High ellipticity of harmonics from molecules in strong laser fields of small ellipticity
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n High ellipticity of harmonics from molecules in strong laser fields of small ellipticity
F. J. Sun , C. Chen , , ∗ , W. Y. Li , X. Liu , W. Li , and Y. J. Chen , † (Dated: January 14, 2021)We study high-order harmonic generation (HHG) from aligned molecules in strong ellipticallypolarized laser fields numerically and analytically. Our simulations show that the spectra and po-larization of HHG depend strongly on the molecular alignment and the laser ellipticity. In particular,for small laser ellipticity, large ellipticity of harmonics with high intensity is observed for parallelalignment, with forming a striking ellipticity hump around the threshold. We show that the inter-play of the molecular structure and two-dimensional electron motion plays an important role here.This phenomenon can be used to generate bright elliptically-polarized EUV pulses. I. INTRODUCTION
In recent decades, high-order harmonic generation(HHG) from atoms [1–3], molecules [4–6] and solids [7–9]has been a hot subject in experimental and theoreticalstudies of strong laser-matter interaction. The HHG hasshown its promising applications as a seed to generateattosecond pulses. It can also be used as a unique tool toprobe the ultrafast dynamics of electrons within atomsand molecules with unprecedent attosecond resolution.The HHG can be well understood with the classical[10] and quantum [11] three-step models. These mod-els describe the HHG as a three-step process: the va-lence electron of atoms or molecules is ionized by thelaser field; the freed electron propagates in the externalfield; the electron is driven by the laser field to returnto and recombine with the parent ions with the emissionof a high energy harmonic along the direction parallelto the laser polarization (parallel harmonic). The quan-tum model [11], frequently termed as strong-field approx-imation (SFA), also predicts the emission of harmonicsalong the direction perpendicular to the laser polariza-tion (perpendicular harmonic). For a linearly-polarizedlaser field and for linear symmetric molecules, the per-pendicular harmonics appear only for molecular targetsaligned along a direction not parallel or perpendicular tothe laser polarization [12]. According to the SFA, paral-lel and perpendicular harmonics are emitted at the sameinstant and have no phase differences between them. Sothe whole harmonics emitted are also linearly polarized.Experimental studies indeed have observed the el-liptical polarization of harmonics emitted by alignedmolecules in a linearly polarized laser field [13]. Thispolarization effect of HHG has attracted great atten-tions in recent years, as it implies that one can acquireelliptically-polarized XUV or EUV ultrashort pulses withHHG in a linearly-polarized laser field [14]. A great dealof experimental and theoretical efforts have been devotedto the complex origins of this HHG polarization [15–21].On the whole, this polarization is associated with theatomic or molecular properties in strong laser fields. It occurs for the harmonics the parallel or perpendicularcomponents of which are subject to certain destructiveinterferences so that different contributions are involvedin the emission of these two components, resulting in ainherent phase difference between them.When a two-dimensional (2D) laser field is used, suchas a elliptically-polarized laser field [22–28], it is naturalto think that the HHG from both atoms and moleculesin such laser fields is also elliptically polarized. However,in this case, how the molecular properties will affect