We have investigated the properties of THz radiation generated inairplasmaby focused bichromatic femtosecondlaserpulses, when one of the pump beams (second harmonic) is an optical vortex. The presence of a phase singularity in the generated THz beam was confirmed by astigmatic transformation of the singular THz beams in the focus of a cylindrical lens, as well as by fully space and time resolved numerical simulations. We report that, in contrast to other nonlinear processes (second harmonic generation, parametric generation, etc.), the THz radi- ation generated by electron currents in a plasma filament can not be characterized as a THz vortex beam in the ‘classical’ sense, such as a pure Laguerre-Gaussian beam. Instead, the intensity of the THz beam is modulated along the beam azimuthal an- gle and contains two minima between two lobes of maximum intensity. This is because the relative phase between two har- monics varies azimuthally when the SH pump pulse is a vortex. Moreover, our numerical simulations demonstrate that trans- verse instabilities in the filamentary pump propagation affect the THz vortex without destroying it. They may introduce sec- ondary phase singularities, which renders the phase topology of produced structured THz fields particularly rich. One of the benefits of THz generation from plasma currents is the large ( > 40 THz) spectral range achievable, contrary to bandwidth limited external THz shaping techniques. We envisage that dif- ferent combinations of the topological charges of the FH and SH pulses open a wide playground for the creation of structured singular THz sources.
2) THz generationin a stratified plasma
It has been predicted that a short laser pulse propagating in a periodically varying (stratified) plasma can generate electromagnetic radiation in the THz domain , if the modulation period is of the order of a few hundred microns. However, such a modulation cannot be created spontaneously in the time between twopulses. The plasma oscillations created by the first pulse are decaying in a time scale less than 1 ps, and they cannot induce any specific large-scale plasma motion. Moreover, the plasma column is expected to be fairly homogeneous along the filament because of the strong clamping effect upon the pulse intensity . Therefore, this undulator effect is not likely at the origin of the THz emission in our experiments.
Broadband ultrashort terahertz (THz) pulses can be produced using plasmagenerationin a noble gas ionized byfemtosecondtwo-colorpulses. Here we demonstrate that, by using multiple-frequency laserpulses, one can obtain a waveform which optimizes the free electron trajectories in such a way that they reach the highest velocity at the electric field extrema. This allows to increase the THz conversion efficiency to the percent level, an unprecedented performance for THz generationin gases. Besides the analytical study of THz generation using a local current model, we perform comprehensive 3D simulations accounting for propagation effects which confirm this prediction. Our results show that THz conversion via tunnel ionization can be greatly improved with well-designed multicolor pulses.
earities (instantaneous and delayed) in filament-driven THz pulse generation. We display numer- ical evidence that four-wave mixing impacts the THz generation process over long propagation distances, even inintensity regimes where the photocurrent mechanism is the dominating THz emitter. We also demonstrate that the Raman-delayed Kerr nonlinearity does not contribute as a THz source. By means of the local current (LC) model , we moreover explain the variations in the THz field strength with respect to the fundamental wavelength of the optical radiation. In this respect we emphasize the role of the electron current component associated with the high- frequency laser pulse, whose fundamental at longer wavelength a ﬀects more significantly the THz spectrum. Importantly, the scaling of the THz field strength with the fundamental pump wavelength is shown to vary with the relative phase between the fundamental and second har- monic, the pulse envelope and duration, so that no universal λ-dependent scaling is achievable with a two-color pulse. Despite this, the conversion e ﬃciencies reported from the LC model show that the THz yield is roughly scaling as λ α with α > 4 for small relative phases between the two colors and α > 2 on the average. For focused pulses , these results are confirmed by direct simulations employing a unidirectional pulse propagator [24,25]. Smaller gain factors are achieved by meter-range two-color filaments due to the generation of weaker plasma densities. The paper is organized as follows: Section 2 proposes a one-dimensional (1D) approach combining known laser-driven THz sources. It recalls that, in the range of intensities reached bytwo-color filaments inair, photoionization and to a lesser extent the Kerr nonlinearity are the principal players in THz generation. Section 3 discusses analytical estimates of the laser- to-THz conversion e ﬃciency when considering a Raman-delayed nonlinearity and when in- creasing the fundamental laser wavelength. Section 4 verifies our analytical statements through three-dimensional (3D) comprehensive numerical simulations for both focused and filamentary pulses.
