Improvements to the signal-to-noise ratio of magnetic resonance detection lead to a strong reduction in measurement time, yet as a sole optimization goal for resonator design, it would be an oversimplification of the problem at hand. Multiple constraints, for example for field homogeneity and sample shape, suggest the use of numerical optimization to obtain resonator designs that deliver the intended improvement. Here we consider the 2D Lenz lens to be a sufficiently broadband flux transforming interposer between the sample and a radiofrequency (RF) circuit and to be a flexible and easily manufacturable device family with which to mediate different design requirements. We report on a method to apply topology optimization to determine the optimal layout of a Lenz lens and demonstrate realizations for both low- (45 MHz) and high-frequency (500 MHz) nuclear magnetic resonance.

Nuclear MR spectroscopy and imaging are powerful tools for determining the molecular structure of chemical substances or for studying the anatomy of organisms. For MR measurements, it is important to achieve a high SNR to obtain a high-resolution spectrum or a highly resolved image, which also leads to a reduction in the overall measurement time. The relationship between the SNR and the magnetic field produced by the coil was derived by

From the statement above, it is self-evident that reducing the size of the coil will improve its filling factor, which leads to the SNR enhancement, since the strength of Faraday induction increases with a reduction in the distance between the coil and the sample. The miniaturization of coils, as for example discussed by

In some cases, the reception coil and its capacitors cannot be placed too close to the sample or specimen, e.g. when performing MRI on small living organisms or for sensitive spectroscopy of small samples. In such scenarios, the improvement of the filling factor is remedied by using a Lenz lens (LL)

If we focus our attention on the LL, its design needs to be further investigated to improve the amplified field uniformity and to increase the field amplification for cases where the geometrical space for the LL is limited. Although it was shown by

In this paper, we now explore the use of computational optimization to “discover”, via inverse design, a novel distributed metallic track arrangement that produces the same effect as a Lenz lens. The computational procedure will aim to maximize the magnetic field flux (i.e. the lensing effect) in the sample and at the same time aim for a flux distribution that is as constant as possible. As will be shown, these two requirements are in conflict, so a supervisor will have to balance these requirements depending on the application. Furthermore, the design will depart considerably from the Lenz lens topology and may require additional constraints to ensure manufacturability.

Topology optimization has been used in various fields for inverse material design, such as for acoustics

In the field of electromagnetics, topology optimization has been explored for applications such as photonic crystals

However, in surface plasmonics

For our problem formulation, we chose the conductivity function as a material property in the domain rather than on the boundary. The conductivity range between which the material property is interpolated was of the order of

Using the methodology presented in Sects.

The obtained geometries were then characterized, and the simulation results were verified by NMR experiments described in Sect.

In this section, we consider the electromagnetic wave equation that governs the behaviour of the device, derive the equations of the material distribution method, and formulate the objective function with constraint equations with which to obtain an optimized geometrical configuration corresponding to a spatial material distribution.

The time-varying magnetic field

Sketch of the computational domain for the topology optimization of the magnetic lens. The domain in light blue represents the target domain (

Figure

After defining the wave equation and the material distribution equations, control equations were defined to meet the requirements of the MR experiments, i.e. to have a uniform

From the above discussion, the optimization problem formulated can be said to be (oxymoronically) a minimization–maximization process, where the minimization of the objective function

The material properties applied to the different domains are represented in Table

Material properties assigned to the different domains for the computation of the optimized geometrical configurations.

The optimization computations were carried out in the commercial software COMSOL MULTIPHYSICS (V5.4) using its AC/DC and Optimization modules. The simulations were computed using an Intel(R) Xeon(R) Silver 4210 CPU with a processing speed of 2.2 GHz and a RAM size of 64 GB on a 64-bit Windows 10 operating system.

The computational domains were meshed with linear elements to exploit the efficiency of the linear discretization of the magnetic vector potential. This reduced the total computational cost of the simulations. The simulations were formulated such that they follow an iterative procedure as represented by the flowchart in Fig.

The material properties were initialized in different domains.

The initial value of the design variable value was set to be 0.5. It was filtered using Eqs. (

The wave equation was solved using the initial value of the design variable, after which it was updated using the method of moving asymptotes (MMA)

The tolerance for the objective function error was set to be a very small number to allow the conditions of Eq. (

In an MR experiment, the net magnetization of the sample, which is aligned along the

The background magnetic vector potential becomes

The dimension of the entire computational domain was

Figure

A magnetic lens was also designed for an 11.7

The background field was set as defined before in Sect.

