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Strong terahertz emission from electromagnetic diffusion near cutoff in plasma

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2015 New J. Phys. 17 043045

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New J. Phys. 17 (2015) 043045 doi:10.1088/1367-2630/17/4/043045

PAPER

Strong terahertz emission from electromagnetic diffusion near cutoffin plasma

M-HCho1,5, Y-KKim2, H Suk3,7, B Ersfeld4, DA Jaroszynski4,7 andMSHur1,6

1 School of Natural Science, UNIST,Ulsan, 689-798, Korea2 School of Electrical andComputer Engineering, UNIST,Ulsan, 689-798, Korea3 Department of Physics and Photon Science, GIST, Gwangju, 500-712, Korea4 Department of Physics, ScottishUniversities Physics Alliance andUniversity of Strathclyde, GlasgowG4 0NG,UK5 Center for Relativistic Laser Science, Institute for Basic Science (IBS), Gwangju 500-712, Korea6 Author towhomany correspondence should be addressed.7 Co-corresponding authors.

E-mail:mshur@unist.ac.kr, D.A.Jaroszynski@strath.ac.uk andhysuk@gist.ac.kr

Keywords: laser-plasma, terahertz radiation, laser wakefield

AbstractAnewmechanism for electromagnetic emission in the terahertz (THz) frequency regime from laser-plasma interactions is described. A localized and long-lasting transverse current is produced by twocounter-propagating short laser pulses inweaklymagnetized plasma.We show that the electro-magnetic wave radiating from this current source, even though its frequency is close to cut-off of theambient plasma, grows and diffuses towards the plasma-vacuumboundary, emitting a strongmonochromaticTHzwave.With driving laser pulses ofmoderate power, the THzwave has a fieldstrength of tens ofMVm−1, a frequency of a fewTHz and a quasi-continuous power that exceeds allpreviousmonochromaticTHz sources. The novelty of themechanism lies in a diffusing electro-magnetic wave close to cut-off, which ismodelled by a continuously driven complex diffusionequation.

1. Introduction

Far-infrared light has been expected to be uniquely advantageous for probing and imaging the structure anddynamics ofmatter comparedwith other light sources. However, the characteristics of the radiation sources inthat frequency regime have not been satisfactory with respect to various requirements. Enormous efforts arenowbeingmade tofill the perceived terahertz (THz) gapwith the development of high power THz sourcesbecause of their potential high-impact applications in science and technology. Diverse sources have beendeveloped using both electronic and opticalmethods, for example the laser-plasma-based schemes [1–20],which promise compact THz sources with notably highfield amplitude over awide range of frequencies, andconventional beamdriven vacuum sources such as gyrotrons, backwardwave oscillators etc.

Linearmode conversion and two-colour schemes are known to yield short-duration andwide-bandTHzpulses with remarkably high amplitude. The latter is especially attractive because a high THz amplitude up to2 GVm−1 can be expected [21]. In this scheme, a slow component of a strong photocurrent is efficiently yieldedby phase-controlled photoionization at each laserfield extremum [22, 23], which is essential in generating abroad-bandTHz pulse.

While the peak power of thesewide-bandTHz sources is increasing rapidly, the average power of availablenarrow-bandTHz sources remains very low. AmonochromaticTHz source based on laser-plasma interactionsandwakefields is feasible [6]. By converting the longitudinal current of thewakefield to a transverse one using anexternalmagnetic field, this Cherenkovwake scheme extends the pulse duration and increases the emissionpower [18–20].However, the field amplitude or power from such amonochromatic THz source is still lowcomparedwithwide-band schemes. Considering the numerous important applications such as high-contrast

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imaging [24–26] or interferometry of Tokamak plasmas [27], it is necessary to provide suitablemonochromaticsources that have high tunability, compactness, and above all, high power.

