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 b1Σg+X3Σg 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.

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, 10μG 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 1012W/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.

FIG. 1.

Remote nonlinear optical magnetometry configuration. The earth's magnetic field is Bo0.5G and δB10μG is the perturbation caused by the underwater/underground object.

FIG. 1.

Remote nonlinear optical magnetometry configuration. The earth's magnetic field is Bo0.5G and δB10μG is the perturbation caused by the underwater/underground object.

Close modal

Molecular oxygen's paramagnetic response is due to two unpaired valance electrons. The ground state of oxygen X3Σg, 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 b1Σg+. It has J=0 and is a spin singlet state S=0 with only one sublevel. The upper state can undergo three radiative transitions, b1Σg+X3Σg(m=±1), b1Σg+X3Σg(m=0), but the latter is insignificant because it is an electric quadruple transition. There is an intermediate state, referred to as a1Δg, into which the excited O2 molecule can decay and is discussed in Appendix  A. The O2 transition line being considered is the b1Σg+X3Σg transition band of oxygen near 762 nm. In the low intensity, long laser pulse, regime, this transition has been investigated theoretically5,6 and experimentally7 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 γc=Nairσvth=3.5×109s1, where σ 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 Ωo=qBo/(2mc)4.5×106rad/s (Ωo=3×109eV), 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 work6 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.

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 (∼1012 W/cm2), 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.

FIG. 2.

Simultaneous transverse focusing and longitudinal compression of a chirped ultrashort laser pulse in air due to nonlinear self-focusing and group velocity dispersion. For the 100 ps pulses that are optimal for magnetometry, longitudinal compression is negligible, but transverse self-focusing can compensate.

FIG. 2.

Simultaneous transverse focusing and longitudinal compression of a chirped ultrashort laser pulse in air due to nonlinear self-focusing and group velocity dispersion. For the 100 ps pulses that are optimal for magnetometry, longitudinal compression is negligible, but transverse self-focusing can compensate.

Close modal

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,η,τ)=(1/2)Ê(r,η,τ)eiωτêx+c.c., where Ê is the complex amplitude, ω is the frequency, r is the radial coordinate, τ=tz/c and η=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 Ê(r,η,τ),8 

[2+2ikηc2kβ22c2τ2+ω2nK4πc|Ê(r,η,τ)|2]Ê(r,η,τ)=0,
(1)

where the wavenumber is k=ω/c. For air at STP and λ=2π/k762nm, the group velocity dispersion is β2=2.2×1031s2/cm, the Kerr nonlinear index is nK=3×1019cm2/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

Ê(r,η,τ)=Eo(η)eiθ(η)e(1+iα(η))r2/R2(η)e(1+iβ(η))τ2/T2(η),
(2)

where Eo(η) is the field amplitude, θ(η) is the phase, R(η) is the spot size, α(η) is related to the curvature of the wavefront, T(η) is the laser pulse duration, and β(η) is the chirp parameter. The quantities Eo, θ, T, R, α, β are real functions of the propagation distance η. The instantaneous frequency spread along the pulse, i.e., chirp, is δω(η,τ)=2β(η)τ/T2(η), where β(η)=T(η)/(2β2)T(η)/η. A negative (positive) frequency chirp, β(η)<0(β(η)>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 τ, the following coupled equations for R and T are obtained:

2Rη2=4k2R3(1εoPNL1T),
(3a)
2Tη2=4β2kεoPNL1R2T2+4β22T3,
(3b)

where εo=P(0)T(0) is proportional to the laser pulse energy and is independent of η, P(η)=πR2(η)I(η)/2 is the laser power, I(η)=cEo2(η)/(8π)=I(0)R2(0)T(0)/(R2(η)T(η)) is the peak intensity, and PNL=λ2/(2πnK) is the self-focusing or critical power. In Eq. (3), the initial conditions are given by α(0)=(kR(0)/2)R(0)/η and β(0)=T(0)/(2β2)T(0)/η=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>PNL3GW.8,9

In the limit that the pulse length does not change appreciably, the laser spot size is given by R(η)=R(0)[12α(0)η/ZR0+(1P/PNL+α2(0))(η/ZR0)2]1/2, where ZR0=kR2(0)/2 is the Rayleigh length. The spot size reaches a focus in a distance η/ZR0=α(0)/(1P/PNL+α2(0)) as long as P<(1+α2(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 λ=762nm. At focus, the laser intensity Ifocus=6×1010W/cm2 and spot size Rfocus=1.3mm are held constant by choosing appropriate initial conditions: wavefront curvature α(0)=37, pulse duration T(0)=100ps, and chirp β(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.

