, the current of the outer and the inner coils has to be set to the same value by connecting them to a single current source. Secondly, the turns of the outer and the inner coils (both are divided by their greatest common divisor), assumed to range from 1 to 5 in this paper, should be few enough to produce a simple structure. Lastly, the aspect ratio of the coils must be low (assumed to be 1 in this paper). Optimization of the nested saddle coils can be expressed in a mathematical form as follows:

With the objective function and multiple constraints, a table of ranking about field uniformity can be obtained. Certain configurations with satisfactory magnetic field uniformity as well as simple structures can be selected in the table of ranking. Since the subjective criterion of structure simplicity and the restricted processing technic, the rank is changeable. Therefore, the table of ranking is not shown in this paper. Although the two nested saddle coil configurations introduced in this paper are not ranked on top, they represent a trade-off between the field uniformity, simplicity and convenience of the structure.

## III. EXPERIMENTAL

### A. Coil test with a flux-gate magnetometer

The two optimized nested coils with a diameter of 126 mm are manufactured by double sided Teflon FPC technology whose thickness of each layer is only 0.1 mm. Then the coils are glued on the surface of a cylindrical frame and tested with a flux-gate magnetometer (Fig. 6) in order to experimentally verify the effectiveness of the designs. The magnetic induction intensities at various points along the *x* axis with a range of $\u22120.6R,0.6R$ are measured with the flux-gate magnetometer while a suitable current is flowing through the coils. In the range of $\u22120.2R,0.2R$, which is the most important and interesting region, the measurements are taken at greater spatial density. Certain steps are taken to ensure the accuracy and reliability of the measurements. Firstly, all measurements are taken late at night in order to avoid electricity peak demand. Secondly, the probe of the flux-gate is placed on a ruler to ensure the fixed position of the probe. Thirdly, the magnetic coils are oriented parallel to the circles of latitude, in order to avoid interaction with the Earth’s magnetic field.

Compared with Hall effect sensors, flux-gate magnetometers have higher measurement accuracy, especially in weak magnetic measurements. The flux-gate magnetometer used in this paper is produced by Bartington Instruments (Mag-03). This flux-gate magnetometer has a measuring range of -100 *μ*T-100 *μ*T and a low noise below 6 pTrms/Hz^{1/2} at 1 Hz. LDC 205C produced by Thorlabs is selected as the current source. LDC 205C is a current source designed for laser diode which can supply up to 500 mA current. The AC modulation and demodulation are introduced to avoid the effect of the fluctuation of environmental magnetic field. Therefore, HF2 LI Lock-in Amplifier produced by Zurich Instruments is used. HF2 LI has an A/D conversion of 14 bit, 210 MSa/s, a input noise voltage 5 nV/Hz^{1/2} (above 10 kHz) and a dynamic reserve of 120 dB. The measurements are taken under a AC modulation 68.34 Hz considering the bandwidth of Mag-03 and the noise power spectrum of environmental noise. An AC current with a frequency of 68.34Hz, an amplitude of 200 mA is applied to the nested coils. Then, the magnetic field signal measured by Mag-03 is demodulated to obtain the amplitude of the magnetic field.

The measurements of the two nested coils are shown in Fig. 7 and Fig. 8. Table II and Table III give the full data along with uncertainties. It can be found that all measurements are slightly less than the theoretical value. This may be caused by the gain attenuation of the magnetometer in the band. Nevertheless, the distribution of the magnetic field along the *x* axis is basically consistent with the theoretical curve.

### B. Application demonstration in a miniature NMR gyro

The optimized nested saddle coil is applied in a miniature NMR gyro to validate the usability of the nested coil. Nuclear magnetic resonance gyros based on spin-exchange optical pumping of noble gases have been developed over several decades.^{13,15} The basis for the NMR gyro is the stability of the Larmor precession frequency of the nuclear spin when nuclei are exposed to a constant magnetic field. A NMR gyro consists of a noble gases cell, two lasers, a three-dimensional magnetic field coil system, multilayer magnetic shields and other indispensable attachments. The theory and structure of NMR gyro are not the emphasised issues of this paper, and details of them can be found in many papers.^{4,13,15}

The optimized nested coil with a diameter of 25 mm is manufactured by FPC technology and glued on the surface of a frame. The Lee-Whiting coil is adopted to provide a sufficiently uniform longitudinal magnetic field for NMR gyro. The three-dimensional magnetic field coil structure and assembled NMR gyro are shown in Fig. 9.

To evaluate the usability of the nested saddle coils, experiments using the free induction decay (FID) of noble gases to obtain the coil constant^{2} can be implemented. By applying different pulse width of transverse magnetic field (*B*_{1} field), a series of initial amplitudes of FID signal of noble gases can be gained. The initial amplitudes can be used to fit a sine curve from which the *π*/2 pulse duration can be calculated. The amplitude of the applied pulsed magnetic field can then be calculated. Therefore, by applying different amplitudes of pulsed current to the transverse nested coil, the coil constants can be calibrated.

