A magnetic field rotates the polarization of light
The rotation is proportional to the magnetic field strength

Interpretation: The superposition of a left circularly polarized light beam and a right circularly polarized beam provides a linearly polarized light beam. The physical interpretation of the Faraday rotation is based on this superposition. For the purpose of this analysis, we consider that the linearly polarized beam consists of the above beams. As mentioned by Wikipedia:

"In circularly polarized light the direction of the electric field rotates at the frequency of the light, either clockwise or counter-clockwise. In a material, this electric field causes a force on the charged particles comprising the material (because of their low mass, the electrons are most heavily affected). The motion thus effected will be circular, and circularly moving charges will create their own (magnetic) field in addition to the external magnetic field. There will thus be two different cases: the created field will be parallel to the external field for one (circular) polarization, and in the opposing direction for the other polarization direction – thus the net B field is enhanced in one direction and diminished in the opposite direction. This changes the dynamics of the interaction for each beam and one of the beams will be slowed down more than the other, causing a phase difference between the left- and right-polarized beam. When the two beams are added after this phase shift, the result is again a linearly polarized beam, but with a rotation in the polarization direction."

Figure 1: Polarization rotation due to the Faraday effect (Wikipedia - by DrBob~commonswiki assumed CC BY-SA 3.0).

## Surface magneto-optic Kerr effect (SMOKE)

Light reflected from a magnetized surface (object) can change in polarization and intensity (Wikipedia).

## Laser/LIDAR remote magnetometry using oxygen and the Faraday rotation effect or fluorescence detection

Remote atmospheric optical magnetometry, magnetometry using atoms and light

Remote magnetometry can be performed using lasers, atmospheric oxygen and measurement of the Faraday effect rotation or alternatively measurement of oxygen fluorescence). The technique can be applied in remote magnetic anomaly detection (MAD) for underwater and underground objects, i.e. the perturbation of the Earth’s magnetic field due to the presence of a metallic object.

In accordance with a report by the Naval Research Laboratory and the corresponding publication abstract, a laser field induces oxygen molecule transitions of a magnetic dipole nature with a spectroscopy line of ~762 nm whose intensity depends on the magnetic field strength. The magnetic dipole transitions create a magnetization field M (the process is termed magnetization ringing). This field depends on the density of the oxygen molecules and the effective magnetic dipole moment associated with the transition. The magnetization field M will induce a response electric field E and as a result an electromagnetic propagating field will be generated behind the pulse i.e. wakefield co-propagating with the pulse. This field will demonstrate Faraday rotation with respect to the incident laser field. In other words, its polarization will be rotated and as this will occur due to oxygen magnetic dipole transitions which are dependent on the magnetic field strength, this Faraday rotation will be dependent on the magnetic field strength.

By measuring the Faraday rotation we can determine the magnetic field strength and by comparing the measurements of different points in the wakefield we can determine if there is an object present perturbing the magnetic field.

In order to deliver laser pulses to a remote location (kilometer scale) in focus, Kerr-focusing is used. An extended discussion of related parameters e.g. laser energy etc. is provided.

The oxygen transition used is the one at approximately 762 nm which has been extensively studied and is a prominent feature of air-glow, night-glow and aurorae. The four states of O2 considered in this study are described below and are illustrated in Figure 4 of the first NRL report.

Figure 2: Four states of oxygen referred to for this technique (Figure 4 from NRL report).

The ground state of oxygen, termed “triplet state” represented by 3Σ−g refers to the state with total spin S=1 such that there are three allowed values of the spin component, ms = −1, 0, and 1 (ref.) and therefore three correponding sublevels, referred to as |1〉|2〉and |3〉respectively. Its total angular momentum is J = 1.

The excited upper state being considered is denoted by 1Σ+g  and is termed “singlet state” as it has total spin S=0 and therefore ms = 0 and only one sublevel referred to as |4〉. Its total angular momentum is J = 0.

The upper state can undergo three radiative transitions 1Σ+g  → 3Σ−g →with ms=-1,0,1 but the m=0 is insignificant.

The excited state |4〉can be populated by right hand polarized (RHP) light from level |3〉or by left hand polarized light (LHP) light from level |1〉.

At appendix C the case of resonant fluorescent excitation (Hanle effect) is discussed which requires sufficiently high laser intensities.

Figure 3: Legend included above (Figure 1 from NRL report).

## Laser/LIDAR remote magnetometry for underwater objects using the Faraday effect and the Surface Magneto-Optical Kerr Effect (SMOKE)

Remote optical magnetometry

Remote magnetometry for underwater objects can be performed using lasers and two magneto-optical effects, the Faraday rotation effect and the Surface Magneto-Optical Kerr Effect (SMOKE). These are applicable in remote magnetic anomaly detection (MAD) i.e. the detection of the perturbation of the Earth’s magnetic field due to the presence of a metallic object.  The use of the two effects is described below based on a report by the Naval Research Laboratory.

