Widefield microwave imaging in alkali vapor cells with sub-100 μm resolution

Atomic vapor cells are one of the most versatile systems for measuring electromagnetic fields [1–3], and are at the heart of the most sensitive dc [4, 5] and rf [6, 7] magnetometers. Our group has recently developed a technique for imaging magnetic fields at microwave frequencies [8–10], and alkali atoms in Rydberg states have been used for imaging microwave electric fields [11–14]. These techniques promise to have a transformative effect on the development, function and failure analysis of microwave devices in science and industry, as there is currently no established and satisfactory technique for imaging microwave fields. There is also significant interest in microwave sensing and imaging for medical applications, such as breast cancer screening [15–17]. However, while providing high field sensitivity, current vapor cell devices are limited to an exploitable spatial resolution on the millimeter scale.

Here we report a new setup based on a $140\;\mu {\rm{m}}$ 'ultrathin' vapor cell for high-resolution imaging, providing $50\times 50\times 140\;\mu {{\rm{m}}}^{3}$ spatial resolution in the cell bulk, and allowing us to image fields as close as $150\;\mu {\rm{m}}$ above surfaces, thanks to a thin external wall. This represents an order of magnitude improvement in exploitable spatial resolution compared to previous vapor cell experiments, and allows us to enter the relevant regime for imaging fields of industrial microwave devices. Our camera-based imaging technique allows us to record widefield 2D images at a rate of 10 Hz, which could be further improved to kHz rates using a faster camera system [18]. This allows us to record live movies of time-dependent processes, which would be rather difficult with a scanning probe system. A particularly promising feature of our system is that it can be configured to also image microwave electric fields [13].

Sub-millimeter spatial resolution has been reported in the vapor cell bulk for a number of sensing techniques [9–13, 19–23], but typical outer dimensions of cells have limited usable spatial resolution to the millimeter-scale or larger. Feature sizes in near-fields are on the order of the distance from the field source, meaning that, for example, micrometer-order spatial resolution cannot be exploited when performing sensing millimeters away from a field source. In order to resolve small structures on objects under investigation, it is crucial to measure fields at similarly small distances above the structures. There are many applications where sub-millimeter spatial resolution is essential, such as integrated microwave circuit characterization [24], corrosion monitoring [25–27], and in lab-on-a-chip environments for microfluidic analytical chemistry and bio-sensing [19, 28–30], and molecular imaging [31–33].

We demonstrate our new high-resolution imaging system through the imaging of microwave magnetic near-fields above a selection of microwave circuits. As a demonstration of the flexibility of our setup, we also present vector-resolved images of the dc magnetic field above a wire loop.

A photograph of our setup and typical microwave field images above a microwave integrated circuit are shown in figure 1. We use a microfabricated glass vapor cell with an inner thickness of $140\;\mu {\rm{m}}$ to position a two-dimensional sheet of atomic rubidium vapor near the microwave device under test (figure 1(a)). The cell features a $150\;\mu {\rm{m}}$ thin side wall (figure 1(b)), which allows us to place the atoms at similarly small distance from the structure. The microwave field of the chip drives Rabi oscillations of frequency ${{\rm{\Omega }}}_{\mathrm{Rabi}}(\vec{r})$ between hyperfine states-of the atom, which depend on the projection of the local microwave field vector onto the direction of an applied uniform static magnetic field. The Rabi oscillations are recorded on a camera through the hyperfine state-dependent absorption of a laser by the atomic vapor. Microwave field images obtained from the observed ${{\rm{\Omega }}}_{\mathrm{Rabi}}(\vec{r})$ are shown in figure 1(c) in several cross-sections of the chip.

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A critical component in this work was the design of the two vapor cells used. The cells, identical in all but internal cell thickness, consist of two optically bonded 0.5 and $1.5\times 20\times 90\;\mathrm{mm}$ pieces of Suprasil glass. As shown schematically in figure 1(b), a cell chamber with a thickness of either 140 μm or 200 μm is etched into one end of the 1.5 mm thick piece. An etched channel connects the cell chamber to a through-hole, around which we attach a glass-to-metal transition with epoxy (Epotek-377). The key advance of our cells is their thin external walls (see figure 1(b)), as thin as $150\;\mu$m. In contrast to typical millimeter-scale vapor cell wall thicknesses, our thin walls allow us to image microwave near fields as close as $150\;\mu {\rm{m}}$ above microwave devices, and for the first time take practical advantage of our high spatial resolution.

