Polarisation control of quasi-monochromatic XUV produced via resonant high harmonic generation
PPolarisation control of quasi-monochromatic XUVproduced via resonant high harmonic generation
M.A. Khokhlova , ∗ , M.Yu. Emelin , M.Yu. Ryabikin , , T. Sato , K.L. Ishikawa , and V.V. Strelkov , Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Straße 2A, Berlin 12489, Germany Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov street, Nizhny Novgorod 603950, Russia Prokhorov General Physics Institute of the Russian Academy of Sciences, 38 Vavilova street, Moscow 119991, Russia The University of Tokyo, School of Engineering, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8654, Japan Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Region 141700, Russia ∗ [email protected] (Dated: February 11, 2021)We present a numerical study of the resonant high harmonic generation by tin ions in anelliptically-polarised laser field along with a simple analytical model revealing the mechanismand main features of this process. We show that the yield of the resonant harmonics behavesanomalously with the fundamental field ellipticity, namely the drop of the resonant harmonicintensity with the fundamental ellipticity is much slower than for high harmonics generated throughthe nonresonant mechanism. Moreover, we study the polarisation properties of high harmonicsgenerated in elliptically-polarised field and show that the ellipticity of harmonics near the resonanceis significantly higher than for ones far off the resonance. This introduces a prospective way tocreate a source of the quasi-monochromatic coherent XUV with controllable ellipticity potentiallyup to circular. I. INTRODUCTION
Extreme ultraviolet (XUV) sources of ultrafast emis-sion have proven themselves to be an immensely im-portant and powerful tools for tracking and controlof electron dynamics in atoms, molecules, and con-densed matter [1, 2]. One of the handles of thesetools is the polarisation of the emitted electromagneticfield. In particular, circularly-polarised (CP) XUVpulses recently have become extremely valuable dueto their broad involvement in a growing number ofexperimental techniques for the study of structural,electronic, and magnetic properties of matter, such aschiral molecules [3, 4] and magnetic materials [5–8], aswell as for deeper investigation of strong-field processesin atoms (such as nonsequential photoionisation) [9] orother curious applications [10].On the one hand, although synchrotrons [5] and free-electron lasers [11] are typically used as a source of CPXUV pulses and perform some progress, they still sufferfrom the low degree of coherence. An alternative groupof sources, which does not only allow to switch fromlarge-scale facilities towards table-top setups, but alsoenables the significant improvement of the coherenceproperties of the emission, is based on the high-orderharmonic generation (HHG) process [1].The naive approach to create CP harmonics — viaHHG by atoms in an elliptically-polarised (EP) field —fails immediately [12]. Although the ellipticity of highharmonics grows with the driver’s (fundamental) ellip-ticity [13], there is a limiting factor: the harmonic yielddrops rapidly (exponentially) with the fundamentalellipticity [12, 14]. Thus, HHG in an atomic gaseousmedium driven by an EP laser field does not appear asa reasonable candidate for EP XUV sources.There have been extensive efforts towards the produc- tion of EP and CP XUV via HHG recently, resultingin the suggestion of numerous nontrivial experimentalschemes. One class of such schemes involves bicircularbichromatic fields [15–19] or other geometries of two-colour fields [20–22]. A second class manipulates thepolarisation of the generated XUV using reflection-[7, 8, 23, 24] and transmission- [25] based polarisers. Anew technique [26] presented recently, combines two in-dependent, phase-locked, orthogonally-polarised HHGsources. One more way to produce EP XUV pulsesis HHG using aligned molecules in gas phase [27–29].These schemes rely on rather sophisticated experimentalsetups, which make them harder to implement. Alsothe common drawback of these schemes, as of based onHHG, is a low intensity of emitted XUV.In this paper we suggest a new way to generate brightXUV with tunable polarisation