In this paper, we propose a newmechanism of generating high-intensitymonochromaticTHz radiation fromlaser-plasma interactions. The fundamental idea is to evoke a strong, localized, long-lasting electron oscillationin plasma, which acts as a radiating antenna emitting a continuous THzwave. In practice, the radiation currentsource can be generated by the strong ponderomotive force from two short laser pulses colliding at a desiredposition inside the plasma [28]. The current generated in this way ismaintained for a long time by the plasmaoscillationmechanism even after the passage of the driving laser pulses. The longitudinal current induced in thisway oscillates electrostatically. To obtain electromagnetic radiation, i.e. the THzwave, some fraction of thelongitudinal electron oscillation is converted to a transverse one by aweak transverse externalmagnetic field.Because the radiation from this current source oscillates at the plasma frequency, the electromagnetic fields areat the cut-off of the ambient plasma and cannot propagate under normal conditions, which is indeed the veryreasonwhy plasma oscillations cannot be easily converted to transverse electromagnetic waves in laser-plasmasystems.However, we have discovered that if the electromagnetic wave is constantly driven near cut-off, as inour case, the field grows as its energy is fed by the driving current. Furthermore, both electric andmagneticcomponents of thefield diffuse strongly into the plasma. As a consequence, the diffusing electromagnetic fieldeventually propagates across the plasma-vacuumboundary and is converted to an electromagnetic wave emittedinto free space. This concept of THz generation is shown schematically infigure 1. Our scheme shares acommon featurewith theCherenkovwake driven by a single pulse in using the externalmagnetic field to converta longitudinal oscillation to a transverse one.However, the stronger, local electron oscillation by collidingpulses, and the diffusion-growthmechanism are two unique features of our scheme.

As the plasma frequency can be easily controlled in the few-THz regime, thismechanism can be utilized as anewTHz radiation source. Aswewill show later through numerical simulations, the resultant amplitude of theTHz emission reaches tens ofMVm−1 and has a sub-nanosecond pulse duration. Such high amplitudes in theregime of narrow-bandTHz pulses with long pulse duration are not readily obtainedwith conventionalmethods in compact systems. Note that fromwide-band THz sources, such as the two-colour scheme, shortTHz pulses with peak amplitudes higher by two orders ofmagnitude are available [21].

This paper is organised as follows. In section 2, we describe the diffusion and growth of the electromagneticfield near cutoff theoretically and using simulations. In section 3, a theoretical scaling law for the THz emissionamplitude as a function of the amplitude of the driving pulse is derived. Then, in section 4we give a summaryand conclusions.

2. Electromagnetic diffusion and growth near cutoff

The diffusion of the electromagnetic field, which is a key factor for enhanced THz emission in our scheme, isapparent in thefield evolutionmodelled by a constantly driven complex diffusion equation. Such diffusion ofthe electromagnetic wave is quite different from thewell-knownmagnetic field diffusion into conductingmaterial, where just the slowly varyingmagnetic component diffuses while the electric component remainsnegligibly small. To describe electromagnetic diffusion near cut-off, we start from thewave equation:

ωμ

∂

∂−

∂

∂= +

∂∂

E

x c

E

t cE

J

t

1. (1)

y y py

y2

2 2

2

2

2

2 0

Similar equations describing thefield evolution in the two-colour scheme can be found in [29]. Equation (1)differs from those in so far as, firstly, Jy originates from the ponderomotive force of the laser pulses rather than

Figure 1. Schematic of terahertz emission from a current source generated by two counter-propagating laser pulses.

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New J. Phys. 17 (2015) 043045 M-HCho et al

fromphotoionization, and secondly, the role of the laser pulses in our scheme is limited to an initial sourcecurrent Jy. Thus Ey in equation (1) includes just the THzfieldwithout the laser field. Note that the first termonthe right-hand side is the self-current induced byEy itself, while the second term is a constant driving current.Assuming that the electric field amplitude evolves slowly, the electricfieldEy can bewritten as ω−Ee ti , where E isa slowly evolving function of time. Since ω ω≃ p, i.e.Ey is nearly at cut-off, the first termon the right-hand sidecancels out the second time derivative of ω−e ti on the left-hand side. Neglecting the second time derivative of Ein accordancewith the slowly varying assumption, we obtain a constantly driven diffusion-like equation of theelectric field:

ωω μ∂

∂+ ∂

∂= −E

x c

E

tJ

ˆ 2i ˆi , (2)

pp

2

2 2 0 0

where J0 is the current amplitude defined by = ω−J J eyt

0i p . The transverse current Jy is induced by the external

magnetic field (B0) from the longitudinal current Jx, which is driven by the beat of two counter-propagating laserpulses. Because the pulse duration is very short, a few tens of femtoseconds, the longitudinal current Jx isgenerated almost instantaneously at themoment of pulse collision. Once Jx is formed, it continues to oscillate bythe regular plasma oscillationmechanism. Since the conversion rate from Jx to Jy is very small forω ω= ≪eB mc p0 , Jx acts as feeding Jy constantly. Thus J0 can be considered temporally quasi-constant, leading

to ω∂ ≃ − ω−J Ji et y pt

0i p . TheTHz pulse generated by such a constantly oscillating current is expected to have a

long pulse duration, as will be confirmed later. Note that equation (2) takes the familiar formof a drivendiffusion equation, except that the diffusion coefficient, which is ω− ci 2 p

2 , is imaginary.By applying Laplace and inverse Laplace transforms to equation (2), the solution can be represented in

integral form as follows:

∫ ∫ω π= + ′ σ

γ

γ η

−∞

∞−

− ∞

+ ∞ +eE x t

mcx j s

s

ˆ ( , ) i 1

8d e d

e, (3)

p

yy st

0¯

i

i i ¯

3 2J

2 2

where the variables are normalized as ω= ∣ − ′∣y x x cp , σ ω σ= c¯J p J , η = s2i , ω=t t¯ p , and =j J en c0 0 0 .In deriving equation (3)we have assumed that the transverse current amplitude J0 takes on a very narrow

Gaussian form, = σ−J J e x0 0

J2 2

. Here σJ is the spatial length of the current source, which corresponds to half thelaser pulse duration.

Approximate solutions of equation (3) can be obtained in certain limits. First, for ≫y t 1, i.e. far from thecurrent source and temporally in the early stage, applying the steepest descentmethod leads to

ωσ π≃ − +eE x t

mc

j t

x

x

t

ˆ ( , ) ¯

2

¯

¯cos

¯

2¯ 4, (4)

p

J 03 2

2

2⎛⎝⎜

⎞⎠⎟

with ω=x x c¯ p . Second, for x=0, i.e. at the oscillation centre of the current, by applyingWatson’s lemma to thefirst order termwe obtain an asymptotic form as a function of t as follows:

ωσ

≃ −eE t

mc

jt

ˆ (0, ) ¯

2(i 1) . (5)

p

J 0

Figure 2(a) are snapshots of the diffusing fields at different times, obtained fromnumerical integration ofequation (2). As thefield diffuses, itsmodulation length shrinkswith increasing distance forfixed time, whilegrowing at afixed position as can be predicted from the cosine term in equation (4). This temporal behaviour ispresented infigure 2(b), where the result of numerical integration of equation (2) is plotted at ω =x c 0p

(current centre), 5, and 10. The central field increasesmonotonically by t as expected from equation (5), while

Figure 2. (a) Spatial profile of the diffusing field at different times and (b) its temporal growth at different positions, where ω0 is thelaser angular frequency.

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New J. Phys. 17 (2015) 043045 M-HCho et al

the off-central field initially follows t 3 2 obeying equation (4), but eventually reduces to t as the central peakedregion of thefield expands.

The simultaneous diffusion and growth of the field can be utilized as amethod to convert the plasmaoscillation into an electromagnetic wave in free space. As the growing field eventually hits the plasma-vacuumboundary through diffusion, radiationwill be emitted into free spacewith temporally increasing amplitude.Indeed, this field growth driven by the diffusionmechanism is themajor advantage of our scheme in producingstrong THz emission. To confirm this scenario of THz radiation, we have performed one- and two-dimensionalparticle-in-cell (PIC) simulations. A trapezoidal shape has been chosen for the plasma density profile, with twodifferent densities in the flat region, ×1.25 1018 and × −5 10 cm18 3 (10 and 20THz, respectively). To reduce anymismatch of radiation impedance, a density ramp-up over 100 μm was added to the flat plasma. An additionalimportant effect of the density gradient is that thefield growth indicated by equation (4) is sustained for a longertime, eventually leading to stronger THz emission. The two counter-propagating pulses are arranged so that theycollide at 25 or 50 μmfrom the knee of the density gradient. Thewavelength of one of the pulses is 870 nm,which is typical for Ti:sapphire lasers, and the other is detuned so that their beat resonantly drives the plasmaoscillation. In the two-dimensional simulation, the pulses focus at the colliding point with 50 μmspot radius.The normalized vector potential of the pulse is 0.05, for which ∼ × −I 5 10 W cm15 2 and ∼P 0.2 TW.