We fill the cells with a 3:1 mixture of Kr and N₂ buffer gasses, with a typical filling pressure of 100 mbar measured at ${T}_{\mathrm{fill}}=22\;^\circ {\rm{C}}$. The heavy Kr acts to localize the Rb atoms, improving our spatial resolution and limiting depolarizing Rb collisions with the cell walls [34]. The N₂ gas is included for quenching effects [35, 36].

A schematic of our cell is shown in figure 2(c). The cell and microwave device are placed inside an oven, with operating temperatures of 130 °C to 140 °C chosen to give an optical depth of ${OD}\approx 1$ [37–39]. The Rb vapor density is controlled by a cold finger wrapped around the end of the glass-to-metal transition, and the 10 °C temperature gradient between the cold finger and the cell helps reduce the deposition rate of Rb and other contaminants on the cell windows. The experiment is surrounded by a cage of Helmholtz coils, which cancel the Earth's magnetic field, and provide a static field of 1–2 G along the X-, Y-, or Z-axes. This field serves as the quantization axis, and the resulting ∼MHz Zeeman splitting of the ⁸⁷Rb hyperfine ground state transitions allows each transition to be individually addressed by tuning the microwave frequency.

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Figure 2(a) shows the hyperfine structure of the D₂ line of ⁸⁷Rb and the relevant microwave and optical transitions involved. We produce images of each of the polarization components of the microwave magnetic field, using Rabi oscillations driven on the $| F=1,{m}_{F}=0\rangle \to | F=2,{m}_{F}=0\rangle$ 'clock' transition of the ⁸⁷Rb hyperfine ground state [8–10]. The Rabi frequency on this transition is given by ${{\rm{\Omega }}}_{\mathrm{Rabi}}(\vec{r})=\displaystyle \frac{{\mu }_{B}}{{\hslash }}{B}_{\mathrm{mw}}(\vec{r}),$ where ${B}_{\mathrm{mw}}(\vec{r})=({\vec{B}}_{\mathrm{mw}}(\vec{r})\cdot \vec{B})/| \vec{B}|$ is the projection of the local microwave field vector ${\vec{B}}_{\mathrm{mw}}(\vec{r})$ onto the direction of an applied uniform static magnetic field $\vec{B}.$ Imaging with the static field pointing in the X-, Y-, and Z-directions allows us to image each polarization component in turn. Using atoms as sensors, our technique avoids the significant calibration problem in other microwave sensors [40], relating the field to a measured oscillation frequency and well-known fundamental physical constants. Data taking is fast, due to the parallel nature of the measurement (imaging as opposed to scanning). By applying an external static magnetic field, we have imaged microwave fields from 2.3 to 26.4 GHz with a single device [41]. The technique is applicable to microwave devices of all types, recently showing success in characterizing and debugging microwave cavities in high-performance miniaturized atomic clocks [10, 42, 43].

We begin an experiment sequence, shown in figure 2(b), by preparing the atoms in the ⁸⁷Rb F = 1 ground state with a 1 ms optical pumping pulse. Through frequent ($\sim {10}^{9}\;{{\rm{s}}}^{-1}$) collisions with the buffer gas, Rb atoms sample the entire velocity space over the course of the optical pumping pulse (and also the subsequent probe pulse). We typically see a 30% reduction in OD due to optical pumping. The optical pumping efficiency is below 100% due to several factors: radiation trapping, collisional broadening of the optical line, absorption due to ⁸⁵Rb, and the detuning of the lasers from the collisionally shifted ⁸⁷Rb and ⁸⁵Rb optical lines [44]. We drive Rabi oscillations by injecting a microwave pulse of length ${\rm{d}}{t}_{\mathrm{mw}}$ into the microwave device under test. We then image the resulting repopulation of the F = 2 state with a ${\rm{d}}{t}_{\mathrm{probe}}=0.3\;\mu {\rm{s}}$ probe pulse using absorption imaging, which selectively detects the F = 2 state [10, 45]. The optical pumping and probing is performed with two separate 780 nm diode lasers, frequency stabilized to the $F=2\to {F}^{\prime }=2,3$ crossover peak of the ⁸⁷Rb D₂ line, red-shifted by an AOM $80\mathrm{MHz}$ from the stabilization point, and with intensities of $120\;\mathrm{mW}\;{\mathrm{cm}}^{-2}$ and $30\;\mathrm{mW}\;{\mathrm{cm}}^{-2},$ respectively. The short probe pulse length ensures that optical pumping due to the probe pulse is minimal. We take reference images to account for short and long term drifts, and combine the images to give an image of OD_mw, the change in optical density (OD) induced by the microwave pulse. An example OD_mw image is shown in figure 3(a). The inhomogeneous microwave field drives Rabi oscillations at different rates across the image, which form patterns in OD_mw following the contour lines of the microwave field. Atoms along the outermost (mostly) red line of figure 3(a) are at the peak of their first Rabi oscillation, corresponding to maximal repopulation of the absorptive F = 2 state. The inner red line corresponds to a region of higher field, where atoms are at the peak of their second oscillation. We take multiple OD_mw images, scanning ${\rm{d}}{t}_{\mathrm{mw}},$ to produce OD_mw movies. A sample of these movies are available online, with the frame rate matching the 10 Hz image acquisition rate of our experiment. The counter on top of the movies indicates the microwave pulse duration. As shown in figure 3(c), each pixel in these movies has an oscillating signal which we can fit to obtain the local microwave field strength.

