In this paper, an analysis of a remote atmospheric magnetometry concept is considered, using molecular oxygen as the paramagnetic species. The objective is to use this mechanism for the remote detection of underwater and underground objects. Kerr self-focusing is used to bring a polarized, high-intensity, laser pulse to focus at a remote detection site where the laser pulse induces a ringing in the oxygen magnetization current. This current creates a co-propagating electromagnetic field behind the laser pulse, i.e., the wakefield, which has a rotated polarization that depends on the background magnetic field. The detection signature for underwater and underground objects is the change in the wakefield polarization between different measurement locations. The coupled Maxwell-density matrix equations are used to describe the oxygen magnetization in the presence of an intense laser pulse and ambient magnetic field. The magnetic dipole transition line that is considered is the $b1\Sigma g+\u2212X3\Sigma g\u2212$ transition band of oxygen near 762 nm. The major challenges are the collisional dephasing of the atmospheric oxygen transitions and the strength of the effective magnetic dipole interaction.

## I. INTRODUCTION

Optical magnetometry is a highly sensitive method for measuring small variations in magnetic fields.^{1–3} The development of a remote optical magnetometry system would have important applications for the detection of underwater and underground objects that perturb the local ambient magnetic field. In our remote atmospheric optical magnetometry model, a high-intensity pump laser pulse is employed to drive wakefields, which have a rotated polarization due to the earth's magnetic field. This can, in principle, provide a means to measure variations in the earth's magnetic field. For a number of magnetic anomaly detection (MAD) applications, $10\u2009\mu G$ magnetic field variations must be detected at standoff distances of approximately one kilometer from the sensor.^{4}

In this paper, we consider molecular oxygen at atmospheric conditions as the paramagnetic species in a remote optical magnetometry configuration depicted in Fig. 1. The propagation of the high-intensity pump laser pulse to remote detection sites is considered. By employing the optical Kerr effect, we show that high laser intensities (below $1012\u2009W/cm2$ to avoid photoionization processes) can be propagated to remote locations. Using a linearly polarized, high-intensity laser pulse, we consider the magnetization currents that are left ringing behind the pump pulse and the resulting co-propagating electromagnetic field. This field is referred to as the wakefield and it undergoes polarization rotation due to the Zeeman splitting of oxygen's ground state. The magnetic field variation is detected by measuring the wakefield's polarization.

Molecular oxygen's paramagnetic response is due to two unpaired valance electrons. The ground state of oxygen $X3\Sigma g\u2212$, commonly referred to as “triplet oxygen,” has total angular momentum $J=1$, total spin $S=1$, and three degenerate sublevels. The excited upper state being considered is denoted by $b1\Sigma g+$. It has $J=0\u2009$ and is a spin singlet state $S=0$ with only one sublevel. The upper state can undergo three radiative transitions, $b1\Sigma g+\u2192X3\Sigma g\u2212(m=\xb11)$, $b1\Sigma g+\u2192X3\Sigma g\u2212(m=0)$, but the latter is insignificant because it is an electric quadruple transition. There is an intermediate state, referred to as $a1\Delta g$, into which the excited $O2$ molecule can decay and is discussed in Appendix A. The $O2$ transition line being considered is the $b1\Sigma g+\u2212X3\Sigma g\u2212$ transition band of oxygen near 762 nm. In the low intensity, long laser pulse, regime, this transition has been investigated theoretically^{5,6} and experimentally^{7} and is a prominent feature of air glow. A high intensity, polarized titanium-doped sapphire laser is considered for the pump laser. These lasers have an extremely large tuning range from 660 nm to 1180 nm and can have linewidths that are transform limited.

A major challenge for this, as well as any remote atmospheric optical magnetometry concept, is collisional dephasing (elastic collisions) of the transitions. The elastic molecular collision frequency, at standard temperature and pressure (STP), is $\gamma c=Nair\sigma vth=3.5\xd7109\u2009s\u22121$, where $\sigma $ is the molecular cross section and $vth$ is the thermal velocity.^{7} On the other hand, the Larmor frequency in the earth's magnetic field is $\Omega o=qBo/(2mc)\u2009\u22484.5\xd7106rad/s$ ($\u210f\u2009\Omega o=3\xd710\u22129eV$), where $m$ and $q$ are the electron mass and charge and $c$ is the speed of light. Since the dephasing frequency is far greater than the Larmor frequency, the parameters are somewhat restrictive for remote atmospheric magnetometry. However, rotational magnetometry experiments based on molecular oxygen at STP and magnetic fields of ∼10 G have shown measurable linear Faraday rotational effects.^{7}

Previous theoretical work^{6} had major issues with atmospheric magnetic field measurements using oxygen, these include: (1) extremely low photon absorption cross sections, (2) a broad magnetic resonance linewidth due to collisions, and (3) quenching of excited-state fluorescence. These issues largely stem from oxygen's small magnetic dipole moment and large collision rate. In our work, however, the wakefield's polarization rotation is the magnetic signature and the laser pulse intensities are approximately six orders of magnitude larger.

