High-Power , High-Intensity Laser Propagation and Interactions

This paper presents overviews of a number of processes and applications associated with high-power, high-intensity lasers and their interactions. These processes and applications include: free electron lasers, backward Raman amplification, atmospheric propagation of laser pulses, laser driven acceleration, atmospheric lasing and remote detection of radioactivity. The interrelated physical mechanisms in the various processes are discussed. _______________________________ Manuscript approved January 24, 2014


I. Introduction
High average power lasers, e.g., free electron lasers (FELs) and solid-state lasers (including fiber lasers) are prime candidates for efficient, directed energy applications.
These include laser power beaming and laser weapons, requiring multi-kWs of CW power operating in the IR regime.
The physical processes associated with ultrashort pulse lasers (USPL) interactions include: photo/collisional ionization, Kerr and rotational Raman effects, self phase modulation, filamentation, group velocity dispersion, optical shocks, frequency chirping, etc.We discuss these interrelated physical processes and some of the applications of these lasers.
One of the important topics to be discussed in connection with both high-power and high-intensity lasers is propagation in a turbulent atmosphere.Laser propagation in atmospheric turbulence can results in beam centroid wander, spreading and intensity scintillation.A phase conjugation technique to mitigate the effects of atmospheric turbulence is described.
In Sec.II we discuss free electron lasers and related topics including optical guiding, efficiency enhancement, electron beam quality, coherent and incoherent x-ray FELs and backward Raman amplification.In Sec.III we discuss ultrashort pulse laser interactions and related topics, including atmospheric propagation of USPLs and laser driven electron acceleration in tapered plasma channels.In Sec.IV we discuss the role of atmospheric turbulence on the propagation of laser beams and also briefly discuss optical phase conjugation as a means to mitigate the effects of turbulence.In Sec.V we discuss various applications of high-power, high-intensity lasers including remote detection of radioactive material using electromagnetic signatures, atmospheric lasing of N 2 molecules.Concluding remarks are presented in Sec.VI.

II. Free Electron Lasers
Free electron lasers can generate high-power coherent radiation from the microwave to the x-ray regime [1][2][3].The FEL mechanism is based on classical physics in which an electron beam induces radiation as it traverses a periodic magnetic field (wiggler) [4][5][6][7][8][9][10][11][12][13].The FEL can operate in the low gain (interference, oscillator regime) [4] or a high gain (exponential, amplifier regime) [5].One of the important characteristics of FELs is that the generated radiation beam can be guided by the electron beam, making possible extended interaction lengths.Also FEL efficiency can be enhanced by various methods such as spatially tapering the wiggler field.Free electron lasers are generally large, complex and expensive.
A schematic of an FEL is shown in Fig. 1.In this configuration, electron bunches from a radio frequency (RF) linac propagate through a wiggler and radiation is amplified by the electron-wiggler interaction.The spent electron beam is deflected either into a beam dump or re-circulated.
is the transverse electron velocity due to the wiggler (radiation) field.This beat wave is referred to as the ponderomotive wave.The phase velocity of the ponderomotive wave where is the normalized wiggler amplitude and b  is the beam plasma frequency.The dispersion relation describes how the wiggler field couples the electromagnetic modes (first factor on the left hand side) to the electron beam modes (second factor on the left hand side).

a) Optical Guiding in FELs
In the FEL the wavefronts of the amplified radiation are such that the radiation tends to focus toward the electron beam [13].Solving the dispersion relation, Eq. ( 1), the radiation wavenumber is found to be given by , where the wavenumber shift when the growth rate is    (amplification).Since the refractive index associated with the radiation is , the radiation undergoes focusing [13].When focusing is balanced by diffraction the radiation beam is optically guided though the interaction region as shown in Fig. 3. Optical guiding is important since the interaction length in the wiggler is generally long and diffraction would, in the absence of guiding, spread the radiation, terminating the FEL interaction.Amplification of the radiation is at the expense of the electron beam kinetic energy.As the electrons slow down the FEL interaction coupling is reduced and efficiency decreases.However, by spatially tapering the wiggler field it is possible to reduce the phase velocity of the trapping (ponderomotive) wave to maintain synchronism with the electrons [11].This is one of a number of methods to enhance the FEL efficiency.Figure 4 is a plot of the output power versus interaction length obtained from a simulation using the code GENESIS [14].Theoretically and experimentally it has been demonstrated that the FEL mechanism can be reversed resulting in a laser driven electron accelerator [15].

