1. Introduction

While a ground-based approach for remote magnetometry has been developed by detecting the sodium atomic fluorescence at resonance [1–5], a broad angular distribution of the emitted fluorescence limits the flux of photons that can be detected at the ground telescope. One idea to overcome this limitation is to use mirrorless lasing by the sodium atoms in the mesosphere [6], but this concept has yet to be implemented in on-sky experiments. Besides mirrorless lasing, an alternative approach is to employ an assisting satellite with a photodetector or a retroreflector, so that the polarization and the intensity of transmitted probe light can be measured directly, obtaining stronger signals. However, this configuration poses new challenges associated with the motion of the satellite.

Thanks to advancing technologies, small and inexpensive satellite platforms such as CubeSat have gained popularity since the 2000s. Hundreds of CubeSats were launched in the past decade due to their cost-effective components and availability of launch opportunities [7,8]. A standard CubeSat unit (1U) is $10\times 10\times 11$ cm$^3$ and weights $1.33$ kg or less [9], with the 3U CubeSat being the most popular as its larger volume allows more components to be incorporated [10]. Despite its small size, a CubeSat can feature attitude control and detumble systems such as reaction wheels and magnetorquers [11,12]. Additionally, optical communication and high-speed data transmission have been demonstrated via optical up- and downlinks, allowing opportunities for real-time measurements [13–15]. CubeSat missions have become increasingly capable and versatile since the first proposal in 1999, making direct measurements of transmitted light possible for remote sky magnetometry.

Building upon these advances, the new proposed system involves a laser that serves as both the pumping and probing beams as well as a trackable miniaturized satellite. A satellite that can be optically tracked is launched into low earth orbit (LEO) at an altitude of 500 km and is equipped with a photodetector and a polarizer nearly orthogonal to the laser polarization. The laser beam first optically pumps mesospheric sodium in the target-measurement region, as the satellite orbits over the measurement region, the laser beam probes the sodium atoms that have been optically pumped. Photodetectors on the satellite detect the polarization of the transmitted light. The nonlinear magneto-optical rotation (NMOR) has a time-dependent component at the Larmor frequency ($\Omega _L$) or its second harmonic, with $\Omega _L$ proportional to the magnetic field strength. A schematic of the configuration is shown in Fig. 1.

 

Fig. 1. A schematic illustration of satellite-assisted remote magnetometry. The inset shows an example of the control sequence of the laser beam.

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2. Measurement procedure

After optical pumping, the Larmor frequency will be measured by detecting the magneto-optical rotation ($\varphi _1$) and/or the transmission ($T$) of the polarized probe-laser light. In general, the geomagnetic field makes an arbitrary angle with the laser beam. However, one can identify two extreme cases, namely measurements near the poles and at the equator, respectively, as shown in Fig. 2 where the analysis is particularly simple. Here, the magnetic field lines run, correspondingly, parallel or perpendicular to the light beam. We emphasize that this technique can be used at arbitrary latitudes.

 

Fig. 2. Measurements of the Earth magnetic field near the poles and at the equator. Orange arrows show atomic polarization from optical pumping relative to the magnetic field.

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Near the poles, we use linearly polarized laser light for pumping, which creates atomic alignment (see, for example, Ref. [16]). The linearly polarized probe laser light will show Faraday rotation of $\varphi _1$ that can be measured with a polarimeter on board the satellite. The analyzer can be configured with a transmission polarizer oriented nearly orthogonal ($\varphi _0$) to the polarization of the light. The intensity at the detection port is then

An estimate of $\varphi _1$ near the poles can be obtained from

At the equator, the laser beam would be nearly perpendicular to the magnetic field of the Earth. Optical pumping can then be achieved with circularly polarized light, and magnetometry is performed by either measuring $\varphi _1$ with linearly polarized probing light, or by measuring the transmission of a circularly polarized probing laser beam. Optical rotation at the equator can be estimated as

If we are limited by photon shot noise [see Sec. 3.], the sensitivity of the polarimeter does not depend on the angle between the polarizer and analyzer $\varphi _0$. This is because the useful signal [see Eq. (4)] is bilinear in the rotation angle in sodium $\varphi _1$ and $\varphi _0$, while the photon shot noise is dominated by the fluctuations of the leading term proportional to the square root of $\varphi _0^2$. If $I_0$ itself fluctuates, for example due to intensity noise, light scattering in clouds, etc., these fluctuations will contribute to noise in proportion to $\varphi _0^2$. It is thus beneficial to keep $\varphi _0$ as small as possible. However, this angle should be much larger than the angle corresponding to finite extinction of the polarizer and analyzer. In conjunction with this, we note that the laser depolarization on the uplink is expected to be negligible [17,18] and that the value of $\varphi _0$ can be adjusted at the launch telescope. The effect of possible polarization variations due to the atmosphere is further reduced by the fact that the detection occurs at 1$\times$ or 2$\times$ the Larmor frequency (typ. 300 kHz), while the perturbations have a time scale on the order of 1 ms.

Another approach in this scenario is to investigate the transmission spectrum of circularly polarized probe light. Although the measurements are scalar, vector information can be obtained as the NMOR resonances depend on the magnetic field direction [19]. In general, the signal from the probe light contains first and second harmonics of the Larmor frequency, whose ratio can give information about the direction of the magnetic field. For linearly polarized light with frequency modulated at $\Omega _m$, the main resonance occurs at $\Omega _m = 2\Omega _L$ when the magnetic field is along the light propagation direction, as in the case of measurements near the poles. For measurements near the magnetic equator in which the magnetic fields may be tilted in the plane perpendicular to the direction of light propagation, an additional resonance appears at $\Omega _m = \Omega _L$. The amplitudes of the resonance corresponds to the tilt of the magnetic field relative to the light propagation direction, allowing quantitative determination of the magnetic field direction.