thispolarization is not very clear, especially when the minorcomponent of the 2D laser field is weak so that it destroysthe symmetry of the laser-driven system but does nothave a remarkable influence on the electron dynamics.In this paper, we focus on the polarization of HHGfrom aligned molecules [29, 30] in strong elliptical laserfields with small ellipticity. When the spectral propertiesof such HHG have been studied widely [31–35], the po-larization properties of relevant HHG are less explored.We pay attention to the issue if it is possible to obtaina bright and ultrashort pulse with large ellipticity usingHHG polarization of molecules in such laser fields.Based on the numerical solution of the time-dependentSchr¨odinger equation (TDSE) and SFA, our simulationsshow that the polarization of HHG from H +2 is stronglydependent on the molecular alignment and the laser ellip-ticity. The HHG of the molecule shows the striking ellip-tical polarization located at some harmonic energy, andthis location shifts towards higher energy as increasingthe alignment angle θ (the angle between the molecularaxis and the main component of the elliptical laser field).In particular, for parallel alignment of the molecule, forwhich the HHG elliptical polarization does not occurin a linearly-polarized laser field, harmonics around thethreshold show large ellipticity for a small laser ellipticity,with forming a striking ellipticity hump located at a wideenergy region. This phenomenon can be attributed to theinterplay of the molecular structure and laser-induced 2Delectron dynamics. The small component of the ellipti-cal laser field destroys the symmetry of the laser-drivensystem at θ = 0 ◦ , allowing the emission of perpendicularharmonics at this angle. At the same time, the parallelharmonics of θ = 0 ◦ around the threshold are subjectto destructive intramolecular interference related to themain component of the elliptical laser field, resulting indifferent phases of parallel and perpendicular harmon-ics. This phenomenon holds as we change the internu-clear distance and the molecular species. Because near-threshold harmonics including both parallel and perpen-dicular components generally have remarkably higher in-tensities than those in the HHG plateau, the ellipticityhump for parallel alignment therefore can be used to ob-tain a strong elliptically-polarized EUV pulse. II. THEORETICAL METHODSA. Numerical method
We assume that the main component of the ellipti-cal laser field is polarized along the direction parallelto the x axis and the minor one is along the y axis.In addition, the molecular axis is located in the xy plane. Then the elliptical electric field can be writtenas E ( t ) = e x E x ( t ) + e y E y ( t ) with E x ( t ) = f ( t ) E sin ω t and E y ( t ) = εf ( t ) E sin( ω t + φ ) and φ = π/ e x ( e y )is the unit vector along the x ( y ) axis (i.e., the major(minor) axis of the polarization ellipse). ε is the laserellipticity, ω is the laser frequency, and f ( t ) is the en-velope function. E ≡ E ( ε ) = E / p (1 + ε ) and E is thelaser amplitude relating to the peak laser intensity I .In the length gauge, the Hamiltonian of the moleculeinteracting with the elliptical laser field can be writtenas H ( t ) = p / r ) + r · E ( t ) (in atomic units of ~ = e = m e = 1 ). Here, V( r ) is the Coulomb po-tential of the molecule. For the H +2 system first ex-plored in the paper, the potential V( r ) used has theform of V ( r ) = − Z/ p ζ + r − Z/ p ζ + r with r =( x ± R cos θ ) + ( y ± R sin θ ) in 2D cases. Here, ζ = 0 . θ is the alignment angle. Z isthe effective nuclear charge which is adjusted such thatthe ionization energy of the model molecule at the