## II. ATMOSPHERIC PROPAGATION OF INTENSE LASER PULSES (FOCUSING AND COMPRESSION)

The magnetometry concept considered here relies on propagation of intense laser beams in the atmosphere. This propagation is strongly affected by various interrelated linear and nonlinear processes.^{8} These include diffraction, Kerr self-focusing, group velocity dispersion, spectral broadening, and self-phase modulation. In general, a laser pulse propagating in air can be longitudinally and transversely focused simultaneously at remote distances (∼km) to reach high intensities (∼10^{12 }W/cm^{2}), as indicated in Fig. 2. Due to group velocity dispersion, pulse compression can be achieved by introducing a frequency chirp on the pulse; however, for the parameters under consideration, pulse compression is not significant. Nonlinear transverse focusing is caused by the optical Kerr effect.

Here, we present the model describing longitudinal and transverse compression of a chirped laser pulses in air.^{8} The laser electric field is given by $E(r,\eta ,\tau )=(1/2)E\u0302(r,\eta ,\tau )e\u2212i\omega \tau \u2009e\u0302x+c.c.$, where $E\u0302$ is the complex amplitude, $\omega $ is the frequency, $r$ is the radial coordinate, $\tau =t\u2212z/c$ and $\eta =z$ are the transformed coordinates, and the propagation distance $z$ and time $t$ are in the laboratory frame. Substituting this field representation into the wave equation results in an extension of the paraxial wave equation for $E\u0302(r,\eta ,\tau )$,^{8}

where the wavenumber is $k=\omega /c$. For air at STP and $\lambda =2\pi /k\u2248762\u2009nm$, the group velocity dispersion is $\beta 2=2.2\xd710\u221231s2/cm$, the Kerr nonlinear index is $nK=3\xd710\u221219cm2/W$, and $1+nKI$ is the refractive index of air.

Equation (1) can be solved by assuming the pulse is described by a form that depends on certain spatially dependent parameters. With this assumption, a set of simplified coupled equations can be derived for the evolution of the spot size, pulse duration, amplitude, and phase of the laser field. Taking the laser pulse to have a Gaussian shape in both the transverse and longitudinal directions, the complex amplitude can be written as

where $Eo(\eta )$ is the field amplitude, $\theta (\eta )$ is the phase, $R(\eta )$ is the spot size, $\alpha (\eta )$ is related to the curvature of the wavefront, $T(\eta )$ is the laser pulse duration, and $\beta (\eta )$ is the chirp parameter. The quantities $Eo$, *θ*, *T*, *R, α, β* are real functions of the propagation distance $\eta $. The instantaneous frequency spread along the pulse, i.e., chirp, is $\delta \omega (\eta ,\tau )=2\beta (\eta )\tau /T2(\eta )$, where $\beta (\eta )=T(\eta )/(2\beta 2)\u2202T(\eta )/\u2202\eta $. A negative (positive) frequency chirp, $\beta (\eta )<0$$(\beta (\eta )>0)$, results in decreasing (increasing) frequencies towards the back of the pulse.

Substituting Eq. (2) into Eq. (1) and equating like powers of *r* and $\tau $, the following coupled equations for *R* and *T* are obtained:

where $\epsilon o=P(0)T(0)$ is proportional to the laser pulse energy and is independent of $\eta $, $P(\eta )=\pi R2(\eta )I(\eta )/2$ is the laser power, $I(\eta )=cEo2(\eta )/(8\pi )=I(0)R2(0)T(0)/(R2(\eta )T(\eta ))$ is the peak intensity, and $PNL=\lambda 2/(2\pi nK)$ is the self-focusing or critical power. In Eq. (3), the initial conditions are given by $\alpha (0)=\u2212(kR(0)/2)\u2202R(0)/\u2202\eta $ and $\beta (0)=T(0)/(2\beta 2)\u2009\u2202T(0)/\u2202\eta =0$. The first term on the right hand side of Eq. (3a) describes vacuum diffraction while the second term describes nonlinear self-focusing, i.e., due to $nK$. Nonlinear self-focusing dominates diffraction resulting in filamentation when $P>PNL\u2248\u20093\u2009GW$.^{8,9}

In the limit that the pulse length does not change appreciably, the laser spot size is given by $R(\eta )=R(0)[1\u22122\alpha (0)\eta /ZR0+(1\u2212P/PNL+\alpha 2(0))(\eta /ZR0)2]1/2$, where $ZR0=kR2(0)/2$ is the Rayleigh length. The spot size reaches a focus in a distance $\eta /ZR0=\alpha (0)/(1\u2212P/PNL+\alpha 2(0))$ as long as $P<(1+\alpha 2(0))PNL$.

Figures 3(a) and 3(b) show the evolution of the laser spot size and the intensity as a function of propagation distance for $\lambda =762\u2009nm$. At focus, the laser intensity $Ifocus=6\xd71010W/cm2$ and spot size $Rfocus=1.3\u2009\u2009mm$ are held constant by choosing appropriate initial conditions: wavefront curvature $\alpha (0)=37$, pulse duration $T(0)=100\u2009\u2009ps$, and chirp $\beta (0)=0$. By changing the laser energy and the initial spot size, the nonlinear self-focusing effect changes the focal point from 0.25 km to 0.75 km (see Fig. 3). Nonlinear laser pulse propagation allows for moving of the detection site location.

