1 Introduction

Nuclear magnetic resonance (NMR) is one of the most versatile and specific analytical techniques with applications in physics, chemistry, biology, and medicine. NMR uses the spins of nuclei as very-sensitive, nanoscopic probes inside a molecule that react to the smallest changes in their environment. Although NMR instrumentation has undergone more than 70 years of development, with major breakthroughs in radio frequency electronics greatly improving the performance of modern NMR spectrometers compared to their predecessors, one challenge that remains only partially solved is the reduction of the receiver dead time after an excitation pulse in pulsed NMR. In systems that use the same coil for transmitting (TX) and receiving (RX), this dead time is directly linked to the (large) energy stored inside the coil when producing the (large) \(B_1\)-fields required for non-selective pulses. In NMR relaxometry, excessive dead times prevent the detection of spins with short relaxation times, making it, e.g., more difficult to assess materials, such as glass-fiber-reinforced plastic or carbon-fiber-reinforced plastics [1] and gray cement [2]. Similarly, in magnetic resonance imaging (MRI), long dead times after the pulse render tissue imaging with short relaxation times, such as lung tissue, bones, and teeth challenging [3]. Finally, in NMR spectroscopy, nearby paramagnetic species can result in nuclear spins with very short relaxation times, frequently preventing the conventional NMR on paramagnetic molecules [4]. Techniques to reduce the dead time after the pulse include the use of geometrically decoupled TX and RX coils [5], Q switches that enable faster discharge after the excitation pulse [5, 6], the use of untuned [7] and/or miniaturized NMR coils [8,9,10], and the active cancelation of the TX signal in the RX path before entering the low-noise amplifier (LNA) [11]. In this paper, we present an entirely new way of inductively detecting the NMR signal that makes use of voltage-controlled oscillators (VCOs). The VCO-based approach for performing pulsed magnetic resonance (MR) experiments has recently been introduced by our group in the context of electron paramagnetic resonance (EPR) [12]. Here, after a brief recap of the VCO-based detection approach and a short discussion of the challenges in applying the VCO-based detection approach to NMR, we present a prototype VCO-based NMR detector with an operating range from approximately 40–100 MHz. Proof-of-concept proton NMR measurements performed at 62 MHz confirm the possibility of performing dead time-free NMR experiments, including the possibility of detecting the spin magnetization during the pulse.

2 VCO-Based NMR

2.1 The Dead Time Problem in Conventional NMR

Fig. 1
figure 1

a Illustration of a conventional setup for pulsed NMR experiments, and b illustration of the intrinsic dead time after an excitation pulse due to the energy stored in the NMR coil (FID—free induction decay)

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Figure 1a shows a conventional pulsed NMR setup. According to the figure, a single TX/RX coil is used to excite the spin ensemble and detect the resulting change in spin magnetization. Since the same coil is used for TX and RX, the above-mentioned dead time problem arises (Fig. 1b). During the pulse, the \(B_1\)-field, which is required to manipulate the spin ensemble, is associated with energy stored inside the coil. The corresponding energy density is approximately given by \(2/(\mu _0) B_1^2\). Therefore, the total stored energy scales linearly with the coil volume and quadratically with the \(B_1\) amplitude. Due to energy conservation, the (typically large) energy stored during the pulse cannot be removed instantaneously from the coil, resulting in a finite time after the pulse during which the LNA cannot be safely connected to the coil. Here, it should be mentioned that one simple solution to the dead time problem lies in the use of miniaturized coils, since they store a lot less energy due to their small volume. However, such miniaturized coils are suboptimal for concentration-limited samples, representing by far the majority of NMR use cases.

If no dedicated measures are taken, the LC resonator, into which the coil is typically embedded, produces a characteristic ring-down, which lasts for Q cycles, where Q is the quality factor of the LC resonator. This introduces a strong trade-off between detection sensitivity-favoring large Q factors—and dead time—benefiting from low Q factors. This trade-off can be mitigated by using so-called Q switches [6, 13]. However, these Q switches require high-voltage switches with a high current capability, typically introducing large parasitic capacitances and, therefore, are challenging to design. Here, high-voltage compliance is required due to the potentially large voltages across the NMR coil and high current capability is needed due to the potentially large initial current flowing through the switch before the energy is dissipated in an ohmic resistance (and/or the resistance of the switch itself). Overall, for most conventional setups, the ring-down signal prevents the LNA from monitoring the NMR signal directly after the pulse, resulting in the above-mentioned dead time.

