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Abstract
Atomic magnetometry is one of the most sensitive field-measurement techniques for biological, geo-surveying, and navigational applications. An essential process in atomic magnetometry is measurement of optical polarization rotation of a near-resonant beam due to its interaction with atomic spins under an external magnetic field. In this work, we present the design and analysis of a silicon-metasurface-based polarization beam splitter that have been tailored for operation in a rubidium magnetometer. The metasurface polarization beam splitter operates at a wavelength of 795 nm and has a transmission efficiency > 83% and a polarization extinction ratio > 20 dB. We show that these performance specifications are compatible with magnetometer operation in miniaturized vapor cells with sub-picotesla-level sensitivity and discuss the prospect of realizing compact, high-sensitivity atomic magnetometers with nanophotonic component integration.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Atomic magnetometers based on optical pumping can provide sub-femtotesla-level sensitivity [1] and are therefore attractive for a host of magnetometry applications including biological sensing [2,3], geo-surveying [4], and magnetic map-based navigation [4,5]. Optically pumped magnetometers (OPMs) typically involve optical pumping of alkali atoms (most commonly rubidium or cesium) with a circularly polarized beam to modify the atomic-spin-dependent optical properties of a medium inside a vapor cell. A linearly polarized probe beam is then used to detect the precession of the atomic spin in the presence of a magnetic field [6]. Typically, this precession causes a polarization rotation of the probe beam which is detected with an optical polarimeter via a balanced polarimetry, a photoelastic, or a Faraday modulation method [7]. Most OPMs have been realized using an orthogonal pump-probe arrangement with balanced optical polarimetry, since it has a simple configuration and if sufficiently balanced, can suppress common-mode noise arising from laser intensity fluctuations [8].
While breakthroughs in microfabricated vapor cells [9] have enabled the miniaturization of OPMs and their integration with chip-scale photonic technologies such as VCSELs [2,10], state-of-the-art OPMs based on balanced polarimetry detection mostly utilize bulk birefringent polarization optics such as waveplates and polarizing beam splitters which still limit the sensor volume, scalability, and their use in many portable applications [11–13]. More recently, various schemes have been proposed [14,15] and experimentally demonstrated [16,17] to further integrate OPMs with nanophotonic components, but the performance limits of nanophotonic-integrated OPMs are still being explored.
A common approach to simplify OPM designs and make them more scalable towards realizing multi-channel OPM arrays is to have the pump and probe beams share a single optical axis. These inline OPM schemes can be achieved by using a single elliptically polarized beam [18] or by overlapping the pump and probe beams at different transitions (e.g., the D1 and D2 lines) with dichroic optics [19]. Inline OPM configurations are also highly compatible with miniaturized magnetometers since microfabricated vapor cells usually allow only one optical access path.
In this work, we propose integration of inline OPMs with metasurface-based polarization components (Fig. 1). Specifically, we have designed an efficient and compact balanced polarimetry scheme using a metasurface-based polarization beam splitter (PBS). The geometry of our metasurface-based PBS can be freely optimized and tailored for integration with other atomic species and single-beam OPM designs. In addition, with advances in MEMS vapor cell fabrication [20] as well as techniques to integrate atomic vapor with integrated photonic circuits [16,17,21–23], it should be possible to directly fabricate metasurfaces on the glass windows of atomic-vapor cells. As illustrated in Fig. 1, pump and probe light can be delivered by fiber and go through a set of linear polarizer and quarter waveplate before transmitting through a microfabricated vapor cell. A metasurface PBS is integrated on the vapor cell window and splits the transmitted beam for balanced detection through a bi-cell or quadrant detector. For the polarimetry subsystem consisting of the beam-splitting optic and photodetectors, we estimate a volume reduction of 15-35% by the metasurface PBS in comparison to using bulk birefringent components such as a cube PBS or Wollaston prism (see Supplement 1).
Fig. 1. Schematic of the proposed atomic magnetometer with a metasurface PBS. LP: linear polarizer; QWP: quarter waveplate; PD: photodetector.