To achieve high focal intensities at ranges from 0.25 km to 0.5 km without relying on atmospheric nonlinearities, i.e., Kerr index, would require focusing optics with diameters from 22 cm to 66 cm.

## III. OPTICAL MAGNETOMETRY MODEL

The four levels of $O2$ being considered in the magnetometry model are shown in Fig. 4. The ground state is split by the Zeeman effect into three levels $|1\u3009$, $|2\u3009$, and $|3\u3009$ and the excited state is denoted by $|4\u3009$. The transition frequency with no Zeeman splitting corresponds to $\u210f\omega A=1.63\u2009eV$ (762 nm). The magnetic quantum number *m* associated with the various levels is indicated in Fig. 4. The excited state, level $|4\u3009$, can be populated by left hand polarized (LHP) light from level $|3\u3009$ or by right hand polarized (RHP) light from level $|1\u3009$. Here, the quantization axis and the direction of the static magnetic field are taken to be along the direction of laser propagation, the z-axis. Circularly polarized radiation carries angular momentum $\xb1\u210f$, which is directed along the propagation direction. The selection rule for allowed transitions is $\Delta m=\xb11$, which will conserve angular momentum.^{10} It should be noted that this transition is strictly magnetic dipole and spin forbidden, but spin-orbit coupling between the $b1\Sigma g+$ and $X3\Sigma g\u2212(m=0)$ states leads to a transition with a magnetic dipole-like nature and a larger than expected dipole moment.^{5,6,11}

A high-intensity pump pulse generates a magnetization current density $JM=c\u2009\u2207\xd7M$, where $M$ is the magnetization field. The current density in turn generates a response electric field and can also modify the pump pulse. The response electric field $E$ is given by $(\u22072\u2212(1/c2)\u22022/\u2202t2)E=(4\pi /c2)\u2202JM/\u2202t=(4\pi /c)\u2009\u2202(\u2207\xd7M)/\u2202t\u2009$ (Gaussian units). The magnetization is represented by a sum of LHP and RHP components $M(z,t)=ML(z,t)e\u0302L+$$MR(z,t)e\u0302R+c.c.$, where $ML(z,t)=N\mu m\rho 43(z,t)$, $MR(z,t)=N\mu m\rho 41(z,t)$, *N* is the density of the oxygen molecules, $\mu m$ is the effective magnetic dipole moment associated with the transitions, $\u2009\rho 43$ and $\u2009\rho 41$ are the off-diagonal coherence of the allowed density matrix elements (see Fig. 4), and $e\u0302L,R=(e\u0302x\xb1i\u2009e\u0302y)/2$ are vectors denoting the polarization direction. The magnetization current density can be written as $JM=\u2212ic\u2009\u2202ML(z,t)/\u2202z\u2009e\u0302L+$$ic\u2009\u2202MR(z,t)/\u2202z\u2009e\u0302R$ $+c.c.\u2009$ In terms of the x and y components,

The density matrix equation is given by $\u2202\rho nm/\u2202t=\u2212i\omega nm\u2009\rho nm+$$\u2009i\u2211l{\Omega nl\u2009\rho lm\u2212\Omega lm\u2009\rho nl\u2009}+relaxation\u2009terms$, where $\omega nm=\omega n\u2212\omega m\u2009$, $\Omega nm$ denotes the interaction frequency, the phenomenological relaxation terms are due to elastic and inelastic collisions, and spontaneous transitions and the magnetic dipole interaction Hamiltonian is $\u2212\mu m\u22c5B$ (Appendix B).^{10,12,13} The off-diagonal coherence elements of the density matrix for the relevant transitions, $|1\u3009\u2192|4\u3009$ and $|3\u3009\u2192|4\u3009$, are given by

where $\gamma c$ is the elastic collision frequency (not population transferring), the full set of density matrix equations are given in Appendix B.

The pump laser field, which induces the magnetization field, is expressed as a sum of LHP and RHP fields $Bpump(z,t)=BL(z,t)e\u0302L+BR(z,t)e\u0302R+c.c.$, where $BL,R(z,t)=B\u0302L,R(z,t)ei\psi (z,t)\u2009$ and $\psi (z,t)=kz\u2212\omega \u2009t$. The interaction frequencies associated with the allowed transitions are $\Omega 43(z,t)=\Omega \u0302L(z,t)ei\psi (z,t)$ and $\Omega 41(z,t)=\Omega \u0302R(z,t)ei\psi (z,t)$, where $\Omega \u0302L,R(z,t)=\mu mB\u0302L,R(z,t)/\u210f$ is half the Rabi frequency associated with the LHP and RHP components of the pump. Note that the Rabi frequency is defined with respect to the peak field.

Although we are considering a magnetic dipole transition, it is convenient to express the Rabi frequency normalized to an electric dipole moment. The magnitude of the Rabi frequency can be written as $\Omega \u0302Rabi=\mu m\u2009B\u0302peak/\u210f$=$(\mu m/\mu e)\u2009(\mu eE\u0302peak/\u210f)=$$(\mu m/\mu e)\u2009(\mu e/\u210f)(8\pi I/c)1/2$, where $I=c\u2009E\u0302peak2/(8\pi )$ is the pump laser intensity and $E\u0302peak$ is the peak electric field. Taking the normalizing electric dipole moment to be $\mu e=q\u2009rB=2.5\xd710\u221218\u2009statC\u2013cm$, where $rB$ is the Bohr radius, the magnitude of the Rabi frequency is $\Omega \u0302Rabi[rad/s]=$$\u20092.5\xd7108\u2009(\mu m\u2009/\mu e)\u2009I[W/cm2]$. As an example, for $I=1011\u2009W/cm2$ and $\mu m/\mu e=10\u22124$, the Rabi frequency is $\Omega \u0302Rabi=8\xd7109\u2009rad/s$.

