High-Order Harmonic Generation in Plasmas

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997 1915

High-Order Harmonic Generation in PlasmasPaul Gibbon

(Invited Paper)

Abstract—The phenomenon of harmonic generation by elec-trons oscillating in high-intensity laser fields is surveyed andassessed as a means of producing short-wavelength radiation.Starting from the seminal early work by Sarachik and Schappert,simple motivatory examples are given of incoherent harmonicgeneration via nonlinear scattering from single electrons. Morerecent studies aimed at observing the coherent version of thiseffect in underdense plasmas are then reviewed and some prob-lems noted in distinguishing these harmonics from those producedvia the analogous nonlinear mechanism from bound electronsin rare gases. Finally, the revival of interest in harmonics re-flected from overdense plasmas is considered. Short-pulse laser-generated “surface” harmonics appear to offer a very promising,compact, and efficient means of upshifting coherent radiation tosub-10-nm wavelengths.

Index Terms—Harmonic generation, relativistic waves in plas-mas.

I. INTRODUCTION

EVER SINCE the development of the laser in the 1960’s,a sizeable research effort has been devoted to finding

ways of generating coherent radiation with much smallerwavelengths than the present working range of 1/4–10m.Given the current widespread use of lasers in almost everyfield of science and technology, it is not difficult to imaginethe potential benefits of a tunable, coherent radiation sourceoperating in the sub-100-nm range. Not too surprisingly, giventhis strong motivation, a number of workers were quick torealize that assuming intensities could be scaled to sufficientlyhigh levels, lasers would eventually be capable of generatingharmonic radiation via nonlinear Thomson scattering from freeelectrons [1], [2].

Parallel to these technological developments, a series ofpapers appeared in the early 1970’s on the theory of pulsars[3], which are believed to emit coherent electromagneticradiation with field strengths V m into a tenoussurrounding plasma. To interpret observations of these objects,it is important to understand how such radiation propagatesaway from the core and interacts with the plasma. A theoreticalbasis for relativistic wave propagation had been provided byAkhiezer and Polovin [4] some years prior to this. What thelaser promised, but took three more decades to deliver, was ameans of investigating these phenomena in the laboratory.

This possibility was nevertheless anticipated by Sarachikand Schappert [2], who duly classified existing and future

Manuscript received June 17, 1996; revised May 7, 1997.The author is with the X-Ray Optics Group, Institute of Optics and Quantum

Electronics, University of Jena, 07743 Jena, Germany.Publisher Item Identifier S 0018-9197(97)07830-5.

radiation sources according to a normalized pump strengthparameter , where isthe classical electron radius, the laser intensity, and

, and are the electron charge, electron mass, lightspeed, and laser wavelength, respectively. Modern conventionrequires that we define the pump strength in terms of theelectron quiver momentum, . It isinteresting to note that the highest pump strengths consideredby these authors were 100–1000, which, though probablydismissed as fanciful at the time, correspond to intensitiesabout to be achieved by multiterawatt femtosecond lasers.Sarachik and Schappert proceeded to calculate in detail theharmonic content of radiation emitted by electrons subjectto such field strengths. These harmonics areincoherent, butone can imagine that summing over many scatterers—as ina plasma, for example—should yield coherent radiation insome preferred directions. This is indeed the case, though wewill see later that this procedure generally leads to somewhatcounterintuitive results.

In this paper, we will concentrate on harmonic generationfrom electrons in pre-ionized matter irradiated by short, in-tense laser pulses. In doing so, we will omit a variety ofother physical processes which are equally important in thefield of short-wavelength generation. These include harmonicgeneration by atoms [5], frequency upshift by ionization fronts[6], and by photon acceleration [7], X-ray [8], and free-electronlasers. A recent review and collection of papers on these topicscan be found in [9]; for a general overview of femtosecondlaser-plasma interaction physics and applications, see [10].

The remainder of this paper is divided into three parts, ina roughly chronological progression from isolated electronsto collective behaviors in gas-like and solid-like plasmas.In Section II, the results for single relativistic electrons arequoted and practical considerations for experimental detectionof harmonics are discussed. In Section III, we review the recentwork on harmonic generation in underdense plasmas, includingthe issues of phase matching and optimization using densitymodulations. Finally, in Section IV, we consider the possibilityof harmonic generation from solid surfaces, a phenomenonoriginally studied in the context of laser fusion as a plasmadiagnostic, but which at high intensities is intimately connectedto the relativistic electron motion central to the two precedingsections.

II. SINGLE ELECTRONS

The starting point for the early investigations on harmonicgeneration was the nonlinear orbit of a single electron in a

0018–9197/97$10.00 1997 IEEE

1916 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997

strong electromagnetic plane-wave. This well-known problemis exactly soluable [11]–[13] and is conveniently described interms of an “average rest” frame, which drifts with a velocity

with respect to the laboratory frame.Although this ground has been well covered in the literature(and is still revisited intermittantly), it proves useful to recallthe main results in order to introduce some notation and toget a feel for the nonlinear nature of the more complicatedlaser-plasma interaction which we will come to in the latersections.

The motion of a particle in an electromagnetic wave isdescribed by the Lorentz equation

(1)

together with an energy equation

(2)

where and is the relativisticfactor.

