CEP-stable Tunable THz-Emission Originating from Laser-Waveform-Controlled Sub-Cycle Plasma-Electron Bursts
T. Balčiūnas, D. Lorenc, M. Ivanov, O. Smirnova, A. M. Zheltikov, D.Dietze, J. Darmo, K. Unterrainer, T. Rathje, G. Paulus, A. Baltuška, S.Haessler
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b CEP-stable Tunable THz-Emission Originating fromLaser-Waveform-Controlled Sub-CyclePlasma-Electron Bursts
T. Balˇci¯unas , D. Lorenc , M. Ivanov , O. Smirnova , A.M. Zheltikov , D. Dietze , K. Unterrainer , T. Rathje , G.Paulus , A. Baltuˇska , S.Haessler ‡ [email protected] Abstract.
We study THz-emission from a plasma driven by an incommensurate-frequency two-colour laser field. A semi-classical transient electron current modelis derived from a fully quantum-mechanical description of the emission processin terms of sub-cycle field-ionization followed by continuum-continuum electrontransitions. For the experiment, a CEP-locked laser and a near-degenerate op-tical parametric amplifier are used to produce two-colour pulses that consist ofthe fundamental and its near-half frequency. By choosing two incommensuratefrequencies, the frequency of the CEP-stable THz-emission can be continuouslytuned into the mid-IR range. This measured frequency dependence of the THz-emission is found to be consistent with the semi-classical transient electron currentmodel, similar to the Brunel mechanism of harmonic generation. ‡ Present address: Laboratoire d’Optique Appliqu´ee, ENSTA-Paristech – Ecole Polytechnique –CNRS, 91762 Palaiseau, France.
EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts
1. Introduction
Since the demonstration a decade earlier by Cook and Hochstrasser [1] of intense THzemission from air ionized with a two-colour ( ω + 2 ω ) laser field, this phenomenonremains in the focus of attention. This is motivated by its potential as a source ofintense single-cycle THz pulses, enabled by the high generated THz-field strength andlarge bandwidth and facilitated by the absence of material damage. The THz-rangecovers a wealth of fundamental resonances (in molecules: vibrational and rotationalresonances, in solids: phonon and plasmon resonances, impurity transitions), whichopens a wide field of possibilities for fundamental material and device research [2, 3]as well as possible sensor applications, including the identification of atmosphericpollutants and use in food quality-control, atmospheric and astrophysical remotesensing, and (medical) imaging with unique contrast mechanisms [4].Another motivation is the the interesting underlying plasma dynamics leadingto the THz emission. Both an ω − (2 ω − ω ) four-wave mixing mechanism basedon stationary or nonstationary third-order susceptibilities [5], as well as a tunnel-ionization-induced micro-current mechanism [6, 7] have been considered. Accordingto the electron current model, the THz emission from plasma originates from sharpquasi-periodic changes of free electron density occurring at the peaks of optical fieldionization in the tunneling regime. The induced electron current rapidly varies on thesub-cycle time scale and yields generation of many harmonics, also known as Brunelharmonics [8, 9]. Similarly to HHG where the mixing of the laser driving field withits second harmonic allows breaking the symmetry between the two quantum pathsand yields generation of even harmonics [10], the THz emission can be considered aneven (0 th -order) harmonic. This particular mechanism of THz emission is thus closelyrelated to two prominent phenomena, Above Threshold Ionization (ATI) and High-order Harmonic Generation (HHG), also based on quasi-periodic sub-cycle ionizationfollowed by electron acceleration in the driving laser field. High-order harmonics aregenerated via transitions of the electron between a continuum state and the groundstate, which can be efficiently controlled on the single-atom and sub-cycle level [11, 12].Here we show that the low-frequency THz emission corresponds to transitionsbetween continuum states and can also be controlled by shaping the laser field. Fromquantum-mechanical continuum-continuum transition matrix elements, we recover,using the Strong Field Approximation (SFA) with a stationary phase analysis, thesemi-classical interpretation in terms of transient electron currents by Brunel [8]and Kim et al. [6]. Furthermore, we experimentally