Laser/LIDAR remote magnetometry (remote sensing of magnetic fields)

Magnetometry using atoms and light

Introduction

Interrogating atoms for their precession frequency using lasers/LIDAR

By measuring the precession frequency (Larmor frequency, fL) of atoms, we can determine the magnetic field in which the atoms are immersed, as this is proportional to fL. An atom excitation signal can be used, such as the 589 nm laser-mediated fluorescence of sodium via optical polarization, due to which an enhanced photon return signal is received when the laser is pulsed at the Larmor frequency of sodium, i.e. at conditions of magneto-optical resonance. By scanning the pulse repetition frequency of the laser while measuring the photon return signal (fluorescence), we can determine the Larmor frequency upon signal maximization i.e. upon (magneto-optical) resonance. Having measured the Larmor frequency, we can calculate the magnetic field strength.

Application: measurement of the Earth's magnetic field at a specific point found at a distance of 100 Km from the Earth surface (mesosphere) using mesospheric sodium.

The fundamental technical issue is the handling of the atomic polarization, as this will determine the optimal backscatter/flux of photons towards the source.

A fundamental theoretical issue is the optical polarization of atoms, that is the understanding that when a sodium atom which has one electron in its outer shell is excited with e.g. left circularly polarized light, it will obtain a specific polarization, i.e. its total angular momentum which qualifies it as a magnetic dipole will point in a specific direction and this polarization will be determined by that electron.

Theoretical background and technical implementation

At a distance of approximately 100 Km from the Earth's surface, within the middle layer of the Earth's atmosphere called the mesosphere, exists a layer of sodium atoms which are deposited due to vaporization of meteorites upon entry in the Earth's atmosphere. The magnetic field of the Earth induces precession of the sodium atoms with a frequency, termed Larmor frequency, which is proportional to the magnetic field strength. The constant of proportionality termed gyromagnetic (or magnetogyric) ratio γ is known with precision for sodium (γNa = 699,812 Hz G−1). Therefore, by measuring the Larmor frequency (fL) of the sodium atom or atoms in general, we can determine the magnetic field (B) in which the atom is immersed, since fL=γ*Β. For instance, if the Larmor frequency for sodium is measured to be 220 kHz at a specific location, then the magnetic field will be approximately 0.31 G at that location.

In order to measure the Larmor frequency, we use a technique that has been established for astronomical analysis of light emitted from stars with the purpose of correcting the effects of atmospheric turbulence on the light path towards the telescope. The technique is called “Laser Guide Star” (LGS) as its purpose is to create an artificial light source or star that will serve as a guide for adaptive optics. By determining the effect of atmospheric distortion on the diffusion of a known artificial light source, we can then subtract this effect from the light coming from the star that we study and compensate the distortion or adapt to it. For instance, if we create a light point in the sky and instead our telescope shows a diffuse extended circular area, we can establish a correction procedure to obtain a point of light from the diffused signal.

In order to create an artificial light source, we excite sodium atoms in the mesosphere using lasers with the purpose of obtaining fluorescence that will be captured by a telescope. The most recent study (Bustos et al. 2018) used a laser of 2 W. It is noted that the emission may constitute a 4% backscatter towards the source. The Laser Guide Star (LGS) technology has been used for adaptive optics initially in astronomy and subsequently in the military (U.S. Air Force) for satellite reconnaissance and tracking (image correction with the purpose of identification of hostile satellites). 

The technology has been adapted for remote magnetometry or remote magnetic field sensing using lasers/LIDAR. The sodium atom (1s2, 2s2, 2p6, 3s1) has eleven electrons, all in closed shells with the exception of a single electron in its outer shell (one valence electron). As a result, the charge distribution is not homogeneous, a non-zero magnetic moment is generated and the sodium atom behaves like a natural magnetic dipole.