Figure

After the designs of OLs were obtained, they were characterized and compared with a wired LL similar to that discussed by

To characterize the magnetic field distribution and its enhancement, a second simulation environment was set up where the background field was replaced by the magnetic field produced by a realistic coil geometry. The boundaries of the OLs were truncated by an IBC, and electromagnetic properties of

An OL is designed to enhance a unidirectional magnetic field. It focuses the magnetic field for coils exhibiting this property (Fig. S2), but for characterization, only a solenoidal coil type was used since the OL properties would be similar to the other coils. Figure

The setup used for characterizing the OL:

To compare the OLs with the LLs, geometries similar to those as shown in the inset of Fig.

The OL was then replaced with an LL. The LL had an outer diameter of 19 mm, with an outer to inner diameter ratio of 5.59. The total magnetic flux calculated for this arrangement was 139.55

The LL was found to achieve better enhancement of the field compared to the OL. However, the field distribution for the OL was more uniform. The field uniformity was calculated as the deviation from the

Comparison summary between the OL and the LL. For the LL, the values in brackets represent the ratio of the outer to the inner diameter.

By increasing the frequency of operation for this particular arrangement to 500 MHz, the LL produced a magnification of 3, which is slightly higher than at 45 MHz, and the OL produced a magnification of 3.9. As can be seen from the design due to the asymmetric material distribution, with the OL the field distribution of the magnetic field was less central. If the region of interest is reduced such that the variation lies below 10 %, the OL at higher frequencies can still be used; therefore, depending on the application, one can also use the magnetic lens designed for 45 MHz at 500 MHz to get a higher amplification if maintaining uniformity is not a concern or a smaller sample volume can be used. In order to get a uniform field distribution following the same protocol for the optimization as described in Sect.

The magnification of the magnetic field produced by the LL depends directly on the ratio of the outer to inner diameter. For 1.05 T (Bruker ICON) measurements, the ratio of the coil size to the sample size was large enough to have a higher amplification by the LL. If we reduce the size of the coil by keeping the sample dimensions the same, as was the case for 11.7 T (Bruker AVANCE (III)) measurements, where a commercially available Bruker's 10 mm saddle coil was used, this leads to a reduction in amplification produced by the LL. The OL designed in Sect.

To verify this, the OL and the LL were analysed in a solenoidal coil arrangement. The outer radius of the coil was 6 mm. The two rings were separated by a distance of 3.2 mm. The coils were excited with an alternating current oscillating at a frequency of 500 MHz.

The total magnetic flux in the volume of the sample calculated using Eq. (

By replacing the OL with an LL whose outer diameter was 7.6 mm and outer to inner diameter ratio 2.24, the total magnetic flux was 322

Therefore, with the reduction in the outer to inner diameter ratio, the total magnification produced by the LL is also reduced. However, the OL was still able to maintain the defined amplification. The maximum variation along the central lines calculated from Eq. (

To summarize the discussion, the LL was found to have a higher magnification, but the field distribution was less uniform. When the ratio of the outer to the inner diameter for the LL is reduced, it produces a lower magnification compared to the optimized magnetic lens albeit the field uniformity of these devices was similar. Table

From the above discussion a question arises: why not set a reference field to achieve an amplification of 5 times rather than a mere factor of 2? The reason a reference field was not set higher is that this leads to the material not being properly distributed and we get undefined conductivity values, i.e. at greyscale values besides 0 or 1.

After processing the designs from the simulations, they were fabricated and verified with NMR experiments using distilled water as a test sample.

The masks for the designs were printed on butter paper using an HP Laserjet Enterprise P3015 dn printer.

Using the mask for UV lithographic patterning, the designs were copied onto a positive photosensitized copper board with FR4 laminate with a PCB thickness of 1.6 mm and a

The board was etched in a sodium persulfate solution (

For NMR characterization, a 0.5

The low-frequency magnetic resonance measurements were acquired by placing the OL in a solenoidal coil, whereas for the high-frequency measurements a saddle coil was used, as shown in Fig.