2.2 VCO-Based NMR

In the previous subsection, we have discussed the dead time problem associated with the conventional NMR setups according to Fig. 1a. In this section, we discuss the VCO-based detection approach as a new detection method for NMR that is intrinsically dead time-free. The VCO-based detection principle for pulsed MR experiments has been introduced in [12] in the context of EPR. Here, after a brief review of its working principle, we will discuss the specific challenges of VCO-based NMR detectors before discussing the details of a proof-of-concept VCO-based NMR detector in Sect. 3.

Fig. 2
figure 2

a Illustration of the VCO-based spin detection principle, and b illustration of a simple VCO-based NMR experiment with an on-resonance excitation pulse (\(f_\textrm{L}\)—Larmor frequency; \(f_\textrm{det}\)—detection frequency)

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Figure 2a illustrates the basic operating principle of a VCO-based NMR detector. The current running through the tank inductor L produces the \(B_1\) field that excites the spin ensemble. The resulting change in spin magnetization \(M_\textrm{s}\) then induces an electromotive force \(v_\textrm{emf}\) in the tank coil, which, in turn, modulates the VCO’s oscillation frequency and amplitude, cf. Fig. 2b, according to [12]

$$\begin{aligned} \Delta \omega _\textrm{osc,spin}(t)\approx & {} \frac{\omega _\textrm{osc,0}}{A_\textrm{osc,0}}\cdot \cos \left( \omega _\textrm{osc,0} t\right) \cdot \frac{\textrm{d}}{\textrm{d}t} \int _{V_\textrm{s}} \vec {B}_\textrm{u}\cdot \vec {M}_\textrm{s} \textrm{d}V \end{aligned}$$
(1a)
$$\begin{aligned} \Delta A_\textrm{osc,spin}(t)\approx & {} \frac{Q_\textrm{coil}}{\alpha _\textrm{od}-1}\cdot \sin \left( \omega _\textrm{osc,0} t\right) \cdot \frac{\textrm{d}}{\textrm{d}t} \int _{V_\textrm{s}} \vec {B}_\textrm{u}\cdot \vec {M}_\textrm{s} \textrm{d}V, \end{aligned}$$
(1b)

where \(\Delta \omega _\textrm{osc,spin}(t)\) and \(\Delta A_\textrm{osc,spin}(t)\) are the spin-induced changes in oscillation frequency and amplitude, respectively. Here, \(\omega _\textrm{osc,0}\) and \(A_\textrm{osc,0}\) are the unperturbed oscillation frequency and amplitude, respectively, \(Q_\textrm{coil}\) is the Q factor of the tank coil, and \(\vec {B}_\textrm{u}\) is the unitary magnetic field of the tank coil, i.e., the B field per unit current. \(\alpha _\textrm{od}\) is the transistor overdrive parameter, which has to be larger than 1 to ensure a stable oscillation [14]. Notably, the frequency and amplitude signals are in quadrature with respect to each other, allowing for a simple way of detecting the complex spin magnetization, even without quadrature demodulation. In the following, we will focus on the frequency signal and leave the details of the amplitude signal for a future publication.

A simple pulse-acquire experiment can be performed by toggling the oscillation frequency from an off-resonance value, i.e., \(\omega _\textrm{osc,0}\ne \omega _\textrm{L}\), where \(\omega _\textrm{L}\) is the Larmor frequency of the spin ensemble under investigation, to on-resonance (\(\omega _\textrm{osc,0} = \omega _\textrm{L}\)) and back. This can be achieved by applying an appropriate waveform to the VCO tuning voltage (Fig. 2b). Importantly, the change in frequency is not associated with a ring-down process, since the energy stored inside the NMR coil remains (mostlyFootnote 1) constant throughout the entire pulse sequence. This not only results in a theoretically zero-dead time after the pulse but also allows for the observation of the spin magnetization (Fig. 2b) [12].