The procedure for the experiment could be as follows. Once the satellite is within the laser spot, pumping is performed for about a millisecond (several ground-state spin relaxation times). The pump beam could be modulated (in amplitude, frequency or polarization) synchronously with the Larmor precession (see [4,20–22]) or it could be a strong continuous light pulse. The pumping interval is followed by a probing interval of comparable duration during which polarization rotation is monitored. Because spins cannot free-precess in the presence of the strong pump light field, they are “locked" by the pump beam (both in the case of cw and synchronous pumping), and the relative phase between pumping and probing is the same for all probed sub-layers of the mesosphere as the phase delay due to propagation is the same for the pump and the probe. For this reason, we expect propagation effects in the 10 km-long layer to be negligible. Note that for the case where pump light is circularly polarized and the probe is linearly polarized, the light polarization must be switched between pumping and probing. After a measurement is complete, the cycle can be repeated as long as the satellite is within the laser-beam. An attractive option is to utilize a tracking telescope [23], which would then allow magnetic measurements along the entire arc traced by the satellite while it is visible by the ground observer.

3. Sensitivity estimates: photon shot noise

For state-of-the-art remote magnetometry, i.e. fluorescence based mesospheric Na magnetometry, the fundamental sensitivity is hard to achieve because only a tiny fraction of fluorescent photons is detected within the small solid angle subtended by ground-based detectors. In this case, the sensitivity is limited by photon shot noise [1]. Likewise, the shot noise of the detected photons is a limiting factor in satellite-borne detection. The photon shot noise of an ideal polarimeter is solely determined by the total number of measured photons $N_p$, such that

A typical guidestar laser emits a power of 20 W at a wavelength of 589 nm (orange), which corresponds to a flux of $\phi _0\approx 6\times 10^{19}$ photons per second. The emitted beam has a typical divergence due to diffraction of

The dimensionless optical pumping saturation parameter for the mesospheric sodium atoms can be estimated as in, for example, Ref. [16], Sec. 9.1, taking into account that the effective ground-state relaxation is dominated by the Larmor precession in the magnetic field of the Earth at a frequency of a few hundred kHz. With full light power and the assumed spot size, this saturation parameter is on the order of $10^2 \gg 1$. This means there is more than enough light to polarize the sodium atoms, but on the other hand, the probe beam power would need to be attenuated (or frequency-detuned), which would, in turn, adversely affect the photon shot noise (signal size). However, if the area of the beam in the mesosphere was to be increased by two orders of magnitude, the repolarization effect would be minimized without affecting the pumping efficiency. We thus choose the beam diameter at the sodium layer as 5 m corresponding to approximately $24\,\rm {m}$ at an altitude of $H_{\text sat}=500$ km. At this altitude, the orbital speed is given by

Because of the limited size of the CubeSat, only a fraction of the total photon flux $\phi _0$ will be detected. Assuming an on-board light-collection system with a diameter of $D_{SD}=10$ cm, the fraction of the detected photon flux is about $1.7\times 10^{-5}$ leading to $\phi _{Sat}\approx 1.0\times 10^{15}\,\rm {s^{-1}}$. With these numbers, we estimate a photon shot-noise limit of $\delta \varphi \approx 4.8 \times 10^{-7}\,\rm {rad}$ per single shot ($\tau =1\,$ms). From Eq. (3), a crude estimate of the shot-noise-limited magnetic sensitivity for $t\sim 1/\gamma _0$ reads

With a bandwidth of $1/4\tau =250\,$Hz, accounting for the $\sim 1\,$ms pumping time, the conversion to a sensitivity per root Hz yields $4.9\,\rm {pT}/\sqrt {\rm {Hz}}$ when measuring at the equator, and half that value at the poles. This is about two orders of magnitude better compared to sensitivities derived for fluorescence-based sky magnetometry [1].

4. Conclusions and outlook

We have presented a concept for magnetometric measurements in the upper atmosphere based on magneto-optical rotation in the mesospheric sodium layer. A laser beam resonant with an atomic sodium transition is launched with a ground-based laser and detected using a polarimeter on board a small satellite. Sensitivity estimates show that this technique, based exclusively on existing well-proven technologies, is highly promising and will complement the existing measurement techniques based on the currently employed balloons and aircraft on the one hand, and space-borne instruments on the other, addressing the range of altitudes of considerable current interest to geophysics [25,26]. Going beyond the estimates presented here, one should perform detailed modeling of the experiment [27], taking into account factors like the magnetic field gradients across the sodium layer, the effect of the wind and the satellite-tracking dynamics, etc., to optimize the operation parameters (the exact timing pattern, laser-frequency tuning with respect to the atomic resonance, hyperfine repumping, etc.), as well as considering various avenues for achieving better performance. For example, we have assumed a relatively small area of 100 cm$^2$ of the satellite-borne detector which results in wasting a lot of photons and thus raising shot noise. The light collection may be improved by several orders of magnitude by deploying an origami-folded collection mirror in orbit along the lines of well-proven space technologies [28–30]. Finally, we note that the proposed method can be used also for other types of measurements in addition to magnetometry. For example, one can extract the density of sodium atoms integrated over the mesospheric layer from the amplitude of optical rotation signal; measurements of electric field may also be possible [31].