not visibly reduce the total THz yield, as confirmed by Fig. 1(e). Besides, an important debate has been the role of Kerr-induced four-wave mixing compared to that of photocurrents inlaser-driven THz pulse generation . To clear up this point, Figs. 2(a,b,c) compare THz spec- tra computed until the indicated distances without the plasma terms. The presence of plasma clearly enlarges the spectra by about 3 orders of magnitude and shifts the peak frequencies down to νTHz < 1 THz. Hence, plasmageneration and related photocurrents — even in- volving rather low electron densities ∼ 10 15−17 cm −3 — remain the key players in THz pulse generation. Fig- ures 2(d,e,h) evidence the broader laser-plasma interac- tion surfaces engaged at longer FH wavelengths. This is due to the λ0-dependent increase in the filament waist [27, 28] and so, in the plasma transverse cross-section, which amplifies in turn the THz yield. Also, the conical emission angle of the radiated THz pulses decreases with the FH wavelength: THz waves indeed diffract with an- gles of 6.2 ◦ and 2.3 ◦ for λ
Laser filamentation is actively studied for its rich variety of applications, from supercontinuum generation to lightning control . Inair, femtosecond filaments result from the self-focusing of ultrashort light pulses that couple to their own plasma channel and stay self-guided upon extended paths at high intensity levels. Driven by strong nonlinearities, these optical structures are able to promote broadband THz-to-far-infrared radiation when using laser fields composed of two colors, e.g., a fundamental and its second harmonic .
Terahertz (THz) time-domain spectroscopy is a promising way to analyze various complex materials, the molecular transitions of which possess unique ﬁngerprints in this spectral range [ 1 – 3 ]. Amongst others, quite
efﬁcient methods to produce THz waves exploit laser-matter interaction processes and they rely on difference frequency generationin c ( ) 2 materials [ 4 ] or on more complex nonlinear processes in gas plasmas [ 5 ]. In the latter situation, a two-color ultrashort laser ﬁeld composed of a fundamental harmonic (FH) frequency and its second harmonic (SH) ionizes the gas, which creates a plasma acting as frequency converter towards the THz range. However, to properly identify a material, the THz spectrum has to be tunable around selected frequencies and /or be broadband with high enough spectral intensity. Such versatility can be obtained from two-color gas plasmas, e.g., by increasing the fundamental laser wavelength [ 6 , 7 ] or by employing incommensurate
3. Experimental results
We first compared the THz emissions obtained bytwo-color and three-color sawtooth wave excitation. For the two-color case, the insertion of the fused silica wedge was optimized for the most intense THz generation [ 3 ]. The energy of the incident 800 nm pump pulse was 1.6 mJ and the second harmonic was 0.2 mJ. The measured THz waveform is presented in Fig. 2(a) (red line). Since both 800 nm and 400 nm optical fields are linearly polarized in the horizontal plane ( Fig. 3(a) ), the resulting THz electric field is observed to be horizontally polarized with a weak vertical component, in agreement with previous results [ 13 ]. For the three-color field excitation, it is observed that both the rotation and azimuth angle of the SFG crystal determine the intensity of the 266 nm field, as well as the polarization state of the exiting three optical fields. Moreover, the distance of the SFG BBO crystal with respect to the focus determines the phase Δ . In general
linearly-polarized (LP) pulses with orthogonally-polarized colors (LP-O). Among the four investigated polarization states (CP-S, CP-C, LP-P, LP-O), only pulses being linearly polarized underwent a strong dependency on the relative phase φ. Meanwhile, the effects of the pump polarization states were also investigated numerically. In [ 40 ] the inﬂuence of circularly-polarized two-colorpulses driven either by four-wave mixing (FWM) or by photocurrents conﬁrmed the previous dependencies of the resulting THz ﬁeld on the two-color phase offset. In reference [ 41 ], the generation and control of elliptically-polarized THz waves was demonstrated from air plasmas driven by few-cycle CP pulses. For in-line laser focusing, the THz polarization state was experimentally found to evolve from linear to elliptical by increasing the plasma length [ 42 ]. More recently, theoretical evaluations based on an extended 3D local-current (LC) model [ 43 ] explored the role of the laser parameters (pulse duration, relative phase, tilt angle of linear polarization) on the radiation characteristics. Kosareva and co-workers [ 44 , 45 ], by analyzing the polarization properties of the pulse harmonics and broadband THz generation from atmospheric plasmas, reported the decrease by about one order of magnitude of the THz yield provided by LP-O pulses compared to LP-P’s. In [ 44 ], abrupt changes in the THz polarization were observed from an angle ∼85 ◦ between the FH and SH polarization axis and a weak ellipticity was sufﬁcient to drive efﬁciently an elliptical THz radiation.
for CO 2 and water absorption . Peak intensities of ∼ 200 TW/cm 2 produce electron densities above 10 17 cm −3 .