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The longitudinal spatial resolution of our imaging setup is set by the 140 or $200\;\mu {\rm{m}}$ thickness of the cell. The buffer gas pressures set a similar transverse spatial resolution, by determining the distance an atom can diffuse over the course of a measurement [34]. At T = 135 °C, we estimate the r.m.s diffusion distance during a ${\rm{d}}{t}_{\mathrm{mw}}={T}_{1}=8.8\;\mu {\rm{s}}$ measurement to be ${\rm{\Delta }}x=\sqrt{2D{\rm{d}}{t}_{\mathrm{mw}}}=50\;\mu$m. Here, T₁ is the hyperfine population relaxation time constant in the cell. The diffusion coefficient, D, is given by $1/D=1/{D}_{\mathrm{Kr}}+1/{D}_{{{\rm{N}}}_{2}},$ where ${D}_{\mathrm{Kr}({{\rm{N}}}_{2})}={D}_{0,\mathrm{Kr}({{\rm{N}}}_{2})}\frac{{P}_{0}}{{P}_{\mathrm{Kr}({{\rm{N}}}_{2})}}{(\frac{T}{{T}_{0}})}^{3/2}$, ${P}_{0}=1$ atm, ${P}_{\mathrm{Kr}({{\rm{N}}}_{2})}$ is the Kr (N₂) pressure, and we have used ${D}_{0,{Kr}}=0.068\;{\mathrm{cm}}^{2}\;{{\rm{s}}}^{-1}$ [46] and ${D}_{0,{N}_{2}}=0.159\;{\mathrm{cm}}^{2}\;{{\rm{s}}}^{-1}$ [47]. These are order-of-magnitude increases in spatial resolution compared to previous imaging experiments [9, 13, 21, 48].

Peak-to-trough feature sizes as small as $70\pm 10\;\mu {\rm{m}}$ can be seen in the OD_mw images, approaching the estimated diffusion-limited spatial resolution. An example is shown in figure 3.

The $6\times 6\;{\rm{mm}}$ cell can be thought of as an array of ${N}_{\mathrm{sens}}=120\times 120$ sensors, with each sensor corresponding to a $50\;\mu {\rm{m}}\times 50\;\mu {\rm{m}}\times 140\;\mu {\rm{m}}$ voxel. The sensor size is given by the diffusion-limited spatial resolution, with a sensor volume $V=1.8\times {10}^{-7}\;{\mathrm{cm}}^{3}.$ To estimate our experimental sensitivity, we examined CCD pixels binned in 2 × 2 blocks, corresponding to an area of $42\times 42\;\mu {{\rm{m}}}^{2},$ slightly smaller than the sensor size. The fitting error to our microwave Rabi data was as low as $21\;$ nT per 2 × 2 pixels, giving an estimated sensitivity of $\delta {B}_{\mathrm{mw}}^{\mathrm{exp}}=1.4\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2}$ per sensor, taking into account the 4440 s measurement time (148 averaged runs). Integrating over a larger volume would give an increase in sensitivity, at the expense of spatial resolution.

