Vector Magnetometry Exploiting Phase-Geometry Effects in a Double-Resonance Alignment Magnetometer

Double-resonance optically pumped magnetometers are an attractive instrument for unshielded magnetic-field measurements due to their wide dynamic range and high sensitivity. The use of linearly polarized pump light creates alignment in the atomic sample, which evolves in the local static magnetic field, and is driven by a resonant applied field perturbation, modulating the polarization of transmitted light. We demonstrate experimentally that the amplitude and phase of observed first- and second-harmonic components in the transmitted polarization signal contain sufficient information to measure the static-magnetic-field magnitude and orientation. We describe a laboratory system for experimental measurements of these effects and verify a theoretical derivation of the observed signal. We demonstrate vector-field tracking under varying static-field orientations and show that the static-field magnitude and orientation may be observed simultaneously, with an experimentally realized resolution of 1.7 pT and 0.63 mrad in the most sensitive field orientation.

Figure 1

An isometric-projection schematic showing the reference frame used. The orientation of the static magnetic field ${\overrightarrow{B}}_0$ is described by the spherical polar angles ${\theta }_V$ and ${\theta }_L$, and the oscillating magnetic field applied on the $z$ axis. The intensity difference between orthogonal components of light transmitted through the cell is measured using a differential photodetector (not shown).

Figure 2

A schematic of the experimental system, showing the external-cavity diode laser (ECDL), the Glan-Thompson linear polarizer (GT), the magnetometer cell, the five-layer mu-metal shield, the three-axis Helmholtz coils, the half-wave plate ($\lambda /2$), the polarizing beam splitter (PBS), the differential photodetector (DPD), the low-noise coil driver (LNCD), and the data-acquisition system [digital-to-analog converter (DAC) and/or analog-to-digital converter (ADC)]. The coordinate basis is shown in the top left-hand corner and shares the same axes as Fig. 1. The data-acquisition system is controlled using a personal computer (PC) (not shown).

Figure 3

A measured and fitted magnetic resonance, taken with $|B_0|=200$ nT applied at ${\theta }_V={118}^{\circ }$, ${\theta }_L={101}^{\circ }$ and $|B_{\mathrm{rf}}|=1.5$ nT. A total of 150 segments of data are taken, with a segment sample time of 20 ms. Left: the amplitude ($R$) and phase ($\varphi$) components of the first-harmonic demodulated signal. Right: the amplitude ($R$) and phase ($\varphi$) of the second-harmonic demodulated signal. The data are fitted with Eqs. (6)–(9), yielding ${\omega }_L=2\pi \times 699.60(2)\mathrm{Hz}$, $\mathrm{\Gamma }=12.1(1)\mathrm{Hz}$, $A_{1f}=107.4(7)\mathrm{mV}$, ${\varphi }_0^{1f}=0.9672(9)\pi \mathrm{rad}$, ${\varphi }_0^{2f}=-0.383(6)\pi \mathrm{rad}$ , and $\mathrm{\Omega }=2.89(8)$ Hz.

Figure 4

The measured magnitude of ${\overrightarrow{B}}_0$, determined from measurements of ${\omega }_L$ at 1646 different ${\overrightarrow{B}}_0$ orientations, evenly covering the full solid angle. Top: the distribution of measured $|B_0|$ around the desired field magnitude of 200 nT. The root-mean-square (rms) spread of $|B_0|$ is 302 pT. Bottom: the angular distribution of $|B_0|$.

Figure 5

The observed and (inset) calculated distributions of the on-resonance signal phase with variation of the ${\overrightarrow{B}}_0$ orientation over the full solid angle. Top: the first-harmonic phase ${\varphi }_0^{1f}$. Bottom: the second-harmonic phase ${\varphi }_0^{2f}$. The calculated distributions are found using Eqs. (12) and (13).

Figure 6

The observed and (inset) calculated distributions of the on-resonance signal amplitude with variation of the ${\overrightarrow{B}}_0$ orientation over the full solid angle. Top: the first-harmonic amplitude $A_{1f}$. Bottom: the second-harmonic amplitude $A_{2f}$. The polarimeter response to optical rotation is 55.6 V/rad for small rotations. The calculated distributions are found using Eqs. (10) and (11).

Figure 7

The measured and set magnetic-field magnitude and orientation for a wide range of ${\overrightarrow{B}}_0$ orientations. Observed values and uncertainties for ${\theta }_V$ and ${\theta }_L$ are calculated from ${\varphi }_0^{1f}$ and ${\varphi }_0^{2f}$ using Eqs. (15) and (16). Uncertainties in $|B_0|$, ${\theta }_V$, and ${\theta }_L$ are displayed as error bars, which in the case of $|B_0|$ are smaller than the marker. The point of best angular resolution is measured at ($|B_0|=199.8445(17)$ nT, ${\theta }_V=100.986(27{)}^{\circ }$, ${\theta }_L=118.198(24{)}^{\circ }$). The inset in the lower-left corner shows the contour described by successive ${\overrightarrow{B}}^{\mathrm{APP}}$ orientations, plotted using the same projection as in Figs. 5 and 6.