demonstrate control over thetiming of sub-cycle optical field ionization relative to the sign and value of the vectorpotential of the pulse [13]. This is achieved by by fixing the carrier envelope phases(CEP) and detuning the frequencies of the two optical driving fields away fromcommensurability [14]. This results in a very wide continuous tunability of the THzfrequency emitted by the laser-generated plasma.This frequency dependence of THz emission received little attention as mostlythe commensurate ω + 2 ω schemes have been used so far. Incommensurate frequencycombinations have been realized by superimposing a broadband 20-fs pulse with itsnarrowband second harmonic; however the tunability range in this case is limited tothe bandwidth of the fundamental [14]. Employing the relatively broad tuning range ofthe OPA, we obtained the initial evidence for the wide frequency-tunability of the THz-emission [15, 16]. Also, a scheme similar to our experimental setup based on a near-degenerate optical parametric amplifier (OPA) superimposed with the fundamental EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts ω p ω s ω p /20 Ω I n t e n s i t y Frequency a) b) c) | g > |p> | p ’> HHGTHz ω p ω s ω s Ω Figure 1. (a) Scheme of mixing two incommensurate frequencies for generatingtunable THz emission. (b) Scheme of the atomic transitions in a strong laser field:transitions between a continuum and a ground state yield HHG, while transitionsbetween continuum states yield low order harmonic generation (including THz).(c) Perturbative four-wave mixing scheme for THz generation. laser pulse has been implemented [17]. However, due to lack of CEP stabilisation ofthe driving laser, field-resolved detection of the generated THz pulse was impossible.Furthermore, in that work only the lowest THz-frequencies ( < ω /ω is not an integer, based on a nearly degenerate OPA pumpedby a CEP-stable Ytterbium-based laser emitting at 1030-nm wavelength. Inspiredby our initial experiments [15, 16], we present extensive experimental data, theoryand calculations that agree with the measurement data. We analyse the spectralproperties of the generated THz emission over a wide range of detuning frequenciesfrom the exact commensurate case. With the CEP stabilised input pulses we are ableto generated CEP-locked THz pulses and detect them in a field-resolved measurement,thus demonstrating their applicability for time-domain spectroscopy probing theamplitude and phase response of a material. Due to the long wavelengths of theconstituent colours (1030 nm and 1800–2060 nm), our scheme is a promising route fora high-efficiency [18], CEP-locked, broadband and widely tunable mid-IR/THz source.
2. Quantum mechanical transient electron-current model for THzemission
A proper quantum mechanical treatment of THz emission is desirable to support thecommonly used semi-classical two-step model by Brunel [8] and Kim et al. [6]. Avery successful fully quantum mechanical model that describes emission of an atomexposed to a strong laser field is based on the SFA [19], where the influence of theatomic potential on the electron in the continuum, as well as the influence of all boundstates except the ground state are neglected. Recently full ab-initio quantum TDSEcalculations were performed to calculate THz emission [20, 21], but the drawback isthat the physical nature of the underlying dynamics is difficult to interpret. Herewe use an analytical SFA model to calculate the atomic emission and derive theexpressions that are identical to those previously attained in a classical descriptionby Brunel of recollision-free harmonic emission.An electron in a continuum interacting with the strong laser field can be describedas a wavepacket, which propagates in phase space:Ψ c ( t ) = − i Z t ˆ U ( t, t b ) ˆ V L ( t b ) ˆ U ( t b , d t ′ , (1) EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts U ( t , t ) and ˆ U ( t , t ) are the field-free and full propagators between times t and t and the operator V L ( t ) describes the interaction with the laser field. Herethe velocity gauge is used. The interpretation of Eq. (1), which has proven bothvery fruitful and accurate, is that t b are the instants of strong-field ionization andΨ c ( t ) is a superposition of many wavepackets emitted into continuum at times t b .Calculations based on semi-classical wavepacket