An elliptically polarized plane-wave travelling inthe ve -direction can be represented by the wave vector

(3)

where and is a polarization parameter suchthat for a linear polarized wave 1,0 and for a circularwave . Using the relations and

we can integrate the parallel and perpendicularcomponents of (1) to obtain

(4)

(5)

where and are constants of motion determining theinitial longitudinal and perpendicular momenta, respectively.In arriving at these expressions, we have introduced thenormalizations ,and . Using the identity andchoosing , we find

(6)

The two frames of special interest are the laboratory frame,corresponding to , and the average rest frame for which

and . In the latter case, (5)and (6) reduce to

(7)

(8)

Noting that , we can integrate again to obtainthe orbits

(9)

(a)

(b)

Fig. 1. Characteristic orbits of free electrons in a plane electromagnetic wavein (a) average rest frame and (b) laboratory frame. The trajectories correspondto light intensities of 1017 W/cm�2 (solid line), 1018 W/cm�2 (dashed line),and 1019 W/cm�2 (dotted line), respectively.

where . For linear polarization, , andeliminating , we obtain the famous figure-of-eight

(10)

In the extreme intensity limit, , we find ,and the orbit becomes a “fat-8” with proportions

. The laboratory frame orbits can likewise beobtained from (5) and (6) by setting , or by transforming(9) back. For linear polarization, we find

(11)

The orbits in each of these frames for linearly polarized lightare depicted graphically in Fig. 1.

Knowing the particle orbits , we can immediately com-pute the harmonic content of the emitted radiation by substitut-ing (8) and (9) into the standard formulae for light emission byan accelerated charge [14]. A special case of this is periodicmotion, for which the radiated power per solid anglecan be decomposed into multiples of the fundamental ,

GIBBON: HIGH-ORDER HARMONIC GENERATION IN PLASMAS 1917

whence

(12)

where is the unit vector from the source to the observerand is the retarded time, which takes into account thefinite time for the emitted light to reach an observer situatedfar from the source.

We can apply this formula directly to the periodic figure-8orbit (10) in the average rest frame. In the laboratory frame,however, the motion is not purely harmonic [as evident fromFig. 1(b)], so we have to either resort to the Lienard–Weichertformula [14], or Lorentz-transform the solution obtained from(12) back from the average rest frame. The latter procedure issomewhat subtle, and is explicitly demonstrated by Sarachikand Schappert [2]. The detailed results for the harmonic powercontent are algebraically somewhat involved, but the mainconclusions can be summarized as follows. For circularlypolarized light in the extreme intensity limit ,the radiation is predominantly forward and confined to anangle from the axis of propagation, with aharmonic cutoff at . The observedharmonics are Doppler-shifted, so that in fact,

, where is the angle of observation withrespect to the laser axis. For linearly polarized light, theangular distribution shifts from sideways to forward due tothe increase in drift velocity .

The total scattered power in the first three harmon-ics—integrated over solid angle in the laboratory frame—canbe expanded in the limit to give the following leadingterms:

(13)

where is a characteristic scattered power perelectron for a given laser frequency . The expression for

is readily identified as the classical Thomson scatteringresult (see [14, Ch. 14]). A more thorough discussion of theSarachik and Schappert work, including an explicit analysisof the angular distribution and some corrections to the higherorder coefficients can be found in [15].

Recent work by Bardsleyet al. [16] and Mohideenetal. [17] has reexamined the single electron dynamics, takinginto account finite pulse shapes, ionization, and space-chargeeffects. The first two effects mainly give rise to irregularities inthe amplitude and phase of the electron orbits which can leadto “residual heating”; the third effect is potentially the mostdamaging with regards to harmonic generation in a plasma.As we will see in Section III, the collective restoring forcedue to charge separation not only suppresses the drift motion,

but produces density nonlinearities which tend to cancel theharmonic-producing relativistic effects.

In effect, the single electron picture is only valid for times, where , where is the electron

density, which in practice means choosing pulse lengths

cmfs (14)

On the other hand, the radial ponderomotive force will kickelectrons away from the focal region before they experiencethe maximum laser field. Mohideenet al. attempted to takethese effects into account by summing the harmonic spectrumdue to test particles in the self-consistent fields of a Gaussianfocus, finding interference effects which enhance the lowerharmonics but which limit the emission at higher harmonicorder.

Despite the enormous increases in laser intensity seen inrecent years, experimental evidence for harmonic generationby free electrons is very limited, apart from an earlier obser-vation of second harmonic photons by Englert and Rinehart[18]. A more effective way of exploiting nonlinear Thomsonscattering is to use relativistic electron beams in place of aplasma [19]. In this case, the harmonic content is modifiedby the relative Doppler shift of the electron beam, an effectwhich can be optimized by scattering a short laser pulse off acounterstreaming e-beam [20].

III. U NDERDENSE PLASMAS

The key to efficient harmonic generation is to bring photonsand electrons together at high number densities for at least afew optical cycles, so that the nonlinearities described earliercan be efficiently exploited. Given the practical difficulties ofachieving these conditions with isolated electron “clouds” orbeams, it is natural to ask what can be achieved with a fullyionized plasma as a scattering medium. In this case, harmonicgeneration can be analyzed in terms of the collectivefluidresponse of the plasma electrons in a uniform ion background,and instead of (12), we calculate

(15)

from which we see that we now require the plasmacurrent. For an axisymmetric pulse propagating through

a uniform plasma, it turns out that the scattering contribu-tions cancel except in the forward direction, where they addcoherently. The calculation of harmonic emission reduces todetermining the current source and using the wave equationto obtain the vector potential.

Such an analysis was first performed by Sprangleet al.[21] in the context of short-pulse laser-plasma interaction, whoshowed that the third harmonic power scales asrather than

, as we would expect by summing over a set of scatterers.The reason for this is that the longitudinal electron motionin (7) sets up a space charge oscillating at twice the laser


frequency. The resulting density perturbation beats with thelaser field to form a nonlinear current source which nearlycancels the relativistic nonlinearity. A tutorial account of thiseffect is given by Mori [22], which we reproduce here in aslightly simplified form. Starting from the wave equation

(16)

we see that the harmonics are driven by the nonlinear currenton the right-hand side. To evaluate this source term, we needfirst the component of (1), which we can rewrite as

(17)

where as before, and we have retained onlynonoscillatory terms in the relativistic factor, i.e., we expandabout a relativistically strong pump. Combining (17) withPoisson’s equation and the continuityequation , we write ,normalize to and neglect terms to obtain

(18)

Expanding to the same order, the (normalized) transversecurrent can be expressed as

(19)

The second and third terms represent the density bunching andrelativistic nonlinearities, respectively; both are .