## IV. FARADAY ROTATION OF WAKEFIELDS DRIVEN BY INTENSE LASER PULSES

The incident pump field is taken to be polarized in the x-direction $E=E\u0302o(z,t)ei(kz\u2212\omega t)\u2009(e\u0302L+e\u0302R)+c.c.$, where $\omega $ is the carrier laser frequency and the complex pulse amplitude $E\u0302o(z,t)\u2009$ can be modulated. Employing the variables $\tau =t\u2212z/c$ and $\eta =z$, $E=E\u0302o(\tau )e\u2212i\omega \tau \u2009ei\Delta k\eta (e\u0302L+e\u0302R)+c.c.$, the corresponding magnetic field in the y-direction is $B=\u2212iE\u0302o(\tau )e\u2212i\omega \tau \u2009ei\Delta k\eta (e\u0302L\u2212e\u0302R)+c.c.$, where $\Delta k=k\u2212\omega /c$ is the wavenumber mismatch. The imaginary part of the wavenumber mismatch $Im[\Delta k]=\Gamma D=(2\pi kN\mu m2\rho 11\gamma c/\u210f)((\omega \u2212\omega A)2+\gamma c2)\u22121$ is obtained from the linear dispersion relation and accounts for absorption. The characteristic wavenumber mismatch for $\lambda =762\u2009nm$ at atmospheric molecular oxygen density $N=5.7\xd71018\u2009cm\u22123$ and an equilibrium population of $\rho 11=1/3\u2009$ is $\Gamma D=1.7\xd710\u22122cm\u20131$ $(1/\Gamma D\u224860\u2009cm)$. To circumvent this short absorption length, the laser frequency can be moved off-resonance. For example, if we detune the laser by $30\gamma c$, which corresponds to a wavelength shift of 0.03 nm, then the absorption length is $1/\Gamma D\u2248500m$.

As the pulse propagates through the atmosphere, it induces a magnetization current, which generates a field polarized in both the x and y directions. The wave equation for the forward propagating, y-component of the complex field amplitude is $(\u2202/\u2202\eta +i\Delta k)E\u0302y(\eta ,\tau )=\u2212i\pi N\mu mk(\rho \u030243(\tau )+\rho \u030241(\tau ))\u2009$, where the Faraday rotated field is $E\u0302y(\eta ,\tau )e\u2212i\omega \tau \u2009ei\Delta k\eta \u2009e\u0302y+c.c.$ The magnetization current is a function of the off-diagonal coherence terms of the density matrix elements $\rho 43(\eta ,\tau )=\rho \u030243(\tau )e\u2212i\omega \tau \u2009ei\Delta k\eta $ and $\rho 41(\eta ,\tau )=\rho \u030241(\tau )e\u2212i\omega \tau \u2009ei\Delta k\eta $. The slowly varying quantities $\rho \u030243(\tau )$ and $\rho \u030241(\tau )$ are given by reduced density matrix equations $(\u2202/\u2202\tau \u2212i\Delta \omega 43)\u2009\rho \u030243(\tau )=i\u2009\Omega \u0302\u200943(\tau )\rho o$ and $(\u2202/\u2202\tau \u2212i\Delta \omega 41)\rho \u030241(\tau )=i\u2009\Omega \u0302\u200941(\tau )\rho o$, where $\Omega \u030243(\tau )=\Omega \u0302L(\tau )=\u2212i\mu mE\u0302o(\tau )/\u210f$, $\Omega \u030241(\tau )=\Omega \u0302R(\tau )=i\mu mE\u0302o(\tau )/\u210f$, $\rho o=\rho 11=\rho 22=\rho 33=1/3$, $\rho 44=0$, $\Delta \omega nm=\omega \u2212\omega nm+i\gamma c$, $\omega 43=\omega A\u2212\Omega o$, $\omega 41=\omega A+\Omega o$ and it has been assumed that $c|\Delta k|/\omega \u226a1$.

In the case of conventional Faraday rotation within a long pump duration, $\u2202/\u2202\tau =0$, the spatial change in the Faraday rotated field is given by $(\u2202/\u2202\eta +i\Delta k)E\u0302y(\eta ,\tau )=2\pi kN\mu m2(E\u0302o/\u210f)\rho o\Omega \u2009o/\gamma c2$. After propagating a distance *L*, the ratio between the Faraday rotated and incident intensities is $Iy/Io=$$\u2009|Ey|2/|Eo|2=$$\u2009(2\pi )4(L/\lambda )2(N\mu m2\rho o/\u210f)2(\Omega o/\gamma c2)2$.