We then evaluate the accuracy and sensitivity of atomic magnetometry using the miniaturized balanced detection scheme. For our OPM platform, we use Spin Exchange Relaxation Free (SERF) magnetometry in rubidium (Rb) with elliptically polarized light [16]. SERF magnetometers typically operate in a high atomic density regime (>${10^{13}}\; c{m^{ - 3}}$) where the spin-exchange collision rate of the alkali metal atoms is much larger than the Larmor precession frequency, resulting in longer ground state Zeeman coherence time and thus extremely high sensitivity [18]. In this case, the circularly polarized component of the elliptically polarized light induces atomic polarization in the ground state of isotopically purified Rb atoms through optical pumping. A balanced polarimetry scheme is then used to measure the optical rotation of the linearly polarized component of the transmitted light at the output of the cell. The balanced polarimeter consists of a metasurface PBS that splits the horizontal and vertical linear polarization components of the light beam propagating through the vapor cell medium which are then measured using two photodetectors which can be integrated into a single bi-cell or quadrant detection device. The detection of the difference of the horizontal and vertical linear polarization intensities provides a direct measure of the optical rotation. The sensitivity of the magnetometer is determined by the signal-to-noise ratio with which the rotation signal is measured.
The paper is structured as follows. In section 2, we introduce our approach to design nanophotonic components for miniaturized balanced detection scheme in in-line OPMs. Optical performance of those components is discussed including transmission losses and polarization extinction ratio. A performance analysis of SERF with metasurface optics is presented in section 3.
2. Design and modeling of metasurface-based PBS
2.1 Design approach
Recently, dielectric metasurfaces made of arrays of elliptical posts have been adopted to design a wide range of nanophotonic components (such as beam splitters, waveplates and optical lenses) offering efficient control of the amplitude, phase, and polarization of light. The elliptical nature of each scattering element provides an effective way to induce birefringence and impart independent phase shifts for polarizations along the two ellipse axes. The majority of efforts have been dedicated to engineering metalenses [24], beam shaping devices [25,26], polarization optics including polarization beam splitters [27], and holograms [28]. These advances have recently been applied to trap and cool atoms on a chip [29]. In this paper, we adapt an approach to design metasurface polarization components for operation at near-infrared (780 - 795 nm) wavelengths and thus compatible with Rb-based sensors. Our designs maintain high-transmission efficiency and polarization extinction ratio. Our approach is based on the formalism described by Arbabi et al. [27]. Specifically, we implement metasurface optical components by arrays of elliptical posts with the same height, but different diameters to locally modify the phase distribution and polarization of any arbitrary input beam.
We consider a transmissive metasurface polarizing beam splitter design (Fig. 2(a)) that can be used in a variety of miniaturized OPMs for balanced polarimetry of the magnetometry signal resulting from the polarization rotation of the probe light. For such a metasurface design, the optical response is fully determined by the phase and amplitude of light in transmission. Figure 2(b) shows a schematic illustration of the unit cell structure of the metasurface, consisting of a single-crystalline silicon elliptical post of height H and post diameters ${r_x}\; $ and ${r_y}$ along the x and y axes on a sapphire substrate. In the design, H is fixed while ${r_x}\; $ and ${r_y}\; $ are tailored to achieve the desired ${\varphi _x}$ and ${\varphi _y}$ phase shifts for x and $y$-polarized waves, respectively. To implement beam splitting based on input polarization of a normally incident beam, the desired phase shifts are set to ${\varphi _x} ={-} kxsin\theta $ for an $x$-polarized beam and ${\varphi _y} = kxsin\theta $ for a $y$-polarized beam, where the wavenumber is $k = 2\pi /\lambda $ and $2\theta $ is the separation angle between the split beams. For this design, the phase shifts are invariant along $y.$ We note that the metasurface splits the incident light into beams with polarizations in the $xz$ plane (from the $x$-polarized component of the incident beam) and y (from the $y$-polarized component of the incident beam). For simplicity, we will refer the $xz$-polarized transmitted beam as the $x$-polarized beam for the remainder of the paper.