in-ternuclear distance R is I p =1 . R = 2 a.u., we have Z = 1.Numerically, the TDSE of i ˙Ψ( t ) = H ( t )Ψ( t ) is solvedwith the spectral method [36]. We use a grid size of L x × L y = 204 . × . x =∆ y = 0 . x and y axis, respectively. To elim-inate the spurious reflections of the wave packet from theboundary, a mask function F ( r ) is used in the boundaryto absorb the continuum wave packet. For the x direc-tion, we have used F ( x ) = cos / [ π ( | x | − x ) / ( L x − x )]for | x | ≥ x and F ( x ) = 1 for | x | In analytical treatments, we assume that the molecularaxis is along the z axis, and the laser field E ( t ) is locatedin the xz plane with an angle θ between its major axisand the molecular axis. Then according to the SFA, thetime-dependent dipole moment can be written as [11] x ( t ) = i Z ∞ dτ (cid:2) ξ ( τ ) d ∗ r ( p st + A ( t )) e − iS ( p st ,t,τ ) × E ( t − τ ) · d i ( p st + A ( t − τ )) (cid:3) + c.c.. (3)Here, τ = t − t ′ is the excursion time of the rescatteringelectron in the driving laser field when it is ionized at thetime t ′ , and ξ ( τ ) = ( πǫ ′ + i τ ) with infinitesimal ǫ ′ . Theterm p st ≡ p st ( t, t ′ ) = − t − t ′ R tt ′ A ( t ′′ ) dt ′′ is the canonicalmomentum, and A ( t ) = − R t E ( t ′ ) dt ′ is the vector po-tential of the external field E ( t ). The term S ( p st , t, τ ) = R tt ′ dt ′′ { [ p st + A ( t ′′ )] + I p } is the semiclassical action. Theterm d i ( p ) is the bound-free dipole transition matrix el-ement between the molecular ground state | i and thecontinuum | p i (which is approximated with the planewave | e i p · r i ) in the ionization step and can be written as d i ( p ) = h p | r | i = (2 π ) − / R d r e − i p · r r h r | i . Similarly,the term d r ( p ) = h p k | r | i = (2 π ) − / R d r e − i p k · r r h r | i is that in the recombination step with the effective mo-mentum p k which considers the Coulomb correction onthe momentum p of the continuum | p i ∝ | e i p · r i in re-combination. Note, this correction changes only the mo-mentum p of the state | p i , with assuming the energy E p = p / | p i unchanged. For linearly-polarized cases, one can use the expression of p k = p | p | p k with p k = p E p + I p ) [37]. We will discuss the formof the effective momentum p k for the present elliptically-polarized cases with small laser ellipticity later.Through fourier transform of x ( t ), the coherent partof the spectrum along the major axis e k of the ellipticallaser field can be written as F l ( ω ) = i Z dt Z ∞ dτ (cid:2) ξ ( τ ) e k · d ∗ r ( p st + A ( t )) × E ( t − τ ) · d i ( p st + A ( t − τ )) e − iS ( p st ,t,τ ) e iωt (cid:3) . (4)The integration in the above expression can be treatedwith solving the saddle-point equation [11, 38][ p st + A ( t ′ s )] / I p = 0;[ p st + A ( t s )] / I p = ω. (5)The first equation describes the tunneling process withthe ionization momentum p si = p st + A ( t ′ s ), and the sec-ond equation describes the recombination process withthe recollision momentum p sr = p st + A ( t s ). These mo-menta generally have complex forms in 2D laser fields.Solving the saddle-point equations, one can get thesaddle-point ionization time t ′ s and saddle-point returntime t s of the rescattering electron, as well as the saddle-point momentum p st ( t s , t ′ s ). So Eq. (4) can be simplifiedas F l ( ω ) ∝ X s [ G ( t s , τ s ) e k · d ∗ r ( p sr ) E ( t ′ s ) · d i ( p si ) S p ( ω )] . (6) Here, G ( t s , τ s ) = ξ ( τ s )[1 /det ( t s , τ s )] / , and det ( t s , τ s )denotes the determinant of the 2 × t and τ [39]. The term τ s = t s − t ′ s is the saddle-point traveltime. The sum in Eq. (6) extends over all possible sad-dle points ( t ′ s , t s ) for the