c) Electron Beam Quality
In the FEL, the gain, growth rate and efficiency are sensitively dependent on the axial velocity spread z   of the electrons.The electron beam quality is measured in terms of emittance and energy spread, the origins of which include i) temperature of the cathode, ii) roughness of the emitting surface, iii) non-uniformity of the emission from the cathode surface, iv) nonlinearity of electric and magnetic fields, and v) self-fields.
The electron beam quality requirement for the FEL is that z   be small compared to the efficiency eff L  /  at the entrance to the wiggler where eff L is the effective interaction length.This requirement can be expressed by the following inequality [2] where the total longitudinal energy spread is In Eq. (2b), the first term represents the fractional energy spread due to longitudinal emittance z ε , the second term arises from transverse rms emittance  , n  , and the third term is due to transverse wiggler gradients.The right hand side of Eq. (2a) is the approximate FEL conversion efficiency, where the effective interaction length, eff L , is the gain length in the case of the amplifier or the wiggler length in the case of the oscillator.If the inequality in Eq. (2a) is well satisfied the spread in the axial velocity of the electrons plays little role in the FEL interaction.As an example, for a MW-class FEL operating in the IR regime, the typical parameters are , and cm 3  w λ [16].For these parameters the inequality is also satisfied.

d) Coherent and Incoherent X-Ray FELs
The FEL can generate coherent, polarized, short pulses of x-rays for numerous applications in research.A number of large-scale electron accelerator facilities throughout the world are being used (or will be commissioned soon) for coherent x-ray generation using a conventional FEL configuration [17][18][19][20].Generation of x-rays at these facilities typically requires electron beam energies in the multi-GeV range with peak currents in the multi-kA range, and wiggler lengths of many tens of meters.An x-ray FEL amplifier can be operated in the self-amplified regime, eliminating the need for a coherent input x-ray source [21][22][23].
The wiggler field in the FEL can be replaced with an intense laser beam propagating anti-parallel to the electron beam [24].In a laser pumped x-ray FEL electron beam energies in the multi-MeV range would be sufficient, leading to a more compact device.However, a number of major challenges would have to be overcome before a laser pumped coherent x-ray FEL would be feasible, e.g., ultra-high electron beam brightness.A compact source of incoherent x-rays can be configured in which microwaves or laser radiation is Compton backscattered off a relativistic electron beam [25][26][27].

e) Backward Raman Amplification
In the backward Raman amplification (BRA) scheme, two counter propagating laser beams are resonantly coupled by a plasma wave [28][29][30].The laser beams have a frequency difference which is approximately equal to the plasma frequency.The higher frequency ( o  ), long pulse pump laser supplies energy to the lower frequency ), short pulse seed laser which undergoes amplification and pulse compression.The phase velocity of the plasma wave is in the same direction as the pump laser.
The BRA in the linear regime is fundamentally the same as the FEL mechanism in the frame of the electron beam.In the electron beam frame the wiggler field is essentially a counter propagating electromagnetic wave.Thus, replacing the wiggler field in Eq.( 1) with a counter propagating laser beam results in the BRA dispersion relation.By setting in the FEL dispersion relation, Eq.( 1), with appropriate redefinition of symbols the BRA dispersion relation is obtained and sketched in Fig. 5.In the BRA process the seed laser grows and is temporally compressed at the expense of the pump laser.This process can generate high intensity ultra-short laser pulses.Experiments have demonstrated amplification of seed intensities by a factor of 10 4 with an efficiency of > 6% [31,32].

III. Ultra Short Pulse Laser Interactions
A great deal of progress has been made in the development of ultra-short pulse lasers (USPL) [33].Table I lists some of the important parameters associated with these lasers.Experiments using ultrashort, high intensity laser pulses have demonstrated atmospheric propagation, air breakdown, filamentation, and white light generation [42][43][44][45].Intense, directed white light pulses have been generated and backscattered from atmospheric aerosols up to altitudes of ~15 km [37].The generation of THz radiation by femtosecond pulses has also been observed and analyzed [46,47].

