This paper analyzes and evaluates a concept for remotely detecting the presence of radioactivity using electromagnetic signatures. The detection concept is based on the use of laser beams and the resulting electromagnetic signatures near the radioactive material. Free electrons, generated from ionizing radiation associated with the radioactive material, cascade down to low energies and attach to molecular oxygen. The resulting ion density depends on the level of radioactivity and can be readily photo-ionized by a low-intensity laser beam. This process provides a controllable source of seed electrons for the further collisional ionization (breakdown) of the air using a high-power, focused, CO2 laser pulse. When the air breakdown process saturates, the ionizing CO2 radiation reflects off the plasma region and can be detected. The time required for this to occur is a function of the level of radioactivity. This monostatic detection arrangement has the advantage that both the photo-ionizing and avalanche laser beams and the detector can be co-located.

Sources of radioactivity range from terrestrial to cosmogenic and man-made.1 In general, radioactive material emits ionizing radiation, for example, gamma rays, which ionize the surrounding air, producing high-energy electrons which cascade down to low energy, thermal electrons.2 These low energy electrons rapidly attach to oxygen molecules forming O2 ions. At ambient levels of radiation, the density of free electrons is much less than the density of molecular oxygen ions.3 

Remote radiation detection concepts have been proposed based on high-power terahertz (THz) radiation pulses that induce avalanche (collisional) air breakdown in the vicinity of the radioactive material.4,5 In this concept, a THz pulse is focused near the radioactive material. In order to initiate avalanche breakdown, at least one electron needs to be in the optical volume for at least an ionization time. At ambient levels of radioactivity, the free (seed) electron density in the optical volume will be small so that the probability of avalanche breakdown is small, i.e., the average breakdown time is long, or breakdown does not occur, depending on the length of the THz pulse. In the presence of radioactive material, the probability of breakdown occurring is higher since the density of free electrons for the avalanche process is higher, i.e., the average breakdown time is shorter. Differences in breakdown probability and average breakdown delay time between a control region and a suspect region are indications of additional radioactivity.

Another proposed concept for the remote detection of radioactivity consists of photo-ionizing the O2 ions in order to enhance the level of seed electrons in the optical volume for the avalanche breakdown process.6 In this concept, a single, high-power IR laser beam is used for both the photo-ionizing and avalanche ionizing beams. In this bistatic detection concept, the electromagnetic (EM) signature for the presence of radioactive material is a frequency modulation on a probe beam caused by the temporally increasing electron density.

In this paper, we propose a monostatic detection concept which uses a low-intensity photo-ionizing laser pulse and a focused high-power CO2 laser to induce air breakdown in the vicinity of the radioactive material. The low-intensity photo-ionizing laser pulse, Nd:YAG, operating at λ=1μm provides a controllable source of free electrons. The free electrons are generated by photo-ionizing the elevated levels of O2 in the vicinity of radioactive materials. The ionization potential of O2 is 0.46 eV, below the photon energy of the photo-ionizing laser. The avalanche breakdown is achieved with a high-power, focused, CO2 laser pulse. The critical electron density corresponding to the λ=10.6μm, CO2 wavelength is Ncrit=1019cm3. When the electron density reaches Ncrit in the breakdown region, the CO2 radiation is reflected. The breakdown delay time is the signature that indicates elevated levels of radioactivity. The ionizing CO2 pulses are focused at various nearby locations making possible a differential comparison of the breakdown time, which is a function of the level of radioactivity. This monostatic detection arrangement has the advantage that both the photo-ionizing and avalanche laser beams and the detector are co-located. A schematic of the overall detection concept is shown in Fig. 1.

FIG. 1.

Schematic of the detection concept. A low intensity Nd:YAG laser pulse photo-ionizes elevated levels of O2 ions that make available a controllable source of free electrons. A high-power CO2 laser pulse is focused in the vicinity of the radioactive material and initiates avalanche ionization breakdown of the air. When the electron density reaches a critical density, the CO2 radiation is reflected. The breakdown timing forms the signature for this monostatic detection concept.