emission of a harmonic ω . Theterm S p ( ω ) has the form of S p ( ω ) = e − i ( S s − ωt s ) , with S s ≡ S ( p st , t s , τ s ). The real parts of the saddle points( t ′ s , t s ) have been considered as the ionization and returntimes of the electron, respectively. The saddle pointshave also been termed as electron trajectories includinglong trajectory, short trajectory and multiple returns.The trajectories are well resolved in the temporal regionand have different ionization and return times.In Eq. (6), these two dipoles e s · d i ( p si ) and e k · d ∗ r ( p sr )are mainly responsible for the angle dependence of HHG,as discussed in [40]. The symbol e s denotes the unitvector along the laser polarization of the electric field E ( t ) at t = t ′ s . We write the product of these two dipolesat the saddle point ( t s , t ′ s ) as M s ( ω, θ ) = | e s · d i ( p si ) | · | e k · d ∗ r ( p sr ) | . (7)With describing the ground-state wavefunction using lin-ear combination of atomic orbitals-molecular orbitals(LCAO-MO) approximation, the dipole moment for H +2 with 1 σ g valence orbital can be written as [37] d σ g ( p ) = N σ g [ − i cos( p · R · d s ( p )] . (8)Here, N σ g is the normalization factor, and R is the vec-tor between the two atomic cores of the molecule. Theterm cos( p · R / 2) denotes interference between these twocores and d s ( p ) denotes the atomic dipole moment of1 s orbital. When the laser ellipticity ε is small so thatthe main component of the elliptical laser field dominatesionization, the ionization momentum p si can be approx-imately expressed with p si ≈ ± iκ = ± i p I p [41], thenEq. (7) can be rewritten as [37, 40, 42, 43] M s ( ω, θ ) ∝| cos( iκ R θ I ) | M rs ( ω, θ ) M as , (9)with M rs ( ω, θ ) = | cos( p ′ sr R θ R ) | . (10)Here, M as ≡| ( e s · d si ( p si ))( e k · d s ∗ r ( p ′ sr )) | , and θ I ( θ R ) is the exit (recollision) angle between the vectors p si ( p sr ) and R [44]. In Eq. (10), we have used theeffective momentum p ′ sr instead of p sr in the recom-bination dipole to consider the Coulomb correction, asdiscussed in Eq. (3). For the elliptical laser field witha small ellipticity ε explored here, the effective momen-tum p ′ sr used has the form of p ′ sr = e k p ′k + e ⊥ p ⊥ with p ′ sr = q p ′ k + p ⊥ . Here, the symbol e ⊥ denotes the unitvector along the minor axis of the elliptical laser field, p ′k = p ω − εI p ] and p ⊥ being the real part of the -11-9-7 20 40 60 800.00.40.8 l og [I n t en s i t y ] short full(a) 406080100 H a r m on i c o r de r ( i n un i t s o f w ) (c) E lli p t i c i t y Harmonic order (in units of w ) (b) 4.5 5.0 5.5406080100 Time (in units of 2 / ) (d) Figure 1: Spectra (a) and ellipticity (b) of harmonics foraligned molecules H +2 at θ = 0 ◦ and ε = 0 . 1, obtained withshort-trajectory (black square) and full (red circle) TDSE sim-ulations. In (c) and (d), we show relevant time-frequencydistributions for short-trajectory (c) and full (d) simulations. e ⊥ component of p sr = p st + A ( t s ) at the saddle point( t ′ s , t s ). Accordingly, the recollision angle θ R has the ex-pression of θ R = arctan( p ⊥ /p ′k ) − θ .As discussed in [37], for H +2 in linearly polarized laserfields, the term M rs ( ω, θ ) in Eq. (9) is the most sensi-tive to the molecular alignment. Our simulations showthat the situation also holds for the cases of small laserellipticity in the paper. In the following, we will com-pare the predictions of M rs ( ω, θ ) of Eq. (10) with theTDSE results to understand the angle-dependent HHGellipticity. III. RESULTS AND DISCUSSIONS In Fig. 1, we show the HHG results of full TDSEsimulations and short-trajectory simulations at ε = 0 . θ = 0 ◦ . As the