As discussed in [12], the frequency signal corresponds to the imaginary part of the complex spin magnetization, i.e., natively displays the shape of a dispersion spectrum. Therefore, for an on-resonance pulse with \(\omega _\textrm{osc,0} = \omega _\textrm{L}\), the frequency deviation is zero during the pulse. While, in principle, it is possible to perform an open-loop MR experiment simply by toggling the VCO tuning voltage \(V_\textrm{tune}\) in an open-loop configuration, such a setup requires very precise control of the VCO temperature, since typical VCOs display a large sensitivity of their oscillation frequency with respect to temperature, rendering averaging over longer time periods or phase-coherent multi-pulse experiments difficult. To solve this problem, one can embed the VCO into a high-bandwidth phase-locked loop (PLL) [12]. Thereby, the VCO frequency can be phase-synchronously derived from an external reference. Within the PLL bandwidth, the spin signal can be conveniently read out at the VCO tuning voltage, which contains an inverted version of the original spin signal (Fig. 3). The closed-loop scheme produces reliable, phase-coherent excitation pulses but introduces a short dead time after each pulse. Here, “dead time” refers to the time during which the PLL is out of lock after a frequency jump due to its finite bandwidth. It should be noted that the spin signal is still present in the combination of the tuning voltage and the VCO frequency during this "dead time" and can, therefore, in principle, be reconstructed, preserving the intrinsic dead time-free detection capability of the VCO-based detector.

Fig. 3
figure 3

Illustration of the closed-loop VCO-based setup, in which the VCO-based detector is embedded in a phase-locked loop (PFD—phase-frequency detector; FB—feedback signal)

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While conceptually, the VCO-based detection approach applies equally to EPR and NMR, there are several important differences, which are discussed in the following. Apart from the vastly different operating frequencies of NMR and EPR detectors imposed by the largely different gyromagnetic ratios, NMR and EPR detectors also have largely different bandwidth, precision, and driving requirements. Here, while EPR spectra frequently occupy a large bandwidth of 10% or even more of the Larmor frequency, most NMR spectra are relatively narrow, occupying bandwidth of only a few 10 or, at most, a few 100 ppm. Due to the very crowded nature of NMR spectra, NMR receivers have to be able to resolve very fine frequency differences, rendering the use of planar, on-chip coils, as they have been employed in [12] for EPR detection, not very practical for VCO-based NMR. To reflect these need NMR coils producing highly homogeneous magnetic fields, the VCO-based NMR detector prototype, which is discussed in detail in the following section, makes use of a conventional, large-diameter solenoidal coil driven by a custom-designed high-voltage oscillator core.

3 Methods

3.1 Custom-Designed VCO-Based NMR Detector

Fig. 4
figure 4

a Schematic of the presented VCO-based NMR detector, including an off-chip NMR coil, off-chip chokes, and an off-chip varactor and ASIC, which contains the VCO core. b Detailed schematic of the ASIC containing the VCO core

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According to the discussion at the end of the previous section, NMR detectors require coils with highly homogeneous \(B_1\) fields and must not introduce significant susceptibility artifacts. To take these requirements into account, the prototype presented in this section uses the topology shown in Fig. 4a. It combines a custom-designed cross-coupled VCO core (ASIC) with an off-chip, solenoidal NMR coil, \(L_\textrm{NMR}\), an additional off-chip tuning capacitor, \(C_\textrm{tune,ext}\), to increase the frequency tuning range of the ASIC toward the lower end, and off-chip chokes to bias the VCO without the need for a perfectly symmetrical, center-tapped NMR coil. The detailed schematic of the cross-coupled pair VCO, including component values, is shown in Fig. 4b. It essentially consists of a cross-coupled pair formed by transistors \(M_\textrm{1a}\) and \(M_\textrm{1b}\), a current mirror \(M_\textrm{2a}\) and \(M_\textrm{2b}\) with a mirror ratio of 1:10 biased from an off-chip reference current \(I_\textrm{ref}\) for maximum experimental flexibility, and a set of analog and digital varactors. The analog varactor allows for embedding the VCO into a conventional PLL, while the 2-bit digital varactor allows for rapid frequency switching of the VCO in an open-loop fashion.