FIG. 1.

Schematic of the detection concept. A low intensity Nd:YAG laser pulse photo-ionizes elevated levels of O2 ions that make available a controllable source of free electrons. A high-power CO2 laser pulse is focused in the vicinity of the radioactive material and initiates avalanche ionization breakdown of the air. When the electron density reaches a critical density, the CO2 radiation is reflected. The breakdown timing forms the signature for this monostatic detection concept.

Close modal

Upon disintegration, many types of radioactive nuclei emit ionizing radiation (gammas) which, through a Compton scattering process, generate high-energy electrons that cascade down in energy. Due to its high electron affinity, the majority of these ions are oxygen molecules. An example of this process is the disintegration of Cobalt-60 (Co60). Upon each disintegration of a Co60 nucleus, two gammas are emitted, each with an energy of 1MeV. The MeV gammas have a range in air of ∼130 m. Each gamma produces ∼30 000 electrons which eventually recombine and/or form negative oxygen ions.2 

As a result of cosmic rays, radioactive substances in the ground and air, the ambient ionization rate is Qrad1030cm3s1.1 The presence of additional radioactive material can significantly increase the radioactive ionization rate to (1+αrad)Qrad, where αrad is the enhancement factor resulting from the additional radioactive material. For example, 50 cm from 10 mg of Co60 (a dirty bomb may contain many hundreds of mg), the enhancement factor in air can be as high as αrad106.6 In general, for an unshielded point source of gammas, the radioactivity enhancement factor falls off like αradexp(R/Lγ)/R2, where R is the distance from the source and Lγ is the effective range of the gammas.

The electron density, ion density, and electron temperature are modeled using the following rate equations:6 

Net=(1+αrad)Qrad+νphotoN+νcollNeηNeβe+N+Ne+βnNnN,
(1a)
Nt=νphotoN+ηNeβ±N+NβnNnN,
(1b)
32(NeTe)t=JE32Ne(TeTeo)τcool,
(1c)

where Ne is the electron density, N is the negative ion density (taken to be O2), N+=Ne+N is the positive ion density, Nn=Nn0N+N is the neutral density, νphoto is the photo-ionization rate, νcoll is the collisional ionization rate, η is the electron attachment rate, βe+ is the electron-ion dissociative recombination rate, βn is the negative ion detachment rate due to collisions with neutrals, β± is the ion-ion recombination rate, Te is the electron temperature, JE is the Ohmic heating rate, Teo=0.025eV is the ambient electron temperature, and τcool is the inelastic electron cooling time. These air chemistry rates and collisional ionization rates are discussed in  Appendix A and are in general very complicated functions of electron temperature. Typical values for some of these rates are η1.4×107s1, βe+2×108cm3s1, βn7.4×1019cm3s1, β±2×107cm3s1, and τcool1011s. In estimating these typical values, the electron temperature was assumed to be Te1eV. The photo-ionization rate for O2 is νphoto[s1]2.3Iphoto[W/cm2] for a 1μm laser wavelength.7 

The effect of radioactivity is contained in the first term on the right-hand side of Eq. (1a), i.e., (1+αrad)Qrad. The steady state electron and negative ion densities can be estimated to be given by Ne(βnNn/η)((1+αrad)Qrad/β±)1/2 and N((1+αrad)Qrad/β±)1/2.6 In steady state, the negative ion density is determined by the ion-ion recombination rate and the level of radioactivity. The ratio of the electron to ion density in the steady state is Ne/NβnNn/η106.

Figures 2 and 3 show the electron density and negative ion at equilibrium as a function of the radioactivity enhancement factor αrad. These are numerical solutions of the rate equations, Eqs. (1a) and (1b), and are in good agreement with the above analytical estimates. The negative ion density is in agreement with the simplified analytic solution and in approximate agreement with experimental results.8,9

FIG. 2.

Numerical solution of the electron density at equilibrium in the absence of laser radiation as a function of the radioactivity enhancement factor αrad. Electron temperature is 0.025 eV. The plot is in good agreement with the analytical estimate Ne(βnNn/η)((1+αrad)Qrad/β±)1/2.