FIG. 3.

Evolution of (a) laser spot size and (b) normalized peak laser intensity as functions of propagation distance for different initial laser energies and spot sizes. The laser energy and initial spot size for the solid, dashed, and dotted lines are εo= 100, 150, and 190 mJ and R(0)= 4.7, 6.7, and 8.2 cm, respectively. By tuning laser parameters, the remote detection region can be moved.

FIG. 3.

Evolution of (a) laser spot size and (b) normalized peak laser intensity as functions of propagation distance for different initial laser energies and spot sizes. The laser energy and initial spot size for the solid, dashed, and dotted lines are εo= 100, 150, and 190 mJ and R(0)= 4.7, 6.7, and 8.2 cm, respectively. By tuning laser parameters, the remote detection region can be moved.

Close modal

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.

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, |2, and |3 and the excited state is denoted by |4. The transition frequency with no Zeeman splitting corresponds to ωA=1.63eV (762 nm). The magnetic quantum number m associated with the various levels is indicated in Fig. 4. The excited state, level |4, can be populated by left hand polarized (LHP) light from level |3 or by right hand polarized (RHP) light from level |1. 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 ±, which is directed along the propagation direction. The selection rule for allowed transitions is Δm=±1, 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 b1Σg+ and X3Σg(m=0) states leads to a transition with a magnetic dipole-like nature and a larger than expected dipole moment.5,6,11

FIG. 4.

Energy levels associated with the ground and excited state of O2. The transition frequency corresponds to ωA=1.63eV (762 nm). The Zeeman splitting of the ground state is caused by the ambient magnetic field.

FIG. 4.

Energy levels associated with the ground and excited state of O2. The transition frequency corresponds to ωA=1.63eV (762 nm). The Zeeman splitting of the ground state is caused by the ambient magnetic field.

Close modal

A high-intensity pump pulse generates a magnetization current density JM=c×M, 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 (2(1/c2)2/t2)E=(4π/c2)JM/t=(4π/c)(×M)/t (Gaussian units). The magnetization is represented by a sum of LHP and RHP components M(z,t)=ML(z,t)êL+MR(z,t)êR+c.c., where ML(z,t)=Nμmρ43(z,t), MR(z,t)=Nμmρ41(z,t), N is the density of the oxygen molecules, μm is the effective magnetic dipole moment associated with the transitions, ρ43 and ρ41 are the off-diagonal coherence of the allowed density matrix elements (see Fig. 4), and êL,R=(êx±iêy)/2 are vectors denoting the polarization direction. The magnetization current density can be written as JM=icML(z,t)/zêL+icMR(z,t)/zêR+c.c. In terms of the x and y components,

JM=i(c/2)(MLMR)/zêx+(c/2)(ML+MR)/zêy+c.c.

The density matrix equation is given by ρnm/t=iωnmρnm+il{ΩnlρlmΩlmρnl}+relaxationterms, where ωnm=ωnωm, Ωnm 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 μmB (Appendix  B).10,12,13 The off-diagonal coherence elements of the density matrix for the relevant transitions, |1|4 and |3|4, are given by

ρ41/t=γcρ41iω41ρ41+iΩ41(ρ11ρ44)+iΩ43ρ31,
(4a)
ρ43/t=γcρ43iω43ρ43+iΩ43(ρ33ρ44)+iΩ41ρ13,
(4b)

where γc 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)êL+BR(z,t)êR+c.c., where BL,R(z,t)=B̂L,R(z,t)eiψ(z,t) and ψ(z,t)=kzωt. The interaction frequencies associated with the allowed transitions are Ω43(z,t)=Ω̂L(z,t)eiψ(z,t) and Ω41(z,t)=Ω̂R(z,t)eiψ(z,t), where Ω̂L,R(z,t)=μmB̂L,R(z,t)/ 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 Ω̂Rabi=μmB̂peak/=(μm/μe)(μeÊpeak/)=(μm/μe)(μe/)(8πI/c)1/2, where I=cÊpeak2/(8π) is the pump laser intensity and Êpeak is the peak electric field. Taking the normalizing electric dipole moment to be μe=qrB=2.5×1018statCcm, where rB is the Bohr radius, the magnitude of the Rabi frequency is Ω̂Rabi[rad/s]=2.5×108(μm/μe)I[W/cm2]. As an example, for I=1011W/cm2 and μm/μe=104, the Rabi frequency is Ω̂Rabi=8×109rad/s.