To proceed further, we expand all quantities as a harmonicseries and let

as before. One can then show that (18) reduces to

(20)

so that in the limit , the third harmonic currentbecomes

(21)

In other words, the relativistic and density-bunching non-linearities for the third harmonic source actually cancel to

. To obtain the power efficiency, we equate (21)with the matching term on the left-hand side of theexpanded wave equation (16), taking care to eliminate termswhich would lead to secular growth [23]. This procedureeventually yields a steady-state amplitude

(22)

Using the fact that the electric field strength of the harmonic,, we finally obtain an overall power scaling of

[24]–[26]

(23)

The third harmonic power has a maximum for , afterwhich it falls off slowly. Typical efficiencies are rather low.For example, a 1-TW Ti-Sa laser with intensity 10W/cm( 0.68) focused into a plasma with density 10cm willyield just 5 kW of third harmonic radiation. If this result byitself is not disappointing enough, it turns out that includingpropagation effects reduces the efficiency even further. In fact,there are two pieces of physics (initially neglected in [21])which are responsible for this.

First is the fact that in a plasma the phase velocities of theharmonics differ from the velocity of the fundamental. This iseasily seen from the dispersion relation for theth harmonic

(24)

Thus

(25)

and the harmonic will be out of phase with the pump aftera length

(26)

For the Ti-Sa example just considered above, with 11m and 0.8 m, the dephasing length of the third

harmonic would be 56 m, which is much smaller than atypical Rayleigh focusing length. The second problem is thatharmonics generated by a Gaussian beam focused in finite-length medium will not be easily detected in the far field due tophase cancellation between the source terms during focusingand defocusing [27].

The dephasing between pump and harmonics was firstconsidered self-consistently in a concise analysis of thirdharmonic generation by Rax and Fisch [24]. Using a La-grangian fluid model, they showed that due to the mismatchin phase velocities, the third harmonic oscillates in magnitudearound an average value given by (22). These authors proposedto get round the dephasing problem by imposing a densitymodulation with period via an ion acoustic wave ora multilayered medium [24], [28]. The idea is to suppressharmonic generation in the lower density regions, using theminstead to allow the third harmonic to “coast” back into phasewith the pump, thereby restoring linear growth.

Following this work, several groups reexamined the problemin more detail in order to determine the asymptotic scaling ofthe higher order harmonics [25] and to address the issue ofthree-dimensional (3-D) effects [26]. Moriet al. [25] showedusing a fully nonlinear analysis that near the critical densityand for large , the th harmonic should scale as

(27)

They also presented the first simulations of harmonic gener-ation in an underdense plasma using a particle-in-cell (PIC)code specially adapted to follow a “window” of plasmaoccupied by the laser pulse. While simulation results largelyagree with theory for the third harmonic, the higher harmonics


were orders of magnitude larger than predicted, and tendto form a plateau reminiscent of harmonic generation ingases [29]. An example of such a cyclic-grid PIC simulation(performed by the author) with parameters close to those usedin [25] is shown in Fig. 2. The third harmonic

5.6 10 oscillates around the theoreticalvalue given by (22). The oscillation amplitude in Fig. 2(b)differs from the ideal case—i.e., from 0 to twice the steady-state value—because of the finite pulse shape.

Further simulations by Moriet al.showed that the harmoniccontent could be enhanced by introducing a density ramp,a result which can be partially understood in the light ofthe density-ripple analysis of Rax and Fisch [28]. Anotherpossibility for enhancement was recently suggested by Zenget al. [30], who found that harmonic amplitudes could beincreased by at least an order of magnitude for laser pulselengths much shorter than a plasma wavelength.

Esareyet al. [26] used an iterative approach—expandingthe fluid equations (1) and (16) to successively higher order,obtaining one-dimensional (1-D) solutions for the third, fifthand seventh harmonics, giving power efficiencies for the lattertwo of

(28)

In a realistic focusing geometry, further phase differences areintroduced between the pump and the harmonics. This effecthas been known for some time in the context of harmonicgeneration via third-order nonlinearities in gases, where it canbe shown thatno harmonic signal emerges from an infinitemedium [27]. Esareyet al. calculated this effect for thirdharmonic generation in plasmas and found that for a semi-infinite medium, the harmonic signal is 1/15 smaller than the1-D result. In practice, this can be arranged by focusing thepulse onto either the leading or trailing edge of the medium(e.g., gas jet or fill).

Another mechanism which we have not yet considered isstimulated backscattered harmonic (SBH) generation [31],which is basically an extension of the nonlinear Ramanbackscatter (RBS) instability [32]. The latter is a three-waveprocess in which the pump wave decays into astationaryplasma wave plus a backward EM wave :

, where .The plasma wave therefore acts like a mirror grating, whichat high pump strengths superimposes the harmonic contentin the nonlinear density perturbation onto the reflected wave.Unlike the forward harmonics, the SBH radiation grows fromthe foot of the laser pulse toward the rear, up to a point wherethe plasma wave becomes saturated by particle trapping. Inpractice, therefore, this radiation will be very sensitive to thebackground plasma temperature. For this reason, it may beeasier to detect using electron beams instead [31].