Fig. 2. (a) Operation of a metasurface polarizing beam splitter, which splits an incident beam into beams of orthogonal xz and y polarizations. Note that for simplicity, we will refer the xz-polarized beam as the x-polarized transmitted beam for the rest of the paper. (b) Illustration of a unit cell comprising of a silicon post on a transparent substrate (here, sapphire). The silicon post has an elliptical cross-section (principal axis lengths ${r_x}\; $ and ${r_y}\; $) and height H. (c)-(d) Calculated normalized transmitted power (“Transmittance”) and phase shift of an incident x- or y- polarized plane wave, based on full-wave simulation of swept elliptical parameters with fixed lattice constant (a = 400 nm) and post height (H = 500 nm). (e) Top view of a section of the final metasurface polarization beam splitter design.
Next, we construct the metasurface by sampling the phase profile using a square lattice of silicon elliptical posts of various diameters that implement the required phase shift at that position. As stated in [27], to avoid non-zero order diffraction and to achieve a better approximation of the phase profile, the sampling lattice constant a should be smaller than the operating wavelength. In our case, we choose $a = 400\; nm$ for the Rb D1-line (795 $nm$).
By varying the geometric parameters ${r_x}\; ,\;{r_y}\; $ of the elliptical posts, we can impose independent phase shifts along the x and y polarization axes. Figures 2(c) and (d) show the simulated transmitted power and phase shift of an array of elliptical posts with different axis radii from 40 nm to 140 nm for an incident linearly polarized ($x$ or $y$) plane wave. We see directly from the figure that this platform provides a complete phase coverage independently over ${\mathrm{\varphi }_\textrm{x}}\; \textrm{and}\;{\mathrm{\varphi }_\textrm{y}}$ while maintaining a relatively high transmission amplitude of $> 87{\%}.$ Elliptical posts of specific radius are selected to match the metasurface phase profile at their position by minimizing the squared error $\epsilon = {|{{t_x} - {e^{i{\varphi_x}}}} |^2} + {|{{t_y} - {e^{i{\varphi_y}}}} |^2}$, where ${t_i}$ is the transmission coefficient of the elliptical post and ${\varphi _i}$ is the calculated phase profile for polarization $i = x,y$, respectively. The resulting metasurface consists of 2500 ${\times} \; $ 2500 posts (a portion of which is shown in Fig. 2(e)) and is designed to achieve a $2\theta = $ 20$^\circ $ split angle between the x and y polarization components. For this design, the maximum $\epsilon $ obtained is 0.0886 while the average $\epsilon $ is 0.0166. These $\epsilon $ values are limited by the mismatch between the desired phase and amplitude profiles and the discretized values in Figs. 2(c) and 2(d), but are nonetheless sufficiently low to ensure good beam splitting performance while maintaining high transmission, as will be shown in the next section.
2.2 Simulated optical performance
To characterize the optical performance of our metasurface beam splitter, we performed finite-difference time domain (FDTD) simulations for a design with a device area of 16 µm × 16 µm using a Gaussian beam source (see Supplement 1 on justification of our simulated beam source, which shows that the metasurface PBS is compatible with any paraxial beam with a near-uniform phase profile at the plane of incidence). Our design can be extended to a much larger (>1 mm2) device footprint, but this representative simulation area is chosen to limit computational time. The incident beam of wavelength $\lambda = 795\; nm$ (D1 transition of ${}_{}^{87}Rb$) is propagating in the $z$-direction at normal incidence. We evaluated the transmission characteristics of the metasurface for different input polarization angles. Figure 3(a) represents the normalized transmitted far-field intensity for an incident linear polarization at 45° (comprising of an equal superposition of x and y polarizations). As expected, the designed metasurface splits the 45° linearly polarized incident beam into $x$- and $y$-polarized beams propagated along different directions. The split beams have a slightly different transmittance with a split ratio of 49% and 47% for x and y polarization respectively. The transmittance among various input polarization angles is shown in Fig. 3(b). The results for $x$-polarized beam (respectively for $y$-polarized beam) demonstrate that the transmitted intensity varies as the square of the cosine (respectively sine) of the polarization angle of the input beam. This is in good consistency with Malus law for linear polarizers.