FIG. 2.

Numerical solution of the electron density at equilibrium in the absence of laser radiation as a function of the radioactivity enhancement factor αrad. Electron temperature is 0.025 eV. The plot is in good agreement with the analytical estimate Ne(βnNn/η)((1+αrad)Qrad/β±)1/2.

Close modal
FIG. 3.

Numerical solution of negative ion, O2, density at equilibrium in the absence of laser radiation as a function of the radioactivity enhancement factor αrad. Electron temperature is 0.025 eV. The plot is in good agreement with the analytical estimate N((1+αrad)Qrad/β±)1/2.

FIG. 3.

Numerical solution of negative ion, O2, density at equilibrium in the absence of laser radiation as a function of the radioactivity enhancement factor αrad. Electron temperature is 0.025 eV. The plot is in good agreement with the analytical estimate N((1+αrad)Qrad/β±)1/2.

Close modal

The photo-ionization rate is proportional to the intensity of the beam since the laser photon energy is greater than the ionization potential of the ions, i.e., 0.46 eV. The photo-ionizing source is taken to be a Nd:YAG laser, with wavelength 1.06 μm (1.17 eV). For this wavelength, the rate of photo-ionization of O2 is νphoto[s1]2.3Iphoto[W/cm2], where Iphoto is the intensity of the photo-ionizing laser field.6 When the photo-ionizing beam is turned on, the electron density will increase until it reaches a new equilibrium. The resulting equilibrium electron density is a function of the level of radioactivity and the beam intensity. The time-dependent behavior and equilibrium level of the electron density is a function of both the level of radioactivity and the photo-ionizing beam intensity. Figures 4 and 5 show several solutions of Eqs. (1a) and (1b) in this parameter space.

FIG. 4.

Numerical solutions of the rate equations, Eqs. (1a)–(1c), showing electron density as a function of time in the presence of a photo-ionizing laser (λ=1μm) beam for various values of intensity Iphoto. The radiation enhancement factor is αrad=103. The solid curve is for the intensity Iphoto=103W/cm2, the dashed curve is for Iphoto=105W/cm2, and the dotted curve is for Iphoto=107W/cm2. The initial electron density is not shown but has the value Ne(0)=0.125cm3 in all three cases.

FIG. 4.

Numerical solutions of the rate equations, Eqs. (1a)–(1c), showing electron density as a function of time in the presence of a photo-ionizing laser (λ=1μm) beam for various values of intensity Iphoto. The radiation enhancement factor is αrad=103. The solid curve is for the intensity Iphoto=103W/cm2, the dashed curve is for Iphoto=105W/cm2, and the dotted curve is for Iphoto=107W/cm2. The initial electron density is not shown but has the value Ne(0)=0.125cm3 in all three cases.

Close modal
FIG. 5.

Numerical solutions of the rate equations, Eqs. (1a)–(1c), showing electron density as a function of time, for various values of the radiation enhancement factor αrad, in the presence of a photo-ionizing laser beam. The intensity of the photo-ionization beam (λ=1μm) is Iphoto=105W/cm2.

FIG. 5.

Numerical solutions of the rate equations, Eqs. (1a)–(1c), showing electron density as a function of time, for various values of the radiation enhancement factor αrad, in the presence of a photo-ionizing laser beam. The intensity of the photo-ionization beam (λ=1μm) is Iphoto=105W/cm2.

Close modal

The high-power CO2 laser has a number of characteristics which make it suitable for an ionizing (collisional) laser. These include operating wavelength, high average power capability, and flexible pulse format. The TEA CO2 laser (λ=10.6μm) can be Q-switched to produce pulses with peak intensity greater than 1010 W/cm2 and pulse duration ∼10–100 ns.10 The focused optical laser volume is taken to be Vlaser=ro2ZR, where ro is the laser waist (minimum spot size) and ZR=πro2/λ is the Rayleigh length. The laser spot size, as measured from the source (z = 0), is r(z)=ro(1+(zL)2/ZR2)1/2. The laser spot size at the source is r(0)=ro(1+L2/ZR2)1/2λL/(πr0)=r03L/Vlaser, where L is the range (distance from the laser transmitter to the detection site). For example, for a focused volume of Vlaser=1cm3, the CO2 laser spot size at the detection site is ro0.1cm and the Rayleigh length (length of the focused volume) is ZR50cm. For a range of 300 m, the laser spot size at the source (transmitter) is r(0)30cm.