The valence (outer shell) electron has orbital angular momentum Le due to its orbit/rotation around the nucleus and spin angular momentum Se due to its spin (self-rotation analog in classical physics) (Figures 1,2). This electron will determine the direction of the vector of the total electronic orbital angular momentum L and the total electronic spin angular momentum S, and by extension their sum, the (total) electronic angular momentum J of the atom. It will also determine the direction of the vector of the atomic angular momentum F (ref. 1) (ref. 2) which is equal to the sum of the nuclear angular momentum (net nuclear spin) I and the electronic angular momentum J (total angular momentum of the electrons). The vector of the atomic angular momentum determines the polarization of the atom. Therefore, the sodium outer shell electron will determine the polarization of the sodium atom.

Figure 1 | On the left: Spin angular momentum S due to the rotation of a sphere upon itself. In the middle: Orbital angular momentum L of a sphere due to its rotation around an origin (the angular momentum has meaning only with respect to a specific origin and its direction is always perpendicular to the plane formed by the position vector r and the linear momentum vectors p.) On the right: Total angular momentum J is the vector sum of L and S (from Wikipedia (selection) - By Maschen - Own work, CC0, included at https://en.wikipedia.org/wiki/Angular_momentum).

Figure 2 | On the left: Spin angular momentum L_spin due to the rotation of a sphere upon itself and orbital angular momentum L_orbital due to the rotation of a sphere around an origin. On the right: Total angular momentum Lorbital is the vector sum of Lorbital and Lspin (from Wikipedia - By Maschen - Own work, Public Domain, - included at https://en.wikipedia.org/wiki/Angular_momentum).

For an atom with resolvable hyperfine structure, the complete specification of a magnetic substate consists of the orbital angular momentum L, the spin S, their combination J, the nuclear spin I, the total angular momentum F, and the magnetic quantum number M (ref) (integers ranging from -F to +F). It is noted that the nuclear spin I for a given isotope e.g. of sodium found in the mesosphere is 3/2.

Note: The orbital angular momentum of an electron is represented by a vector L which is a special kind of vector. Although we cannot determine the direction of L in space, we can determine a component of it in space i.e. its projection along an axis, conventionally the z axis (called the quantization axis). It is mentioned that this vector is special because its projection is quantized to values of one unit of angular momentum apart. More details at this reference: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2.

The 3s shell where the valence electron is found has an L of 0 and is split in two substates or energy levels, depending on whether the electron is in the spin-up configuration i.e. S=½ or in the spin-down configuration i.e. S=-½. The electron angular momentum J which is the sum of L and S will be respectively J=½ or J=-½. And the atom angular momentum F which is the sum of I=3/2 and J will be respectively F=3/2+½=2 or F=3/2-½=1. Therefore, the 3S½ state can be either F=2 or F=1.

For the 3S½ F=2 ground state the quantum numbers (L, S, J, I, F) are (0, 1/2, 1/2, 3/2, 2). This F=2 state has 5 magnetic sublevels corresponding to values of M ranging from -2 to +2. 

Let us consider a most important sodium transition which is used in Laser Guide Stars and remote magnetometry, the 3S½ (F=2) →  3P³/₂ (F'=3).

For the 3P3/2  F'=3 excited state, the quantum numbers are (1, 1/2, 3/2, 3/2, 3). 

When a sodium atom is illuminated with electromagnetic energy represented by a photon of energy E and frequency f (E = hf = ħω), corresponding to the energy difference/gap between two energy levels of an electron (Ε = E2 - E1), e.g. 589 nm for the 3S½ to 3P3/2 transition (termed D2 transition), the photon will be absorbed by the electron found on the 3S½ level which will be excited and will transition to the 3P3/2 level. It is said that 589 nm light is resonant with the specified energetic transition. It is noted that a transition of a frequency/wavelength corresponding to the visible, UV and IR spectrum (spectra seen by living organisms) is termed an optical transition. Shortly after, following a time interval equal to the excited state lifetime which is approximately 16 ns for sodium, the absorbed photon will be emitted via fluorescence (resonant scatter) and the electron will relax or transition to the ground state. 