spectra of full simulations show thecomplex interference structure, the short-trajectory oneis more smoothing, as seen in Fig. 1(a). Accordingly,the ellipticity of short-trajectory harmonics is also moreregular than the full-simulation one, with showing a re-markable ellipticity hump around the threshold, as seenin Fig. 1(b). This ellipticity hump at θ = 0 ◦ whichdisappears in a linearly-polarized laser field is the mainphenomenon we will discuss in the paper. In Fig. 1(d).the time-frequency analysis of TDSE dipole acceleration[45] also clearly shows the complex interference betweendifferent electron trajectories in full TDSE simulations,as the interference structure is basically absent in short-trajectory results in Fig. 1(c). In the following, we focuson the short-trajectory results which allow a clear iden-tification of the angle-dependence polarization of HHG -14-10-6-14-10-6-14-10-6 20 40 60 80-14-10-6 20 40 60 80parallel(a)q=0(cid:176) perpendicular e=0.0 e=0.1 e=0.2 e=0.3(e)q=0(cid:176)(b)q=30(cid:176) (f)q=30(cid:176)(c)q=45(cid:176) (g)q=45(cid:176)(d)model atom Harmonic order (in units of w ) (h) l og [I n t en s i t y ] model atom Figure 2: Spectra of parallel (a-d) and perpendicular (e-h)harmonics for aligned molecules H +2 at θ = 0 ◦ (a,e), 30 ◦ (b,f),45 ◦ (c,g) and for model atom with similar I p to H +2 (d,h),with the laser ellipticity of ε = 0 (black dotted), ε = 0 . ε = 0 . ε = 0 . from molecules in the elliptical laser field.To understand the polarization of HHG from alignedmolecules in elliptical laser fields, in Fig. 2, we plot theTDSE short-trajectory spectra of parallel versus perpen-dicular harmonics of H +2 at different angles θ and laserellipticity ε . For comparison, the results of a model atomwith I p = 1 . θ = 0 ◦ in Fig. 2(e) and for theatom case in Fig. 2(h), the perpendicular harmonics dis-appear for ε = 0 corresponding to a linearly-polarizedlaser field, due to the symmetry of the laser-driven sys-tem in the cases [12]. In addition, results in Fig. 2 alsoshow that for lower harmonic orders below the threshold(about H19), the yields of parallel and perpendicular har-monics are remarkably higher than those in the plateauregions of the spectra and are not sensitive when chang-ing the laser ellipticity. In particular, they are compara-ble at small laser ellipticity, suggesting the possibility of E lli p t i c i t y e=0 (a) e=0.1 (d) e=0.2 =0 =30 =45 (g) l og [I n t en s i t y ] (b) (e) (h) l og [ A m p li t ude ] (c) Harmonic order (in units of w ) (f) (i) Figure 3: Ellipticity (the first row), spectra of parallelharmonics (second) and the corresponding function curves M rs ( ω, θ ) of Eq. (10) (third) for aligned molecules H +2 atdifferent angles of θ = 0 ◦ (black curves), θ = 30 ◦ (red), and θ = 45 ◦ (blue) and different laser ellipticity of ε = 0 (the firstcolumn), ε = 0 . ε = 0 . θ = 0 ◦ (solid), θ = 30 ◦ (dashed) and θ = 45 ◦ (dotted) are also plotted withgray curves for comparison. a high harmonic ellipticity according to Eq. (2). Withthe help of the horizontal arrows in each panel, a care-ful comparison for the molecular cases also shows thatboth of the parallel and perpendicular spectra differ forthe molecular alignment and the parallel spectra in theplateau region are stronger at θ = 0 ◦ than other cases.We will return to this point later.In Fig. 2, our discussions mainly concentrate on the de-pendence of HHG yields of parallel versus perpendicularharmonics on the laser ellipticity for a certain alignmentangle. In Fig. 3, we present the comparison betweenHHG ellipticity, the corresponding spectra and dipolesof Eq. (10) at different angles for a certain laser ellip-ticity. Firstly, for ε = 0 of the linearly-polarized case inFig. 