Fig. 3. (a) The 3D far-field normalized intensity scattering patterns for the 45-linearly polarized incident Gaussian beam under normal incidence. (b) The normalized transmission in far field for different polarization angles of the input beam. Each data point for the x and $y$-polarized beam is obtained by integrating the optical power within their respective scattered spot, then normalize it to the overall transmission power. The solid lines indicate sinusoidal fit to the data.
The performance shown here is for input beams of wavelength $\lambda = 795\; nm$ at normal incidence—in which the simulated transmittance is over 80%—exhibits a slight linear dichroism (83.3% for $x$-polarized beam and 85.8% for $y$-polarized beam). Polarization extinction ratio (PER) is another important performance metric for PBS devices. We define the PER for each output port as the light intensity of the main polarization mode divided by the light intensity of the orthogonal polarization mode (see Supplement 1). For normal incidence, the PER is 20.55 dB for x polarization and 28.3 dB for y polarization. The unbalanced PER values and linear dichroism between the polarizations arise from the inherent asymmetry in the desired phase profile for the two polarizations, which scatters the incident $x$-polarized light to the $- x$ direction and $y$-polarized light to the $+ x$ direction. The phase-matching algorithm minimizes the sum of the matching errors for the x and y polarizations at each elliptical post, which can lead to one polarization being matched better than the other. The PER values of this design are further limited by the discrete sampling of the metasurface and the residual error ($\epsilon $ in Section 2.1) when matching the geometric parameters of each elliptical post with the desired phase response.
In reality, the transmission efficiency and PER of the metasurface structure may vary with the incident angle as it has been reported in earlier works [15]. These deviations are significant considerations in an OPM, which may have axial misalignments between an input beam and the atomic cell. We have analyzed the transmission and polarization extinction of our design for different angles of incidence to evaluate the alignment requirements for achieving sensitive balanced polarimetry measurements. Figures 4(a) and 4(b) show the transmittance and polarization extinction ratio at different incident angles from $0$ to 10$^\circ $ tilted around the y axis. The PER is observed to improve, especially for the x polarization, as the incident angle increases to around ${3^ \circ }$. This improvement is due to the displacement of leakage light from the orthogonal polarization which falls out of the collected region (see Supplement 1). Both the PER and transmission suffer from degradation as the incident angle exceeds $5^\circ $, which is again attributed to the higher-order diffraction as the incidence condition deviates from the requirement for zero-order diffraction: $\lambda \gg \frac{a}{{cos\theta }}$ (where $\theta $ is the incident angle). Nevertheless, the polarization extinction ratio maintains a level of $> 17\; dB$ over a range of 10$^\circ $, an alignment condition that can be readily achieved in the actual experiment.
Fig. 4. Simulated intensity transmission and polarization extinction ratio as function of the incident angle for (a) $x$-polarized beam and (d) $y$-polarized beam.
3. Sensitivity performance analysis
In this section, we evaluate how the optical performance of our proposed metasurface PBS affects balanced polarimetry and thereby the sensitivity and accuracy of an OPM. Our PBS design is compatible with millimeter-scale, microfabricated Rb cells involving separate non-parallel [15] or inline pump and probe beams. For reasons specified in Section 1, we focus on the magnetometry schemes that are compatible with inline pump-probe configurations such as the Mx [30], Bell-Bloom [31], SERF [32], and nonlinear magneto-optical rotation (NMOR) [33] OPMs. Of the above schemes proposed, SERF is an attractive candidate since it provides the highest sensitivities among OPMs and has been demonstrated to achieve sub-picotesla sensitivity in microfabricated cells [32]. Moreover, while the shot-noise-limited sensitivity of SERF magnetometry is expected to degrade with decreasing cell size [34], SERF operation relies on a high buffer gas pressure which is also necessary in very small ($\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel\prec\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }$ mm3) vapor cells to prevent decoherence from collisions of the atoms with the cell wall. A high buffer gas pressure would broaden atomic transitions and preclude the operation of NMOR [35]. Finally, SERF using elliptically polarized light has shown very promising results with device scale down to 5 mm [16,30] and can require only a single optical beam to operate [18,19], and thus highly compatible with nanophotonic integration.