Filamentation in Air
Hot spots on the intensity profile of a laser beam can grow as a result of a filamentation instability.Filamentation, i.e., transverse break-up of a laser beam is due to the interplay between diffraction and nonlinear self-focusing (Kerr effect).Consider a laser beam propagating in a neutral gas for which the nonlinear focusing power is where K n is the Kerr effect contribution to the refractive index and the transverse laser intensity profile is slightly perturbed by a small, localized hot spot.The spatial growth rate of this filamentation instability [40] is where I is the laser intensity and  x is the characteristic transverse dimension of the filament, i.e., spot size.As a function of the dimension of the filament, the growth rate , and decreases inversely with  x as  x increases.At maximum growth rate the power within the filament is roughly equal to K P .It is therefore expected that a laser beam with a power P will break-up into N filaments where . As an example, the nonlinear focusing power associated with air for a where W is the photoionization rate,

b) Laser Driven Acceleration
The extremely large field associated with USPLs propagating in plasmas can generate plasma waves which trap and accelerate electrons [54][55][56][57][58][59][60][61][62][63][64][65][66][67][68].In the standard laser wakefield accelerator (LWFA), a short laser pulse, on the order of a plasma wavelength long, excites a trailing plasma wave that can trap and accelerate electrons to high energy (see Fig. 9).Numerous analyses and experiments have been performed to characterize and advance the LWFA concept.However, there are a number of issues that must be resolved before a practical high-energy accelerator can be developed.These include Raman, modulation and hose instabilities that can disrupt the acceleration process [68].
In addition, extended propagation of the laser pulse is necessary to achieve high electron energy.
In the absence of optical guiding the acceleration distance is limited to a few which is far below what is necessary to reach GeV electron energies [54].The diffraction limitation can be mitigated by employing a plasma channel in which the plasma density increases as r 2 .The physics of laser beams propagating in plasma channels has been studied [56,64,68,69] and there is ample experimental confirmation of extended guided propagation in plasmas and plasma channels [70][71][72][73][74][75][76].In addition, dephasing of electrons in the wakefield can limit the interaction length to cm ) / (   p p    and, therefore, limit the energy gain.By increasing the plasma density as a function of distance, the phase velocity of the accelerating field behind the laser pulse can be made equal to the speed of light.Thus, electron dephasing in the accelerating wakefield can be postponed and energy gain increased by spatially tapering the plasma channel.Staging LWFAs can be used to overcome the laser energy loss limitation.Injection, trapping and acceleration in the wakefield continues to be of interest [77]. associated with the laser pulse generates large amplitude plasma waves ( > 100 GV/m) that can accelerate self-trapped or injected electrons to high energies.
The plasma density wave (wakefield) is driven by the ponderomotive force and satisfies a driven harmonic oscillator equation the laser field.
The accelerating wakefield (space charge field) is given by The accelerating gradient

i) Optical Guiding in LWFA
The laser spot size ) (z R as a function of axial position in a plasma with refractive index n is given by [64] r where ... is an average performed using the laser intensity as the weighting function.
The full refractive index associated with the laser plasma interaction having a parabolic density variation in the radial direction is The expression for the refractive index is obtained by noting that is the plasma frequency, where is the plasma density in the channel, ) (r is the depth of the plasma channel and e r is the classical electron radius (see Fig. 10).Note that in the absence of a plasma wave and plasma density channel, i.e., uniform plasma density, optical guiding is possible through a relativistic effect.Here the laser power must equal the relativistic focusing power, which is 2 ) / ( 4 .17 Extended optical guiding is important for achieving significant energy gain in the LWFA.
Laser wakefield experiments at LBNL employing optical guiding of a 40 TW laser pulse in a gas jet configuration resulted in observation ~ GeV electrons [78].

ii) Raman and Modulational Instabilities
The dynamics of a laser pulse propagating in a uniform plasma channel is affected by Raman and modulational instabilities.In the broad beam limit ) ( , however, the nonlinear terms in the wave equation that lead to Raman and modulation instabilities tend to cancel [79].This can result in pulse propagation over extended distances, limited only by dispersion with minimal pulse distortion due to instabilities.