3(a), harmonics at θ = 30 ◦ and θ = 45 ◦ show largeellipticity, with forming an ellipticity peak around someharmonic energy. The position of the ellipticity peakshifts with the increase of the angle and corresponds tothe minimum in the relevant parallel spectrum in Fig.3(b). These spectral minima are associated with two-center interference and are well described by the corre-sponding minima in the function curves of Eq. (10) inFig. 3(c). For simplicity, we will call the function curveof Eq. (10) the dipole below. A careful comparison tellsthat the spectrum at θ = 0 ◦ in Fig. 3(b) also shows a minimum at about H21 which is near the threshold, inagreement with the prediction of M rs ( ω, θ ) in Fig. 3(c).However, the harmonics at this angle do not show the el-lipticity due to the absence of perpendicular harmonics.For the case of a small laser ellipticity of ε = 0 . θ = 0 ◦ , which hasbeen indicated in Fig. 1. For this laser ellipticity in Fig.3(d), harmonics at θ = 30 ◦ and θ = 45 ◦ also show theremarkable ellipticity peak, the magnitudes of which aresomewhat smaller than the corresponding ones in Fig.3(a). When comparing with the spectra in Fig. 3(e),one can also observe that large ellipticity of harmonicsgenerally appears at the harmonic orders at which theharmonic spectrum shows a striking minimum, as dis-cussed in the first column of Fig. 3. The positions of thespectral minima in Fig. 3(e) also basically agree with thepositions of the dipole minima, as shown in Fig. 3(f). Incomparison with the results in Fig. 3(c), one can alsoobserve from Fig. 3(f) that the minima in dipoles for in-termediate angles of θ = 30 ◦ and θ = 45 ◦ shift somewhattowards lower energy, while the minimum in the dipoleof θ = 0 ◦ shifts somewhat towards higher energy. Thisis due to the influence of the minor component of theelliptical laser field, which changes the recollision angleof HHG. We will address the question in detail in Fig. 4.Comparing the spectra at different angles in Fig. 3(e),it is clear that the spectrum of θ = 0 ◦ is somewhat lowerthan those of θ = 30 ◦ and θ = 45 ◦ at lower harmonicorders near and below the threshold, but is remarkablyhigher than those in the HHG plateau region. Since theyields of below-threshold harmonics for θ = 0 ◦ are oneorder or several orders of magnitude higher than thosein the plateau region for intermediate angles of θ = 30 ◦ and θ = 45 ◦ , one can expect that EUV pulses with highellipticity obtained with the ellipticity hump of θ = 0 ◦ atlower harmonic orders will also be remarkably brighterthan those obtained with the plateau harmonics of inter-mediate angles. We will discuss this point in Fig. 5.With further increasing the laser ellipticity to ε = 0 . p ′ sr used here is more applicable for small ellipticity.Inaddition, the spectral minima can be influenced by thecontributions of excited states to HHG which are not in-cluded in SFA. By comparison, the intersections of thespectra at different angles match better with those of rel-evant dipoles in Fig. 3. Similar phenomena have beendiscussed in [37, 40]. 20 40 60 800.00.2 20 40 60 80 -0.20.00.20.00.40.8 0.00.40.820 40 60 800.00.40.8 20 40 60 80 0.00.20.40.6 q R ( i n un i t s o f p ) q=0(cid:176)(a) q R ( i n un i t s o f p ) Harmonic order (in units of w ) =0.0 =0.1 =0.2 =0.3(b) q=30(cid:176) E lli p t i c i t y (c) R=2.0 a.u. E lli p t i c i t y e=0.1 e=0.2 e=0.3(d) model atom(e) R=1.7 a.u.Harmonic order (in units of w ) (f) R=2.3 a.u. Figure 4: Recollision angles of HHG short trajectories fromaligned molecules H +2 at θ = 0 ◦ (a) and θ = 30 ◦ (b) fordifferent laser ellipticity ε . In (c)-(f), we show the ellipticityof harmonics for H +2 with