2.1 Setup overview
Our setup consists of a single channel OPM, an excitation coil for MNP alignment and a 3D Helmholtz coil system for the generation of defined background magnetic fields. The setup is positioned within a three-layer magnetic shielding barrel [27]. A MNP sample is placed on top of the OPM vapor cell, surrounded by the excitation coil (Fig. 1).
2.2 MNP excitation circuit
The MNP’s magnetic moments are aligned by a magnetic field of about 1 mT (unless otherwise noted), generated by a 65-turn, 48 mm diameter coil (Fig. 1). After 200 ms (unless otherwise noted), this excitation field is shut-off within a few μs, which is achieved by a fast de-energizing of the coil. This is ensured by a low coil inductance and a fast-switching high-voltage MOSFET, combined with a high-voltage rated transient-voltage-suppression (TVS) diode (Fig. 2). The TVS diode clamps the back EMF to a (maximum) fixed level, protecting the MOSFET and decreasing the magnetic field linearly over time.
2.3 Intensity-modulated OPM
A schematic overview of the OPM is shown in Fig. 3. The OPM employs a microfabricated Cs vapor cell with nitrogen buffer gas (38 mbar) and a sensing volume of approx. 50 mm3. The cell consists of a 4 mm thick silicon body with a cylindrical cavity, closed by anodically bonded glass faces [28]. The vapor cell is operated at 70∘C by electrical thin-film heaters glued to the side faces of the Si body and driven by a 10 kHz ac current. Circularly polarized light from a distributed Bragg reflector (DBR) laser diode at the Cs D1 transition (895 nm) is intensity-modulated at the chopping frequency \(f_{\text{c}}\) resonantly tuned to the Larmor frequency \(f_{\text{L}}\) of the Cs atoms. It should be noted that \(f_{\text{c}}\) is not feedback-controlled to avoid an increased settling time and transient signals after switching-off the MNP’s excitation field. In this all-optical Bell–Bloom configuration, the magnetometer sensitivity is optimized when operated in static magnetic fields oriented perpendicular to the laser beam propagation direction [29, 30]. The laser light power transmitted through the vapor cell is detected by a Si photodiode.
2.4 Data acquisition and preprocessing
The OPM photodiode signal is passed through a current voltage converter (I/U) and amplifier. Then, the signal is mixed with the OPM laser chopping frequency \(f_{\text{c}}\) by the lock-in amplifier (LIA), and the resulting in-phase component (LIA-X) and quadrature component (LIA-Y) are directly digitized by the LIA after 4th order low-pass filtering with a \(-3~\mbox{dB}\) bandwidth of 10 kHz. A sample rate of 107.1 kHz is used. Additionally, the MNP excitation coil current is recorded by the same device, which serves as trigger for the data processing. In order to remove spurious frequency components emerging from the electrical ac heating, the OPM data is preprocessed in software by low-pass filtering with a cutoff-frequency of 1 kHz for liquid MNP and 100 Hz for MNP immobilized in gypsum. The digital filter is realized as bessel filter to preserve sharp edges in the data.
2.5 MNP and MRX model
Magnetic nanoparticles with a hydrodynamic diameter of 100 nm (plain BNF-Dextran) from Micromod (Micromod Partikeltechnologie GmbH, Rostock, Germany) were used in this experiment. The MNP are dextran based, water suspended and their iron-oxide core with a diameter of 45 nm consists of multiple 15–20 nm iron-oxide crystallites [31]. Two samples of \(140~\upmu \mbox{l}\) were prepared for the experiment. One sample is composed of \(100~\upmu \mbox{l}\) liquid (factory supplied) MNP and diluted in \(40~\upmu \mbox{l}\) distilled water, whereas the second sample consists of \(100~\upmu \mbox{l}\) MNP, embedded and thus immobilized in gypsum, resulting in a total sample volume of \(140~\upmu \mbox{l}\). The undiluted iron concentration is about 15 mg/ml, resulting in an iron amount of 1.5 mg in each sample.