iii) LWFA in a Tapered Plasma Channel
By employing a tapered plasma channel electron-wakefield dephasing, i.e., slippage, can be postponed [80,81].The phase of the wakefield behind the laser pulse is and the phase velocity is given by where  is the time measured in the frame of the pulse with group velocity ) ( v z g and L  is the laser pulse length.The phase velocity of the wakefield varies with distance behind the pulse.The phase velocity increases (decreases) with distance from behind the pulse for an increasing (decreasing) plasma density.The location behind the pulse, c  , for which the phase velocity equals the speed of light in vacuum is given by The solid curve in Fig. 11 shows the energy of a test electron along a tapered plasma channel (increasing plasma density) that is trapped and accelerated to ~ 4 GeV.The dashed curve shows an energy gain of ~ 1GeV in an untapered channel.

IV. Laser Propagation in Atmospheric Turbulence
Atmospheric turbulence, absorption, scatterings, scintillation and thermal blooming can have an important effect on laser propagation [83].The following discussion deals primarily with laser propagation through atmospheric turbulence.The refractive index of turbulent air is ) , , , ( 1 , where The strength of the turbulence is characterized by the structure parameter 2 n C which typically is in the range of at ground level [83][84][85]. Turbulence leads to an increase in spreading of the laser beam spot size, s R .In addition, turbulence leads to wandering of the laser beam centroid, w R Figure 12 shows the laser beam spot (small circles) at several instants in time.Over a time scale that is long compared with that associated with wander one obtains the broadened spot shown by the large dotted circle in Fig. 12.
The time averaged laser spot size , (11) where  Figure 13 shows the laser intensity contours for three levels of atmospheric turbulence.

In limit of extremely weak turbulence (
, Fig. 13(a)) the laser intensity is well-defined with a relatively small spot size [86].In moderate turbulence 13(b)) the laser profile is distorted and the spot size is larger.In strong turbulence ( , Fig. 13(c)) the laser profile is highly distorted.For sufficiently strong turbulence a single discernable high-intensity region in the cross-section may not be obtained.
Figure 13.Comparison of the intensity contours in the x-y plane at a range of 0.5 km for three levels of turbulence using the propagation code HELCAP [86].
Effective beam control of high energy lasers (HELs) is a key component of a laser directed energy system.Strong turbulence can have a significant deleterious effect on the propagation of the HEL beam.By appropriately modifying the wavefront of the transmitted beam the laser power can be more effectively focused on target [87,88].A beacon beam can be used to record the atmospheric phase aberrations used to modify the amplitude and phase of the transmitted high power beam; see Fig. 14 for a simplified view of the process.The information needed for modifying the amplitude and phase is obtained by phase conjugating the beacon beam, i.e., ) exp( ) exp( . Phase conjugation can be achieved by using deformable mirrors or by a nonlinear optical mixing mechanism, e.g., 4-wave Brillouin mixing [89,90].
transmitted beam is phase conjugated target initial wave front turbulent air Figure 14.Using phase and amplitude information provided by a beacon beam the distortion of the transmitted high power laser beam is gradually undone as it propagates to the target.

V. Applications of High-Power, High-Intensity Lasers
Advances in FELs and solid-state lasers (including fiber lasers) have made them prime candidates for high-average power applications.These include power beaming and directed energy applications.In this section we discuss remote detection of radioactive material and remote atmospheric lasing for chemical/biological detection, employing high-intensity lasers.