We record data for all of the sensors in our array simultaneously. Compared to creating an image by scanning a single sensor, this improves our data-taking speed by a factor of at least ${N}_{\mathrm{sens}},$ or four orders of magnitude. The effective sensitivity is therefore significantly improved by our parallel imaging, and a single, scanned sensor would require a sensitivity of at least $\delta {B}_{\mathrm{mw}}^{\mathrm{exp}}/\sqrt{{N}_{\mathrm{sens}}}=12\;\mathrm{nT}\;{\mathrm{Hz}}^{-1/2}$ to produce an image with the same sensitivity. Parallel imaging is also more suitable than scanning for applications requiring high temporal resolution over an image.

We can compare our experimental sensitivity with the photon shot noise limited sensitivity. Assuming ${{\rm{\Omega }}}_{\mathrm{Rabi}}\;{\rm{d}}{t}_{\mathrm{mw}}\ll \pi ,$ we have [44]

$$\begin{eqnarray}&&\delta {B}_{\mathrm{photon}}=\sqrt{\displaystyle \frac{{\rm{d}}{t}_{\mathrm{run}}}{{N}_{\mathrm{shots}}}}\displaystyle \frac{2}{{\rm{d}}{t}_{\mathrm{mw}}}\displaystyle \frac{{\hslash }}{{\mu }_{B}}\displaystyle \frac{{{OD}}_{\mathrm{min}}}{{{OD}}_{\mathrm{mw}}^{\mathrm{max}}}\;\mathrm{exp}({\rm{d}}{t}_{\mathrm{mw}}/{\tau }_{2}).\end{eqnarray} \tag{ 1 }$$

We first calculate $\delta {B}_{\mathrm{photon}}$ for conditions matching our experiment parameters: an experiment run of ${N}_{\mathrm{shots}}=150$ shots taking a time ${\rm{d}}{t}_{\mathrm{run}}=30\;{\rm{s}};\;$ ${\rm{d}}{t}_{\mathrm{mw}}=22.5\;\mu$s; an atomic coherence lifetime ${\tau }_{2}=7.8\;\mu$s; a measured operating temperature of ${T}_{\mathrm{res}}=140\;^\circ {\rm{C}}$; total buffer gas pressure of ${P}_{\mathrm{fill}}=100\;\mathrm{mbar};$ optical pumping resulting in 1/3 of the atomic population residing in each of the F = 1 ground states, such that OD${}_{\mathrm{mw}}^{\mathrm{max}}=\frac{1}{3}{{OD}}_{87}=0.24,$ where OD₈₇ is the OD of the ⁸⁷Rb in the cell; and a photon shot noise limited ${{OD}}_{\mathrm{min}}=\sqrt{2}{[\;Q\;{I}_{\mathrm{probe}}\;{{\rm{e}}}^{-{OD}}\;A\;{\rm{d}}{t}_{\mathrm{probe}}/({\hslash }{\omega }_{L})]}^{-1/2}=1.0\times {10}^{-2},$ where ${\omega }_{L}$ is the laser frequency, Q = 0.27 is the camera quantum efficiency, ${I}_{\mathrm{probe}}=30\;\mathrm{mW}\;{\mathrm{cm}}^{-2}$ is the probe intensity, ${\rm{d}}{t}_{\mathrm{probe}}=0.3\;\mu {\rm{s}}$ is the probe duration, and the 2 × 2 pixel area is $A=42\times 42\;\mu {{\rm{m}}}^{2}.$ This gives us $\delta {B}_{\mathrm{photon}}=0.45\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2}.$ The exact operating temperature was unclear, however, with measurements of the OD indicating that the operating temperature may have been closer to ${T}_{\mathrm{res}}=130\;^\circ {\rm{C}}$, which would give $\delta {B}_{\mathrm{photon}}=0.28\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2}.$ We therefore conclude that our measured $\delta {B}_{\mathrm{mw}}^{\mathrm{exp}}=1.4\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2}$ is 3−5 times the photon shot noise limit determined by our experiment parameters. Analysis of OD_mw noise in the absence of a microwave field indicates that half of the $\delta {B}_{\mathrm{mw}}^{\mathrm{exp}}$ in excess of $\delta {B}_{\mathrm{photon}}$ is caused by imaging noise, due to factors such as camera readout noise and fluctuations in the intensities and frequencies of the lasers. Sources for the second half of the excess noise include fitting errors and timing jitter in the experiment sequence. We also note that we perform the imaging without magnetic shielding.