Both OLs featured self-resonance frequencies in the

For the NMR experiments, all acquisitions were done in single shot without any averaging. The coils were positioned at the iso-centre of the

Next, the OL was introduced in the coil. The shimming profile had to be re-adjusted in order to obtain a similar spectral line width for both experiments. With the same volume of the sample, acquisition time, and power to the coil, a second nutation spectrum experiment was acquired to determine the change in 90

The relative intensities of the two arrangements were determined from the areas under the spectrum for a 90

Values calculated from the nutation spectra of water at Larmor frequencies of 45 and 500 MHz, respectively. The values represent the ratio of the SNR,

The measurements at 45 MHz proved difficult, mainly due to the large magnetic field drift experienced for the ICON system, exacerbated by the lack of a lock channel on the device. The strategy was to acquire nutation spectra as quickly as possible, i.e. at a rather large step size, to estimate the 90

It is hardly a surprise that the quest for more signal-to-noise from an existing NMR detector arrangement without changing other conditions like sample volume, radiofrequency power applied, coil geometry, etc., is a matter of numerical optimization of a passive element which can improve the filling factor of the coil. Topology optimization offers a feasible pathway with which to reach optimal designs that goes beyond mere intuition, and we could show, using a commercial finite-element tool, that it is possible to find practical Lenz lens arrangements that, when implemented, achieve their set goals. The topologies found form a compromise between signal enhancement and field uniformity. Of course, it would have been possible to extend the Pareto front (set of all Pareto-efficient solutions,

We only discussed the use of optimization to enhance the magnetic field of an MR coil, but of course the methodology could be further extended to design a self-resonant structure in order to avoid the tedious task of matching to the characteristic impedance of the coaxial cable (usually 50

An important aspect is the ability to achieve manufacturable designs. For the case where a design is essentially a 2D metal patch on a dielectric sheet, printed circuit boards are an inexpensive route towards implementation, easy to manufacture, and lead to satisfying results for arbitrary embedded topologies. However, for the case of 3D topologies, i.e. to find the material distribution in 3D space, the situation is quite different, and not all geometries found will be manufacturable. For example, to minimize eddy current losses, by breaking the continuity of the induced current (due to varying magnetic fields) away from the sample region, designs tend to evolve towards tiny disconnected islands of metal, arranged in a dielectric background or metal patch suspended in air. This would require very advanced 3D printing, and in some cases the designs might not even be pragmatic. We did not pursue such designs in this contribution, but the message should be clear. Optimization must include manufacturing constraints in order to achieve feasible designs.

Beyond manufacturability, a design must be practicable in use, which further limits the design freedom, because the sample must be provided with a convenient way in and way out of the sensitive volume of the detector. We found no problems with 2D designs, but 3D designs posed a challenge, resulting in designs that left too little space.

The general conclusion is that all aspects of a design must be mathematically expressible as a goal function in order to be considered, but, as more terms are added to the optimization goal, the numerical convergence process slows down, eventually reaching a standstill.

Because computational electromagnetics is scale invariant, the topology optimization methodology is applicable to resonator arrangements beyond the range of typical nuclear magnetic resonance frequencies, applications such as the design of magnetic or electric resonators used in electron paramagnetic resonance (EPR), the optimization of individual wavepath components including capacitors and strip lines, and even for wireless energy transfer.

Simulation and measurement data used for the figures are
available at

The supplement related to this article is available online at:

The concept was initialized by SW, DM and JGK. Topology optimization and post-processing simulations were set up by SW, YD and JGK. Fabrication was done by SW. Initial measurements were performed by SW and ON. Measurements were performed by MJ and SW. Funding request and supervision were done by JGK. Writing, review, and editing were done by SW, MJ, YD, DM and JGK.

The authors declare that they have no conflict of interest.

The authors would like to thank the Karlsruhe Institute of Technology for its continued support and also for providing a safe working
environment during the COVID-19 pandemic and their respective financial
sources. We would also like to use this opportunity to express our gratitude
to the

This research has been supported by the Deutsche Forschungsgemeinschaft (METACOILS (grant no. KO 1883-20)), the Deutsche Forschungsgemeinschaft (grant no. KO 1883/29-1), the European Union's Horizon 2020 research and innovation programme (TISuMR (grant no. 737043)), the Alexander von Humboldt-Stiftung (grant no. 1197305), the KIT (Virtual Materials Design, VIRTMAT), the Deutsche Forschungsgemeinschaft (DFG) (OptiMuM (grant no. KO 1883/39-1)), and the DAAD.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Perunthiruthy Madhu and reviewed by two anonymous referees.