The incident pump field is taken to be polarized in the x-direction E=Êo(z,t)ei(kzωt)(êL+êR)+c.c., where ω is the carrier laser frequency and the complex pulse amplitude Êo(z,t) can be modulated. Employing the variables τ=tz/c and η=z, E=Êo(τ)eiωτeiΔkη(êL+êR)+c.c., the corresponding magnetic field in the y-direction is B=iÊo(τ)eiωτeiΔkη(êLêR)+c.c., where Δk=kω/c is the wavenumber mismatch. The imaginary part of the wavenumber mismatch Im[Δk]=ΓD=(2πkNμm2ρ11γc/)((ωωA)2+γc2)1 is obtained from the linear dispersion relation and accounts for absorption. The characteristic wavenumber mismatch for λ=762nm at atmospheric molecular oxygen density N=5.7×1018cm3 and an equilibrium population of ρ11=1/3 is ΓD=1.7×102cm1(1/ΓD60cm). To circumvent this short absorption length, the laser frequency can be moved off-resonance. For example, if we detune the laser by 30γc, which corresponds to a wavelength shift of 0.03 nm, then the absorption length is 1/ΓD500m.

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 (/η+iΔk)Êy(η,τ)=iπNμmk(ρ̂43(τ)+ρ̂41(τ)), where the Faraday rotated field is Êy(η,τ)eiωτeiΔkηêy+c.c. The magnetization current is a function of the off-diagonal coherence terms of the density matrix elements ρ43(η,τ)=ρ̂43(τ)eiωτeiΔkη and ρ41(η,τ)=ρ̂41(τ)eiωτeiΔkη. The slowly varying quantities ρ̂43(τ) and ρ̂41(τ) are given by reduced density matrix equations (/τiΔω43)ρ̂43(τ)=iΩ̂43(τ)ρo and (/τiΔω41)ρ̂41(τ)=iΩ̂41(τ)ρo, where Ω̂43(τ)=Ω̂L(τ)=iμmÊo(τ)/, Ω̂41(τ)=Ω̂R(τ)=iμmÊo(τ)/, ρo=ρ11=ρ22=ρ33=1/3, ρ44=0, Δωnm=ωωnm+iγc, ω43=ωAΩo, ω41=ωA+Ωo and it has been assumed that c|Δk|/ω1.

In the case of conventional Faraday rotation within a long pump duration, /τ=0, the spatial change in the Faraday rotated field is given by (/η+iΔk)Êy(η,τ)=2πkNμm2(Êo/)ρoΩo/γc2. After propagating a distance L, the ratio between the Faraday rotated and incident intensities is Iy/Io=|Ey|2/|Eo|2=(2π)4(L/λ)2(Nμm2ρo/)2(Ωo/γc2)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 τp, can be comparable or longer than the damping time 1/γc. 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 ρ43 and ρ41, 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.

FIG. 5.

Pump pulse train and induced polarization rotated wakefields. The envelopes of both a train of x-polarized laser pulses and the x-component of the induced electric wakefield are shown with dashed-red and solid-green lines, respectively. The wakefield's y-component, the rotated signal, and the magnetization current are not shown for simplicity. The wakefield and magnetization current have a similar polarization and temporal form.

FIG. 5.

Pump pulse train and induced polarization rotated wakefields. The envelopes of both a train of x-polarized laser pulses and the x-component of the induced electric wakefield are shown with dashed-red and solid-green lines, respectively. The wakefield's y-component, the rotated signal, and the magnetization current are not shown for simplicity. The wakefield and magnetization current have a similar polarization and temporal form.

Close modal

The general form of the off-diagonal coherence elements is (/τiΔωnm)ρ̂nm(τ)=iΩ̂nm(τ)ρo with solution ρ̂nm(τ)=iρo0τdτΩ̂nm(τ)exp(iΔωnm(ττ)) within the pump pulse. The solution behind the pump pulse is ρ̂43(τ)=ρ̂43(τp)exp(iΔω43(ττp)) and ρ̂41(τ)=ρ̂41(τp)exp(iΔω41(ττp)). The reduced wave equations for the x and y components of the wakefields are