We thus choose SERF magnetometry using a single elliptical beam as the platform for evaluating the sensitivity of a metasurface-integrated magnetometer [18]. In this scheme, an elliptically polarized light close to the ${}_{}^{87}Rb\; $ D1 transition interacts with a ${}_{}^{87}Rb$ vapor cell of optical path length l and Rb number density n to generate atomic spin polarization. This same optical beam is used to detect spin polarization via balanced polarimetry. An amplitude-modulating field is applied along the direction of magnetic field measurement (along $x$) which must be transverse to the light propagation direction; all other field components are nulled. The optical signal is then detected using lock-in detection at the modulation frequency (${\omega _{mod}}$). The theory of operation for SERF magnetometry with elliptical light was worked out in [18] for the case of an ideal PBS with transmittance $T = 1$ and infinite polarization extinction ratio. In this scenario, the balanced polarimetry gives the polarization angle rotation $\phi $ via the differential signal of the two outputs from the PBS:
where $\phi $ is proportional to the spin polarization along the optical axis (see Supplement 1 for derivation), ${E_0}$ is the electric field amplitude, ${e^{ - \sigma nl}}$ is related to light absorption by Rb atoms (whereby $\sigma $ is the photon absorption cross section for unpolarized Rb atoms), $\beta $ is the ellipticity of the light and is set to $\pi /8$ [18]. In the low polarization limit, the signal $\mathrm{{\cal D}}$ is linear in terms of spin polarization ${P_z}$. This optical rotation will allow us to determine the transverse magnetic field.3.1 Effects of polarization-dependent transmittance and PER on magnetometry
Note that the differential signal $\mathrm{{\cal D}}$ shown in Eq. 1 only includes the polarization rotation term induced by circular birefringence. Such simplicity cannot be assumed for our metasurface PBS which has a non-negligible transmittance difference for the two orthogonal linear polarization and polarization leakage. We thus modify the theory to incorporate these non-idealities. Let ${T_x},\; {T_y}$ be the transmittance for the x and y polarizations, ${b_x}$ and ${b_y}$ be the percentage of the transmitted optical power that is leaked into the other output of the PBS for a pure x and y polarization input, respectively. b is related to PER through $b = \frac{1}{{PER + 1}}$. Then the outputs of the PBS become:
We will now compare the magnetometry signal of a metasurface-integrated magnetometer with one based on ideal optics (namely, a PBS with near unity transmittance, ultra-high PER, and no linear dichroism). We use identical excitation and cell conditions as in [18], in which the Rb vapor cell is filled with 300 Torr helium and 100 Torr nitrogen and heated to 200 $^{\circ}{C}\; $. The laser frequency detuning is set to ${v_0} - \nu = 45\; GHz$. These conditions lead to a linewidth $\mathrm{\Delta }\nu $ of 7.97 GHz, optical pumping rate of 880 Hz, and a spin relaxation rate of 1200 Hz. Figure 5 shows the differential signal $\mathrm{{\cal D}}$ as a function of the transverse magnetic field ${B_x}$ for magnetometers based on ideal optics (red) and our metasurface design (blue) for a vapor cell with $l = 5\; mm$. We can see from the Fig. 5 that the incorporation of the metasurface PBS into the magnetometer does not change the dispersive character of the signal and only modifies the signal amplitude by a scaling factor $\xi $ that is given by:
Fig. 5. Simulated differential signal $\mathrm{{\cal D}}$ with lock-in detection at the modulating frequency ${\omega _{mod}}$ as a function of ${B_x}$ with an ideal PBS (red) and metasurface PBS (blue).