If we illuminate the sodium atom with light of 589 nm, we will obtain upon examination with low resolution the D2 spectroscopy line, which upon higher resolution is shown to be split into two hyperfine lines, the D2a and D2b corresponding to transition from the F = 2 and F = 1 ground state, respectively. The D2a transition is resonant with 589.158 nm light.

For LGS and remote magnetometry we are interested in the D2a transition from F = 2 to F' = 3 and particularly in the D2a transition from the F=2 magnetic substate with magnetic number 2 termed |F = 2, m = 2> to the F' = 3 magnetic substate with magnetic number 3 termed |F' = 3, m = 3> substate (Figure 3, strongest red arrow).

It is noted that this is a transition where the magnetic number changes by +1 or in other words ΔM=+1. Transitions with ΔM=+1 or ΔM=-1 can only be mediated with circularly polarized light,  σ+ (left) or σ- (right) respectively. Therefore, for this transition we need left-circularly polarized light which additionally must be of 589.158 nm as this is resonant with the specific transition. (Transitions with ΔM=0 can only be mediated with linearly polarized light.)  

This type of light will transfer the atoms to the |F = 2, m = 2> ground state and will initiate a cycling between this state and the |F' = 3, m = 3> state.

Figure 3:  Energy level diagram of the Na D2 line, showing transitions induced by circularly polarized light resonant with the D2a transition. The widths of the arrows indicate the relative transition strengths. It is noted that the transition from |F = 2, m = 2> to |F' = 3, m = 3> state is the strongest (Figure 1 from Rochester et al 2018).

We are interested in this transition because, first, it is the strongest D2 transition, which means that there will be increased absorption (increased effective absorption cross section) of the provided laser light. Also, use of circularly polarized light will induce polarization in the direction of light propagation (Figure 4). Due to this, the atoms will have enhanced backscatter in the axis of polarization. The emitted fluorescence from |F' = 3, m = 3> will be directed along the polarization axis and therefore along the light beam. This will induce an enhanced return signal or an enhancement of photon flux observed at the location of the light source. 

Figure 4: The atomic polarization P is generated initially along the light wave vector k, corresponding to the direction of propagation of circularly polarized light. Upon the effect of the magnetic field B, the atomic polarization P will precess around the direction of the field (reproduction based on Figure 2 from Rochester et al 2018).

If the magnetic field has a transverse component in the direction of polarization, it will induce Larmor precession of the atomic polarization around the magnetic field direction (Figure 4). This will have a depolarizing role as the precession can be compared to a certain cancelling or dampening effect which will tend to eliminate the atomic polarization generated by light absorption, as we will have the case of a top that wobbles while losing energy until it eventually stops. This can be avoided if the polarization is continuously generated with light pulses and the Larmor precession is sustained (no wobbling towards stopping effect). Under these conditions the polarization will tend to be averaged while precessing around the magnetic field direction (cf. Figure 4). 

Once the atomic polarization precesses at the Larmor frequency, if electromagnetic radiation (light) is provided at this frequency or with a pulse repetition frequency (PRF) identical to the Larmor frequency, the atoms will become optimized for enhanced fluorescence backscatter once per Larmor cycle along the atomic polarization axis, which corresponds to the light propagation axis and this will be observed at the location of the source. Given this, by slowly scanning the laser pulse repetition frequency (PRF) of the transmitted light near the expected Larmor frequency, we will observe a maximal signal when the Larmor frequency is reached. This would be indicative of magneto-optical resonance resonance (D2a optical transition). Having obtained the Larmor frequency, we can calculate the strength of the magnetic field. 

The most recent study on laser remote magnetometry using mesospheric sodium has been conducted by researchers in Canada, Europe and the United States obtaining an accuracy of 0.28 mG Hz−1/2 (Bustos et al. 2018 - Press release). The researchers were among other from the European Southern Observatory (headquarters in Germany - Laser Guide Star Unit was at the ESO Paranal site in Chile), University of British Columbia and University of Heinz (Germany) with the pioneer of the field Prof. D. Budker being also affiliated with the University of Berkeley. 