(/η+iΔk)Êx(η,τ)=(2π/c)ĴMx(τ)=CokρoWx(τ)Êo,
(5a)
(/η+iΔk)Êy(η,τ)=(2π/c)ĴMy(τ)=iCokρoWy(τ)Êo,
(5b)

where k=ω/c, ω|/τ|,c|Δk| and Co=2π(Nμm2/)/γc6×107 is a unitless parameter. In estimating Co we have taken the magnetic dipole moment to equal μm=μe×104=2.5×1022statCcm, the collision frequency to be γc=3.5×109s1 and the O2 density to be N=5.7×1018cm3. The current densities are ĴMx(τ)=(Nμmω/2)(ρ̂43(τ)ρ̂41(τ)) and ĴMy(τ)=i(Nμmω/2)(ρ̂43(τ)+ρ̂41(τ)). When the collision rate is much larger than the Larmor frequency or detuning γcΩo,ωωA, the current densities behind the pulse (ττp) are given by

ĴMx(τ)Nμm2ÊoρoωγcWx(τ),
(6a)
ĴMy(τ)Nμm2ÊoρoωγcWy(τ),
(6b)

where the time dependence of the wakefield is captured by

Wx(τ)=eγc(ττP)[cos(Ωo(ττp))eγcτpcos(Ωoτ)(Ωo/γc)(sin(Ωo(ττp))eγcτpsin(Ωoτ))],
(7a)
Wy(τ)=eγc(ττP)[sin(Ωo(ττp))eγcτpsin(Ωoτ)+(Ωo/γc)(cos(Ωo(ττp))eγcτpcos(Ωoτ))].
(7b)

When the laser detuning is larger than the collision rate ωωAγc, there is a phase shift from Eqs. (6), but, more importantly, the magnitude of the current is suppressed by a factor of γc/(ωωA).

Figure 6 shows the wakefield time dependence, Eqs. (7), for pump pulse durations of τp= 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 6×109W/cm2 to 6×1010W/cm2. Equations (5) indicates that Êx,y/Ê0 is proportional to Wx,y(τ), if Δk is neglected. For the parameters in Fig. 6, the normalized peak wakefield amplitudes are |Êx/Êo|0.5, 1.5, and 1.6 and |Êy/Êo|1×104, 1.2×103, and 2×103. There is a tradeoff between driving the wakefields with a higher intensity pump (Êoτp1/2) versus driving it for a longer duration (Wx,yτp). As a result, for τp>3/γc, the wakefield amplitude begins monotonically decreasing

FIG. 6.

(a) x-component and (b) y-components of the wakefield response functions Wx,y(τ), as defined in Eqs. (7), behind the pulse for Ωo=4.5×106rad/s. The pulse durations for the solid, dashed, and dotted lines are τp= 0.1 ns, 0.5 ns, and 1 ns, respectively.

FIG. 6.

(a) x-component and (b) y-components of the wakefield response functions Wx,y(τ), as defined in Eqs. (7), behind the pulse for Ωo=4.5×106rad/s. The pulse durations for the solid, dashed, and dotted lines are τp= 0.1 ns, 0.5 ns, and 1 ns, respectively.

Close modal

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+δB. 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 |I1I2|/I1=|δI|/I12|δÊy/Ê1y|, where Ê1y is the y-component of the wakefield amplitude and δÊy is the difference in the wakefield amplitudes between the two locations. Figure 7 shows the fractional wakefield intensities for various values of δB. For the values shown, |δI|/I1103.

FIG. 7.

Fractional difference in the wakefield intensity for various fractional differences in the magnetic field δB/Bo. The pump pulse has duration τp= 0.5 ns, spot size 1 mm, and energy 100 mJ. The differences in magnetic field and corresponding intensities are from two nearby locations.

FIG. 7.

Fractional difference in the wakefield intensity for various fractional differences in the magnetic field δB/Bo. The pump pulse has duration τp= 0.5 ns, spot size 1 mm, and energy 100 mJ. The differences in magnetic field and corresponding intensities are from two nearby locations.

Close modal

The pump pulse energy is I(π/2)R2τp, where R is the spot size. For a pulse of duration τp=0.5nsec2/γc, R=1mm and intensity I=1010W/cm2, the pump pulse energy is 80mJ/pulse. For a pulse train, rep-rated at fp=1kHz, the average pump laser power is P=fpI(π/2)R2τp=80W.

It is worth noting that at sufficiently high intensities, the upper level, level |4, can be populated resulting in a laser induced florescence signal to lower energy levels, i.e., levels |1 and |3. 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 JMeγcτcos(ωAτ)[cos(Ωoτ)êxsin(Ωoτ)êy].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/Iycot2(Ωoτ). Note that the ratio is independent of the collision rate as long as the individual intensities are greater than the inherent intensity fluctuations.