Experimentally there is still—as with free elec-trons—woefully little data to compare these results with,largely because of the difficulty in excluding the harmonics

(a)

(b)

Fig. 2. (a) Harmonic spectrum generated by a short pulse with amplitudea0 = 0.2 propagating through an underdense plasma with!0=!p = 5. (b)Time-evolution of third (solid), fifth (dashed), and seventh (dotted) harmonics.

generated by bound electrons [5]. Even at focused intensitiesof 10 W/cm , multiphoton effects can occur near theionization threshold in the foot, tail, and wings of thepulse, which is typically 10 –10 W/cm dependingon the gas used. In an experiment designed to cover thetransition between atomic and ionized media, Liuet al.[33] determined an upper bound for the efficiency of thirdharmonic generation in a plasma jet of 10 foran intensity of 10 W/cm m and plasma density of10 cm , which is consistent with theory after taking intoaccount phase-matching and 2-D effects. This value, whichwas basically detection-limited, was arrived at by using adouble-pulse arrangement in which the first pulse ionized aH gas jet, and the harmonic signal was measured from a theinteraction of a delayed second pulse with the plasma. Evenif plasma harmonics could be observed this way, however,accurate interpretation of a pump-probe arrangement wouldbe complicated by radial plasma motion due to ponderomotiveexpulsion and thermal expansion.

IV. OVERDENSE PLASMAS

From the previous sections, one might conclude that har-monic generation from free electrons is an inherently ineffi-cient process compared to other means of frequency up-shift.The two-fold limitation indicated by (23) and (28) due tothe intensity saturation and the restriction thatcannot easily be overcome in a tenous medium. However,laser interaction withoverdenseplasmas also offers a means offrequency conversion. This type of interaction—in which the


Fig. 3. Schematic picture of harmonic generation via linear mode-coupling in a plasma density gradient.

light is primarily reflected near the critical surface —hasbeen studied for 25 years in the context of laser fusion.It has been known for some time that low-order harmonicscan be generated in such plasmas via resonance absorption,parametric instabilites [34], and transverse density gradients[35]. Integer and noninteger harmonics generated in laser-ablated plasmas have been extensively studied because theycan yield useful diagnostic information on the movement ofthe plasma near the critical surface.

Resonance absorption is of particular significance herebecause it provides an efficient means of converting an elec-tromagnetic wave into a localized electrostatic mode

[36]. At the critical surface , and these twowaves can mix to produce a second harmonic via the current

. Thus . This wave is mainlyreflected, but part of it can propagate up the density profile to

, where it excites a plasma wave at . This in turngenerates a third harmonic, which is resonant at, and soon—see Fig. 3. Note that unlike in the underdense case, we cangenerate both odd and even harmonics in an overdense profile.This simple mode-coupling picture was suggested by a numberof early experiments using long pulse lasers. Burnettet al. [37]found odd and even harmonics up to the 11th using a 2-ns COlaser with intensities up to 10 W/cm . Shortly afterwards,McLeanet al. [38] measured five harmonics with high spectralresolution using a 75-ps Nd:glass system, demonstrating thatthe harmonic efficiencies were consistent with those measuredby Burnettet al. provided one assumed an scaling.

Interest in this phenomenon really took off when Carmanetal. detected up to 46 harmonics in a series of experiments withthe powerful CO laser systems at Los Alamos [39], [40]. Aspart of a detailed analysis of their results, Carmanet al. used2-D PIC simulations to model the laser-plasma interaction ina self-consistently steepened density profile. Assuming thatthe maximum density was determined by pressure balancebetween the laser and dense plasma, i.e.,

W/cm keV(29)

they reasoned that according to the mode-coupling picture, the

harmonic spectrum should exhibit a cutoff given by

(30)

This scaling appeared to be supported by the simulations, sug-gesting that the maximum harmonic could be used to obtain theupper plasma density. Applying this rule to their later results,however, the authors inferred densities of 500–2000, whichaccording to (29) would have implied unusually low plasmatemperatures of a few tens of electronvolts. However, theyargued that intensity amplification by self-focusing [41] couldaccount for the simultaneously high densities and (more typicalkilovolts) temperatures.

The challenge of finding a more quantitative explanation ofthese results was taken up by Bezzerideset al., who formulateda nonrelativistic Lagrangian fluid model incorporating anidealised step-profile [42]. In this case, it is possible to obtainclosed solutions for the oscillation amplitude and accelerationin the overdense region, which after Fourier analysis intoharmonic components and substitution into an expressionsimilar to (12) for incoherent emission gives a lower boundfor the efficiency of the th harmonic

(31)

which is flat up to , after which it rolls off sharply.This problem was analyzed independently by Grebogiet al.

[43], who used a similar fluid model but took into accountmore realistic density profiles and wavebreaking. The latterprocess (signalled by crossing of fluid elements) restricts theintegration time to , making it difficult to analyze thelong-time behavior of the plasma oscillations. Nevertheless,they were able to determine the power spectrum numericallyand also found a cutoff corresponding to the upper shelfdensity.

With the replacement of large COlasers by Nd:glassand KrF systems for the fusion programmes in the 1980’s,interest in harmonic generation from high-density plasmas laymore or less dormant until the development of short pulsesystems capable of delivering intensities matching theachieved in the early experiments. In a logical extension ofthe relativistic mechanism in underdense plasma, Wilkset


Fig. 4. Lorentz boost technique to reduce 2-D oblique-incidence system (L-frame) to 1-D system in which the EM wave is normally incident (S-frame).

al. analyzed the harmonics generated by the forceexperienced by electrons in a density step-profile [44]. Asin underdense plasmas, this mechanism produces only oddharmonics at normal incidence

(32)

Assuming this current source is confined to a collisionless skindepth, one obtains a power scaling of

(33)

This relation is in good agreement with PIC simulations per-formed by the same authors, which also show that higher orderharmonics are generated for irradiances 10 –10W/cm m .

Unlike the mode-coupling mechanism, the effectshares the property with the single-particle model of Section IIthat there is no cutoff at the upper plasma density. For short-pulse interactions in which there is little or no time for profilesteepening, the cutoff rule (30) predicts that the minimumattainable wavelength,

(34)

which is independentof the laser wavelength and intensity. Bycontrast, the factor in the mechanism ensures that the

harmonic source terms can always be generated fromthe th harmonic. On the other hand, Wilkset al. went onto demonstrate that not only do the lower harmonics start tosaturate, but other effects can ultimately reduce the efficiency.Hole boring [45] due to the radial pressure differential from afinite focal spot size creates a dimpled surface which permitsgeneration of both even and odd harmonics, but also leads toenhanced absorption via fast electrons driven directly by thelaser field on the sides of the hole [46].