Fig. 5
figure 5

a Photograph of the PCB-based probehead containing the solenoidal NMR coil and the ASIC with the VCO core, and b micrograph of the VCO ASIC manufactured in a 0.35 \(\upmu \hbox {m}\) high-voltage CMOS technology

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An annotated photograph of the PCB-based probe head that implements the architecture is shown in Fig. 5a together with a micrograph of the VCO core in Fig. 5b. The rest of the PCB contains two buffers and a differential amplifier to amplify the VCO output and turn it from a differential to a single-ended signal for better compatibility with measurement equipment. The ASIC has been implemented in a 0.35 \(\upmu \hbox {m}\) high-voltage CMOS technology. The NMR coil is implemented as a conventional 5-turn solenoid with an inner diameter of 3 mm. The choke biasing of Fig. 4a removes the need for a highly symmetrical, center-tapped NMR coil. This is because, in contrast to an on-chip inductor realization, it is very difficult to produce a perfectly symmetrical center-tapped solenoid using the conventional manufacturing techniques. Here, it should be noted that any asymmetry in the center-tapped inductor can degrade the VCO performance. High-voltage CMOS technology was chosen, because it allows for large supply voltages of up to 20 V, which are required to produce large currents through the NMR inductor to allow for short excitation pulses despite the relatively large NMR coil. In the so-called current-limited regime, the current through the tank inductor is limited by the bias current \(I_\textrm{bias}\) of the cross-coupled pair. With external reference currents, \(I_\textrm{ref}\), ranging from 0.1 mA to 2mA, the chip can support nominal bias currents \(I_\textrm{bias}\) between 1 mA and 20 mA. In the current-limited regime, the current through the off-chip tank inductor is approximately given by \(2/\pi \cdot I_\textrm{bias}\cdot R_\textrm{eq}\), where \(R_\textrm{eq} \approx Q^2\cdot R_\textrm{coil}\) is the equivalent tank resistance at the oscillation frequency, Q is the quality factor of the tank inductor, and \(R_\textrm{coil}\) is its AC series resistance. In the voltage-limited regime, the current is limited by the supply voltage. By varying the supply voltage and/or the reference current \(I_\textrm{ref}\), this prototype offers great experimental freedom for adjusting the excitation \(B_1\) field.

3.2 Measurement System

Most of the measurement system was realized digitally using the Liquid Instruments Moku:Pro platform using its Multi-Instrument Mode, which allows for the deployment of four independent instruments simultaneously (Fig. 6).

Fig. 6
figure 6

Block diagram of the measurement setup consisting of the VCO-based detector and an all-digital backend

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In the first instrument slot, the Waveform Generator module was inserted to synthesize the PLL reference signal, including the frequency jumps. The second slot is used to embed the VCO in a digital PLL using the Lock-in Amplifier module. The tuning voltage is sent both to the physical VCO as well as to the third instrument slot, containing the Digital Filter module with a second-order Butterworth low-pass filter with a cut-off frequency of 500 kHz. The final slot, containing the Oscilloscope module, is used to record the data, using the signal modulation function as the acquisition trigger.

3.3 Experimental Details

The ASIC is glued onto a custom copper heat spreader using thermally conductive epoxy. A 15 \(\times\) 15 \(\hbox {mm}^{2}\) Peltier element with a passive aluminum heat sink is attached to the heat spreader. Thermally conductive paste is used at all interfaces. The temperature of the heat spreader is stabilized to \({27}\,^\circ \hbox {C}\) using a Pt1000 temperature sensor in combination with a Meerstetter Engineering TEC-1161 controller with \({0.001}\,^\circ \hbox {C}\) precision. The supply voltages of the ASIC are supplied by a Keysight E3631A Power Supply, while \(I_\textrm{ref}\) is supplied from a Keysight B2912B Source-Measure Unit.