When increasing the pump wavelength and pump energy (P cr ∝ λ 0 2 ), longer filaments are promoted and they start to self-focus at shorter propagation distances. Figure 1 (a) illustrates the resulting THz energy yields below 10 THz that reach the 0.1 mJ level for 3.9-µm pumps and several mJ for 10.6-µm lasers. Figures 1 (b,c,d) detail the THz spectra computed at the distance of maximum THz field production, i.e., where the on-axis THz fields attain their maxima. Because an important question has been the role of Kerr-induced four-wave mixing compared to that of photocurrents , Figs. 1 (b,c,d) also compare the spectra computed with or without the plasma terms before the linear focus. The presence of plasma clearly increases the THz spectral intensityby at least three orders of magnitude and shifts the frequency centroid down to 0.2 THz for the three pump wavelengths. Hence, plasmageneration – even when it involves low electron densities ∼ 10 15−17 cm −3 – is here the key player in THz pulse generation.
L. Berg´e et al.
Fig. 3: (a) Mechanisms generating THz waves by intense two- colorlaserpulses, distributed according to the optical intensity. The first region involves the Kerr effect (four-wave mixing) and photoionization. The second region accentuates the contribu- tion of photoionization in the tunnel regime (photocurrents) and involves plasma waves created by ponderomotive forces. (b) Photocurrent process: The two-color electric field gener- ates free electrons in stepwise increase via tunneling ionization occurring near the field extrema at t = t n . This builds a slow
Figure 2. Experimental images and spectra of the generated THz beam versus energy of FH. Energy of SH is constant 60 µJ, time delay is 0 ps. See text for details.
Spectra of THz beam versus energy of FH is shown in Fig. 2 . Energy of SH kept constant at 60 µJ. Column ‘spectra’ shows spectra of the generated THz beam at the specified energy levels of FH. Insets to the left from this column show intensity distribution of the generated THz beam at specified energy levels. Two rightmost columns additionally show spectral maps with original (non-normalized) scale (‘original values’) and normalized data (‘normalized’). These maps are automatically generated in an OriginLab software using data from the column ‘spectra’. The data points between these six experimentally measured data sets (those at 6, 4.5, 3, 1.5, and 0.5 mJ) are created automatically by the OriginLab software for smooth transition between these data sets. As is visible from from Fig. 2 , increasing the energy of FH from 0.5 mJ to 6 mJ broadens the spectrum of the generated THz wave from 13.5 THz to 28.5 THz at FWHM, correspondingly, and produces higher THz yield. At any incident energy level of FH the spectrum of the generated THz wave peaks at 13 and 25 THz (Fig. 2 ). Additionally, from the image insets of the THz intensity distributions, one can notice slight decrease in the radius of the generated THz radiation with increase of the energy of FH.
MHz are stabilized in order to fix the difference of the rep- etition rates at ⌬f rep = 10 kHz. This offset allows for an au- tomatic scan of the measurement window 共f rep −1 = 1.2 ns兲 by the probe pulse in ⌬f rep −1 = 100 s without mechanical delay line. The time resolution achieved by this technique lies be- low 150 fs. Experiments are performed in reflection geom- etry. The average pump and probe powers did not exceed a total of 110 mW, with typical probe powers of about 10 mW. The spot sizes on the membrane were about 25 m in di- ameter, and the wavelength used for pump and probe beams was 810 nm. The samples were made from the 关100兴- oriented standard silicon-on-insulator 共SOI兲 wafers produced via the smart cut process. 15 To obtain the free-standing mem-
factor (1 + E e /E w ) 2 , which accounts for a coherent addition of electric currents driven by the ponderomotive
force and the external field. Also the length of the plasma filament L has to be replaced by the distance D between the electrodes.
In order to verify the dependence on the filament length, we performed the measurements of radiation pattern for the distances D = 15 and 60 mm. As a result, the radiation angle θ changed from 25° in Fig. 2(a) to 17° in Fig. 2(b). According to the transition-Cherenkov model 5, 6 , the angle of the most intense THz
polarization in the beam transverse plane. They may possess phase singularities in the transverse plane at which the field amplitude vanishes. VVBs can be expanded in terms of orbital angular momentum (OAM) carrying, i.e. twisted, beams. Twisted beam carry an OAM value of ℓ per photon, where is the redueced Plank constant, and ℓ indicates the number of twists in the helical wavefront in one wavelength which its sign determines the chiral- ity of the helix. We produce VVBs using a birefringent plate enclosing a patterned liquid crystal layer, known as a q-plate. The liquid crystals in the q-plate have an optic axis whose orientation depends on the azimuthal coordinate, thereby forming a pattern defined by a topological charge q consisting of either a full or a half-integer value. As light propagates in the q-plate, spin angular momentum associated with light polarization is coupled to photon OAM of ℓ = ±2q . The conversion efficiency is determined by the q-plate’s optical retardation, which can be controlled by an externally applied electric field.