The optimal photon shot noise limited sensitivity, $\delta {B}_{\mathrm{photon}}^{\mathrm{opt}}=0.08\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2},$ is reached for ${T}_{\mathrm{res}}=130\;^\circ {\rm{C}}$, ${P}_{\mathrm{fill}}=60\;\mathrm{mbar},$ and with the laser tuned to the buffer-gas-shifted ⁸⁷Rb $F=2\to {F}^{\prime }=2$ line. Assuming that we can reach the photon shot noise limit, by reducing the excess noise from the above sources, we could expect a factor of 17.5 improvement in sensitivity with only minor modifications to our setup.

An improvement in sensitivity of several orders of magnitude is possible with more involved modifications. We are operating 5 × 10⁵ above the atomic projection noise limit, the ultimate sensitivity limit of an atom-based sensor [2]. Both $\delta {B}_{\mathrm{mw}}^{\mathrm{exp}}$ and $\delta {B}_{\mathrm{photon}}$ are limited by the camera readout speed and data-saving time, which give a poor experiment duty cycle (10 OD_mw images per second) and result in the atoms sitting uninterrogated for the vast majority of the time. This could be dramatically sped up with a different camera and camera operation mode, and we note that 50 × 50 pixel imaging of ultracold atoms has been reported with a continuous frame rate of 2500 fps [18]. Approaching the atomic projection noise limit will ultimately require moving to a quasi-continuous measurement scheme, likely based on Faraday rotation [18, 49], and perhaps replacing the CCD camera with an array of photodiodes.

In order to characterize and demonstrate our imaging system, we created three demonstration structures. The structures, shown in figures 4 and 5, respectively, are: a coplanar waveguide (CPW); a waveguide making several bends across its substrate, which we dubbed the 'Zigzag' chip; and a split-ring resonator (SRR). All of the microwave field measurements were made using the $140\;\mu {\rm{m}}$ cell.

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For imaging, the chip is generally placed perpendicular to the end of the vapor cell, as shown in figure 2(b). For chips built on a transparent or reflective substrate, operation in a second mode is also possible, with the chip placed in front of and parallel to the vapor cell, as shown in figure 6(a).

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We use the program Sonnet to perform a simulation of the microwave propagation on our structures using the method of moments. This technique is well suited for our mostly planar structures, excited at a single frequency. The program outputs the current distribution on the chip, from which we compute the magnetic near-fields using the Biot–Savart law. The only free parameters in comparisons with measurement were the amplitude of the input microwave signal and the exact position of the cell relative to the chip.

5.1. The coplanar waveguide

5.2. The Zigzag chip

5.3. The split-ring resonator

Our imaging technique can be adapted to measure dc magnetic fields. We use a Ramsey sequence [10], where the single microwave pulse of the above Rabi sequence is replaced by two $\pi /2$ pulses separated by a time ${\rm{d}}{t}_{\mathrm{Ramsey}}.$ Driving oscillations on the magnetic field sensitive $| F=1,{m}_{F}=1\rangle \to | F=2,{m}_{F}=1,2\rangle$ transitions, the oscillation frequency of the Ramsey fringes is equal to the detuning of the microwave from resonance, allowing us to measure the Zeeman shift induced by the applied dc magnetic fields. We can then use the Breit–Rabi formula to obtain the dc field of interest.

To detect individual vector components of a field of interest $\vec{B},$ we apply a second dc magnetic field of strength $C\gg B.$ In this way, we are primarily sensitive to the component of $\vec{B}$ that is parallel to $\vec{C}.$ For $\vec{C}$ along the X-axis, the measured field, ${B}_{\mathrm{meas}},$ is [2]

$$\begin{eqnarray}&&{B}_{\mathrm{meas}}=\sqrt{{(C+{B}_{X})}^{2}+{B}_{Y}^{2}+{B}_{Z}^{2}}\approx C+{B}_{X}.\end{eqnarray} \tag{ 2 }$$

We can obtain C in a separate reference measurement, and subtract this from ${B}_{\mathrm{meas}}$ to obtain B_X. The full vector magnetic field can be obtained by imaging with the C-field applied along each of the X-, Y-, and Z-axes.

Figure 7 shows images of the dc field above a 2 mm diameter wire loop, taken using the $200\;\mu$m thick cell. Again, we see a solenoid-like field, with a strong, uniform X-component, and the field turning outwards in the Y- and Z-components. Following the discussion on microwave sensitivity in section 4, fitting uncertainties give a sensitivity as small as $\delta {B}_{\mathrm{dc}}^{\mathrm{exp}}=1.6\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2}$ for a $40\times 40\times 200\;\mu {\rm{m}}$ sensor. As discussed in section 4, the dominant limiting factor is our poor experiment duty cycle, the improvement of which promises an increase in sensitivity by several orders of magnitude.