via HHG more effi-ciently, avoiding overcomplicated experimental schemes.Our approach is based on the process of resonantHHG [30–34]: if the harmonic energy is close to theone of the transition between the ground state and anautoionising state (AIS) of the generating particle, witha high oscillator strength, then the intensity of this“resonant” harmonic becomes significantly boosted by,up to a couple of orders of magnitude. Since the AIS ismuch less localised, in contrast to the ground state, itbecomes possible for the electron in an EP laser fieldto be captured into the AIS for higher fundamentalellipticities than in the nonresonant case, with the sub-sequent recombination to the ground state accompaniedby the enhanced XUV emission. Moreover, resonantharmonics can carry higher ellipticities, which makethem a prospective candidate for a source of quasi-monochromatic EP or even CP XUV radiation withrelatively high intensity. Our approach allows efficientgeneration of XUV with high ellipticity in a potentially a r X i v : . [ phy s i c s . op ti c s ] F e b wide spectral range, in contrast to similar approach togenerate quasi-CP XUV, using Rydberg states [35] orshape resonances [36] instead of AIS, which is limitedto either near-threshold harmonics or appears to be apotentially much weaker effect.To substantiate our proposed scheme, we study nu-merically the resonant HHG in an EP laser field by solv-ing the three-dimensional time-dependent Schrödingerequation (3D TDSE) for the singly ionised tin atom(SnII) in a laser field of different wavelengths andvarying ellipticity. Our results show that the efficiencyof the resonant harmonics exhibits much slower decreasewith the fundamental ellipticity than the efficiency ofnonresonant harmonics [12, 37]. Moreover, the resonantharmonic yield sometimes shows anomalous behaviour,which is expressed in its constancy or even growth forsome laser parameters. We also study the behaviour ofthe harmonic ellipticity, which shows the strong impactof the resonance on it. Not only the ellipticity of theresonant harmonic itself appears to be rather high,even close to unity, but the ellipticity of nonresonantharmonics is affected by the resonance, including thechange of the ellipticity sign as well. These numericalresults lay in line with our analytical toy model. II. NUMERICAL METHODS
We study numerically the resonant HHG in an EPlaser field by solving fully 3D TDSE in a single-activeelectron (SAE) approximation.The TDSE was integrated numerically using the fastFourier transform-based split-operator technique [38].The calculations were performed using a multithreadednumerical code we have created on the basis of librariesimplementing the POSIX Threads standard.The model potential reproducing the interaction ofthe active electron with the nucleus and with the rest ofthe electrons is chosen in the form suggested in [32]. Itcan be written as a combination of a soft-core Coulombpotential and a barrier, thus allowing for a quasi-stablestate with positive energy, which models the AIS: V ( r ) = − Q + 1 (cid:112) a + r + a exp (cid:34) − (cid:18) r − a a (cid:19) (cid:35) . (1)Here Q = 1 is the charge state of SnII, and a = 0 . , a = 1 . , a = 3 . , a = 2 . are constants chosen toreplicate the properties of the generating particle (SnII):the ground state energy (or − E ground = I p = 14 . eV),the resonant energy ( ∆ E = E AIS − E ground = 26 . eV),the AIS width ( Γ AIS = 0 . eV) and the oscillatorstrength ( gf = 1 . ) of the transition between theground state and the AIS (see [39] for details).The laser pulse used for our calculations is trape-zoidal with 4 cycles of constant intensity and 2 cyclesof turning-on and turning-off each. We extend thecalculation time by the AIS lifetime (corresponding to Γ AIS ) to account for the longer emission of the resonantharmonic.From the TDSE solution, we obtain time-dependentdipole moment components d x ( t ) and d y ( t ) , directedalong the polarisation ellipse axes of the laser field. Wethen calculate the Stokes parameters S , S , S and S , S = | d x ( ω ) | + | d y ( ω ) | ,S = | d x ( ω ) | − | d y ( ω ) | ,S =2 Re (cid:2) d x ( ω ) d ∗ y ( ω ) (cid:3) ,S = − (cid:2) d x ( ω ) d ∗ y ( ω ) (cid:3) , (2)using the spectral representation for dipole components, d x ( ω ) and d y ( ω ) . Thus, the harmonic spectrum in thiscase is S , and the harmonic polarisation properties areobtained from the Stokes parameters as ψ = 12 arctan (cid:18) S S (cid:19) , (3)for the rotation angle, and (cid:15) = − tan (cid:34)