1) Surface Magneto-Optical Kerr Effect (SMOKE)

The presence of an underwater object induces magnetization of the sea surface above it due to the alteration of the dielectric properties of water. Light incident on and reflected from the surface will have its polarization rotated to a degree that is proportional to the magnetic field. This is termed the Surface Magneto-Optical Kerr Effect (SMOKE).

Light will penetrate water and will be reflected off the object itself. Its propagation and its reflection will be influenced by the altered magnetic field and as a result its polarization will be rotated due to the Faraday effect to a degree that is proportional to the magnetic field.

The purpose of the above report is to estimate the polarization rotation angle which is proportional to B addressing the issue of whether it is large enough to be measured.

The reflected field which has its polarization modified by both the SMOKE and Faraday effect is obtained (analytical expression). It is concluded that the contribution of SMOKE in the polarization rotation is small.

In order to interpret these effects, it could be noted that the electric field of light exerts a force on the electrons of water. The displacement of electrons of water modelled as a harmonic oscillator motion induces an additional magnetic field, other than the Earth's magnetic field, which influences the polarization of the electric field of light. We refer to a change in the refractive index of the water.

It is suggested to alternatively measure the un-rotated component of the reflected laser field as this can be significantly larger in amplitude than the polarization-rotated component.

Figure 4 : Remote magnetometry for underwater objects (Figure 1 from report by the Naval Research Laboratory)

## Circular Dichroism

https://en.wikipedia.org/wiki/Dichroism

A dichroic (two-color) prism is a prism that splits in two colors or two wavelengths (cf. beam splitter for two different wavelengths).
Dichroism refers to the differential absorption of light in different polarization states when travelling in a material.
When the polarization states in question are right and left-handed circular polarization, we then refer to circular dichroism.

https://en.wikipedia.org/wiki/Circular_dichroism
Left-hand circular (LHC) and right-hand circular (RHC) polarized light represent two possible spin angular momentum states for a photon, and so circular dichroism is also referred to as dichroism for spin angular momentum [3].

"Interaction of circularly polarized light with matter"
"When circularly polarized light passes through an absorbing optically active medium, the speeds between right and left polarizations differ (cL ≠ cR) as well as their wavelength (λL ≠ λR) and the extent to which they are absorbed (εL≠εR). Circular dichroism is the difference Δε ≡ εL- εR[5]. The electric field of a light beam causes a linear displacement of charge when interacting with a molecule (electric dipole), whereas its magnetic field causes a circulation of charge (magnetic dipole). These two motions combined cause an excitation of an electron in a helical motion, which includes translation and rotation and their associated operators."

A circular dichroism experiment will induce excitation and light absorbance (photoabsorption). "(...) the two types of circularly polarized light are absorbed to different extents. In a CD experiment, equal amounts of left and right circularly polarized light of a selected wavelength are alternately radiated into a (chiral) sample. One of the two polarizations is absorbed more than the other one, and this wavelength-dependent difference of absorption is measured, yielding the CD spectrum of the sample."

CD spectroscopy has a wide range of applications in molecular structure studies in biology, chemistry and material sciences.

## Photoexcitation Circular Dichroism

A new CD technique termed "Photoexcitation Circular Dichroism" is presented at the paper "Photoexcitation Circular Dichroism in Chiral Molecules"

https://www.nature.com/articles/s41567-017-0038-z.
The corresponding arXiv.org paper is found at: https://arxiv.org/abs/1612.08764https://arxiv.org/abs/1612.08764.

This study initially notes the helical motion described above.

First, the coherent excitation of electronic states leads to a charge displacement in the light propagation direction. Hence, a dipole moment, a macroscopic dipole and the corresponding electron density, a chiral density (asymmetric pattern), are created in the excited states, with a chiral current oscillating out of phase for the two enantiomers.

The resulting chiral dynamics can be probed/detected with photoelectron circular dichroism arising from the ionization of excited molecules by linearly polarized light pulses (PXECD).

PXCD could be used to drive molecular reactions in chiral systems in a stereospecific way, by imprinting a chiral torque via the helicity of the exciting circularly polarized pulse.

As mentioned in a relevant article (in french), the electrons are ejected from the molecules following two different directions.

(From the arXiv.org paper: "The photoelectrons are accelerated and projected by an electrostatic lens onto a set of dual microchannel plates and imaged by a phosphor screen and a CCD camera.")