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We have demonstrated a new setup for high-resolution imaging of electromagnetic near fields, through the imaging of microwave fields above a variety of microwave devices, and the dc magnetic field above a wire loop. Microwave imaging is performed with a 120 × 120 array of $50\times 50\times 140\;\mu {{\rm{m}}}^{3}$ sensors, with the sensor size given by atomic diffusion during a measurement and the $140\;\mu {\rm{m}}$ cell thickness. The sensitivity per sensor, $\delta {B}_{\mathrm{mw}}^{\mathrm{exp}}=1.4\;\mu {\rm{T}}\;{\mathrm{Hz}}^{-1/2},$ is primarily limited by the experiment duty cycle, and improvements of several orders of magnitude should be achievable. We obtained a similar sensitivity for dc magnetic field imaging in a $200\;\mu {\rm{m}}$ thick cell. The setup allows us to image fields as close as $150\;\mu {\rm{m}}$ above surfaces, resulting in an order of magnitude increase in the resolution of surface features compared to previous vapor cell sensors. To our knowledge, this is the first vapor cell with such thin walls, and it should serve as a model for future vapor cells used in near-field sensing.

We currently perform imaging with the microwave device exposed to temperatures around 140 °C, which would be a barrier to the testing of temperature-sensitive devices. In future setups, we will move to locally heating the vapor cell with a $1.5\;\mu {\rm{m}}$ laser [50], significantly reducing the heat exposure of the device under test. If required, a further reduction in operating temperature could be achieved by using LIAD techniques to modulate the Rb vapor density [51].

Our microwave detection technique is not limited to ⁸⁷Rb, and can be applied to any system comprised of two states coupled by a microwave transition with optical read-out of the states, including the other alkali atoms, and solid state 'atom-like' systems, such as NV centers [52]. NV center–based imaging systems provide nanoscale resolution and typically work in scanning mode. They are thus complementary to our widefield imaging technique which is well adapted to image features on the micrometer scale with temporal resolution.

The full characterization of a microwave near field requires measurements of both the electric (E_mw) and magnetic (B_mw) components, as there is no straightforward relationship between the components. Alkali atoms in Rydberg states have proven to be excellent sensors of E_mw[11–13], but Rydberg states are quickly destroyed in collisions with buffer gas atoms. The vapor cell requirements for B_mw and E_mw imaging would therefore seem somewhat incompatible: we require high buffer gas pressures to prevent wall relaxation and provide spatial resolution for B_mw imaging, but require that there is little to no buffer gas present for E_mw imaging. However, with the addition of a $480\;\mathrm{nm}$ laser to excite Rb Rydberg states, our control over the buffer gas inside our ultrathin cells would allow us to perform an E_mw measurement without buffer gas, then fill the cell with buffer gas and image B_mw. Our setup would therefore be ideal for measurements of both components, and we would avoid the errors that using two different cell would bring, such as in cell alignment.

Microwave sensing and imaging (MSI) is an emerging field that has shown promise in a range of applications, particularly for breast cancer screening [15–17]. Current microwave detection systems consist of an array of microwave antennas sensitive to ${E}_{\mathrm{mw}}.$ Optimal image reconstruction requires a high sensor density; however, the density is limited by cross-talk between antennas, and by their perturbations of the microwave field. Sensor calibration is also a significant concern [17]. Atomic sensors are not affected by any of these problems. Following the success of vapor cell magnetometers in diagnostic imaging of the heart [53, 54] and brain [55–58], microwave imaging with vapor cells may also prove to be an attractive medical tool.

Our spatial resolution, sensitivity and distance of approach are now sufficient for characterizing a range of scientific and industrial microwave devices operating at 6.8 GHz. However, frequency tunability is essential for wider applications, with industry particularly interested in imaging techniques for frequencies above 18 GHz. It is possible to use a large dc magnetic field to Zeeman shift the hyperfine ground state transitions to any desired frequency, from dc to 100s of GHz. Using a 0.8 T solenoid, we have demonstrated microwave detection up to 26.4 GHz in a proof-of-principle setup, which will be presented in a subsequent paper [41].

This work was supported by the Swiss National Science Foundation. We acknowledge helpful discussions with C Affolderbach, G Mileti, P Appel, M Ganzhorn, and P Maletinsky.