R = 2 a.u. (c), R = 1 . R = 2 . θ = 0 ◦ and for model atom with similar I p to H +2 (d). The laser ellipticity used for obtaining the curvesis as shown. From the results in Fig. 3, one can conclude that thelarge ellipticity of harmonics for a small laser ellipticityis also closely associated with the effect of intramolecularinterference, which plays an important role in the emis-sion of parallel harmonics and has a relatively small rolein the perpendicular one, resulting in a phase differencebetween parallel and perpendicular harmonics. From theperpendicular spectra of gray curves in the second row ofFig. 3 (also see Fig. 2), one can observe that the perpen-dicular spectra do not show the striking hollow structurerelating to two-center interference, as seen in the corre-sponding parallel spectra there. On the whole, the inten-sities of perpendicular spectra gradually decrease withthe increase of harmonic order. For the case of parallelalignment of θ = 0 ◦ , this interference between these twoatomic centers just occurs at lower harmonic energy nearthe threshold, leading to a strong ellipticity of lower har-monic orders. The influence of two-center interferenceon ellipticity of HHG from aligned molecules for inter-mediate angles in a linearly-polarized laser field has beendiscussed in [15, 19–21]. Here, we focus on the case ofparallel alignment and elliptically-polarized laser fields.Next, we further discuss the mechanism of the angle-dependent HHG ellipticity observed in Fig. 3. Since theinterference term M rs ( ω, θ ) of Eq. (10) in the recombina-tion dipole gives a good description for the dependenceof HHG ellipticity on the molecular alignment and the -7 -7 -9 -9 -9 -9 -7 -7 T i m e p r o f il e H e=0.1 q=0 H9-H17 (a) T i m e p r o f il e I || (t) I (t)H e=0.1 q=30 H33-H41 (b) H e=0.0 q=30 H33-H41 (c) Time (in units of optical cycle) atom e=0.1 H9-H17 (d) Figure 5: Trains of pluses I k ( t ) and I ⊥ ( t ) obtained from spe-cific HHG spectral region for aligned molecule H +2 at θ = 0 ◦ (a) and θ = 30 ◦ (b,c) and for model atom with similar I p toH +2 (d). In each panel, the laser ellipticity used for obtainingthe HHG spectrum and the corresponding specific spectralregion are as shown. laser ellipticity, we further analyze the implication of thisterm. In Figs. 4(a) and 4(b), we show the recollision an-gles θ R , associated with short electron trajectories anddefined in Eq. (10), for θ = 0 ◦ and θ = 30 ◦ at differ-ent ε . For the case of parallel alignment in Fig. 4(a),on the whole, the recollision angle gradually decreases asincreasing the harmonic order for a certain laser ellip-ticity and increases with the increase of laser ellipticityfor a certain harmonic order. In particular, for ε = 0,the recollision angle is zero and agrees with the align-ment angle of θ = 0 ◦ . For the intermediate angle of θ = 30 ◦ , the situation is different, as shown in Fig. 4(b).In the case, the absolute value of the recollision anglebasically decreases as increasing the laser ellipticity andthis decrease is more striking for lower harmonic orders.According to Eq. (10), in the region of | θ R | ∈ [0 , π/ θ R , the value ofcos θ R decreases, and the minimal value of the function M rs ( ω, θ ) will appear at larger p sr . Therefore, for θ = 0 ◦ ,the increase of the laser ellipticity which gives rise to theincrease of θ R will induce the shift of the minimum of Eq.