propagators or quantum-trajectoriesexplicitly utilize this picture by representing quantum mechanical amplitudes as sumsover contributions of wavepackets moving along classical trajectories [22]. Therefore,we express Eq. (1) as a sum over such wavepackets ψ ( r ( t, t b ) , t ), emitted by the parentneutral system at times t b and propagated to t :Ψ c = X t b ψ t b [ r ( t, t b ) , t ] . (2)Note that we formulate these equations for a linearly polarized laser field, so theelectron dynamics are limited to a single dimension along that polarization direction.The emission of electromagnetic radiation is proportional to the dipoleacceleration, i.e. the time-derivative of the dipole velocity ˙ d cc ( t ) ≡ h Ψ c | ˆ p | Ψ c i , whereˆ p = i ∇ is the momentum operator (atomic units are used throughout the paper). With(2), the dipole velocity of the whole wavepacket is given by the sum of dipole velocitiesfor all birth times: ˙ d cc ( t ) = P t b1 ,t b2 h ψ t b2 | ˆ p | ψ t b1 i . Expressing the wavefunction of thefree electron born at instant t b as a plane wave, ψ t b [ r ( t, t b ) , t ] = p ρ ( t b , t, r ) e iS ( t b ,t,r ) ,where S = pr − E ( t − t b ) is the action and ρ is the electron density, it can be shown(see Appendix), that the diagonal terms ( t b1 = t b2 ) are the stationary points in phasespace and thus dominate this sum. Wavepackets emitted at different times, t b1 = t b2 ,are separated in space, i.e. have a reduced overlap and lead to rapid phase oscillations.We can thus simplify the dipole velocity to a single sum over birth instants:˙ d cc ( t ) = X t b h ψ t b | ˆ p | ψ t b i , (3)and express the continuum-continuum transition dipole velocity matrix elements as: h ψ t b | ˆ p | ψ t b i = Z d rρ ( t b , t, r ) v ( r, t, t b ) + δ ˙ d ( t ) , (4)with the velocity v = ∂S/∂r . Here, the first term describes the case when thewavepacket overlaps with itself and δ ˙ d ( t ) provides corrections. These may arise ifsome wavepackets emitted at different times have met at some place in the phasespace later.This is an important difference as compared to HHG, where the continuumand bound parts of the two wavepackets evolve differently and accumulate a phasedifference that depends on frequency and ultimately leads to chirped emission ofattosecond pulses. In C-C transitions, however, the requirement for the wavepacketsto overlap yields the Brunel harmonic radiation which is unchirped.We now concentrate on the first term in Eq. (4) and rewrite:˙ d cc ( t ) = X t b Z ρ ( t b , t, r ) d r | {z } W ( t,t b ) R ρ ( t b , t, r ) v ( r, t b , t ) d r R ρ ( t b , t, r ) d r | {z } v ( t,t b ) = X t b W ( t, t b ) v ( t, t b ) (5)as a sum over all ionization instants t b of the product of W ( t, t b ), the norm of thewavepacket created at time t b , and v ( t, t b ), the average velocity of the wavepacket. EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts t b : W ( t, t b ) = Γ( t b ) δt b , where δt b is the saddle point region and Γthe instantaneous ionization rate. Thus we have:˙ d cc ( t ) = X t b Γ( t b ) δt b v ( t, t b ) = Z t Γ( t b ) v ( t, t b )d t b . (6)Here we see that ˙ d cc ( t ) is the driving term for the Brunel harmonics [8] andwould correspond to the current density in the quasi-classical model of Kim et al.[6] if multiplied by the atomic density in the generating medium. The emittedelectromagnetic field is proportional to the dipole acceleration¨ d cc ( t ) = Γ ( t ) v ( t b = t ; t ) + Z t Γ ( t b ) ˙ v ( t b , t ) d t b . (7)The first term of this equation is proportional to the small initial velocity aftertunneling: v ( t b = t ; t ) ≈
0, which is neglected in the usual three-step picture, and thesecond one is the force F [ t, r ( t, t b )] = ˙ v ( t b , t ) acting at time t on the wavepacket at theposition r ( t, t b ) (center of mass of the wavepacket). The force is given mainly by thelaser field and the Coulomb potential of the ion F [ t, r ( t, t b )] = F las ( t ) + F Coul [ r ( t, t b )].In practice, the laser field usually dominates for electrons ionized in the strong-fieldregime, so the dipole acceleration is simply given by:¨ d cc ( t ) = F las ( t ) Z t Γ ( t b ) d t b . (8)The emission is simply proportional to the driving laser field multiplied by theionization steps R t Γ ( t b ) d t b . The spectrum of the emission is obtained by Fouriertransforming Eq. (8):¨ d cc ( ω ) = F las ( ω ) ⋆ F (cid:20)Z t Γ ( t b ) d t b (cid:21) ( ω ) , (9)which results in the convolution of the laser field spectrum and the spectrum of theionization steps.Now we apply the formalism derived above to the calculation of the emissionspectrum in the case of a two-colour mixing scheme, which allows generation of evenharmonics, including the THz sideband. Simulating our experimental conditions, thefield in the calculation is composed of a strong fundamental wave at the frequency ω p and a weaker signal wave at around half the frequency of the fundamental wave E ( t ) = E p ( t ) e iω p t + E s ( t ) e iω s t , where E p ( t ) and E s ( t ) are the envelopes of the pumpand signal waves respectively. The direction in which the electron is emitted is givenby the sign of the velocity v d ( t b ) = v ( t → ∞ , t b ) at which the electron drifts afterthe laser pulse is over. Fig. 2 b) illustrates the asymmetric case of the two colourfield where all of the electrons ultimately fly in one direction. The build-up of thisdirectional electron current is responsible for the emission of the low-frequency-THzwaves (see Fig. 2 c,d).Tuning of the central frequency occurs when two incommensurate frequencies aremixed. This is illustrated in the right column of Fig. 2 for a 0 . ω + ω –combination,and compared to the commensurate case (0 . ω + ω ) in the left column. While in thelatter case, the growth of a directional net-current (Fig. 2b) leads to the emission of alow-THz-frequency field (Fig. 2c,d), the modulation of the current direction resultingfrom the detuning of the signal-frequency by ∆ ω leads to a 2∆ ω -shift of the center of EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts −30 −20 −10 0 10 20 30−1−0.500.51 E l ec t r i c fi e l d ( a r b . u . ) Time (cycles) ω + 0 . ω (a)−30 −20 −10 0 10 20 30−101 ˙ d ( t )( a r b . u . ) Time (cycles) (b)−30 −20 −10 0 10 20 30−1−0.500.51 L P [ ¨ d ( t ) ] ( a r b . u . ) Time (cycles) (c)0 0.5 1 1.510 −5 Sp ec t r a li n t e n s i t y ( a r b . u . ) Frequency ( ω/ω p ) (d) −30 −20 −10 0 10 20 30−1−0.500.51 E l ec t r i c fi e l d ( a r b . u . ) Time (cycles) ω + 0 . ω (e)−30 −20 −10 0 10 20 30−101 ˙ d ( t )( a r b . u . ) Time (cycles) (f)−30 −20 −10 0 10 20 30−1−0.500.51 L P [ ¨ d ( t ) ] ( a r b . u . ) Time (cycles) (g)0 0.5 1 1.510 −5 Sp ec t r a li n t e n s i t y ( a r b . u . ) Frequency ( ω/ω p ) (h) Figure 2.
Simulation of the induced current, using Eqs. 6–9 withΓ( t b ) = exp[ − I p ) / / (3 | F las ( t b ) | )] and the classical expression v ( t, t b ) = − R tt b F las ( t ′ )d t ′ , for two-colour fields with two frequency ratios: the left andright-hand sides correspond to the commensurate and incommensurate frequencycase, respectively. Shown are the considered driving electric fields (first row), thedipole velocity (Eq. 6), corresponding to the induced current (second row, blueline; the red line shows a low-pass filtered version), the low-pass filtered (retainingonly the THz-sideband) dipole acceleration (Eq. 8), corresponding to the emittedTHz-field (third row), and the spectral intensity of the emitted radiation (squareof Eq. 9) (fourth row). The considered ionization potential is that of nitrogenmolecules, I p = 15 . ω p -component of the driving field has a peak fieldstrength of F max = 27 GV/m (corresponding to 1 × W cm − ), and the ω s -component has F max = 13 . π/ the emitted THz-spectrum (Fig. 2g,h). The shift of lowest order sideband frequency isthen compared to the experimentally measured THz-emission frequency. Higher ordersidebands are beyond the EO sampling detection range and were not measured in thisexperiment.This tunability is further analysed in Fig. 3 b). The THz-emission appears as azeroth order sideband alongside higher-order sidebands [9]. The THz sideband startsclose to zero frequency at degeneracy ( ω s = 0 . ω p ) and its frequency increases as thefrequency of the signal wave is tuned. This effect can be interpreted as a temporalphase modulation, shifting the THz sideband towards higher frequencies as illustratedin the inset of the Fig. 1a). Although the described THz emission mechanism is EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts Emission frequency Ω /ω p S i g n a l w a v e f r e q u e n c y ( ω s / ω p ) Ω = ω s − ω p Ω = ω s − ω p l og [ s p ec t r a li n t e n s i t y ] ( a r b . u . ) Figure 3.