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 is14ΔI/I=π3/2δn2(L/o)(λ/o)2 where denotes an ensemble average, ΔI is the depolarized intensity, δn2 is the mean square refractive index fluctuation due to turbulence, L is the propagation range, and o is the inner characteristic scale length associated with the turbulence. As an example, we consider the typical parameters λ=762nm, o=1mm, L=1km, and δn21/2=106. For these parameters, ΔI/I1013 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 (b1Σg+X3Σg) 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.

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).

Oxygen's abundance in the earth's atmosphere, approximately 21% (N=5.7×1018cm3) 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 X3Σg, 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=1atm, total number density Nair=2.7×1019cm3, and temperature T=23.5meV), the ground state is fully populated because the next excited electronic state's energy, Ea=0.98eV 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, a1Δg, is referred to as “singlet oxygen” and only has one spin state (S,m)=(0,0). This state has an energy of EaX=0.98eV, a1Δg can undergo spontaneous emission via a magnetic dipole transition to the ground state O2(a1ΔgX3Σg) or aX. The aX transition has a wavelength of 1.27μm. This transition is dominantly due to the orbital angular momentum and has spontaneous emission rate of AaX=2×104s1.15 

FIG. 8.

Electron occupancy energy levels of O2 as two oxygen molecules are brought together.

FIG. 8.

Electron occupancy energy levels of O2 as two oxygen molecules are brought together.

Close modal

The second excited state of oxygen b1Σg+ (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; b1Σg+X3Σg(m=±1), b1Σg+X3Σg(m=0), and b1Σg+a1Δg, 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 bX,1, bX,0, and ba, respectively. The bX transitions have an energy of EbX=1.63eV, wavelength λbX=762nm, and frequency ωbX=2.5×1015rads1. The calculated spontaneous emission rates of the bX,1 and bX,0 transitions are AbX,1=0.087s1 and AbX,0=1.6×107s1, respectively.15 The radiation from the bX,1 transition can be seen in air-glow, night-glow, and aurorae.15 The bX,1 transition is magnetic dipole- and spin-forbidden and it is dominant over the ba and bX,0 transitions, which are electric quadrupole transitions.11 This can be explained by a large spin-orbit coupling between the b1Σg+ state and the X3Σg(m=0) state. The spin-orbit coupling results in a mixing of the levels and the bX,1. The ba transition has an energy of Eba=0.65eV, wavelength λba=1.9μm, frequency ωba=9.9×1014rads1 and spontaneous emission rate of Aba=1.4×104s1.15 

Interaction of an oxygen molecule with radiation is governed by Schrödinger's equation i|ψ/t=H|ψ, where H=H0μmB(t) is the full Hamiltonian, H0 is the electronic Hamiltonian after Zeeman splitting, and μmB(t) is the magnetic dipole interaction energy. The state |ψ(t)=nCn(t)|n can be decomposed into the orthogonal energy eigenstates of O2, |n. The probability amplitudes Cn(t) are related to the density matrix elements ρnm(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., ρnm/t=iωnmρnm+il{ΩnlρlmΩlmρnl}+relaxationterms. The interaction frequency is given by Ωnl=n|μmB(t)|l.

In our model, molecular oxygen is treated as a closed four level atom composed of the ground state O2(X3Σg) and the upper level O2(b1Σg+). The ground state has three spin sublevels m=1,0,+1, which are referred to as states |1, |2, and |3 respectively. The excited upper level is referred to as state |4. The complete set of coupled equations for the density matrix elements, assuming a closed system, are given by

ρ11/t=(Γ12+Γ13)ρ11+Γ21ρ22+Γ31ρ33+Γ41ρ44+i(Ω14ρ41Ω41ρ14),
(B1)
ρ22/t=Γ12ρ11(Γ21+Γ23)ρ22+Γ32ρ33+Γ42ρ44,
(B2)
ρ33/t=Γ13ρ11+Γ23ρ22(Γ31+Γ32)ρ33+Γ43ρ44+i(Ω34ρ43Ω43ρ34),
(B3)
ρ44/t=(Γ41+Γ42+Γ43)ρ44+i(Ω41ρ14Ω14ρ41)+i(Ω43ρ34Ω34ρ43),
(B4)
ρ41/t=γ41ρ41iω41ρ41+iΩ41(ρ11ρ44)+iΩ43ρ31,
(B5)
ρ43/t=γ43ρ43iω43ρ43+iΩ43(ρ33ρ44)+iΩ41ρ13,
(B6)
ρ13/t=γ13ρ13iω13ρ13+iΩ14ρ43iΩ43ρ14.
(B7)