Although the mechanism appears to be highly promis-ing as a short-wavelength source, exhibiting good scalingproperties without the cutoff, it is nonetheless inherently lessefficient than harmonics driven by a-polarized laser fieldwith a component parallel to the density gradient. Again, weappeal to the single electron figure-8 orbit to appreciate this:depending on the angle of incidence, transverse excursionsare typically four times larger than the longitudinal ones,regardless of . Hence we expect the radiated power, which

is proportional to , to be an order of magnitude larger for-polarized light. The problem is that the linear mode-coupling

picture of Fig. 3 breaks down at high intensity in very steepgradients. The reason for this is that the oscillation amplitude

becomes comparable to the density scalelength, namely

(35)

which is typically satisfied for pump intensities above 10W/cm . It is evident that nonlinear oscillations still occur,but wavebreaking now takes place after only half a lasercycle [47], and it is not clear how to replace the appealingmode-coupling description.

One possible direction was indicated by Bulanovet al. [48],who interpreted PIC simulations of low-order harmonics gen-erated by obliquely incident pulses in terms of an acceleratingcharge sheet. These simulations were performed using a boosttechnique to reduce the standard 2-D periodic system normallyrequired for oblique-incidence modeling, to a 1-D system inwhich the wavevector of the laser is normally incident (Fig.4). This trick was first pointed out by Bourdier [49] in ananalytical treatment of resonance absorption and later appliedto PIC simulation by Gibbon and Bell [50] to study absorptionin steep density profiles.

In the context of harmonic generation, the technique al-lows order-of-magnitude improvements in spatial and temporalresolution over traditional 2-D codes, and thus enables thehigh harmonics to be retained. This resolution deficiencyin earlier simulations was acknowledged by Carmanet al.[39], who were restricted to 10 harmonics at that time.In a reexamination of this problem, Gibbon [51] was ableto exploit the spectral advantage of the boost technique byshowing that large numbers of harmonics could in principlebe generated this way, with no cutoff at the upper plasmadensity. As Fig. 5 shows, the harmonic efficiencies for p-polarized light are 1–2 orders of magnitude higher than fornormally incident or -polarized light. In the latter case, onlythe relativistic mechanism is present, whereas density-bunching tends to dominate for-polarized light. To see thismore clearly, it is helpful to examine the current sources forthe first two harmonics. Denoting the velocity componentsnormal and perpendicular to the laser electric field asand

, respectively, and expanding the density in a Fourier series, we have for the two lowest order


Fig. 5. Simulated power spectra for a 1019 W/cm�2 � �m2 pulse in-cident on a plasma with densityNe=Nc = 10 for oblique p-polarizedlight (solid), normally incident light (dashed), and obliques-polarized light(squares—even; circles—odd) The spectra are normalized to the totalreflectedpower P0 =

mPm.

harmonic sources

(36)

Symmetry arguments dictate that for a perfectly flat target,specularly reflected harmonics will be driven by the compo-nent of the current source parallel to the surface. At normalincidence, we have , and we only getodd harmonics. For -polarized light, the term vanishesbecause the electric field is always perpendicular to the densitygradient, but we can still generate even harmonics at obliqueincidence due to the component of parallel to the gradient.

The harmonic efficiencies for-polarized light at 45 inci-dence can be summarized by an empirical relation valid forhigh orders :

W/cm m(37)

This scaling means that the highest harmonic order (downto the detection threshold) is simply determined by ,although we also expect some weak dependence on densityand incidence angle. Thus, contrary to (34), a short-wavelengthpump beam will generate shorter wavelength harmonics forthe same irradiance. Indeed, the simulations in [51] explicitlypredict that a KrF pump with intensity of 1.6 10 W/cmwill generate harmonics with efficiencies into the“water-window” of 2.3–4.4 nm (which is reached by the 56th).

More detailed analyses of this problem have recently beenmade by Lichterset al. [52] and by von der Linde andRzazewski [53], who developed “moving mirror” models ofharmonic generation for arbitrary angles of incidence andpolarization. From this model, a rigorous set of selection rulescan be derived which generalize the heuristic observationsexpressed by (36). For example, whereas a-polarized pumpproduces harmonics which are all-polarized, for -light theodd and even harmonics areand -polarized, respectively.Employing a simple periodic ansatz to represent the surfaceoscillations, the authors proceed to compute the harmonicspectrum by taking into account the retarded time in ananalogous fashion to [2]. For small angles of incidence,

Fig. 6. High-order harmonic spectrum measured from interaction of a 2.5-psNd:glass laser with a plastic target at 1019 W/cm�2 ��m2. (reproduced from[53].)

Lichters et al. find excellent agreement with PIC simulationsafter adjusting only one parameter, namely the maximumoscillation amplitude. For large angles, agreement is not asgood, presumably because kinetic effects start to becomesignificant and the fluid model again breaks down.

A. Short-Pulse Experiments

As mentioned earlier, the technological strides in short pulselasers over the last decade have rekindled experimental interestin the phenomenon unveiled by Carmanet al. [40]. In the firstof a series of experiments in the past year, Kohlweyeret al.sucessfully demonstrated “high” harmonic generation with a100-fs Ti-Sa (794 nm) system in Lund, Sweden [54]. Usingfocused laser intensities up to 10W/cm , up to sevenharmonics were unequivocally observed on various targets.In a very similar experiment, von der Lindeet al. observedup to 15 harmonics from the Ti-Sa system at LOA, France[55]. They estimated the efficiency of the highest harmonicto be 10 , which is about 50 times lower than the valuegiven by (37).