The basic theory of the avalanche breakdown of air in a laser field is well documented.8,11 For a laser field above the breakdown threshold intensity, free electrons are accelerated and collisionally ionize molecules at a rate greater than the rate of attachment. The estimated threshold intensity for CO2 radiation in clean air is IBD109W/cm2, see  Appendix B. To initiate avalanche breakdown, there must be at least one free electron in the optical volume long enough to collisionally ionize a neutral gas molecule and produce another electron, which can then continue this process. The optical breakdown volume is the volume for which the focused laser, i.e., CO2, has an intensity greater than the breakdown threshold intensity.

During breakdown, the electron temperature in our model reaches a steady-state. It can be solved for using the temperature rate equation, Eq. (1c). The assumption is made that during this phase, the electron density is given by Ne(t)Ne(0)exp[(νcollη)t], which is the approximate solution to Eq. (1a) when collisional ionization is the dominant electron source term. The cooling time τcool and the collisional ionization rate νcoll are functions of temperature, making an analytical solution not possible. The function Teq(I0) can be obtained numerically under the assumption that the neutral density is the ambient value, Nn=Nno. This is valid for much of the breakdown phase until the electron density becomes comparable to the ambient neutral density.

The rates νcoll and η are shown in Fig. 6. These rates are functions of temperature and have been evaluated at the steady-state temperature Teq(I0). Under the previous assumption, the criterion for breakdown to occur is for νcoll to exceed η. The intensity at which this occurs in this model is IBD4×109W/cm2. This is approximately the experimental value of the clean-air breakdown threshold intensity for the CO2 laser.12 The effective electric field ( Appendix B) for this intensity and its corresponding steady-state temperature Teq(IBD) is Eeff=32kV/cm.

FIG. 6.

Collisional ionization and dissociative attachment rates as a function of the CO2 intensity, during breakdown in steady-state. These rates correspond to solutions of Eq. (1c) in steady-state at the stated laser intensity. The intersection of these curves yields the approximate breakdown threshold intensity, i.e., IBD4×109W/cm2. At the threshold intensity, the electron temperature was Te=1.4eV. This intensity and temperature correspond to an effective laser field of Eeff=32kV/cm (see  Appendix B).

FIG. 6.

Collisional ionization and dissociative attachment rates as a function of the CO2 intensity, during breakdown in steady-state. These rates correspond to solutions of Eq. (1c) in steady-state at the stated laser intensity. The intersection of these curves yields the approximate breakdown threshold intensity, i.e., IBD4×109W/cm2. At the threshold intensity, the electron temperature was Te=1.4eV. This intensity and temperature correspond to an effective laser field of Eeff=32kV/cm (see  Appendix B).

Close modal

The time required for breakdown to occur is divided into two parts: the statistical delay time τs and the formation delay time τf.13 The sum of these is the breakdown time, τb=τs+τf. The statistical delay time τs is the time for an electron to first appear in the optical volume in which the laser intensity is above the breakdown intensity IBD. The formation delay time τf is the time it takes the electron density to reach the critical density. It can be estimated by making the assumption that the electron density increases exponentially at a rate νionνcollη, i.e., Ne(t)Ne(0)exp(νiont), where Ne(0) is the initial electron density. The formation delay time is the time for which Ne(τf)=Ncrit, i.e., τf(1/νion)ln(Ncrit/Ne(0)).