A 3 mm glass capillary (Hilgenberg GmbH) filled with de-ionized water (Th.Geyer GmbH) is used as sample. The VCO PCB is inserted into a 1.45 T permanent magnet (stripped-down Bruker MiniSpec) to perform the experiment.

4 Results and Discussion

Proof-of-principle NMR experiments were performed by toggling the VCO frequency by 30 kHz and directly observing the signal, both during and after the frequency jump. The resulting measured data (after 20 averages) can be seen in Fig. 7. When matching the VCO frequency during the pulse exactly to the Larmor frequency, there is no signal visible on the tuning voltage due to the dispersion nature of the frequency signal; cf. Sect. 2.2 and [12]. After switching back to a non-resonant frequency, FID signals can be observed at 30 kHz; see Fig. 7. After shifting both the excitation and detection frequencies by \(\Omega /2\pi \approx {18} \,\textrm{kHz}\), one can observe both Rabi oscillations during the pulse, as well as the FID, now shifted to around 12 kHz. The observed Rabi oscillations have a frequency corresponding to the effective Rabi frequency \(\omega _\textrm{eff}\) of an off-resonant excitation given by \(\omega _\textrm{eff}=\sqrt{\omega _\textrm{Rabi}^2+\Omega ^2}\), where \(\Omega = \omega _\textrm{L} - \omega _\textrm{exc}\) is the offset frequency, i.e., the difference between the Larmor frequency \(\omega _\textrm{L}\) and the excitation frequency \(\omega _\textrm{exc}\) and \(\omega _\textrm{Rabi}\ = \gamma B_1\) is the on-resonance Rabi frequency. Since \(\gamma B_1\) corresponding to the bias current of \(I_\textrm{bias}=\)1.8 mA is negligible compared to the offset frequency of approximately \(\Omega /2\pi \approx {18}\,\textrm{kHz}\) (see Fig. 8), the Rabi frequency observed in Fig. 7 corresponds almost exactly to the offset frequency \(\Omega\).

Fig. 7
figure 7

VCO-based pulsed NMR experiment for on- (\(\omega _\textrm{osc,0}=\omega _\textrm{L}\), blue lines) and off-resonant (\(\omega _\textrm{osc,0}=\omega _\textrm{L} - 2\pi \cdot 18\) kHz, orange lines) excitation. a Complete time trace of an FID experiment. The VCO-frequency jump was 30 kHz; the trace was averaged 20 times. b FFT of the left trace during a pulse. c FFT of the left trace after the pulse

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The overshoots observable in the signal after each frequency jump are a consequence of the PLL settling behavior. Through tuning of the PLL parameters, we achieved a settling time of approximately 5 \(\upmu \hbox {S}\). While the finite PLL bandwidth prevents a straightforward observation of the spin magnetization during the settling period, assuming the system is made sufficiently stable through temperature stabilization, the settling behavior is completely reproducible, it is, in principle, possible to reconstruct the spin magnetization from the observed tuning voltage by deconvolving the measured tuning voltage using the known PLL settling behavior.

To calibrate our \(\pi /2\) pulses, we have performed Rabi nutation experiments at two different VCO bias currents (Fig. 8). The lower bias current of \(I_\textrm{bias}=\)1.8 mA, used for most of the presented experiments, corresponds to a Rabi frequency of \(\omega _\textrm{Rabi}/2\pi \approx 2.75\) kHz or a \(\pi /2\) pulse length of \(\tau _{\frac{\pi }{2}}\approx 90\) µs. Increasing the bias current to \(I_\textrm{bias}={10}\,\textrm{mA}\), we can reach a \(\tau _{\frac{\pi }{2}}\) of approximately 20 \(\upmu\)s. This is not the highest current possible with the presented chip. However, in the current experimental setup, we were limited by the input voltage range of the buffers used for signal amplification on the PCB. Replacing these components, we have estimated that the coil current can be increased by at least a factor of two.

