Wide-field microwave magnetic field imaging with nitrogen-vacancy centers in diamond

The schematics of the home-built fluorescence microscope we use to perform wide-field MW magnetic field imaging is shown in Fig. 1(a). A 532 nm laser is pulsed with an AOM before being appropriately focused on the back aperture of a 20$\times$ 0.4 NA objective, to achieve a uniform illumination on a FOV of $\sim 340\times 340$ $\upmu$m². The PL emitted by the NVs is then collected by a CMOS camera after being filtered with a 650 nm long-pass filter. Before analysis we apply 4x4 pixel binning, resulting in a $\sim 1.2\times 1.2$ $\upmu$m² pixel size. However, the lateral resolution in this experiment is likely limited by the NV layer thickness and the diamond-DUT stand-off distance. Two permanent magnets mounted on motorized translation stages for angle and radial control are used to apply a bias magnetic field $B_{0}$ aligned to the [111] crystallographic NV axis; the orientation with respect to the $\left\{x,y,z\right\}$ laboratory reference frame ($\theta=55^{\circ}$, $\phi=97^{\circ}$) is also shown in the inset of Fig. 1(a). The diamond sensor, consisting of a 4 $\upmu$m-thick NV layer overgrown on a 500 $\upmu$m-thick diamond plate, is glued on the MW loop (operated at a power of 40 dBm) that is used to drive the $|m_{s}=0\rangle\leftrightarrow|m_{s}=\pm 1\rangle$ spin transitions. The diamond/MW loop system is in turn mounted on a 5 axis ($x,y,z,\theta,\phi$) translation stage, that we use to bring the diamond in proximity of the DUT, as shown in Fig. 1(b). An optical picture showing the MW loop with the diamond glued on it is reported in Fig. 1(c), whereas Fig. 1(d) shows an optical picture of the DUT, consisting of a 100 $\upmu$m wide, 200 nm thick Cu stripline deposited on a Al₂O₃ substrate. More details on the experimental setup are reported in the SI[29].

The negatively-charged NV electronic ground state is a $S=1$ spin system, with the sublevels $|m_{s}=0\rangle$ separated from the doubly-degenerate $|m_{s}=\pm 1\rangle$ sublevels by the zero-field splitting $D_{GS}\simeq 2.87$ GHz. Through the Zeeman effect, the bias magnetic field $B_{0}$ removes the degeneracy of the $|m_{s}=\pm 1\rangle$ states by a factor $2\gamma_{NV}B_{0}$, where $\gamma_{NV}\simeq 28$ kHz $\upmu$T^-1 is the NV gyromagnetic ratio. This allows us to use either $|0\rangle\leftrightarrow|+1\rangle$ or $|0\rangle\leftrightarrow|-1\rangle$ as an isolated two-level system, as shown in Fig. 1(e). Due to selection rules, the two separate transitions $\sigma_{\pm}$ ($|0\rangle\leftrightarrow|\pm 1\rangle$) are only excited by a circularly polarized MW fields[28]. Thus, the two transitions $\sigma_{\pm}$ are only sensitive to a single polarization component $B_{\pm}$, with $B_{-}$ and $B_{+}$ being respectively the left- and right-handed circularly polarized component of the MW field, with the polarization axis parallel to the NV axis. A MW field resonant with either transition will then result in population oscillations between the relevant spin sublevels, i.e. Rabi oscillations, at a frequency of $\Omega_{\pm}=2\pi\gamma_{NV}B_{\pm}$. In the following, we only use $\sigma_{-}$ for demonstration of MW magnetic field imaging. In particular we use a bias field $B_{0}\sim 10.7$ mT leading to a resonant frequency $\omega_{-}=2\pi\cdot 2.57$ GHz (measured with optically detected magnetic resonance spectrum). We emphasize that $\omega_{-}$, the frequency of the MW field we are sensitive to, can be changed simply by adjusting $B_{0}$. For simplicity in the notation, for the remaining of this manuscript we will suppress the “-” subscript as, unless otherwise stated, we are referring to only the $\sigma_{-}$ transition.