a) Remote Detection of Radioactive Material
A recently proposed radioactivity detection concept is based on a high power THz pulse inducing avalanche breakdown and spark formation in the vicinity of the radioactive material [91,92].In this detection concept the time delay for spark formation and the breakdown rate, which are functions of the initial ion and electron densities, can provide a direct signature for the presence of radioactivity.
We discuss a new concept which is based on the detection of electromagnetic signatures in the vicinity of radioactive material and can enable stand-off detection [93].O ions, providing electrons for an avalanche (collisional) ionization process which increases the electron density.A probe beam in the presence of a temporally increasing electron density undergoes a frequency modulation which becomes a spectral signature of radioactivity.
To obtain the frequency modulation on a probe pulse it is necessary to follow the time evolution of the electron and negative ion density, which are sensitive functions of air chemistry processes and electron heating by the laser radiation.The source terms for the electrons include radioactivity, detachment, photo-detachment and photo-ionization, while the loss terms include various attachment and recombination processes including aerosols.The expressions for the rate of change of electron density e N and negative ion density  N , i.e.,   O in the present context, are [93], where e S represents the various electron source terms, e L is the electron loss terms,  S represents the ion sources,  L is the ion loss terms and is the radiation enhancement factor.For a detailed discussion of the various terms in Eq.(12a,b) see Ref [93].
Since the free electron density is generally much less than the ion density, .Here the plasma frequency is given by The maximum fractional frequency shift occurs for cm 1 / In the absence of a radioactive source, Fig. 16(a), the ionizing laser intensity is just below the breakdown level, i.e., the electron density is low, and there is virtually no frequency modulation on the probe beam.The frequency modulation on the probe millimeter wave beam is shown in Fig. 17.
In the absence of radioactive material there is no frequency modulation on the probe.
However, for    rad  the fractional frequency modulation is significant and equal to % ~ , which is readily detectable. .The laser parameters are the same as in Fig. 16.

b) Atmospheric Lasing
Nitrogen lasers are typically based on collisional excitation using electrical discharges [94][95][96].In this section we discuss a remote atmospheric lasing mechanism [97] in which an ultrashort pulse laser forms a plasma filament (seed electrons) by tunneling ionization and a heater pulse thermalizes the seed electrons.The thermal electrons collisionally excite nitrogen molecules and induce lasing in the ultraviolet.A remote atmospheric lasing configuration based on collisionally exciting the 2 N lasing line at nm 337  λ in a heated plasma filament, as illustrated in Fig. 18.The lasing gain is sufficiently high to reach saturation within the length of the plasma filament.
A remotely generated ultraviolet source may have applications for standoff detection of To analyze the lasing process the density matrix and Maxwell equations [97] together with the electron heating and ionization equations are solved.The N 2 lasing model consists of an open two level system denoted by levels 3 (upper) and 2 (lower) respectively.Since levels 3 and 2 are weakly excited the population of level 1 (ground) is taken to be fixed.The population of the excited levels are given by

VI. Concluding Remarks:
A number of topics associated with high-power, highintensity lasers have been discussed.The aim has been to briefly illustrate the varied physical processes and phenomena that are manifested in the interaction of these lasers with matter.We have also described some of the actual and potential applications of high-power as well as high-intensity lasers.This paper is not intended to be an exhaustive overview of the various topics; important references on the topics may have been omitted.

Figure 1 .Figure 2 .
Figure 1.FEL amplifier configuration slightly less than the speed of light and comparable to the electron axial velocity z v .The relativistic electrons can therefore transfer kinetic energy to the E resulting in amplification.The electrons eventually become trapped in the ponderomotive wave and the radiation field saturates.Greater insight into the FEL mechanism is provided by the one dimensional dispersion relation.The electron density wave, together with the electron wiggle motion, generate a transverse current having the proper phase, i.e., spatialtemporal, dependence to amplify the radiation field.The amplified radiation in turn enhances the electron density wave further amplifying the radiation.Considering spatial variations in the z direction only the FEL dispersion relation is[2]

Figure 4 .
Figure 4. GENESIS simulation of FEL power versus interaction length without (solid curve) and with (dashed curve) tapering.The insets show the electron phase space (energy vs. ponderomotive phase) at various positions.

Figure 5 .
Figure 5. Dispersion diagram describing the backward Raman amplification process.This dispersion diagram also describes the FEL process in the beam frame.

.Figure 8 .
photon and tunneling processes; avalanche ionization is not significant for USPLsThe length of the plasma filament is determined by the electron attachment and recombination coefficients and from Eq.(4) and is given bym ) /( ~   e e filament N c  

Figure 9 .
Figure 9. Schematic of laser wakefield accelerator mechanism and can be ~ 10 3 times greater than accelerating gradients in RF accelerators.