In both of these experiments, the highest harmonic observedis actually consistent with the linear cutoff model, assum-ing densities of 2–4 10 cm for aluminum targets.However, this model was severely stretched by a spectacularexperiment with the Vulcan Nd:glass CPA system at RAL,U.K. In this experiment, the laser delivered moderate contrast-ratio pulses of 2.5-ps duration, focused to 10W/cm ,creating plasma conditions probably not too different to thosein the Los Alamos experiments, albeit at 100 times higherdensity. In the event, Norreyset al. [56] found up to 68harmonics in first-order diffracted signal (75 in second order),with efficiencies exceeding 10, implying power conversionsof (MW) into spectral lines from 14 to 15 nm (see Fig. 6).According to the cutoff theory [39], [42], the highest harmonicobserved implies a barely credible electron density of 5600,or 17 times solid density for the CH-coated glass targets used.

A number of features in the measured spectra were ingood agreement with the (1-D) PIC simulations of Gibbon[51], particularly the efficiency scaling with intensity andharmonic order. This is somewhat surprising given that theinteraction was probably dominated by 2-D effects such assurface rippling and hole-boring, a conclusion arrived at bothfrom the lack of difference between signals measured with-and -polarized light and from the almost isotropic harmonic


emission into steradian. In particular, Norreyset al. es-timate the power conversion into the 38th harmonic at 27.7nm to be 24 MW, corresponding to an absolute efficiency of3 10 , which is about a factor of three lower than thesimulation result for the same intensity.

The coherence properties of harmonics generated fromlaser-solid interaction were studied in a follow-up paper onthe same experiment by Zhanget al. [57]. They estimate asource size of 12 m and an intensity-dependent bandwidth of0.02–0.4 ps, both of which are competitive with other coherentXUV sources such as X-ray lasers. The broadening of theharmonic lines at high intensity is attributed to relativisticself-phase modulation in the underdense plasma in front ofthe target, suggesting that the bandwidth could be improvedby using pulses with sufficiently high contrast to avoid pre-plasma formation. On the other hand, as Sauerbrey has recentlypointed out [58], going to shorter and cleaner pulses mayintroduce Doppler broadening of reflected harmonics as aresult of (hydrodynamic) acceleration of the critical surface.

V. CONCLUSION

An overview has been given of the various methods ofgenerating harmonics from plasmas. Common to nearly allthese mechanisms is the relativistic “figure-8” motion of freeelectrons in an electromagnetic plane wave, the physics ofwhich was already laid down some 30 years ago. The initialexpectation that the nonlinear Thomson scattering analyzedby Sarachik and Schappert [2] could be carried over fromsingle electrons into an efficient, coherent version in plasmaswas dispelled by a series of recent papers showing that therelativistic nonlinearities are essentially cancelled by the col-lective plasma response. Typical efficiencies for the low-orderharmonics prove to be , although self-consistent1-D simulations of this process indicate that the harmonicspectra may exhibit a similar “plateau-like” structure to themultiphoton mechanism in gases. Exploitation of these effectshas now begun in earnest with the availability of short-pulse lasers capable of delivering intensities in excess of 10W/cm . Experimental detection of harmonics generated inunderdense plasmas must overcome the double hurdle of lowefficiency and exclusion of gas-generated harmonics.

A more promising development is the resurgence of inter-est in harmonics reflected from overdense plasma surfaces.Theoretical and numerical studies over the past year haveshown that the “cutoff” inferred previouslyby Carmanet al. [39], disappears at relativistic irradiances.Effiencies are also high, , with an approximatelyquadratic intensity-dependence, features which have also beenverified experimentally. In the near future, we should see manymore experiments with ultrashort pulse lengths (10–100 fs)and higher intensities. The challenge will be to generate high-order ( 50–100) harmonics with good spatial and temporalcoherence properties. Whether this can be done with relativelylow contrast ratios and can be extended to shorter wavelengths(e.g., 1/4 m) remains to be seen, but the plasma-surfacesource is set to become a serious alternative to gas-generatedXUV harmonic radiation and X-ray lasers.

ACKNOWLEDGMENT

The author is grateful to T. Auguste, P. Monot, and J.-M. Rax for introducing him to the subject of harmonicgeneration. Special thanks are due to P. Norreys and M. Zepffor permission to reproduce the spectrum in Fig. 6 from [56].

REFERENCES

[1] L. S. Brown and T. W. B. Kibble, “Interaction of intense laser beamswith electrons,”Phys. Rev., vol. 133, pp. A705–A719, 1964.

[2] E. S. Sarachik and G. T. Schappert, “Classical theory of the scatteringof intense laser radiation by free electrons,”Phys. Rev. D, vol. 1, pp.2738–2753, 1970.

[3] J. P. Ostriker and J. E. Gunn, “On the nature of pulsars. I. Theory,”Astrophys. J., vol. 157, pp. 1395–1417, 1968.

[4] A. I. Akhiezer and R. V. Polovin, “Theory of wave motion of an electronplasma,”Sov. Phys. JETP, vol. 3, pp. 696–705, 1956.

[5] A. L’Huillier, T. Auguste, Ph. Balcou, B. Carre, P. Monot, P. Salieres,C. Altucii, M. B. Gaarde, J. Larsson, E. Mevel, T. Starczewski, S.Svanberg, C. G. Wahlstrom, R. Zerne, K. S. Budil, T. Ditmire, andM. D. Perry, “High-order harmonics—A coherent source in the XUVrange,”J. Nonlin. Opt. Phys. Mater., vol. 4, pp. 647–665, 1995.

[6] W. B. Mori, “Generation of tunable radiation using an underdenseionization front,” Phys. Rev. A, vol. 44, pp. 5118–5121, 1991.

[7] S. C. Wilks, J. M. Dawson, W. B. Mori, T. Katsouleas, and M. E. Jones,“Photon accelerator,”Phys. Rev. Lett., vol. 62, pp. 2600–2603, 1989.