We drive both the MW loop and the DUT with a MW current having same frequency $\omega=(2\pi)\cdot 2.57$ GHz and measure the Rabi oscillations with the pulse sequence reported in Fig. 1(f), extended from Horsley et al [26]. The sequence is divided in three sections: the first “MW-loop” where a MW pulse is only supplied to the MW loop line; the second “MW-loop + DUT” where two MW pulses of same duration are sent in both MW loop and DUT; and the third “reference” where the bare fluorescence of the $|0\rangle$ state is measured without applying any MW pulse. For each section, the sequence starts with a laser pulse (15 $\upmu$s duration) that both initializes the NVs into the $|0\rangle$ state and reads out their spin state. The laser pulse is followed by the MW pulse (or pulses) of duration $\tau$ that drives the Rabi oscillation. This sequence is repeated $N_{R}=250$ times inside a single camera exposure time (6 ms) to accumulate PL counts. For each value of $\tau$, a total of $N_{P}=100$ images are collected per each section and then averaged to further increase the signal to noise ratio. This whole sequence is repeated as we scan the MW pulse duration $\tau$ in the interval [0,1.2 $\upmu$s] to measure the Rabi oscillations. Keeping the camera open during several repetitions of the laser and MW pulse sequence leads to a decrease in the contrast, as the spin readout and the NV center repolarization to the $|0\rangle$ state occur simultaneously [30]. However, this allows us to accumulate more experiment repetitions and to gate the camera at a timescale much longer than the characteristic Rabi oscillation time. At the end of the sequence we obtain three separate PL vs. time measurements for the three MW pulse combinations described above. The “reference” measurement is subtracted to the other two measurements to remove the background fluorescence and correct any possible non-uniformity in the background. We then obtain two sets of measurements that we refer to respectively $B_{LOOP}$ and $B_{LOOP+DUT}$.

Using the pulse sequence from Fig. 1(f), we obtain a series of images where each pixel contains a Rabi oscillation that we fit with the equation:

with $\tau$ being the MW pulse length while the fitting parameters are fluorescence intensity $F_{0}$, Rabi fluorescence contrast $A$, Rabi decay lifetime $\tau_{R}$, and Rabi frequency $\Omega$. We can then assign to each pixel the corresponding value of magnetic field $B$ through the relation $\Omega=2\pi\gamma_{NV}B$. The results for the wide-field measurement obtained for a MW power in the DUT of 17.8 dBm are shown in Fig. 2(b) and (c) for $B_{LOOP}$ and $B_{LOOP+DUT}$. Figure 2(e) shows a typical example of Rabi oscillations for $B_{LOOP+DUT}$ (green curve) and $B_{LOOP}$ (red curve) for a single pixel (red star in Fig. 2(b) and (c)). We notice that the MW field generated by the DUT is not strong enough to be detected directly. This can be observed from the yellow curve in Fig. 2(e), showing a single-pixel $\tau$ scan measured when the MW pulse is sent only in the DUT (not shown in Fig. 1(f)), where no Rabi oscillations can be observed. Measuring $B_{DUT}$ alone does not carry useful information, as it cannot be fit through Eq. 1. To isolate the contribution of the DUT from the total magnetic field $\vec{B}_{Tot}(t)=\vec{B}_{LOOP}(t)+\vec{B}_{DUT}(t)$ we use the following equation

where we assumed that $|\vec{B}_{DUT}|\ll|\vec{B}_{LOOP}|$ and the $\sqrt{2}$ factor results from the root mean square (RMS) of the projection of $\vec{B}_{DUT}$ on $\vec{B}_{LOOP}$. More details on the derivation of Eq. 2 can be found on the SI [29]. We then take the pixel-wise difference reported in Eq. 2 of the two images for the $B_{LOOP+DUT}$ and $B_{LOOP}$ measurements to obtain $B_{DUT}$ as shown in Fig. 2(d). This result shows that our measurement protocol based on the differential measurement brings information on the MW fields generated by the DUT even if amplitude of the MW generated by the DUT alone does not drive Rabi oscillations, thus extending the range of MW power detectable through Rabi oscillations measurement.