L’étude a été réalisée en modélisant deux rectangles de verre dans une géométrie axisymétrique 2D. Le maillage est plus resserré le long de l’axe de révolution, qui correspond également à l’axe de propagation du faisceau laser dans le matériau, avec des triangles de dimensions maximales de 0,2 µm autour de l’axe, et de dimensions maximales de 32 µm aux extrémités de la géométrie. Dans l’idéal, il aurait fallu implémenter des conditions aux limites permettant d’assimiler la géométrie à un solide inﬁni. Cependant, des telles conditions aux limites ne sont pas disponibles pour les études thermiques et il est usuel d’utiliser une condition d’isolant thermique. Ainsi, aﬁn que la condition aux limites ne modiﬁe pas le résultat de la simulation, la géométrie a été ﬁxée de façon a ce qu’elle soit suﬃsamment grande pour que la tempé- rature aux limites ne soit jamais élevée après diﬀusion thermique. Ainsi la simulation a été eﬀectuée dans un cylindre de 50 µm de rayon et de 500 µm de haut autour de la source de chaleur. La température initiale étant 293 K. Les ﬁgures 5.1, 5.2 et 5.3 présentent l’élévation de température dans le verre suite à l’absorption d’une unique impulsion laser et après diﬀusion de la chaleur dans le verre. La ﬁgure 5.1 correspond à la cartographie de température obtenue pour l’absorption d’une impulsion sur un diamètre de 8 µm et une profondeur de 300 µm, la ﬁgure 5.2 à l’absorption sur un diamètre de 10 µm et une profondeur de 250 µm et la ﬁgure 5.3 à l’absorption sur un diamètre de 15 µm et une profondeur de 300 µm.
An aspheric lens (16x with 0.25 NA, f = 11mm, Newport) focused the vortexpulses on to a crystalline (100) p-type silicon mounted on a precision 3-axis (x, y, z) motion control system. The position of the laser focus relative to the sample's surface was determined accurately (±5 μm) by imaging the back-reflected light from the silicon surface. Glass plate directed the back- reflected light towards a charge-coupled-device (CCD) camera (MCE-B013-US, Mightex) after propagating through a focusing lens (Thorlabs, f = 100 mm, plano-convex lens). The Si surface morphology was analyzed using a field emission scanning electron microscope (Zeiss Gemini SEM 500) with both In-Lens detection mode (top-view images) and secondary electrons detection mode (side–view images) and a Park NX10 atomic force microscope in a non-contact mode.
On the other hand, the gas plasma produced by the fs laser pulse is a finite conducting structure with a lifetime largely exceeding the fs time scale. Thus, one can expect that the gas plasma features plasmonic resonances which may have a strong impact on the THz emission propertie . However, no direct evi- dence of plasmonic effects inlaser-induced gas-plas- mas was observed so far: To make an evidence of plasmonic effects, those need to be distinguished from nonlinear propagation effects. Also from the theoreti- cal point of view capturing plasmonic effects is not trivial: plasmonic effects require at least a full two-di- mensional Maxwell-consistent description, and re- duced models like the unidirectional pulse propaga- tion equation , which are frequently used to de- scribe plasma-based THz generation [5,7,10], are by construction not capable of capturing such resonant effects.
fication technique delivers an output beam at 800 nm with super-Gaussian spatial profile and energy up to 300 mJ on target with rms energy stability of ~2.5% at 10 Hz repetition rate. The pulse duration is ~30 fs Full Width at Half Maximum (FWHM) as measured using a self-referenced spectral interferometric device (Wizzler, Fastlite). The p-polarized beam is focused on a 6 mm thick polished (roughness < 1 µm) rotating molybdenum disk with a f/4 silver-coated off-axis parabolic mirror (OAP). A 1 mm thick fused silica wafer with broadband antireflec- tion coating protects the OAP from debris produced during laser-target interaction. It is positioned perpen- dicular to the laser beam and as close as possible to the OAP to minimize the development of nonlinear effects. Considering the group velocity dispersion of fused silica (361 fs²/cm @ 800 nm), note that the pulse duration is not affected. Nevertheless, even if we cannot completely rule out the development of nonlinear effects at the maximum intensity used in the experiments (1.3 × 10 19 W/cm²), we assume that they do not significantly affect
as a result of thermal melting.
To further determine the mechanism of the melting, we varied the temporal profile of the optical excitation. First, we stretched the pump pulse and found the amorphization process was not very sensitive to the pulse width, shown in Fig. 1(b). Secondly, we excited the sample with two identical but weaker pump pulses. As shown in Fig. 1(c), we found similar degrees of amorphization still occurred when the two pumps were separated by 100 ps, which is sufficient for the majority of excited carriers near the surface to diffuse into the bulk region. Therefore, this phase transition was likely a thermal process rather than the non-thermal melting induced by high carrier density. In the case of homogenous melting, the lattice temperature was ramped much higher than the melting point (723K) in order to overcome the heat of fusion within a few picoseconds  and then entered the amorphous phase after fast quenching.