[8] D. L. Matthews and M. D. Rosen, “Soft-x-ray lasers,”Scientific Amer-ican, pp. 86–91, Dec. 1988.

[9] W. B. Mori, “Guest editorial: Special issue on the generation of coherentradiation using plasmas,”IEEE Trans. Plasma Sci., vol. 21, pp. 1–4,1993.

[10] P. Gibbon and E. F¨orster, “Short pulse laser-plasma interactions,”Plasma Phys. Control Fusion, vol. 38, pp. 769–793, 1996.

[11] L. D. Landau and L. M. Lifshitz,Classical Theory of Fields, 2nd ed.Reading, MA: Addison-Wesley, 1962.

[12] J. E. Gunn and J. P. Ostriker, “On the motion and radiation of chargedparticles in strong electromagnetic waves. I. Motion in plane andspherical waves,”Astrophys. J., vol. 165, pp. 523–541, 1971.

[13] J. H. Eberly and A. Sleeper, “Trajectory and mass shift of a classicalelectron in a radiation pulse,”Phys. Rev., vol. 176, pp. 1570–1573, 1968.

[14] J. D. Jackson,Classical Electrodynamics, 2nd ed. New York: Wiley,1975.

[15] C. I. Castillo-Herrera and T. W. Johnston, “Incoherent harmonic emis-sion from strong electromagnetic waves in plasmas,”IEEE Trans.Plasma Sci., vol. 21, pp. 125–135, 1993.

[16] J. N. Bardsley, B. M. Penetrante, and M. H. Mittleman, “Relativisticdynamics of electrons in intense laser fields,”Phys. Rev. A, vol. 40, pp.3823–3835, 1989.

[17] U. Mohideen, H. W. K. Tom, R. R. Freeman, J. Bokor, and P. H.Bucksbaum, “Interaction of free electrons with an intense focused laserpulse in Gaussian and conical axicon geometries,”J. Opt. Soc. Amer.B, vol. 9, pp. 2190–2195, 1992.

[18] T. J. Englert and E. A. Rinehart, “Second-harmonic photons from theinteraction of free electrons with intense laser radiation,”Phys. Rev. A,vol. 28, pp. 1539–1545, 1983.

[19] W. Leemans, “Interaction of relativistic electrons with ultrashort laserpulses: Generation of femtosecond X-rays and microprobing of electronbeams,” this issue, pp. 1925–1934.

[20] S. K. Ride, E. Esarey, and M. Baine, “Thomson scattering of intenselasers from electron-beams at arbitrary interaction angles,”Phys. Rev.E, vol. 52, pp. 5425–5442, 1995.

[21] P. Sprangle, E. Esarey, and A. Ting, “Nonlinear interaction of intenselaser pulses in plasmas,”Phys. Rev. A, vol. 41, pp. 4463–4469, 1990.

[22] W. B. Mori, “Overview of laboratory plasma radiation sources,”PhysicaScripta, vol. T52, pp. 28–35, 1994.

[23] D. Montgomery and P. Tidman, “Secular and nonsecular behavior forthe cold plasma equations,”Phys. Fluids, vol. 7, p. 242, 1964.

[24] J. M. Rax and N. J. Fisch, “Third harmonic generation with ultrahigh-intensity laser pulses,”Phys. Rev. Lett., vol. 69, pp. 772–775, 1992.

[25] W. B. Mori, C. D. Decker, and W. P. Leemans, “Relativistic harmoniccontent of nonlinear electromagnetic waves in underdense plasmas,”IEEE Trans. Plasma Sci., vol. 21, pp. 110–119, 1993.

[26] E. Esarey, A. Ting, P. Sprangle, D. Umstadter, and X. Liu, “Nonlinearanalysis of relativistic harmonic generation by intense lasers in plasmas,”IEEE Trans. Plasma Sci., vol. 21, pp. 95–104, 1993.


[27] J. F. Ward and G. H. C. New, “Optical third harmonic generationin gases by a focused laser beam,”Phys. Rev., vol. 185, pp. 57–72,1969.

[28] J. M. Rax and N. J. Fisch, “Phase-matched third harmonic generationin a plasma,”IEEE Trans. Plasma Sci., vol. 21, pp. 105–109, 1993.

[29] A. L’Huillier and Ph. Balcou, “High-order harmonic generation in raregases with a 1-ps 1053-nm laser,”Phys. Rev. Lett., vol. 70, pp. 774–777,1993.

[30] G. Zeng, B. Shen, W. Yu, and Z. Xu, “Relativistic harmonic generationexcited in the ultrashort laser pulse regime,”Phys. Plasmas, vol. 3, pp.4220–4224, 1996.

[31] P. Sprangle and E. Esarey, “Stimulated backscattered harmonic gener-ation from intense laser interactions with beams and plasmas,”Phys.Rev. Lett., vol. 67, pp. 2021–2024, 1991.

[32] A. S. Sakharov and V. I. Kirsanov, “Theory of raman scattering for ashort ultrastrong laser pulse in a rarefied plasma,”Phys. Rev. E, vol. 49,pp. 3274–3282, 1994.

[33] X. Liu, D. Umstadter, E. Esarey, and A. Ting, “Harmonic generation byan intense laser pulse in neutral and ionized gases,”IEEE Trans. Plas.Sci., vol. 21, pp. 90–93, 1993.

[34] J.-L. Bobin, “High-intensity laser plasma interaction,”Phys. Rep., vol.122, pp. 173–274, 1985.

[35] J. A. Stamper, R. H. Lehmberg, A. Schmitt, M. J. Herbst, F. C. Young,J. H. Gardner, and S. P. Obenshain, “Evidence in the second-harmonicemission for self-focusing of a laser pulse in a plasma,”Phys. Fluids,vol. 28, pp. 2563–2569, 1985.

[36] V. L. Ginzburg,The Propagation of Electromagnetic Waves in Plasmas.New York: Pergamon, 1964.