For laser-induced breakdown, the critical density is the electron density for which the electron plasma frequency equals the laser frequency. In practical units, the electron density is Ncrit[cm3]=1.1×1021/λ2[μm] and for a CO2 laser (λ=10.6μm) is Ncrit1×1019cm3.14 When the electron density reaches this critical density, the breakdown CO2 laser beam is reflected off the electron density. The reflected radiation can be observed at the laser source and allows for accurate measurement of the total breakdown delay time τb.

The detection method proposed in this paper uses the formation delay time as a signature for the presence of radioactivity. Due to the photo-ionization of negative ions, the statistical delay time does not play a relevant role in this detection concept. The breakdown timing is mainly determined by the parameters of the breakdown pulse, such as peak intensity and pulse format.

In this section, a specific example is given of the proposed detection concept. Figure 7 shows two numerical solutions of the rate equations Eqs. (1a)–(1c). The curves represent the electron density as a function of time during an avalanche breakdown, for two different values of the radioactivity enhancement factor αrad. These are numerical solutions of the rate equations (Eqs. (1a)–(1c)). The solid curve represents a control area, with no excess radioactivity (αrad=0). The dotted curve represents a region with excess ionization due to radioactivity, in this case αrad=104. In both cases, the region is first photo-ionized by a 10 ns laser pulse with intensity Iphoto=105W/cm2 and wavelength λ=1μm. Then, a CO2 beam with a square profile and intensity I0=5.5×109W/cm2 is turned on, and the electron density is allowed to saturate. The difference in the breakdown times is Δτ2ns, a measurable signature of radioactivity. In situ, this total breakdown time can be calculated by measuring the time required for the breakdown region to begin reflecting the CO2 laser radiation.

FIG. 7.

Electron density vs time during breakdown. The intensity of the breakdown CO2 laser (λ=10.6μm) is I0=5.5×109W/cm2. The dashed curve is αrad=0, and the solid curve is αrad=104. The difference in the breakdown times is ΔτBD2ns. The initial condition is the state after photoionization by a λ=1μm laser with Iphoto=105W/cm2. The duration of photo-ionization was 10 ns.

FIG. 7.

Electron density vs time during breakdown. The intensity of the breakdown CO2 laser (λ=10.6μm) is I0=5.5×109W/cm2. The dashed curve is αrad=0, and the solid curve is αrad=104. The difference in the breakdown times is ΔτBD2ns. The initial condition is the state after photoionization by a λ=1μm laser with Iphoto=105W/cm2. The duration of photo-ionization was 10 ns.

Close modal

The breakdown time is shown in Figure 8 as a function of the radiation enhancement factor, for several values of the intensity of the photo-ionizing laser. Note that varying this intensity has little qualitative effect on the curves aside from a constant shift in delay time.

FIG. 8.

Breakdown delay time as a function of radioactivity enhancement factor. The intensity of the breakdown CO2 laser (λ=10.6μm) is I0=5.5×109W/cm2. The initial condition for each point is the state after photoionization by a λ=1μm laser. The solid curve had Iphoto=104W/cm2, the dashed curve had Iphoto=105W/cm2, and the dotted curve had Iphoto=106W/cm2. The duration of photo-ionization was 10 ns.

FIG. 8.

Breakdown delay time as a function of radioactivity enhancement factor. The intensity of the breakdown CO2 laser (λ=10.6μm) is I0=5.5×109W/cm2. The initial condition for each point is the state after photoionization by a λ=1μm laser. The solid curve had Iphoto=104W/cm2, the dashed curve had Iphoto=105W/cm2, and the dotted curve had Iphoto=106W/cm2. The duration of photo-ionization was 10 ns.

Close modal

The difference in delay times can be approximated as Δτfνion1ln(Ne,test(0)/Ne,0(0)), where Ne,test(0) is the initial electron density of the region under inspection and Ne,0(0) is the initial electron density of the control region. In the state after photo-ionization of the negative ions, the approximate density ratio is Ne,test(0)/Ne,0(0)(1+αrad,test)1/2. The delay time difference is then Δτf(1/2νion)ln(1+αrad,test), the quantity which we would like to maximize. Figure 6 shows νcoll and η as a function of CO2 beam intensity. By setting the intensity as close as is practical to the breakdown threshold intensity, the rate νionνcollη can be minimized.