12 arctan (cid:32) S (cid:112) ( S + S ) (cid:33)(cid:35) , (4)for the ellipticity. III. ANALYTICAL TOY MODEL
The rapid decrease of the harmonic intensity withthe fundamental ellipticity in the absence of resonancescan be understood in the framework of the three-stepmodel [40–42]: in an EP field, the electron wavepacketreturning to the parent ion is transversely shiftedrelative to the parent ion, and at some threshold ellip-ticity [12, 43] it starts missing the parent ion completely. xy a AIS Δ x ‖ Δ x ┴ free ρ Figure 1. Illustration of the model. The diffused free elec-tronic wavepacket Ψ free ( ∆ x (cid:107) and ∆ x ⊥ are its uncertainties)moves back towards the parent ion, which has an AIS ofeffective size a , with the impact parameter ρ . In the case of resonant HHG, the mechanism behindthe anomalous dependence of the resonant harmonicon the fundamental ellipticity can be explained usingthe four-step model [32]. The first two steps (electronionisation and propagation of the electron wavepacketin the driving field) follow the three-step model, butinstead of immediate radiative recombination into theground state, the returning electron gets captured intoan AIS of the parent ion (new third step), from whichit then relaxes into the ground state emitting the XUVphoton (fourth step). Since the AIS is much less tightlylocalised than the ground state, an electron wavepacket,which would miss the ground state in nonresonant HHG,continues to hit the parent ion AIS at much higherellipticities of the driver (see Fig. 1). This leads toa slower decrease or even to an anomalous (constantor even locally increasing) behaviour of the resonantharmonic efficiency (yield) with fundamental ellipticityin comparison to nonresonant harmonics.Further we consider the influence of the AIS size onthe HHG in an EP laser field. Formally, the contributionof the resonance to the yield I q of the resonant q -thorder harmonic can be understood as a product of theform: I q ≈ w i SR , where w i is the ionisation probability(which can be found within the ADK [44] or PPT [45]formulae, for instance), S is a factor responsible for thetrapping of the free electron into the AIS (depending,in particular, on features of the free-electronic motionbetween the detachment and the return, see below), and R is the recombination factor describing the transitionfrom the AIS to the ground state.The factor S = |(cid:104) Ψ AIS | Ψ free (cid:105)| , which carries thedifference between the resonant and nonresonant cases,is essentially the squared absolute value of the overlapintegral of the AIS wavefunction Ψ AIS and the freeelectron wavepacket Ψ free returning to the parent ion.Here Ψ AIS is the spatial part of the AIS wavefunction,for simplicity, modelled by the hydrogenic set of p -orbitals Ψ AIS = (cid:80) Ψ m , where Ψ = 14 √ π (cid:18) Za (cid:19) / e − Zr a r cos θ , Ψ ± = 18 √ π (cid:18) Za (cid:19) / e − Zr a r sin θe ± ϕ (5)with an effective radius a (see Fig. 1) and effectivecharge Z [46]. The symmetry and size of the chosenstates here are the same as in our numerical SAE TDSEcalculations.The free electron wavefunction Ψ free , before captur-ing, is described as a Gaussian wavepacket (see Fig. 1) Ψ free = (cid:32) π ∆ x (cid:107) (cid:33) / e − ( r sin θ sin ϕ )22∆ x (cid:107) π ∆ x ⊥ e − ( r cos θ )22∆ x ⊥ × e − ( r sin θ cos ϕ − ρ )22∆ x ⊥ e i √ qω − I p ) r sin θ sin ϕ , (6)where ∆ x (cid:107) and ∆ x ⊥ are uncertainties of the electronwavepacket in the longitudinal and perpendicular di- Figure 2. Harmonic spectra calculated within the SAETDSE for the fundamental wavelength 800 nm, and fun-damental ellipticities from 0 to 0.4. rections (see [43, 45] for details), correspondingly, withrespect to the direction of the laser field (see Fig. 1), ρ is the impact parameter (or the shift of the centreof the electron wavepacket with respect to the parention) calculated from the classical equation of motion,the wavenumber is k = (cid:112) qω − I p ) , and I p is theionisation potential of the generating particle.Calculating the properties of the returning electronicwavepacket for different laser ellipticities and harmonicorders, we find the factor S and thus the harmonicintensity. IV. RESULTSA. Harmonic yield