A measurement of *B*_{1} field is demonstrated by applying a sine pulse current whose amplitude is 7.922 mA and frequency is the Larmor precession frequency of ^{129}Xe or ^{131}Xe. The FID signal of ^{131}Xe, ^{129}Xe and the fitted sine curve used to calculate the *π*/2 pulse duration of ^{129}Xe are showed in Fig. 10. Although the quadrupole of ^{131}Xe does not change anything about the discussion of the coils, the FID signal of ^{129}Xe is used to calculate the coil constants. Then, by applying different levels of current to the transverse nested coil, corresponding amplitude of the *B*_{1} field can be achieved, showed in Table IV and Fig. 11. Therefore, a coil constant of (191.990±1.157) nT/mA is gained. The coil constant of the longitudinal coil, which can be achieved from the frequencies of the FID signals, is a bonus in these measurements. The actual coil constant of the longitudinal coil is (838.675±0.085) nT/mA. However, the theoretical values of the coil constant of the nested coil and the Lee-Whiting coil are 131.635 nT/mA and 718.489 nT/mA respectively. The measured value is 1.459 times of the theoretical value for transverse nested coil, and that is 1.167 times for longitudinal Lee-Whiting coil. Similar magnifications are also measured in a larger experimental prototype whose coil diameter is 50 mm. In this larger prototype, the magnifications for transverse nested coil and longitudinal Lee-Whiting coil are 1.364 and 1.141. These magnifications may be caused by the couple of the coils and the innermost magnetic shielding layer. The diameter of the innermost magnetic shielding layer is 30 mm in the miniature NMR gyro, and that is 64 mm in the experimental prototype. Different intervals between the coils and the innermost magnetic shielding layer may lead to different magnifications. The magnifications are summarized in Table V.

A finite element analysis with Ansys Maxwell has been done to study this enhancement. A reasonable conjecture is that if the magnetic shield and its internal coil are scaled equally, then the magnetic shields should have the same enhancement for the coil constants. This conjecture has been verified by finite element analysis. However, the larger magnetic shield and coil mentioned in this paper are not a proportional amplification of the smaller magnetic shield and coil. Not only has a different size ratio, but also has a different coil structure. The small coil adopts the Nested saddle coil 1 configuration, and the large coil adopts the Nested saddle coil 2 configuration. Therefore, it can be foreseen that even if they have the same size ratio, they may have different enhancements. In Fig. 12, the first model shows the structure and proportion of the small coil, the second model scaled the first model to the size of large coil, the last model shows the structure and proportion of the large coil. The first two model adopt the Nested saddle coil 1 configuration, and the last model adopts the Nested saddle coil 2 configuration. The enhancement of the first model is 1.487, the enhancement of the second model is 1.488, and the enhancement of the last model is 1.422. The first two simulation results are consistent, which proves the previous conjecture. The enhancement of the last model is smaller than the first one, this is also consistent with the experiments. It is worth noting that the first and last simulation results are larger than the measurement values. This may be due to the fact that the meshing is not sufficiently detailed in the finite element analysis. Some mechanism and numerical relationship should be further researched.

The ratio of the Larmor precession frequency of ^{129}Xe and ^{131}Xe is another valuable result obtained in this experiment. In the same experiment, the Larmor precession frequency of ^{129}Xe is (130.420±0.001) Hz and the Larmor precession frequency of ^{131}Xe is (38.659±0.001) Hz. In this experiment, the ratio of the Larmor precession frequency of ^{129}Xe and ^{131}Xe is 3.3736 which is slightly larger than the theoretical value of 3.3734. The spin exchange process between Alkali metal and ^{129}Xe is different with that between Alkali metal and ^{131}Xe. This may be the cause of a slight difference between the measured and theoretical values. In addition, the measurement error of the Larmor precession frequency is another cause of the error. The *T*_{1} time and *T*_{2} time for ^{129}Xe and ^{131}Xe can be further measured. This will be used to determine whether the *T*_{2} time is limited by the non-uniformity of the magnetic field or by the *T*_{1} time.

## IV. CONCLUSION

In this work, we propose a novel coil structure with two pairs of saddle coils nested and an analytical optimization study for making the coil arrangement producing a uniform transverse magnetic field in its center under spatial constraints. Appropriate configurations of this novel kind of coil are obtained via Taylor series expansion of the magnetic field and optimized with the remainder terms of the expansion. With this novel structure, the aspect ratio of the coil can be reduced by 50% without a corresponding reduction in field uniformity. Although the uniform performances of the magnetic field of the nested coils are somewhat reduced compared to the ideal saddle shaped coil, the nested coils are still good choices for small applications considering their smaller aspect ratios. For miniature atomic sensors, such as NMR gyros or atomic magnetometers, this improvement is of great importance. The optimized nested saddle coils are demonstrated in a miniature NMR gyro successfully. In addition, when magnetic shields are needed, nested configurations with three saddle coils have potential for designing coils without magnetic moment which can reduce the coupling of magnetic coils and the magnetic shields.

## ACKNOWLEDGMENTS

This research was supported by the National Natural Science Foundation of China (NSFC) (61627806 and 61227902); by the National Key R&D Program of China (2016YFB0501601).