In this paper, a concept is proposed and analyzed for the remote detection of ionizing radioactivity by observation of electromagnetic signatures. The presence of ionizing radiation results in an elevated density of negative ions. A low-intensity laser beam photo-ionizes these ions, producing a high electron density which depends on the level of radioactivity. Next, a CO2 laser beam is focused at a high intensity, causing avalanche breakdown of the air. When this process saturates at a critical density, the beam is reflected from the elevated electron density. The time required for this to occur becomes shorter as the level of radioactivity increases. A reduced breakdown time in one area compared to another is a signature of the presence of a greater amount of radioactivity.

The material presented in this paper assumes that the air is clean, devoid of aerosols. Electrons can attach to and be detached from particulates in the air, in much the same way as they attach to neutral molecules to form negative ions. In this sense, they can be accounted for in the rate equations, Eqs. (1a)–(1c). This would not, however, encompass the full effect that aerosols have on the mechanism of laser-induced avalanche breakdown. In the presence of aerosols, the threshold intensity for breakdown can be substantially reduced.15 

In the proposed detection method, it is necessary to repeat the breakdown process and measurements of the delay time at a high rep-rate. Solutions of the rate equations Eqs. (1a)–(1c) indicate that the time it takes to return to equilibrium after breakdown is exceedingly long, i.e., on the order of minutes. Return to equilibrium involves the exchange of electrons, which is a very slow process at low electron densities, but not slow enough for differences in αrad to have a detectable effect in the time between repetitions. A solution to this problem is to quickly move the physical location of the laser focus so that each repetition of the breakdown process is performed in air which has not had its negative ion density artificially elevated.

Future theoretical work in this area will be to model the saturation process in a hybrid code, with particle electrons, and calculate the characteristics of the laser radiation reflected from the breakdown region. Experimentally, it should be verified that photoionization with an Nd:YAG laser can be used as a source of seed electrons for laser-induced avalanche breakdown, and that the presence of radioactivity reduces the time required for this process.

Proof-of-principle experiments could be based on a commercial source of alpha particles, i.e., 210Po which produces 5 MeV alpha particles. Using a radioactive source of alphas, instead of a gamma ray source, avoids the safety issues associated with radioactive material. The 5 MeV alphas have a short range (∼3.5 cm) in air. The commercial 210Po source produces 20 mCi of radioactivity inside an open metallic tube. Each 5 MeV alpha generates ∼1 × 105 electrons which attach to O2 forming the desired O2 ions. The valence electrons can then be photo ionized by laser radiation. The experimental diagnostics would include visible and extreme ultraviolet spectrometers. The EM signature could be optically guided back to the detector in a laser induced air waveguide.16 This would greatly enhance the level of the return signal at the detector.

The authors acknowledge useful discussions with Professor H. Milchberg. This research was supported by a DTRA, C-WMD Basic Research Program.

In this Appendix the various air chemistry rates, ionization rates and other functions used in this paper are discussed and expressed as functions of electron temperature.6,17–19 It should be noted that these are approximate expressions, but should be able to capture the general behavior of the mechanism.

1. Electron loss terms

Electrons are depleted mainly through attachment to O2. Free electrons can also recombine with positive ions, a rate which becomes important at large electron densities. The electron attachment rate to neutral oxygen is

η[s1]=3.64×1031Te[eV]exp(100Tg[K])exp(6.1×102Te[eV])nO22,
(A1)

where Tg is the ambient gas temperature and Te is the electron temperature. For Te=1eV and weak ionization, η1.4×107s1. The electron-ion recombination rate is

βe+=1.45×108Te0.7(1Θ(TeTe,0))+2×108Te0.56Θ(TeTe,0)cm3s1,
(A2)

where Θ(TeTe,0) is a step function, and Te,0=0.1eV. For Te=1eV, βe+2×108cm3s1.