Figure 3(a) is an image of the THz emission obtained from two-dimensional PIC simulations. The diffusionand growth of thefieldmeasured on the axis,figure 3(b), takes on a very similar shape to the theoreticalmodel.To observe the long-time behaviour of the signals, several one-dimensional PIC simulations have also beenperformedwith the same parameters, varying the distance from the plasma edge to the pulse collision point asshown infigure 3(c). In thisfigure, THz emission grows as t 3 2 initially, but soon evolves into a t dependence,which is exactly the same feature as infigure 2(b).When the pulse collision occurs further into the plasma, ittakes longer for the emission to grow, but eventually it reaches a comparable level (red and blue). Note that, dueto the density gradient, a strong, but short duration emission by linearmode conversion [10] emergessimultaneously in the early stage. Though not fully plotted in thefigure, the emission usually lasts up to an orderof a hundred pico-seconds, which produces quite amonochromatic frequency spectrum as shown infigure 3(d). In the PIC simulations, collisions between charged particles are neglected, since the collisional rate isvery small comparedwith the plasma frequency for the given density. Furthermore, we did not considercollisions between charged andneutral particles, assuming the plasma is prepared one hundred percent ionizedby discharge orfield ionization. If the ionization is not complete, collisions can reduce the lifetime of the currentoscillation, leading to a shortened THz pulse.

Figure 3. (a)Diffusion of the transverse electromagnetic field inside amagnetized plasma ( =B 2 T) andTHz radiationwith= × −n 5 10 cm0

18 3, a0 = 0.05, and the pulse duration σ τ μ= =c( ) 4 m. The snapshot was captured at 1.3 ps after the pulse collision.(b) The fieldmeasured on the axis. (c) Temporal growth and saturation of the THzfield for different collision points (L), and (d) thecorresponding spectra of the THz electricfield, with = × −n 1.25 10 cm0

18 3, σ μ= 8 m (31.4 fs at full-width-half-maximum(FWHM), red, blue), and = × −n 5 10 cm0

18 3, σ μ= 4 m (15.7 fs at FWHM, green). The normalization factor of the electric field,ω = −mc e 3.7 TV m0

1.

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New J. Phys. 17 (2015) 043045 M-HCho et al

3. Scaling of the THz amplitude

To obtain a scaling law for thefield strength of the THz emission as a function of the driving pulse amplitude, wecalculate the radiation current source, i.e. the J0 term in equation (2). Since the current is driven by the beat oftwo counter-propagating pulses, the electron oscillation is spatially fast-varying, so that a linear portion of theoscillation is averaged out. However, the remaining smooth, nonlinear component can still be strong [31],forming a bunched longitudinal current oscillating at the plasma frequency. Once this average longitudinalcurrent is generated, it induces a transverse current oscillation via the externalmagnetic field.

If the right- and left-going laser pulses are represented by = ω+ +

−+ +a a e k z ti i and = ω− −

− −− −a a e k z ti i ,

respectively, their beat is proportional to Δω+ −+ −e k k z ti( ) i . Thus, they exert a ponderomotive force on the electrons

which yields a density perturbation = +Δω+ −+ −( )n n e c.c.k k z t1

1

2 1i( ) i . Because the spatial average of n1

disappears, the linear current −en v0 1does not contribute significantly to the electromagnetic radiation.However, a non-vanishing second-order current remains, given by

ω= − + = −( )J

en v n v

e

knn

4ˆ ˆ ˆ ˆ

8ˆ . (6)

p2 1 1

*1* 1

01

2

Here, we have used the continuity relation between n1 and v1, and assumed resonant driving, i.e. Δω ω≃ p. Thisspatially smooth nonlinear current, togetherwith the linear self-current, drives the longitudinal electric field

oscillations via the relation π∂ = − +E J J4 ( )t z,smooth 1,self 2 . Calculating the self-current using the linearized

equation ofmotion for the electron plasma, we obtain

ωπ ω∂

∂+ = ∂

∂tE

e

kn tn

2ˆ . (7)p z

p2

22

,smooth0

12⎛

⎝⎜⎞⎠⎟

Note that ≃ =+ −k k k , so + ≃+ −k k k2 . To obtain the solution of equation (7)we evaluate the temporalevolution of n1 driven by the beat of the counter-propagating pulses