We consider the HHG by tin ions (SnII) in an EPlaser field with varying ellipticity by solving TDSEnumerically (see Sec. II for details). Three different fun-damental wavelengths (800 nm, 520 nm, and 1300 nm)are used, the fundamental ellipticity varies between0 and 0.5, and the fundamental intensity in all ourcalculations is kept constant, · W/cm .We start with the most typical fundamentalTi:Sapphire wavelength, 800 nm, where the resonantharmonic is H17. Fig. 2 demonstrates the evolutionof the HHG spectrum with increasing laser ellipticity.One can see that there are two types of behaviour:the intensity of nonresonant harmonics rapidly dropsdown with increasing fundamental ellipticity, while theintensity of the resonant one does not decrease in thisway.More explicitly, this difference in the behaviour ofthe yield for the resonant (H17) and for one of thenonresonant harmonics from the middle of the plateau(we choose H29), as a function of the fundamentalellipticity, is presented in Fig. 3(a). Here it is shown thatthe yield of the nonresonant H29 harmonic decreasesquickly with the fundamental ellipticity, so that atfundamental ellipticity 0.2 the H29 yield loses one orderof magnitude, and then two orders for the fundamentalellipticity 0.3.In contrast, the yield of the resonant H17 harmonicnot only starts from significantly higher value forthe linearly-polarised field, but remains on an almostconstant level up to the fundamental ellipticity 0.3,increasing slightly at low fundamental ellipticities andreaching a local maximum at 0.15. Note that aslightly anomalous behaviour of nonresonant harmonicsexplained by the quantum interference effects was shownin [37], which is smoothened out by propagation effects.The exponential decrease of the H17 yield with the samerate as for H29 starts at the fundamental ellipticity0.35. We note that the black diamond in Fig. 3(a)shows the numerical result for three times more spatiallydense grid, which allows to resolve the higher angularmomenta appearing for high fundamental ellipticities.The convergence of the results for the lower elliplicitieshas been reached.We also present the results for smaller (520 nm)and larger (1300 nm) fundamental field wavelengths inFig. 3, graphs (b) and (c), where H11 and H25 are theresonant harmonics, correspondingly. The behaviourof the yield in these cases is similar to the case of800 nm, with either a slightly slower (520 nm) or slightlyfaster (1300 nm) decrease with increasing fundamentalellipticity; this can be explained within the recollisionpicture for a less or more diffused electron wavepacket,respectively. The resonant yield of the H11 in Fig. 3(b)decreases very slowly, and for H25 in Fig. 3(c) it evenreverses its decrease briefly to reach a local maximum atellipticity 0.2. Results for relatively high fundamentalellipticities [ > . / . for panels (b)/(c)] are obtainedwithin a less dense spatial grid, and therefore are notprecise.We compare our numerical results with the toy model(see Sec. III), which is shown with green lines inFigs. 3(a-c). The analytical results only capture theresonant (four-step) mechanism of HHG, while the fullresonant harmonic yield also includes the nonresonant(three-step model) contribution. This leads to thedifference between results obtained with analytical toymodel and numerical simulations. However, the ap-pearance of local maxima can still be explained by thegeometry of the AIS, which was chosen as the simplestgeometry ( p -orbitals) allowing the AIS – ground statetransition.Also the analytical model reproduces much slowerdecrease of the resonant harmonic intensity with thelaser ellipticity for 520 nm and 800 nm fundamentaland comparable decreases of resonant and non-resonantharmonics for 1300 nm fundamental wavelength. Figure 3. Harmonic intensity as a function of the fundamen-tal ellipticity for fundamental wavelengths (a) 800 nm, (b)520 nm and (c) 1300 nm. Pairs of resonant (red) and oneof nonresonant (blue) harmonic intensities calculated withinthe SAE TDSE are presented for each wavelength. Greenlines show analytical resonant contribution to harmonicintensity.