For our design, $\xi \cong 0.837$. The metasurface polarization extinction ratios are still sufficiently high, such that ${b_x},\; {b_y} \ll 1$. Additionally, there is with no significant linear dichroism $|{{T_x} - {T_y}} |\ll 1$ and the laser detuning is sufficiently larger than the pressure-broadened linewidth. Under these conditions, the signal amplitude is simply attenuated by the mean transmittance of $\hat{x}$ and $\hat{y}$ polarization: $\xi \cong \frac{{{T_x} + {T_y}}}{2}$. Finally, it may be possible to further improve the metasurface design using adjoint optimization methods [36–38] that minimize the figure of merit (FOM) defined as $FOM = \left|{\frac{{{T_y}({1 - 2{b_y}} )- {T_x}({1 - 2{b_x}} )}}{{{T_x}({1 - 2{b_x}} )+ {T_y}({1 - 2{b_y}} )}}} \right|$. This analysis can also serve as a useful guideline for designing nanophotonic components for other quantum sensors.
3.2 Impact of metasurface PBS on magnetometer noise
We now discuss the impact of the metasurface PBS on the various noise sources accompanying the atomic magnetometer signal. It is shown in [32] that in a single-beam, millimeter-scale magnetometer based on absorption measurement, the laser intensity noise dominates. The laser intensity noise can arise from laser current fluctuations or light polarization drifts that are converted into amplitude noise by a linear polarizer at the input. The SERF scheme we adopted for miniaturization in this paper uses balanced polarimetry as well as spin modulation with lock-in detection, which are capable of suppressing the intensity noise [18,39]. A sensitivity of 7 $ fT/\sqrt {Hz} $ was reported in a centimeter-scale cell in [18], with the dominant noise source being the Johnson noise from the magnetic shield.
The primary way that the incorporation of the metasurface optic will affect magnetometer noise is through noise sources associated with detection, particularly photon shot noise. In photon-shot-noise-limited magnetometers, we estimate the metasurface PBS to increase photon shot noise by 9.6% in comparison to an ideal PBS (See Supplement 1). In this case the overall magnetic field sensitivity will be reduced by both the reduction of differential signal amplitude ($\xi \cong 0.837$) and the increase in photon shot noise, leading to a 24% reduction in SNR and sensitivity [40].
4. Conclusion and outlook
We have demonstrated a metasurface-based balanced polarimeter design for a compact Rb magnetometer based on SERF and have connected its optical performance to the magnetometer accuracy and sensitivity. Modeling of balanced polarimetry using simulated transmittance and PER shows that the nanophotonic component can be integrated into SERF magnetometry with only a 24% degradation in sensitivity. Our analysis of the magnetometry dependence of an imperfect PBS may inform future efforts to design and integrate nanophotonic polarization optics into atomic magnetometers and other atom-based sensors.
Our metasurface design can be realized with electron-beam lithography and reactive-ion etching techniques on single-crystal silicon films grown on transparent substrates. Millimeter to centimeter-scale devices are achievable with advances in metasurface manufacturing [41–45] and it may be possible to incorporate the metasurface PBS as part of the glass wafer anodically bonded to the silicon housing during MEMS fabrication of mm-scale vapor cells [10]. Both the anodic bonding and SERF operation require that the metasurface PBS be able to withstand high temperatures up to $300{\; ^ \circ }C.$ While the simulated metasurface PBS performance is robust against refractive index changes in silicon under such temperatures (see Supplement 1), thermomechanical design of metasurface-integrated vapor cells will need to account for mismatch in the coefficients of thermal expansion between silicon and the substrate.
Funding
Office of Naval Research (N00014-20-1-2598).
Acknowledgments
This work is supported by the Office of Naval Research and the Wisconsin Alumni Research Foundation. The authors thank Mikhail Kats and Thad Walker for helpful discussions on the metasurface design and its suitability to atomic magnetometry platforms. We also thank Jinsheng Hu for a careful reading of our manuscript and providing valuable feedback on our analysis.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.
Supplemental document
See Supplement 1 for supporting content.
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