Figure 5: Study by Bustos et al. 2018 

Bibliographic references

Section on Laser Guide Stars

http://www.information-book.com/physics/laser-guide-stars/

"Physicists propose beaming laser at atmospheric sodium to measure global magnetic field" - "Magnetometry with mesospheric sodium" (2011)

https://www.sciencedaily.com/releases/2011/02/110214155449.htm

https://news.berkeley.edu/2011/02/14/lasers-vie-with-satellites-to-map-earth%E2%80%99s-magnetic-field/

James M. Higbie, Simon M. Rochester, Brian Patton, Ronald Holzlöhner, Domenico Bonaccini Calia, and Dmitry Budker. Magnetometry with mesospheric sodium. PNAS, February 14, 2011 DOI: 10.1073/pnas.1013641108

What is the magnetometric sensitivity of the technique?

Excerpt from the publication above and also the book "Optical Magnetometry, edited by Dmitry Budker and Derek F. Jackson Kimball, Cambridge University Press (2013) - Location 7069

"The calculated magnetometric sensitivity, which is limited by currently available laser power and may therefore be expected to improve with advances in laser technology, is useful for the envisaged geophysical applications which require measurement of fields, e.g., in the 1–10 nT range for ocean circulation [2] and the tens of nanotesla range for the solar-quiet dynamo [1]; moreover, the dynamic range of the measurement is not subject to any simple physical limit, as the resonance technique works well and with similar sensitivity at any magnetic-field strength. Moreover, the sensitivity of this technique could be further enhanced by as much as five orders of magnitude by reflecting a laser from a rocket- or satellite-borne retroreflector, instead of being limited to the small fraction of fluorescence emitted toward the detecting telescope."

The nanotesla range corresponding to 10^-9, when enhanced by 5 orders of magnitude is 10^-14, i.e. one order below the femtotesla.
 

Please note that the brain magnetic field is in the order of picotesla and femtotesla (cf. 

http://www.bemri.org/publications/natural-fields/113-environmental-and-biological-magnetic-flux-densities/file.html) and

https://en.wikipedia.org/wiki/Orders_of_magnitude_(magnetic_field))

"Modeling of pulsed laser guide stars for the Thirty Meter Telescope project" (2012)

https://www.osapublishing.org/josab/abstract.cfm?uri=josab-29-8-2176

https://arxiv.org/pdf/1203.5900.pdf - Please refer to Figures 1 and 2

"Sodium guide star at Larmor frequency extends geomagnetic studies" - "Sodium guide star laser pulsed at Larmor frequency" (2017)

Figure 2: Facebook post from Laser Focus World 

https://www.facebook.com/LaserFocusWorld/photos/a.10152444584165298.966977.126899915297/10159992102510298/?type=3&theater.

"Researchers at the Shanghai Institute of Optics and Find Mechanics (SIOM) have succeeded in developing a high-power 589 nm sodium laser pulsed at the Larmor frequency in the 200–350 kHz range for remote magnetometry, which translates to analysis of 100-km-scale variations at the ground: https://goo.gl/X4miT4"

Cited study at https://www.ncbi.nlm.nih.gov/pubmed/?term=29088161

"Laser-created guide stars, developed for astronomical adaptive optics, could measure the Earth’s magnetic field at a scientifically crucial, previously inaccessible range" (2018 May/October)

Remote magnetometric capabilities using light (lasers) are reported to have increased by an order of magnitude in 6 months. 
Sensitivity reported to be at 28 nT/Hz½ (noise-equivalent power)

https://twitter.com/OPNmagazine/status/1049284770511966209

> "Laser Remote Magnetometry Using Mesospheric Sodium" (2018 May)

https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2018JA025178

https://arxiv.org/ftp/arxiv/papers/1610/1610.05385.pdf

> "Scientists develop a new way to remotely measure Earth's magnetic field" (2018 October)

2018-10-04

https://www.sciencedaily.com/releases/2018/10/181004095948.htm

Remote sensing of geomagnetic fields and atomic collisions in the mesosphere. Nature Communications, 2018; 9 (1) DOI: 10.1038/s41467-018-06396-7

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