.
In the short pulse limit the laser spot size dynamics in a plasma channel

Figure 10 .
Figure 10.Guided propagation of Gaussian laser beam in a parabolic-profile plasma density.In a plasma channel the guiding condition is independent of wavelength.

Figure 11 .
Figure 11.Energy gain as a function of propagation distance normalized to the Rayleigh range Z R. .Dashed curve is for an untapered channel (maximum energy ~ 1 GeV) while the solid curve is for a tapered channel (maximum energy ~ 4 GeV).

or
is the transverse coherence length associated with turbulence and jitter  is the mechanical jitter angle of the beam.far as laser beam propagation is concerned the effective strength of turbulence is not determined by  n C alone, but also depends on the range of propagation and the wavelength.The relevant combination is the Rytov variance, defined by < 0.3), beam wander and spreading are distinct and tip-tilt compensation can be used to correct for beam centroid wander.On the other hand, for strong turbulence (  R  > 0.3) the beam breaks up into multiple beams and beam wander and spreading are not distinct, hence, tip-tilt compensation is less effective.The Rytov variance is a measure of the intensity scintillation level on axis at the target, i.e.,

Figure 12 .
Figure 12.Schematic showing the effects of turbulence on beam propagation, w R is the centroid displacement (wander), and S R is the increase in spot size (spreading), where     / ) ( s

Figure 15 .
photo-detached electrons, in the presence of laser radiation, initiate avalanche ionization which results in a rapid increase in electron density.The rise in electron density induces a frequency modulation on a probe beam which becomes a direct spectral signature for the presence of radioactive material[93].High-power, short-pulse lasers propagating in the atmosphere can come to a focus in the vicinity of the radioactive material by making use of longitudinal compression and transverse focusing, see Sec.IIIa and[40].The negative ions produced by the radioactive material have an electron affinity resulting in a low ionization potential of 0.46 eV and can be photo-detached by laser radiation ( μm 1 8 .0 ~ ).The detection concept is based on a probe radiation beam undergoing a frequency modulation while propagating in a temporally increasing electron density.The frequency modulation on the probe beam becomes a spectral signature for the presence of radioactive material.A schematic of the detection concept is shown in Fig.15.

T
ion detachment rate due to collisions with neutrals.In the absence of radioactive material ( and ion source and loss terms in Eqs.(12a) and (12b), in particular the collisional ionization rate, are functions of the electron temperature.The electron temperature is determined by the collisional electron heating (Ohmic heating) by the laser radiation and the cooling effect resulting from excitation of vibrational modes of the air molecules.The equations for the electron temperature is the electron temperature, E J  is the Ohmic heating rate, propagating through a region of space in which the density is changing with time will undergo a frequency change.The electron density in the vicinity of the radioactive source and under the influence of the laser radiation varies in space and in time.The frequency of an electromagnetic probe beam propagating in such plasma will vary in space and in time.As an illustration we consider the case where the rise in electron density is exponential in time and spatially uniform within a region L z   

.Figure 16 .
Figure 16.Electron density as a function of time, (a) background radiation only ( 0  rad  Figure 16(b)  shows the electron density as a function of time in the presence of radioactive material ( electron density at the end of the ionizing laser pulse approaches the value of order of magnitude less than the critical electron density.

Figure 17
Figure 17.Fractional frequency shift [%] / o    versus time in the presence of

Figure 18 .
Figure 18.Schematic diagram of remote lasing configuration.An ultrashort pulse laser creates a plasma filament of seed electrons that is heated by a secondary heater pulse.In the energy level diagram for N 2 (right), the energetic electrons collisionally excite the N 2 molecules and induce lasing.

,
z is the position within the filament, n N , are densities of the n th level, de-excitation rate from level i to j (i, j = 1, 2, 3), , I is the UV laser intensity, ω is the lasing frequency stim rates are dominated by electron excitations while the de-excitation rates are dominated by molecular collisions.The lasing intensity is given by is the spatial damping rate, ε is a geometric filling factor associated with the seed radiation within the plasma filament, is the radiative lifetime from level 3 to 2 including the effects of collisions.

Figure 19 plots.
Figure 19 plots the temporal profile of the UV radiation and population inversion at various positions along the filament.Here, the heating pulse duration is nsec 4 .0 