In the present model, the pump pulse consists of a pulse train, as shown in Fig. 5 in which the duration of the individual pulses, denoted by $\tau p$, can be comparable or longer than the damping time $1/\gamma c$. However, the time separation between the pulses $T$ is taken to be long compared to a damping time. With this ordering of timescales, the individual pump pulses excite the density matrix elements $\rho 43$ and $\rho 41$, which generate a magnetization current that decays behind the individual pump pulses (Fig. 5). The magnetization current is oscillating at the transition frequencies, which are shifted from 762 nm by the Larmor frequency. The frequency shifts lead to a polarization rotation of the magnetization current. This generates a Faraday rotated electric wakefield, co-propagating with and behind each pump pulse.

The general form of the off-diagonal coherence elements is $\u2009(\u2202/\u2202\tau \u2212i\Delta \omega nm)\u2009\rho \u0302nm(\tau )=i\u2009\Omega \u0302nm\u2009(\tau )\rho o$ with solution $\rho \u0302nm(\tau )=i\u2009\rho o\u222b0\tau d\tau \u2032\u2009\Omega \u0302nm\u2009(\tau \u2032)\u2009exp(\u2212i\Delta \omega nm(\tau \u2032\u2212\tau ))$ within the pump pulse. The solution behind the pump pulse is $\rho \u030243(\tau )=\rho \u030243(\tau p)\u2009exp(i\Delta \omega 43(\tau \u2212\tau p))$ and $\rho \u030241(\tau )=\rho \u030241(\tau p)\u2009exp(i\Delta \omega 41(\tau \u2212\tau p))\u2009$. The reduced wave equations for the x and y components of the wakefields are

where $k=\omega /c$, $\u2009\omega \u226b|\u2202/\u2202\tau |,\u2009\u2009c|\Delta k|$ and $Co=2\pi (N\mu m2/\u210f)/\gamma c\u22486\xd710\u22127\u2009$ is a unitless parameter. In estimating $Co$ we have taken the magnetic dipole moment to equal $\mu m=\mu e\xd710\u22124=2.5\xd710\u221222\u2009statC\u2013cm$, the collision frequency to be $\gamma c=3.5\xd7109\u2009s\u22121$ and the $O2$ density to be $N=5.7\xd71018cm\u20133.$ The current densities are $J\u0302Mx(\tau )=(N\mu m\omega /2)(\rho \u030243(\tau )\u2212\rho \u030241(\tau ))$ and $J\u0302My(\tau )=i(N\mu m\omega /2)(\rho \u030243(\tau )+\rho \u030241(\tau ))$. When the collision rate is much larger than the Larmor frequency or detuning $\gamma c\u226b\Omega o,\omega \u2212\omega A$, the current densities behind the pulse ($\tau \u2265\tau p$) are given by

where the time dependence of the wakefield is captured by

When the laser detuning is larger than the collision rate $\omega \u2212\omega A\u226b\gamma c$, there is a phase shift from Eqs. (6), but, more importantly, the magnitude of the current is suppressed by a factor of $\gamma c/(\omega \u2212\omega A)$.

Figure 6 shows the wakefield time dependence, Eqs. (7), for pump pulse durations of $\tau p=$ 0.1, 0.5, and 1 ns, pump pulse energy of 100 mJ, and spot size of 1 mm. These choices in pulse duration, for a fixed pulse energy, result in a range of pump intensities from $6\xd7109\u2009W/cm2$ to $6\xd71010\u2009W/cm2$. Equations (5) indicates that $E\u0302x,y/E\u03020$ is proportional to $Wx,y(\tau )$, if $\Delta k$ is neglected. For the parameters in Fig. 6, the normalized peak wakefield amplitudes are $|E\u0302x/E\u0302o|\u2248$$0.5$, $1.5$, and $1.6$ and $|E\u0302y/E\u0302o|\u2248$$1\xd710\u22124$, $1.2\xd710\u22123$, and $2\xd710\u22123$. There is a tradeoff between driving the wakefields with a higher intensity pump ($E\u0302o\u223c\tau p\u22121/2$) versus driving it for a longer duration ($Wx,y\u223c\tau p$). As a result, for $\tau p>3/\gamma c$, the wakefield amplitude begins monotonically decreasing

For remote magnetic anomaly detection, small spatial differences in the magnetic field must be measured. Here, we consider measuring the differences in wakefield intensities at two nearby locations (∼1 m). The locations are referred to as (1) and (2) and have local magnetic fields $Bo$ and $Bo+\delta B$. The intensity of the wakefield's y-component at location (1) and (2) is $I1$ and $I2$, respectively. The fractional change in its intensity of the y-polarized wakefield is $|I1\u2212I2|/I1=|\delta I|/I1\u22482|\delta E\u0302y/E\u03021y|$, where $E\u03021y$ is the y-component of the wakefield amplitude and $\delta E\u0302y$ is the difference in the wakefield amplitudes between the two locations. Figure 7 shows the fractional wakefield intensities for various values of $\delta B$. For the values shown, $|\delta I|/I1\u223c10\u22123$.

The pump pulse energy is $I(\pi /2)\u2009R2\tau p$, where $R$ is the spot size. For a pulse of duration $\tau p=0.5\u2009nsec\u2009\u22482/\gamma c$, $R=1\u2009mm$ and intensity $I=1010W/cm2$, the pump pulse energy is $80\u2009mJ/pulse$. For a pulse train, rep-rated at $fp=1\u2009kHz$, the average pump laser power is $\u3008P\u3009=fpI(\pi /2)\u2009R2\tau p=80W$.