The population level of state |n is given by ρnn while the coherence between the states are given by ρnm=ρmn*. The transition frequencies are defined as ωmn=ωmωn, where ωn is energy of the nth state. For example, the state frequencies are ω1=Ωo, ω2=0, ω3=Ωo, and ω4=ωA, and the transition frequencies are ω41=ωA+Ωo, ω13=2Ωo, and ω43=ωAΩo, where ωA is O2(b1Σg+X3Σg) transition frequency in the absence of a magnetic field. The Larmor frequency is given by Ωo=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., (ρ11+ρ22+ρ33+ρ44)/t=0 (closed system). The populations are additionally normalized unity, i.e., Tr(ρ)=1. The interaction frequency between states m and state n is Ωmn=Ωnm*. Specifically, Ω̂43(z,t)=μmB̂L(z,t)/ and Ω̂41(z,t)=μmB̂R(z,t)/, where μm is the effective magnetic dipole moment between triplet oxygen and the upper state and B̂L,R corresponds to the left (right) handed polarization of the pump field. The rate equation for ρ42 is not considered since it does not couple to the those in Eqs. (B1)(B7).

The rates γ41 and γ43 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 γ41=γ43=γ31=γc. In the absence of the pump field and at equilibrium, we have ρ11=ρ22=ρ33=ρo and ρ44=0. This implies that Γ21=Γ12, Γ31=Γ13, and Γ23=Γ32 and we take these rates, which include inelastic collisions and spontaneous emission, to equal Γo. In addition, the rates Γ41, Γ42, and Γ43 consist of inelastic collisions and spontaneous emission and we take these rates to be equal to ΓU. Taking the inelastic collision rates to be equal, i.e., Γo=ΓU, the density matrix equations become

ρ11/t=γo(ρ11ρ11eq)+i(Ω14ρ41Ω41ρ14),
(B8)
ρ22/t=γo(ρ22ρ22eq),
(B9)
ρ33/t=γo(ρ33ρ33eq)+i(Ω34ρ43Ω43ρ34),
(B10)
ρ44/t=γo(ρ44ρ44eq)+i(Ω41ρ14Ω14ρ41)+i(Ω43ρ34Ω34ρ43),
(B11)
ρ41/t=γcρ41iω41ρ41+iΩ41(ρ11ρ44)+iΩ43ρ31,
(B12)
ρ43/t=γcρ43iω43ρ43+iΩ43(ρ33ρ44)+iΩ41ρ13,
(B13)
ρ13/t=γcρ13iω13ρ13+iΩ14ρ43iΩ43ρ14.
(B14)

The phenomenological inelastic damping rate is given by γo=3Γo=3ΓU108s1.6 The equilibrium populations for the ground state are ρ11eq=ρ22eq=ρ33eq=1/3 and for the upper state ρ44eq=0.

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 fields16 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 Epump=Êo(τ)eiωAτ(êL+êR)+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 at the expense of levels |1 and |3. The pump pulse duration τp 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 ρ44 to states ρ11 and ρ33 is described by the off-diagonal coherence of the molecular density matrix elements ρ43=iΩ̂Lτpρ44ei(ω43iγc)τ and ρ41=iΩ̂Rτpρ44ei(ω41iγc)τ, where ΩL=iμmEo/, ΩR=iμmEo/, ω43ω43=ωAΩo, and ω41 = ω41=ωA+Ωo. The magnetization left behind the pump pulse is M=Moeγcτ(eiω43τêLeiω41τêR)+c.c. where Mo=Nμm2(Eo/)τpρ44. The associated current density is JM=Moeγcτ[ω43{cos(ω43τ)isin(ω43τ)}êL+ω41{cos(ω41τ)isin(ω41τ)}êR]+c.c., where ω43,ω41γc. The current density has components in the x and y directions10 which, for ωAΩo, are given by

JM=2MoeγcτωAcos(ωAτ)[cos(Ωoτ)êxsin(Ωoτ)êy].
(C1)

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/Iycot2(Ωoτ). The ratio is independent of the collision rate as long as the individual intensities are greater than the inherent intensity fluctuations.

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