[37] N. H. Burnett, H. A. Baldis, M. C. Richardson, and G. D. Enright,“Harmonic generation in CO2 laser target interaction,”Appl. Phys. Lett.,vol. 31, pp. 172–174, 1977.

[38] E. A. McLean, J. A. Stamper, B. H. Ripin, H. R. Griem, J. M. McMahon,and S. E. Bodner, “Harmonic generation in Nd:glass laser-producedplasmas,”Appl. Phys. Lett., vol. 31, pp. 825–827, 1978.

[39] R. L. Carman, D. W. Forslund, and J. M. Kindel, “Visible harmonicemission as a way of measuring profile steepening,”Phys. Rev. Lett.,vol. 46, pp. 29–32, 1981.

[40] R. L. Carman, C. K. Rhodes, and R. F. Benjamin, “Observation ofharmonics in the visible and ultraviolet created in CO2-laser-producedplasmas,”Phys. Rev. A, vol. 24, pp. 2649–2663, 1981.

[41] E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Self-focusing and guidingof short laser pulses in ionizing gases and plasmas,” this issue, pp.1879–1914.

[42] B. Bezzerides, R. D. Jones, and D. W. Forslund, “Plasma mechanismfor ultraviolet harmonic radiation due to intense CO2 light,” Phys. Rev.Lett., vol. 49, pp. 202–205, 1982.

[43] C. Grebogi, V. K. Tripathi, and H.-H. Chen, “Harmonic generationof radiation in a steep density profile,”Phys. Fluids, vol. 26, pp.1904–1908, 1983.

[44] S. C. Wilks, W. L. Kruer, and W. B. Mori, “Odd harmonic generationof ultra-intense laser pulses reflected from an overdense plasma,”IEEETrans. Plasma Sci., vol. 21, pp. 120–124, 1993.

[45] S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon, “Absorptionof ultra-intense laser pulses,”Phys. Rev. Lett., vol. 69, pp. 1383–1386,1992.

[46] S. C. Wilks and W. L. Kruer, “Absorption of ultrashort, ultra-intenselaser light by solids and overdense plasmas,” this issue, pp. 1954–1968.

[47] F. Brunel, “Not-so-resonant, resonant absorption,”Phys. Rev. Lett., vol.59, pp. 52–55, 1987.

[48] S. V. Bulanov, N. M. Naumova, and F. Pegoraro, “Interaction of anultrashort, relativistically strong laser pulse with an overdense plasma,”Phys. Plasmas, vol. 1, pp. 745–757, 1994.

[49] A. Bourdier, “Oblique incidence of a strong electromagnetic wave on acold inhomogeneous electron plasma: Relativistic effects,”Phys. Fluids,vol. 26, pp. 1804–1807, 1983.

[50] P. Gibbon and A. R. Bell, “Collisionless absorption in sharp-edgedplasmas,”Phys. Rev. Lett., vol. 68, pp. 1535–1538, 1992.

[51] P. Gibbon, “Harmonic generation by femtosecond laser-solid interaction:A coherent water-window light source?”Phys. Rev. Lett., vol. 76, pp.50–53, 1996.

[52] R. Lichters, J. Meyer-ter-Vehn, and A. Pukhov, “Short-pulse laser har-monics from oscillating plasma surfaces driven at relativistic intensity,”Phys. Plasmas, vol. 3, pp. 3425–3437, 1996.

[53] D. von der Linde and K. Rzazewski, “High-order optical harmonicgeneration from solid surfaces,”Appl. Phys. B, vol. 63, pp. 499–506,1996.

[54] S. Kohlweyer, G. D. Tsakiris, C.-G. Wahlstrom, C. Tillman, and I.Mercer, “Harmonic generation from solid-vacuum interface irradiatedat high laser intensities,”Opt. Commun., vol. 117, pp. 431–438, 1995.

[55] D. von der Linde, T. Engers, G. Jenke, P. Agostini, G. Grillon, E.Nibbering, A. Mysyrowicz, and A. Antonetti, “Generation of high-orderharmonics from solid surfaces by intense femtosecond laser pulses,”Phys. Rev. A, vol. 52, pp. R25–R27, 1995.

[56] P. A. Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, M.Bakarezos, C. N. Danson, A. Dyson, P. Gibbon, P. Loukakos, D. Neely,F. N. Walsh, J. S. Wark, and A. E. Dangor, “Efficient XUV harmonicsgenerated from picosecond laser pulse interactions with solid targets,”Phys. Rev. Lett., vol. 76, pp. 1832–1835, 1996.

[57] J. Zhang, M. Zepf, P. A. Norreys, A. E. Dangor, M. Bakarezos, C. N.Danson, A. Dyson, A. P. Fews, P. Gibbon, P. Lee, P. Loukakos, S.Moustaizis, D. Neely, F. N. Walsh, and J. S. Wark, “Coherence andbandwidth measurements of harmonics generated from solid surfacesirradiated by intense picosecond laser pulses,”Phys. Rev. A, vol. 54, pp.1597–1603, 1996.

[58] R. Sauerbrey, “Acceleration in femtosecond laser-produced plasmas,”Phys. Plasmas, vol. 3, pp. 4712–4716, 1996.

Paul Gibbon was born in Port of Spain, Trinidad& Tobago, in 1964. He received the B.Sc. degreein physics from Bristol University, U.K., in 1985and the Ph.D. and D.I.C. degrees from ImperialCollege, London, U.K., in 1988, specializing inplasma physics.

Since then, he has held post-doctoral positionsin France and Germany, working on a variety ofproblems in the field of short-pulse laser-matterinteraction. He is currently based at the Universityof Jena, Jena, Germany, in the Institute for Optics

and Quantum Electronics. His research interests also include hard X-raygeneration using short-pulse lasers, plasma-based particle acceleration, andnovel plasma simulation techniques.

High-Order Harmonic Generation in Plasmas

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