2. Collisional ionization rate

The rate of collisional ionization is derived by assuming the electrons take on a Maxwell-Boltzmann distribution.19 For each neutral species

νcoll(X)=νX(Te/Ux)3/2(UX/Te+2)exp(UX/Te),
(A3)

where νX and Ux are a characteristic frequency and the ionization potential of species X. For molecular nitrogen and oxygen, UN2=15.6eV, UO2=12.1eV, νN2=3.5×108nN2[cm3]s1, and νO2=1.9×108nO2[cm3]s1.

3. Electron Heating

Electrons in the presence of a laser field gain energy at an average rate JE=(ωp2/8π)Eeff2/νe, where ωp is the plasma frequency, and the effective electric field Eeff is defined in  Appendix B. This is the result of solving the electron momentum equation with a simple collision rate of momentum transfer νe, and then taking a time average. Expressed in more convenient units and variables, the rate of electron energy density increase is

32(NeTe)t=JE[eVcm3/s]=1.9×1013Ne[cm3]λ2[μm]I0[W/cm2]νe[s1]/(1+νe2/ω2),
(A4)

where λ and ω are the wavelength and angular frequency of the laser field, and I0 is the peak intensity.

4. Electron cooling

The primary mechanism of electron cooling is collisional excitation of the vibrational modes of molecular nitrogen N2. The total rate of electron cooling has been calculated using the CHMAIR code.18 The cooling term takes the form

32(NeTe)tcooling=32Ne(TeTeo)τcool.
(A5)

The characteristic cooling time τcool is a function of the electron temperature and the neutral density Nn

τcool[s]=32Te[eV]Nn[cm3]1Qcool[eV-cm3-s1],
(A6)

and the cooling quotient17 is

Qcool[eV-cm3s1]3.5×108exp(5/(3Te[eV]))+6.2×1011exp(1/(3Te[eV])).
(A7)

The threshold laser intensity necessary to initiate breakdown can be approximately estimated by equating the effective laser electric field to the DC breakdown field in clean air which is EBD35kV/cm. The laser field is given by E(z,t)=(E0/2)exp(i(kzωt))+c.c., where ω=ck is the frequency and Eo is the amplitude. The laser intensity is I0=cE02/8π and in practical units is I0[W/cm2]=1.33×103E02[V/cm]. The effective laser electric field11 is given by

Eeff=(1+ω2/νe2)1/2E0,
(B1)

where νe=νen+νei is the electron collision frequency, νen is the electron-neutral collision frequency and νei is the electron-ion collision frequency. Setting Eeff=EBD, the threshold laser breakdown intensity is given by

IBDc8πω2νe2EBD2,
(B2)

where we have assumed that ωνe.

To estimate the threshold intensity we take the electron collision frequency to be equal to the electron-neutral collision frequency νen[s1]=107NnTe1/2[eV],20 where Nn=2.7×1019cm3 is the air density and Te is the electron temperature. The threshold intensity in units of W/cm2 is given by

IBD[W/cm2]1.63×106ω2νe2.
(B3)

The threshold intensity for a CO2 laser (λ=10.6μm), ω=1.8×1014s1, is

IBD[W/cm2]7×109Te[eV].
(B4)

To initiate breakdown some of the electrons most have energies that are at the ionization potential of air, which for O2 is 12eV and for N2 is 15eV. Since, it is the high energy electrons, in the tail of the electron distribution function, which are responsible for the ionization, we take Te5eV. Using this value we find that the threshold intensity in clean air, for CO2 radiation, is IBD1.4×109W/cm2. This value is in good agreement with experiments.12 

The electron oscillation energy in the presence of the laser field is Eosc=q2Eo2/(4mω2) which in units of eV is Eosc[eV]=9.35×1014λ2[μm]IBD[W/cm2] and is much less than the ionization energy. In addition, the mean free path of an electron is δ=vth/νe, where νe[s1]=107Nn[cm3]Te1/2[eV] is the collision frequency and vth[cm/s]=3Te/me=7.3×107Te1/2[eV] is the electron thermal velocity. At the ambient temperature, Te=0.025eV and atmospheric densities, Nn=2.7×1019cm3, the mean free path is δ0.3μm.

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