Simulated power spectrum of the emission (square of Eq. 9) as afunction of the frequency ω s of the tunable signal wave, for the same conditionsas in figure 2. a phenomenon of intrinsically non-perturbative nonlinear optics, the resultant THzfrequency tunability follows the law Ω = 2 ω s − ω p (Fig. 3b), as is also the case inthe perturbative four-wave mixing picture, illustrated in Fig 1c) . We also predictthe feasibility of generating weaker sidebands corresponding to 6, 8, etc. photonprocesses. This process clearly enables a tunability through the complete THz andmid-IR spectral regions [14].
3. Experimental implementation
An Ytterbium-based laser amplifier and a near-degenerate OPA were used to generatetwo-colour pulses that were used for plasma excitation (Fig. 4). The Yb:KGWregenerative amplifier is actively CEP stabilized [23] and is used to pump a two-stagenear-degenerate OPA. First, part of the laser energy is split into three parts. One partis used to generate a broadband super-continuum by focusing it in a sapphire plate.The part of the so-generated spectrum around 680–720 nm is used as a seed for thefirst OPA stage. The second part of the beam is frequency-doubled and serves as apump for the first OPA stage. The resulting difference-frequency (idler) wave fromthe first stage (1800 nm to 2100 nm) is then amplified as a signal wave in the secondOPA stage, pumped by the third part of the fundamental laser beam. The resultingCEP of the OPA output is thus equal to the pump laser pulse CEP phase [24], up toa constant: ϕ s = ϕ p + const . .At the output of the OPA, the pump wave at λ p = 1030 nm and the signal waveat λ s =1800–2100 nm are separated using dichroic mirrors and then re-combined afteradjusting the divergence of the beams and the relative delay. The polarisations of eachof the two colour-component pulses are linear and set parallel to each other. The phasedelay between the two constituent colour waves is τ φ = ϕ p ( λ s − λ p )(2 πc ) − + const . ,and can thus be controlled via the fundamental laser CEP ϕ p . EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts Yb:KGW laser
Two-color pulse combiner
M MM M MWLGSHG
Near-degeneratefs OPACEP stable CPA
SMPM
THz generationin plasma
BS BS MMM MM BS LL λ=2.06 - 1.8 μm λ=1.03 μm MM MM M
Stereo ATI spectrometer
Left MCP Right MCP
M OPA MM MM MMMGaSeQWPP
PDPD
Figure 4.
Scheme of the experimental setup. THz emission was detected viaelectrooptic (EO) sampling setup based on GaSe crystal. The probe pulseswere generated in an additional OPA and compressed down to 20 fs in a prismcompressor.
We verified that the two-colour pulses used for THz generation induce a directionalelectron drift by measuring the photoelectron spectra produced via above thresholdionization (ATI) with a stereo-ATI spectrometer [25, 26]. The electron spectrometerconsists of a small xenon-filled gas cell ( < ≈
50 cm) and amicro-channel plate (MCP) electron detector. The linear polarisation of each of thetwo-colour driving wave is in the direction of the TOF detectors. The signals fromthe two TOF detectors are then acquired with multi-scaler card and processed usinga computer.The ATI spectrum in one direction measured for different CEP phases of thelaser pulse is shown in Fig. 5 a).This measurement is done with the OPA tuned atdegeneracy ( λ s = 2060 nm) so that the maximum directional drift is produced, asillustrated in the left column of figure 2. The asymmetry of the low-energy directelectrons is opposite to that of high-energy rescattered electrons (not considered infigure 2) and partially reduces the asymmetry of the total electron yield. However,because the number of direct electrons is much higher than of the rescattered ones, EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts Relative CE phase φ / π E l e c t r on ene r g y , e V −2 −1 0 1 2020406080 a) −2 −1 0 1 20.860.880.90.920.940.960.981 Relative CE phase φ / π E l e c t r on c oun t, a r b . un i t s LeftRight b) Figure 5.
