Phase projection errors in rf-driven optically pumped magnetometers

TL;DR

This work addresses dynamic phase projection errors (DPPE) in rf-driven optically pumped magnetometers by deriving and solving a modified Bloch equation for the true-scalar M_x geometry and validating the model experimentally with a paraffin-coated Cs vapor cell. Using a rotating-frame analysis and the rotating wave approximation, the authors obtain analytic expressions for stationary and transient spin components and connect these to lock-in demodulated signals, revealing a transient phase shift when the static field direction tilts. The key contribution is the identification and quantification of DPPE, with explicit formulas showing how the transient phase depends on tilt angle, rf drive strength $G_{rf}$, pumping rate $\gamma_p$, and relaxation $\gamma_{tot}$, and how the effect persists even in the geometrically robust true-scalar configuration. The findings have practical implications for PLL-based operation and bandwidth in mobile or dynamically varying magnetic fields, and motivate alternative sensor designs (e.g., pulsed free precession OPMs) that are less susceptible to DPPE. Overall, the work highlights fundamental limits on the speed and accuracy of rf-driven OPMs in dynamically reoriented fields, informing both instrument design and field-deployment strategies for geomagnetic surveying and related applications.

Abstract

We investigate the phase relationship between the oscillating (rf) excitation field and the detected (light) power modulation in scalar rf-driven optically pumped magnetometers (OPMs), in particular in the $M_x$ configuration. While the static dependence of the demodulation phase on the direction of the external static magnetic field vector can be largely mitigated by aligning the oscillating rf field along the light propagation direction, we demonstrate that a dynamic (transient) phase response arises under magnetic field tilts. We analytically solve the corresponding modified Bloch equation and confirm agreement with experimental observations obtained using an $M_x$ magnetometer incorporating a paraffin-coated Cs vapor cell. The results reveal fundamental limitations of $M_x$ magnetometers regarding response time and accuracy, in particular when employed with active electronic feedback, such as a phase-locked loop. Therefore, this work is highly relevant to important magnetometry applications where the direction of the quasi-static magnetic field of interest is unknown \textit{a priori} or varies over time, or in measurements requiring a large detection bandwidth. Such conditions are encountered in applications such as geomagnetic surveying, particularly with mobile platforms.

Figures (9)

• Figure 1: Geometry of the true scalar magnetometer (TSM). The static magnetic field $\vec{B}_0$ is oriented along the $z$-axis, $\vec{k}$ lies in the $zx$-plane, at an angle $\alpha$ to $\vec{B_0}$. The rf magnetic field $\vec{B}_\mathrm{rf}(t)$ oscillates parallel to $\vec{k}$.
• Figure 2: Transformation of the coordinate system and magnetic field components for a rotation of $\vec{B}_0$ around the $y$-axis.
• Figure 3: Transformation of the TSM geometry under a rotation of the static magnetic field $\vec{B}_0$ around the $x$-axis. (a) Initial rotated frame where $\vec{k}$ is misaligned with the TSM plane. (b) Additional rotation restoring $\vec{k}$ to the effective TSM geometry.
• Figure 4: Schematic of the TSM magnetometer head. The key components are illustrated; the plastic housing is not shown for clarity. CL – collimating lens, CP – circular polarizer, HC – Helmholtz coils, FL – focusing lens, PD – photodiode detector. See main text for details.
• Figure 5: Experimental setup schematic. A detailed view of the TSM head is shown in Fig. \ref{['fig:TSM-Real']}. The magnetic field is generated by a triaxial coil system in a Helmholtz configuration. The cylindrical coil for $B_z$ has 15 windings and a diameter of 220 mm, while the square coils for $B_x$ and $B_y$ have side dimensions of $d_x$ = 150 mm and $d_y$ = 170 mm, each with 15 windings.