Fig. 8
figure 8

a Rabi nutation curves measured with two different currents flowing in the VCO coil. Points are measured data; solid lines are approximate fits. b SSFP time trace, blue line is measured data and orange line is guide to the eye

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Here, one should note that increasing the VCO bias current not only increases the \(B_1\) field but also increases the stability of the VCO oscillation signal. This manifests itself in a reduced phase noise but also a reduced sensitivity of the VCO output frequency to the induced emf due to the precessing spin magnetization. Since, in general, both effects can depend nonlinearly on the bias current, the exact dependence of the signal-to-noise ratio (SNR) on the bias current has to be evaluated via simulations or by experiment. For the presented prototype system, we have observed a reduction in SNR when increasing the current from \(I_\textrm{bias}=1.8\) mA to \(I_\textrm{bias}=10\) mA from \(\textrm{SNR}\approx 14\) to \(\textrm{SNR}\approx 9\).

Finally, we have verified that we can perform coherent spin manipulation over multiple pulses by performing a steady-state free-precession (SSFP) experiment [15]. To this end, we have applied a simple repetitive pulse train of 90 \(\upmu\)s \(\pi /2\) pulses with \(\tau _{\textrm{rep}}={2}\,\textrm{ms}\). The resulting signal (Fig. 8) shows the expected FID decay, as well as a buildup of spin coherence before the next pulse, a typical signature of SSFP. If our spin manipulation pulses were not phase-coherent, such an endless pulse train would not produce an SSFP signal, as the spins would be periodically rotated around random axes. This verifies the possibility of coherent spin manipulation, opening up the pathway for implementing more complex pulse sequences for various applications.

From the presented measurements, we have estimated the spin sensitivity using the equation

$$\begin{aligned} N_\textrm{min}=3\frac{N_\textrm{spins}}{\textrm{SNR}\cdot N_\textrm{avg}\sqrt{\textrm{BW}}}, \end{aligned}$$
(2)

where \(N_\textrm{spins}\) is the total number of spins in the sample, \(N_\textrm{avg}\) is the number of averages, and \(\textrm{BW}\) is the measurement bandwidth corresponding to the applied digital filter. For better comparison with the conventional systems, we have also calculated the concentration sensitivity \(c_\textrm{min}=\frac{N_\textrm{min}}{V\cdot N_\textrm{A}}\), where V is the sample volume and \(N_\textrm{A}\) the Avogadro constant. To account for different operating magnetic fields and sample volumes between different spectrometers, one can calculate the normalized sensitivities by multiplying \(N_\textrm{min}\) and \(c_\textrm{min}\) with the square of field strength \(B_0^2\) and dividing by the coil diameter d. The obtained sensitivity values are listed in Table 1. Although these normalized numbers are approximately 2–3 orders of magnitude worse than the current state-of-the-art in chip-based NMR, we note that the reported results should be seen as proof-of-concept experiments for VCO-based detection in the context of NMR and that our measurement setup was not fully optimized for sensitivity. As shown theoretically in [16], under certain mild assumptions, the VCO-based detection approach has a theoretical limit of detection (LOD) equal to that of conventional MR detectors.

Table 1 Measured spin sensitivity of the system
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5 Conclusion

In this work, we have introduced VCO-based NMR as a new detection approach that allows for the real-time observation of the spin magnetization during an excitation pulse as well as its dead time-free detection after the pulse. After a brief review of the principle of pulsed VCO-based MR detection, we have discussed a prototype VCO-based NMR detector consisting of a custom-designed VCO core and an off-chip solenoidal NMR coil. The proof-of-concept NMR experiments with this prototype performed at an operating frequency of 62 MHz both validate the possibility of detection during the excitation pulse as well as the possibility for dead time-free detection after the pulse. In our prototype, the experimental dead time was limited by the used PLL to approximately 5 \(\upmu\)s. Using a high-voltage CMOS technology for the VCO core, the current design achieves competitive \(90^\circ\) pulse lengths of 20 \(\upmu\)s with a 5-turn, 3 mm-diameter solenoid. Overall, the proposed VCO-based NMR approach opens up the path toward improved dead time-free NMR experiments, including spectroscopy, relaxometry, and imaging. As our next steps, we will work on improving the LOD, increasing the PLL bandwidth, as well as means for extracting the spin information outside the PLL bandwidth [17].