Stereo-ATI electron spectrometry measurement results obtained withtwo commensurate-frequency pulses: λ p = 1 . µ m and λ s = 2 . µ m. (a) Electronenergy spectrum (log scale) measured on one side of the stereo-ATI spectrometeras function of the fundamental laser CEP. (b) Total electron count in left and rightdirections of the stereo-ATI spectrometer obtained by integrating the electronspectrum of the left and right detectors. the total yield still exhibits a modulation depth higher than 15%, as shown Fig. 5 b).These findings prove that in our experimental conditions, the induced electron currentis highly directional and we are working in the tunnelling regime, as assumed for thetheoretical model. We now demonstrate the phase-locked temporal modulation of this induced electroncurrent to achieve tunability of resulting CEP-locked THz emission.For THz-generation, the combined two-colour driving waveform is focused usingan f = 7 . f max in the EO sampling detection scheme depends on the probe pulse duration f max ≈ / (2 τ probe ) and it is the main limitation in our setup. The probe pulse of τ probe ≈
20 fs duration was generated by amplifying the spectral portion at around 900nm of the white-light super continuum in a separate OPA stage and then compressingthe pulse in a prism compressor near transform limit (see figure 4). The probe pulsewas characterised using SHG Frequency Resolved Optical Gating (FROG) with theresult shown in Fig. 6.The phase of the THz wave according to the frequency dependence is ϕ THz =2 ϕ s − ϕ p . Therefore, the degenerate scheme where fundamental and frequency-doubledpulses are mixed (with CEPs ϕ ω = 2 ϕ ω + const . ) provides a passive stabilizationof the THz-pulse’s CEP, ( ϕ T Hz = 2 ϕ ω − ϕ ω = const), similar to difference-frequency generation based on a second-order nonlinearity ( χ ) [24]. Since in our case, ϕ s = ϕ p + const . , the CEP of the THz-pulse is the same as that of the fundamentallaser pulse: ϕ T Hz = ϕ p + const. Because of the field-sensitive EO sampling of the THzemission, active stabilization of the laser CEP is thus mandatory. In other words, fornon-CEP-stabilized driving pulses, the THz emission is still emitted, but their CEPis random from pulse to pulse.This is why, as shown in Fig. 7a), without active CEP stabilization of the laser EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts Figure 6.
Temporal characterisation of EO sampling probe pulse using SHGFROG. a) measured FROG trace, b) reconstructed spectrum and group delay ofthe pulse c) measured temporal profile and calculated Fourier limit.
Detuning Δν (THz)Time (ps) T H z a m p li t u d e , a r b . u . Time (ps) (c) (b)(d)
Figure 7.