To assess if the differential measurement for $B_{DUT}$ correctly described the MW field generated by the MW current in the DUT, we fit the MW field along a line-cut (white dashed line in Fig. 2(d)) with the analytical expression that can be found for $B$ assuming an infinitely thin and long stripline carrying a uniform current density $J$ flowing only parallel to the stripline[29]. This equation for $B$ has the stand-off distance $d$ and the DUT current density $J$ as fitting parameters. The result is shown in Fig. 2(f), where a good agreement with the experimental data and the model is obtained for $d=(30\pm 2)$ $\upmu$m and $J=(41\pm 3)\cdot 10^{7}$ A/m². The error bar on magnetic field values of Fig. 2(f) are obtained starting from the standard deviation of the fit parameter $\Omega$ from Eq. 1 for the two separate measurements then propagated for Eq. 2.

To study the dynamic range accessible with the protocol developed here, we perform the same wide-field Rabi measurements for different MW power in the DUT, ranging from -2.3 dBm to 27.7 dB, with all the other same nominal experimental conditions. The results obtained through Eq. 2 are shown in Fig. 3(a), (c) and (e) for respectively -2.3, 17.8, and 27.7 dBm MW power in the DUT. Figures 3(b), (d) and (f) show the magnetic field data along a line-cut perpendicular to the strip-line fit with the theoretical model. From the fitting parameters we can notice that the resulting stand-off distance $d$ is consistent in all three separate measurements. Moreover, it can be observed that the ratios between the estimated current densities $J$ are consistent with the ratios of the input power, thus proving the good quality of the magnetic field measurement. We emphasize that in this experiment, the DUT is driven with significantly weaker MW power compared to the MW loop. At higher power levels, as shown in the SI[29], the DUT is capable of driving Rabi oscillations on its own. However, the primary aim of this work is to develop a differential Rabi imaging technique to detect weak MW signals and show how it allows to extend the MW power range that can be detected.

To estimate the magnetic field sensitivity we firstly calculate the shot-noise limited sensitivity $\eta_{sn}$ through the following equation [31]:

with $F_{0}$, $C$, and $\tau_{R}$ being respectively the fluorescence of the bright state $|0\rangle$, the contrast and the decay time of the Rabi oscillation as shown in Fig. 4. For the typical values of these parameters we have in our experiment we obtain the value for the shot noise limited sensitivity of $\eta_{SN}\sim$ 670 nT Hz^-1/2. We can also estimate the measured sensitivity through the full experimental Rabi oscillation defined as $\eta_{meas}=\delta B\sqrt{T}$, where $\delta B$ is the smallest measurable magnetic field while $T$ is the total measurement time. We estimate $\delta B$ as the minimum measurable change in the Rabi frequency in $B_{LOOP+DUT}$, i.e. the standard deviation of the Rabi fit for $\Omega$. The value we extract from our data set is $\delta B\simeq 0.5$ $\upmu$T that combined with the total measurement time of $T=357$ s lead to $\eta_{meas}\simeq 9$ $\upmu$T Hz^-1/2. As expected, this value is larger than the shot-noise limited sensitivity as it is evaluated from the measurement of a full Rabi oscillation. Eq. 3 instead results from fixing the MW pulse duration relative to the point of maximal slope in the Rabi oscillation, that leads to the maximal change in fluorescence following a magnetic field variation, thus resulting in optimal sensitivity[32]. A better comparison can be obtained by considering that the total measurement time $T=N_{\tau}T_{single}$, where $N_{\tau}=128$ is the number of points acquired during the measurement of a full Rabi oscillation, and $T_{single}$ is the measurement time of a single point, which is limited by the camera frame rate. After normalizing by the number of points we obtain a per-point magnetic field sensitivity $\eta_{meas}\sim$ 0.7 $\upmu$T Hz^-1/2 which is comparable with the shot noise limited sensitivity.

In this section we demonstrate the MW imaging capabilities of the differential Rabi protocol developed in this work with another type of nontrivial DUT. The new DUT consists of a 50 $\upmu$m wide, 200 nm thick copper microstructure, deposited on a Al₂O₃ substrate in the shape shown in Fig. 5(a). The result for the MW magnetic field imaging obtained with the differential Rabi protocol when the DUT is operated at 17.8 dBm is shown in Fig. 5(b).