It is worth noting that at sufficiently high intensities, the upper level, level $|4\u3009$, can be populated resulting in a laser induced florescence signal to lower energy levels, i.e., levels $|1\u3009$ and $|3\u3009$. This process is known as the Hanle effect and is briefly discussed in Appendix C. The magnetization current resulting from the induced florescence of an x-polarized pump laser is $JM\u221d\u2009e\u2212\gamma c\tau \u2009\u2009cos(\omega A\tau )[cos(\Omega o\tau )\u2009e\u0302x\u2212sin(\Omega o\tau )e\u0302y\u2009]$.^{10} Using polarization filters, the intensity on a detector due to the x- and y-components of the current density can be measured separately. Taking the ratio of the intensities from the x- and y-components of $JM$ gives $Ix/Iy\u221dcot2(\Omega o\tau )$. Note that the ratio is independent of the collision rate as long as the individual intensities are greater than the inherent intensity fluctuations.

## V. DISCUSSION AND CONCLUDING REMARKS

Remote magnetometry has important applications, such as detection of underwater and underground objects. Detection of the spatial magnetic field fluctuations caused by such an object is important to the US Navy's missions. In the laboratory, under a controlled environment, conventional magnetometry techniques can be used to measure extremely small magnetic field perturbations (pT).^{2} Limitations on remote detection include effects from the laser propagation such as slight variations in the focal intensity due to air turbulence.

Polarized laser light propagating through atmospheric turbulence will develop small fluctuations in polarization. The ratio of the depolarized light intensity to the polarized light intensity is^{14} $\u3008\Delta I/I\u3009=\pi \u22123/2\u3008\delta n2\u3009(L/\u2113o)(\lambda /\u2113o)2$ where $\u3008\u2009\u2009\u3009$ denotes an ensemble average, $\Delta I$ is the depolarized intensity, $\u3008\delta n2\u3009$ is the mean square refractive index fluctuation due to turbulence, $L$ is the propagation range, and $\u2113o$ is the inner characteristic scale length associated with the turbulence. As an example, we consider the typical parameters $\lambda =762\u2009nm$, $\u2113o=1mm$, $L=1\u2009km$, and $\u3008\delta n2\u30091/2=10\u22126$. For these parameters, $\u3008\Delta I/I\u3009\u224810\u221213$ and depolarization due to turbulence is negligible compared to the polarization rotation of the wakefields.

The paramagnetic species considered here is the oxygen molecule, which has an effective magnetic dipole transition ($b1\Sigma g+\u2212X3\Sigma g\u2212$) near 762 nm. We considered an intense pump laser to induce a polarization rotation of the wakefield. This transition is resonantly driven by a linearly polarized pump laser pulse. Our examples suggest that the intensity of the rotated component of the wakefield can be measured.

Numerous issues remain to be considered, these include signal detection configuration, i.e., monostatic or bistatic, signal-to-noise ratio limitations, magnetic field orientation relative to the optical axis, and pump laser absorption in the atmosphere.

## ACKNOWLEDGMENTS

We would like to acknowledge Dr. S. Potashnik for useful discussions. This work was supported by the Office of Naval Research (ONR) and the Naval Engineering Education Center (NEEC).

### APPENDIX A: TRANSITIONS IN OXYGEN MOLECULE

Oxygen's abundance in the earth's atmosphere, approximately 21% $(N=5.7\xd71018\u2009cm\u22123)$ and its paramagnetic response make it a possible candidate species for a remote optical magnetometer.^{4–7} Molecular oxygen $O2$ has two unpaired electrons in the upper level of the ground state, giving it a paramagnetic response. The ground state of oxygen $X3\Sigma g\u2212$, commonly referred to as “triplet oxygen,” has total spin $S=1$ and three degenerate sublevels (see Fig. 4). In atmospheric conditions near the surface of the earth (pressure $P=1\u2009\u2009atm$, total number density $Nair=2.7\xd71019\u2009cm\u22123$, and temperature $T=23.5\u2009meV$), the ground state is fully populated because the next excited electronic state's energy, $Ea=0.98\u2009eV$ is much greater than the thermal energy.

The electronic configuration of molecular oxygen is shown in Fig. 8. As seen in Fig. 4, the first excited electronic state of oxygen, $a1\Delta g$, is referred to as “singlet oxygen” and only has one spin state $(S,m)=(0,0)$. This state has an energy of $Ea\u2212X=0.98\u2009eV$, $a1\Delta g$ can undergo spontaneous emission via a magnetic dipole transition to the ground state $O2(a1\Delta g\u2212X3\Sigma g\u2212)$ or $a\u2212X$. The $a\u2212X$ transition has a wavelength of $1.27\u2009\mu m$. This transition is dominantly due to the orbital angular momentum and has spontaneous emission rate of $Aa\u2212X=2\xd710\u22124\u2009s\u22121$.^{15}