ω∂∂

= − + −n

t

n c ka a

ˆi ˆ ˆ , (8)

p

1 02 2

*

where n0 represents the unperturbed plasma density. In deriving equation (8), we neglect themagnetic forceterm, becausewe consider a veryweaklymagnetized plasma, i.e. ω ω≪c p. In the case where the plasmaoscillation is driven by the beat of counter-propagating pulses, as in equation (8), the transverse component ofthe ponderomotive force is commonly neglected [31]. The transverse force is proportional to a w2 2, wherew isthe pulse spot size, while the longitudinal one by the counter-beat is ∼k a2 2. Since ≫k w1 under usualcircumstances, the longitudinal ponderomotive force is dominant. For laser pulses assumed to be longitudinallyGaussianwith equalmaximumamplitude a0 and pulse duration τ, the pulse amplitudes can bewritten as

τ= − ∓ ±±a a t z L cˆ exp ( ( ) )02 2⎡⎣ ⎤⎦. Note that it is arranged for the peaks of the pulses to overlap at z=0 after

propagation by distance L. Then, applying theGreen functionmethod to equation (7), we obtain

∫ Φ ξ ξ= ω ξτ−∞

∞− +( )E A z( ) e ( )d , (9)z

t L c,smooth

i 2p⎡⎣ ⎤⎦

where

πτ

ωτ= −( )A z

en c k az c( )

2iexp 4 ,

p

2 3 04 3

04 2

22 2 2

and Φ ξ ξ ξ= − +( )( ) exp (1 erf( )) 22 . Finally, the spatially averaged amplitude of the longitudinal oscillation

becomes

ωπ σ= ω τ σ− −eE

mck a

16e e , (10)

z z,smooth

0

2 204 0.084 4p

2 2 2 2

where ω0 is the laser frequency and σ τ= c is the spatial length of the laser pulse. An interesting feature of thisequation is that the longitudinal field depends onlyweakly on the plasma density via the exponential term, whichis quite different from the single-pulse-drivenwakefield.

By applying an externalmagnetic field in the transverse direction, the longitudinal electric field given byequation (10) can be partially converted to a transverse field oscillatingwith the same frequency ωp. Here theratio of the transverse oscillation to the longitudinal one is given by ω ω=E Ey c p z,smooth, where ω = eB mc 0 ,leading to

5

New J. Phys. 17 (2015) 043045 M-HCho et al

ωπ ω

ωσ= ω τ−

eE

mck a

16e . (11)

y c

p0

2 204 0.084 p

2 2

The radiation source current is then calculated from the relation ε= − ∂J Ey t y0 .If the plasma-vacuumboundary is sharp, and the pulse collision point is located exactly at the edge of the

plasma, the transverse field given by equation (11) is coupled to the vacuumwithout experiencing the diffusionand growth. Thus equation (11) gives the scaling of theminimumTHz amplitude as a function of the drivingpulse amplitude. An interesting point in this scaling is that the THz amplitude is proportional to a0

4,corresponding toP2, where P is the power of the driving laser pulse. This is significantly different from the single-pulse driven systems, where the emission amplitude is proportional just toP [30]. From this different scaling,and due to the k2-factor, one of which is replaced by amuch smaller value σ−1 in the single-pulse case, we expectthe counter-propagating pulse scheme to generate amuch stronger THzwave than the single-pulse drivenschemes, evenwith verymoderate laser power.More detailed comparison between equation (11) and the single-pulse scaling shows that the effect of colliding pulses dominates the single-pulse effect when >a a0 th, where

ωω

∼aN

0.421

. (12)p

th0

HereN, which represents the number of oscillations of the laser fieldwithin the pulse duration, is usually >10,and ω ω ≪ 1p , thus this condition is satisfied even for small a0.

Figure 4 shows theoretical curves of the THz amplitudes from the two different schemes for = ×n 5 10018

and = × −n 1.25 10 cm018 3, corresponding to 20 and 10THz, respectively.