The MNP relaxation signal \(B(t)\) can be described in several ways. For an MNP system with uniform particle size and non-interacting particles, a single exponential decay of the associated magnetic flux density at the OPM’s location is expected [32]:
$$ B_{\text{ideal}}(t) = B_{\text{R}} \exp \biggl(-\frac{t}{\tau_{\text{R}}} \biggr), $$
(2)
with the relaxation amplitude \(B_{\text{R}}\), the relaxation time constant \(\tau_{\text{R}}\) and the time t after the start of the relaxation. However, real MNP systems usually show a non-uniform particle size distribution. Assuming a magnetometer bandwidth of 500 Hz and a typical MNP anisotropy of \(10^{4}~\mbox{J}/\mbox{m}^{3}\), particles with a core diameter in the interval \([17.5~\mbox{nm}, 21~\mbox{nm}]\), while having a hydrodynamic diameter \({\ge}100~\mbox{nm}\), contribute to the MRX signal and therefore can be detected [2]. For MNP ensembles with an equally distributed diameter in the detectable size range, the relaxation can be described by [3, 26]:
$$ B_{\text{unif}}(t) = B_{\text{R}} \ln \biggl( 1 + \frac{\tau}{t} \biggr), $$
(3)
with the amplitude \(B_{\text{R}}\) and a time constant τ which depends on the excitation parameters and the MNP’s anisotropy. The relaxation signal can also be described phenomenologically by a stretched exponential [32]:
$$ B_{\text{ph}}(t) = B_{\text{R}} \exp \biggl[- \biggl( \frac{t}{\tau_{\text{R}}} \biggr)^{\beta} \biggr]+B_{\text{offset}}, $$
(4)
with the relaxation amplitude \(B_{\text{R}}\), the relaxation time constant \(\tau_{\text{R}}\) and the stretching parameter β. The offset \(B_{\text{offset}}\) is introduced to compensate for magnetometer heading error, background magnetic fields and static MNP magnetization (e.g. due to the applied background magnetic field).
In immobilized MNP, Brownian motion is suppressed, while in water suspended samples both, Brownian motion and Néel motion may contribute to the MRX signal. Therefore, it reasonable to fit for both fractions [32]:
$$ B(t) = B_{\text{b}} + B_{\text{ub}} + B_{\text{offset}}, $$
(5)
with the contribution of bound (immobilized) MNP \(B_{\text{b}}\) and unbound MNP \(B_{\text{ub}}\), where for each, one or a linear combination of the previously presented approaches may be well suited.
When fitting the measured data to such a model, the high number of parameters leads to an ill-conditioned problem. Even for simple models, the parameter variance and mutual interdependence is often very high, not only due to the contribution of environmental noise. To overcome this uncertainty, first a model is fitted to the experimental data and the absolute amplitude difference at two fixed time points is used as measure for the relaxation amplitude.
A robust parameter for the relaxation time is the integral relaxation time [33], often called correlation time [34]. It is denoted as the area under the amplitude-normalized relaxation curve. For the estimation of this parameter no curve-fitting was involved.
2.6 MNP relaxation data processing
Since the relaxation signal is of an exponential form, a high OPM bandwidth is required to capture early parts of the relaxation. Latter parts of the relaxation, however, do not require high bandwidths, but would benefit from a high magnetometer sensitivity. To satisfy both requirements, adaptive filtering or resampling can be implemented [32]. With a similar effect, in this work, the data is weighted exponentially during curve fitting. The fits of the data to the relaxation model, which was selected as the sum (5) of two stretched exponentials (4), were performed using the trust-region-reflective least squares algorithm [35] provided by Matlab®. For the extraction of the relaxation amplitude and the integral relaxation time, time intervals are selected as \([12~\mbox{ms}, 280~\mbox{ms}]\) for liquid MNP and \([0.12~\mbox{s}, 6.5~\mbox{s}]\) for immobilized MNP, respectively. This data analysis is repeated for several subsequent MRX sequences at a fixed background magnetic field. Additionally, the analysis is performed on three or seven times averaged MRX data for immobilized or liquid MNP, respectively.