The second excited state of oxygen $b1\Sigma g+$ (see Fig. 4) will be referred to as the upper state. It is also a spin singlet state with only one sublevel. The upper state can undergo three radiative transitions; $b1\Sigma g+\u2212X3\Sigma g\u2212(m=\xb11)$, $b1\Sigma g+\u2212X3\Sigma g\u2212(m=0)$, and $b1\Sigma g+\u2212a1\Delta g$, where the first and second transitions are between the different magnetic sublevels of the ground state and are referred to as the A band.^{11} The transitions will be referred to as $b\u2212X,1$, $b\u2212X,0$, and $b\u2212a$, respectively. The $b\u2212X$ transitions have an energy of $Eb\u2212X=1.63\u2009eV$, wavelength $\lambda b\u2212X=762\u2009nm$, and frequency $\omega b\u2212X=2.5\xd71015\u2009rad\u22c5s\u22121$. The calculated spontaneous emission rates of the $b\u2212X,1$ and $b\u2212X,0$ transitions are $Ab\u2212X,1=0.087\u2009s\u22121$ and $Ab\u2212X,0=1.6\xd710\u22127s\u22121$, respectively.^{15} The radiation from the $b\u2212X,1$ transition can be seen in air-glow, night-glow, and aurorae.^{15} The $b\u2212X,1$ transition is magnetic dipole- and spin-forbidden and it is dominant over the $\u2009b\u2212a$ and $b\u2212X,0$ transitions, which are electric quadrupole transitions.^{11} This can be explained by a large spin-orbit coupling between the $b1\Sigma g+$ state and the $X3\Sigma g\u2212(m=0)$ state. The spin-orbit coupling results in a mixing of the levels and the $b\u2212X,1$. The $b\u2212a$ transition has an energy of $Eb\u2212a=0.65\u2009eV$, wavelength $\lambda b\u2212a=1.9\u2009\mu m$, frequency $\omega b\u2212a=9.9\xd71014\u2009rad\u22c5s\u22121$ and spontaneous emission rate of $Ab\u2212a=1.4\xd710\u22124s\u22121$.^{15}

### APPENDIX B. DENSITY MATRIX EQUATIONS

Interaction of an oxygen molecule with radiation is governed by Schrödinger's equation $i\u210f\u2202|\psi \u3009/\u2202t=H|\psi \u3009$, where $H=H0\u2212\mu m\u22c5B(t)$ is the full Hamiltonian, $H0$ is the electronic Hamiltonian after Zeeman splitting, and $\u2212\mu m\u22c5B(t)$ is the magnetic dipole interaction energy. The state $|\psi (t)\u3009=\u2211nCn(t)|n\u3009$ can be decomposed into the orthogonal energy eigenstates of $O2$, $|n\u3009$. The probability amplitudes $Cn(t)$ are related to the density matrix elements $\rho nm(t)=Cn(t)Cm*(t)$. The macroscopic electromagnetic fields are driven by a statistical ensemble of molecules, not a single molecule, and therefore it is necessary to use the density matrix equations and to introduce phenomenological relaxations terms, i.e., $\u2202\rho nm/\u2202t=\u2212i\omega nm\u2009\rho nm+i\u2211l{\Omega nl\u2009\rho lm\u2212\Omega lm\u2009\rho nl\u2009}+relaxation\u2009terms$. The interaction frequency is given by $\u210f\Omega nl=\u3008n|\mu m\u22c5B(t)|l\u3009$.

In our model, molecular oxygen is treated as a closed four level atom composed of the ground state $O2(X3\Sigma g\u2212)$ and the upper level $O2(b1\Sigma g+)$. The ground state has three spin sublevels $m=\u22121,\u20090,+1$, which are referred to as states $|1\u3009$, $|2\u3009$, and $|3\u3009$ respectively. The excited upper level is referred to as state $|4\u3009$. The complete set of coupled equations for the density matrix elements, assuming a closed system, are given by

The population level of state $|n\u3009$ is given by $\rho nn$ while the coherence between the states are given by $\u2009\rho nm=\rho mn*\u2009$. The transition frequencies are defined as $\omega mn=\omega m\u2212\omega n$, where $\u210f\omega n$ is energy of the n^{th} state. For example, the state frequencies are $\omega 1=\u2212\Omega o$, $\omega 2=0$, $\omega 3=\Omega o$, and $\omega 4=\omega A$, and the transition frequencies are $\omega 41=\omega A+\Omega o$, $\omega 13=\u22122\Omega o$, and $\omega 43=\omega A\u2212\Omega o$, where $\omega A$ is $O2(b1\Sigma g+\u2212X3\Sigma g\u2212)$ transition frequency in the absence of a magnetic field. The Larmor frequency is given by $\Omega o=qBo/(2mc)$, where $q$ is the electric charge, $Bo$ is the static background magnetic field, and $m$ is the electron's mass. Equations (B1)–(B7) imply conservation of population levels, i.e., $\u2009\u2202(\rho 11+\rho 22+\rho 33+\rho 44)/\u2202t=0$ (closed system). The populations are additionally normalized unity, i.e., $Tr(\rho )=1$. The interaction frequency between states $m$ and state $n$ is $\Omega mn=\Omega nm*$. Specifically, $\Omega \u030243(z,t)=\mu mB\u0302L(z,t)/\u210f$ and $\Omega \u030241(z,t)=\mu mB\u0302R(z,t)/\u210f$, where $\mu m$ is the effective magnetic dipole moment between triplet oxygen and the upper state and $B\u0302L,R$ corresponds to the left (right) handed polarization of the pump field. The rate equation for $\rho 42$ is not considered since it does not couple to the those in Eqs. (B1)–(B7).