Experimental THz-generation results with ω p + ω s two-colour drivingfields ( ω s = ( ω p + ∆ ν ) / ν , created by combiningthe fundamental wave ( λ p = 1 . µ m, 250 µ J) with the tunable signal wave( λ s = [2 . , . , . , . µ m, 20 µ J). (a) Electric field of the THz transientmeasured using EO sampling in case of locked CEP or free running laser, forzero detuning, ∆ ν = 0, i.e. exactly commensurate frequency ratio. (b) Temporaltraces of the EO sampling signal for different detunings ∆ ν , (c) THz emissionspectra of plasma calculated from the EO signal, represented on linear scale. (d)The same spectra as in (c), represented on logarithmic scale. Each spectrum andtemporal trace was normalised to its maximum. The arrows in panel (d) showthe theoretically predicted central frequency of the emission. pulse, the emitted THz waveforms for each laser pulse are averaged out in our EOsampling. By locking the laser CEP, we create reproducible THz fields that let usstudy the two-colour-driven plasma dynamics.The detected THz emission spectra as function of the signal wavelength λ s ≤ λ p EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts ≈
40 THz weremeasured as the result of detuning the OPA from degeneracy. The frequency of theTHz-sideband shifts towards higher frequencies due to temporal current modulationwhen the OPA is detuned from degeneracy, as illustrated by the numerical simulationsshown in figure 2. We can detect only the lowest THz-sideband in the spectrum dueto the limited bandwidth of the THz detection apparatus. At higher frequencies,the probe pulse duration is significantly longer than the half-cycle of the field to bemeasured. The decreasing sensitivity of the THz detection setup at higher frequenciesalso contributes to the broadening of the spectrum for higher frequency detuning.The experimentally obtained frequency dependence agrees well with the semi-classicaltransient electron-current model simulations in figures 2 and 3: we observe a shift by ≈ ∆ ν for a signal wavelenegth tuned to ω s = ( ω p + ∆ ν ) /
4. Conclusion
The tunability and CEP-stability of the THz emission from a laser-induced plasma-spark is demonstrated by mixing two incommensurate CEP-stable optical drivingfields, forming a parametric waveform synthesizer producing shot-to-shot reproduciblestrong-field multi-colour waveforms. The measured frequency dependence of the THzemission is consistent with the transient electron current model we derived based oncontinuum-continuum transition matrix elements.
Acknowledgements
These studies were supported by the ERC project CyFi 280202 and Austrian ScienceFund (FWF) via the SFB NextLite F4903-N23. S. Haessler acknowledges support bythe EU-FP7-IEF MUSCULAR and the by the Austrian Science fund (FWF) throughgrant M1260-N16.
Appendix: Stationary phase analysis
An evolution of a continuum electron wavepacket can in general be described bythe propagator (1). For convenience we will replace the integral by a sum of smallwavepackets corresponding to different birth times t b : ψ t b ( r, t, t b ) = − i ˆ U ( t, t b ) ˆ V ( t b ) ˆ U ( t b , δt b (10)where δt b is associated with the width of the saddle point region around t b thatsatisfies the stationary phase condition: ~r = ~r + Z tt b ~v ( t ′′ , t ) dt ′′ . (11)Here ~r is within the size of the initial state Ψ and is therefore negligible comparedto the oscillation of the electron in the strong laser field described by the secondterm. This condition is valid beyond the SFA [22]. By coherently adding up thesmall wavepackets of different birth times we get the wavepacket of the electronin the continuum, Eq. (2). The evolution of ψ t b ( t ) is governed by both theionic potential and the laser field. To calculate the emission, the dipole velocity˙ d cc ( t ) = h Ψ c | ˆ p | Ψ c i = P t b1 ,t b2 h ψ t b2 | ˆ p | ψ t b1 i has to be calculated. In case of a strong EP-stable Tunable THz-Emission from Controlled Plasma-Electron Bursts p is given by thederivative of the action S which is a rapidly oscillating function: i ∇ ψ t b ( t ) ≈ − p ρ ( t, t b ) ∇ S ( t, t b ) e iS ( t,t b ) = − ∂S∂r ψ t b ( t ) (12)Then the dipole velocity can be written as:˙ d cc ( t ) = X t b1 ,t b2 (cid:28) ψ t b1 (cid:12)(cid:12)(cid:12)(cid:12) ∂S ( t, t b2 ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) ψ t b2 (cid:29) (13)= X t b1 ,t b2 Z dr √ ρ √ ρ e i [ S ( r,t,t b2 ) − S ( r,t,t b1 )] v ( r, t, t b1 , t b2 ) (14)where v = ∂S/∂r is the velocity of the electron at point r , ρ ≡ ρ ( t, t b1 , r )and ρ ≡ ρ ( t, t b2 , r ) are the densities of the two wavepackets. The non-negligiblecontribution to the sum is given by the diagonal terms t b1 = t b2 that satisfy thestationary phase condition. By taking into account only these diagonal terms andcalculating the acceleration of the dipole, we finally arrive at equation (8) which is thesame as the one used in photocurrent calculations. References [1] Cook, D. J. & Hochstrasser, R. M. Intense terahertz pulses by four-wave rectification in air.
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