Infigure 4, the simulation data (circles) exhibits saturation, which arises fromwave-breaking of the spatiallyfast varyingwave.Onemajor effect of wave-breaking is the suppression of the plasma current by kineticdetuning [32]. From the fact that it occurs when the electronfluid velocity, which is π ωσ≃v aˆ 8 21 0

2

according to the linearized continuity equation, exceeds the phase velocity of the plasmawave, we estimate thelaser amplitude for saturation to be

πω

ω σ=a

k

8. (13)

psat

1 4

0

⎜ ⎟⎛⎝

⎞⎠

In addition, though not as apparent in the figure, the scaling for very low a0, less than ath, obeys a02 rather than

a .04 As a consequence, operatingwith a0 between ath and asatmay yield optimumenergy conversion from laser to

THz emission.Asmentioned above in the discussion offigure 3(b), the density gradient helps the diffusing field to grow and

to yieldmuch stronger THz emission than for the sharp boundary case. This feature is verified by the PICsimulations, as shown infigure 5, where a significant enhancement due to the density gradient is apparent.Indeed, the positive effect of the density gradient is one of the advantages of the field diffusionmechanism.Because every realistic plasma from a gas-jet or capillary discharge used for laser-plasma interactions has anatural density ramp-up, the experimental conditions can be greatly relaxed in our diffusion-growth scheme.Furthermore, even though a varying density is employed, the radiation frequency is not influencedmuch by thatsince our scheme is based on the local oscillation of the plasma.Note that for a density gradient the THzfrequency is usually chirpedwhen driven by a single pulse, leading to broad-band emission.

Figure 4.Amplitude of the terahertz emission as a function of a0 for the counter-propagating pulse scheme and the single-pulseschemewithmagnetic field =B 2 T. (a) The case for = × −n 5 10 cm0

18 3 (20 THz) and σ μ= 4 m (15.7 fs at FWHM). (b) The casefor = × −n 1.25 10 cm0

18 3 (10 THz) and σ μ= 8 m (31.4 fs at FWHM). The vertical bars represent the theoretical wave-breakingpoints from equation (13).

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New J. Phys. 17 (2015) 043045 M-HCho et al

4. Conclusions

In summary, we have proposed and investigated a novelmethod for obtaining strong THz emission based on adriven-diffusionmechanism.We obtain, analytically, the temporal growth of the diffusing electromagnetic fieldby solving the constantly driven complex diffusion equation. For the driving termof the diffusing field a longlasting localized current source for the radiation can be produced by two counter-propagating laser pulsescolliding in aweaklymagnetized plasma. From analytic theory, wefind that the THz amplitude scales withP2,where P is the power of the driving pulse, which is verified by one-dimensional PIC simulations. This scalinggivesmuch stronger emission than for single-pulse driven systems, for which it is proportional just toP. Suchsignificant enhancement of the THz amplitude arises from two factors: one is the stronger ponderomotive forceproduced by counter-propagating pulses comparedwith a single pulse for a given total energy of the pulses. Theother is the growth of the emission by a driven-diffusionmechanism of the electromagnetic field.

Finally, we have confirmed the feasibility of the proposed scheme for experimental realization by showingthat it does not impose any strict condition on the plasma or lasers such as a sharp plasma edge or the exact pulsecollision point.Moreover, we have discovered that the density gradient positively enhances the emissionamplitude. From these results, we note that the experimental conditions can be significantly relaxed.We suggestthe following proof-of-principle experimental parameters: a couple ofmillimetres plasmawith density of order

−10 cm18 3 and density-ramp-up over μ∼100 m, which is readily available fromgas jet or capillary discharge. Theintensity of the laser pulses with 30 fs FWHMpulse duration used in our simulation is × −5 10 W cm15 2, whichyields a THz amplitude of 50 MVm−1. Thereforewe expect that pulses with a fewTWpower and 200 μm spotradius (pulse energy∼0.2 J)may yield a THz power of order 0.1MW.

Acknowledgments

This researchwas supported by the Basic Science Research Program through theNational Research Foundation(NRF) of Korea funded by theMinistry of Science, ICT and Future Planning (Grant numberNRF-2014M1A7A1A01030175 andNRF-2013R1A1A2006353, NRF-2014M1A7A1A01030173).We acknowledgethe support of theUKEPSRC (grant no. EP/J018171/1), the ECs LASERLAB-EUROPE (grant agreement no.284464, Seventh Framework Programme), EuCARD-2 (grant no. 312453, FP7) and the Extreme LightInfrastructure (ELI) European Project.We greatly appreciate Prof Z -MSheng for useful comments onourwork.

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