The rates $\gamma 41$ and $\gamma 43$ consist of contributions from (i) elastic collisions (soft, dephasing collisions with no population transfers) and (ii) inelastic collisions (population transferring) and spontaneous emission. The elastic collision rate is taken to be the dominate rate and we set $\gamma 41=\gamma 43=\gamma 31=\gamma c$. In the absence of the pump field and at equilibrium, we have $\rho 11=\rho 22\u2009=\rho 33=\rho o$ and $\rho 44=0$. This implies that $\Gamma 21=\Gamma 12$, $\Gamma 31=\Gamma 13$, and $\Gamma 23=\Gamma 32$ and we take these rates, which include inelastic collisions and spontaneous emission, to equal $\Gamma o$. In addition, the rates $\Gamma 41$, $\Gamma 42$, and $\Gamma 43$ consist of inelastic collisions and spontaneous emission and we take these rates to be equal to $\Gamma U$. Taking the inelastic collision rates to be equal, i.e., $\Gamma o=\Gamma U$, the density matrix equations become

The phenomenological inelastic damping rate is given by $\gamma o=3\Gamma o=3\Gamma U\u2248108s\u22121$.^{6} The equilibrium populations for the ground state are $\rho 11eq=\rho 22eq=\rho 33eq=1/3$ and for the upper state $\rho 44eq=0$.

### APPENDIX C: RESONANT FLUORESCENT EXCITATION (HANLE EFFECT)

At sufficiently high intensities, laser induced fluorescence, i.e., Hanle effect, can be considered. The Hanle effect refers to the depolarization of resonant fluorescence lines by an external magnetic field.^{1,2,10} It provides a sensitive experimental technique for a number of measurements, including remote measurement of planetary magnetic fields^{16} and spontaneous emission rates,^{10} and spin depolarization rates.^{17} It is also the basis of one of the most sensitive methods for measuring the lifetime of excited levels of atoms and molecules.^{18} In the presence of a magnetic field, the Zeeman sublevels of the ground state are split, resulting in a difference in the resonance frequencies for LHP and RHP light. The resulting phase difference between LHP and RHP light, which is dependent on the ambient magnetic field, alters the polarization of fluorescing radiation.

To discuss this mechanism in more detail, we consider a short intense laser pulse polarized in the x-direction $\u2009Epump=E\u0302o(\tau )e\u2212i\omega A\tau \u2009(e\u0302L+e\u0302R)+c.c.$ This is just one of many orientations and configurations of the pump polarization and magnetic field direction in which the Hanle effect can occur.

The pump pulse is intense enough to excite level $|4\u3009$ at the expense of levels $|1\u3009$ and $|3\u3009$. The pump pulse duration $\tau p$ is short compared to the collision time which in turn is short compared to a Larmor period. As the short duration, high-intensity polarized pump pulse sweeps by it leaves behind an excited state, which fluoresces with polarization components different than that of the pump. The fluorescence from the excited state $\rho 44$ to states $\rho 11$ and $\rho 33$ is described by the off-diagonal coherence of the molecular density matrix elements $\rho 43=\u2212i\Omega \u0302L\tau p\rho 44e\u2212i(\omega 43\u2212i\gamma c)\tau $ and $\rho 41=\u2212i\Omega \u0302R\tau p\rho 44e\u2212i(\omega 41\u2212i\gamma c)\tau $, where $\Omega L=\u2212i\mu mEo/\u210f$, $\Omega R=i\mu mEo/\u210f$, $\omega 43$$\omega 43=\omega A\u2212\Omega o$, and $\omega 41$ = $\omega 41=\omega A+\Omega o$. The magnetization left behind the pump pulse is $M=\u2212Mo\u2009e\u2212\gamma c\tau (e\u2212i\u2009\omega 43\tau \u2009e\u0302L\u2212e\u2212i\u2009\omega 41\tau \u2009e\u0302R)+c.c.\u2009$ where $Mo=N\u2009\mu m2\u2009(Eo/\u210f)\u2009\tau p\rho 44$. The associated current density is $JM=\u2212Mo\u2009e\u2212\gamma c\tau [\u2009\omega 43\u2009{\u2009cos\u2009(\omega 43\tau )\u2212i\u2009\u2009sin\u2009(\omega 43\tau )\u2009}e\u0302L+\omega 41\u2009{\u2009cos\u2009(\omega 41\tau )\u2212i\u2009\u2009sin\u2009(\omega 41\tau )\u2009}e\u0302R\u2009]+c.c.$, where $\omega 43,\omega 41\u226b\u2009\gamma c$. The current density has components in the x and y directions^{10} which, for $\omega A\u226b\Omega o$, are given by

By using polarizer filters, the time average intensity on a detector due to the x and y components of the current density can be measured separately. Taking the ratio of the intensities from the x and y components of $JM$ gives $Ix/Iy\u2009\u221d\u2009cot2(\Omega o\tau )$. The ratio is independent of the collision rate as long as the individual intensities are greater than the inherent intensity fluctuations.