B. Harmonic polarisation properties
Here we study the polarisation properties — therotation angle and the harmonic ellipticity — of high
Figure 4. The harmonic rotation angle as a function of theharmonic order for the fundamental wavelength 800 nm andfundamental ellipticities from 0.05 to 0.2 calculated withinthe SAE TDSE.Figure 5. Harmonic ellipticity as a function of the harmonicorder for the fundamental wavelength 800 nm and funda-mental ellipticities from 0.05 to 0.2 calculated within theSAE TDSE. harmonics, generated by tin ions in the EP field, asdiscussed in the previous section.Figure 4 shows the rotation angle (3) of the polari-sation ellipse of resonant and nonresonant harmonics,generated by the field with 800 nm wavelength, asa function of the harmonic order for fundamentalellipticities 0.05 to 0.2. Here one can see that therotation angle of harmonics around the resonance (H17)behaves irregularly with the harmonic order, and therotation angle value becomes relatively large, while forharmonics far form the resonance, this behaviour isdemonstrated to be smooth [43] and the rotation angletypically decreases for higher orders.However, the most interesting and practically im-portant characteristic of emitted harmonics is theirellipticity. It is calculated using (4), and presented
Figure 6. Resonant (solid) and nonresonant (dashed) har-monic ellipticity as a function of the fundamental ellipticitycalculated for the fundamental wavelengths 520 nm (blue),800 nm (green) and 1300 nm (red) within the SAE TDSE. in Fig. 5 for the 800 nm fundamental wavelength asa function of the harmonic order. Here it is shown thatthe harmonics around the resonance (H17) have higherellipticity absolute values than nonresonant ones in thefull absence of resonance [43, 47]. It is important tonote that the ellipticity of several harmonics above theresonance appear to be also affected by the presence ofthe resonance. This means that a number of harmonicsaround the resonance can be used for the creation of EPXUV short pulses.Note that there is one more region of high ellipticitiesin Fig. 5 lying around H27. This irregular behaviourlooks like due to a Cooper minimum in the recom-bination cross section of the generating particle [43].However, strictly speaking, in our calculations therecannot be any interference of recombination pathwaysas far as the ground state is s -orbital. The possiblereason can be either the quantum-path interferenceeffects or an influence of the model potential used inour calculations, which is similar to the Ramsauer-Townsend effect.Figure 6 presents the behaviour of the ellipticityof resonant-nonresonant harmonic couples for each ofwavelengths under consideration. Nonresonant har-monic ellipticities are presented up to the fundamentalellipticity, at which the harmonic yield drops by twoorders of magnitude with respect to the yield at zerofundamental ellipticity. Their absolute values increaseslowly with the fundamental ellipticity, as was shownin [13, 43], for all considered wavelengths.For the resonant harmonic ellipticities, on the con-trary, the behaviour is much richer: the resonantharmonic ellipticity grows faster in absolute value thanthe nonresonant one for low fundamental ellipticities;moreover, its absolute value increases with the funda-mental wavelength. Then the resonant harmonic ellip-ticity performs an anomalous behaviour: after reachingrather high harmonic ellipticity absolute values, it startschanging to high values of opposite sign. The maximumvalues of the resonant harmonic ellipticities presentedin our calculations are 0.52 (at fundamental ellipticity0.3), 0.81 (at fundamental ellipticity 0.4), and 0.78(at fundamental ellipticity 0.3) for fundamental fieldwavelengths 520 nm, 800 nm, and 1300 nm, respectively. V. CONCLUSIONS AND OUTLOOK
To summarise, in this study we conduct numericalintegration of the 3D TDSE for resonant HHG in an EPlaser field and also present a toy model of this processexplaining our numerical results. Our calculations arecarried out for three different wavelengths and showsimilar behaviour for all of them.We show that the resonant harmonic yield behavesanomalously as opposed to the rapid decrease of thenonresonant harmonic with the fundamental ellipticity.As a result, for the ellipticities above the threshold one,only the resonant harmonic is generated. This can beexplained within the toy model: as far as the AIS ismuch less localised than the ground state, the electronreturning back to the parent particle is much more likely to be trapped into the AIS than directly recombine intothe ground state in the EP field, especially for laserellipticities larger than the threshold one.Moreover, the ellipticity of resonant harmonics ishigher than the nonresonant ones, without compromis-ing the harmonic yield. This offers us a new route forthe generation of quasi-monochromatic XUV with highellipticity.As a future direction of studies, we would suggestto search for better generating particles, providing abetter (wider or in a different spectral region) rangeof enhancement and higher yield, as well as to studythe effect of the detuning from the resonance, whichshould provide a handle to control the ellipticity of theresonant harmonic, as well as nonresonat ones aroundit, and correspondingly, of the emitted XUV pulses.
ACKNOWLEDGMENTS
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