(10) towards somewhat larger harmonic energy. This sit-uation reverses for θ = 30 ◦ . These analyses explain theresults in the third row of Fig. 3 and shed light on thespectral and polarization results in other rows of Fig. 3.In comparison with intermediate angles, the minimumfor θ = 0 ◦ of H +2 will appear at lower harmonic energyat which the harmonic spectra usually have larger am-plitudes. This effect also holds for other internuclear dis-tances of H +2 and for other molecular targets such as N with other symmetries. We will discuss the case of N later. Therefore, the parallel alignment of molecules ispreferred for obtaining a bright EUV pulse with high el-lipticity. As a comparison, in Figs. 4(c) to 4(f), we showthe HHG polarization results of H +2 at different inter-nuclear distances and laser ellipticity, and we also showthe results of the model atom. One can observe thatthe remarkable polarization phenomenon at lower har-monic energy appears in all of the cases. However, whenthe results of molecules show an ellipticity hump aroundthe threshold on the whole, the ellipticity curves of themodel atom are somewhat shaper. From the perspectiveof shaping a short EUV pulse with a broad energy regionof harmonics, the ellipticity hump is also preferred, asshown in Fig. 5.In Fig. 5, we plot trains of pulses I k ( t ) and I ⊥ ( t ),obtained from the HHG spectra of H +2 at θ = 0 ◦ . Wealso compare the results to cases of H +2 at θ = 30 ◦ andto the model atom. The expressions of I k ( t ) and I ⊥ ( t )are as follows [46, 47]: I k ( ⊥ ) ( t ) = | Z ω u ω d F k ( ⊥ ) ( ω ) e − iωt dω | . (11)Here, F k ( ⊥ ) ( ω ) are the parallel and perpendicular HHGspectra of Eq. (1). For H +2 at θ = 0 ◦ and for the modelatom, we consider the case of ε = 0 . 1, and our calcula-tions are performed for the spectral region from H9 toH17 (i.e., ω d = 9 ω and ω u = 17 ω ), at which the har-monics have large ellipticity and high intensities. For H +2 at θ = 30 ◦ , the cases of ε = 0 and ε = 0 . +2 at θ = 0 ◦ with ε = 0 . 1, the obtainedparallel and perpendicular pulses are comparable for in-tensities with a remarkable time delay, as seen in Fig.5(a). For the case of θ = 30 ◦ with ε = 0 . ε = 0 in Fig. 5(c), the synthesizedparallel and perpendicular pulses show comparable in-tensities and a small time delay. Note, for ε = 0, theHHG from H +2 at θ = 0 ◦ does not show the elliptical-polarization effect due to the absence of perpendicularharmonics. However, the intensities of the pulses shownin Figs. 5(b) and 5(c) for θ = 30 ◦ are one order of magni-tude lower than those in Fig. 5(a) for θ = 0 ◦ . The resultsof the model atom in Fig. 5(d) are somewhat similar tothose in Fig. 5(b), with the perpendicular pulse showingsmall intensities. The results in Fig. 5 support our previ-ous discussions that harmonics of molecules with parallelalignment are preferred for obtaining a bright elliptically-polarized EUV pulse.To confirm our discussions above, we have also per-formed calculations for the N molecule with 3 σ g va-lence orbital, which can be operated more easily in ex-periments. To simulate the HHG of N with I p =0.57a.u., we have used the model potential V( r ) [48] withthe form of V( r ) = − [( Z − Z ) e − ρ r + Z ] / p ζ + r − [( Z − Z ) e − ρ r + Z ] / p ζ + r . The expressions of r are as for H +2 . The relevant parameters used are Z = 5, -11-9-7 02x10 -8 -8 -8 10 20 300.00.20.40.6 6.00 6.25 6.50 6.75 01x10 -8 -8 -8 l og [I n t en s i t y ] N q=0(cid:176) N q=30(cid:176) N q=45(cid:176) model atom (a) I || (t) I ^ (t) T i m e p r o f il e (c) N e=0.1 q=0(cid:176) H23-H31 E lli p t i c i t y Harmonics order (in units of w ) (b) Time (in units of optical cycle) (d) model atom e=0.1 H23-H31 Figure 6: Spectra of parallel harmonics (a) and ellipticityof harmonics (b) for aligned molecules N at θ = 0 ◦ (blacksquare), θ = 30 ◦ (red circle), θ = 45 ◦ (blue triangle), and formodel atom with similar I p to N (green star), obtained withshort-trajectory TDSE simulations. In (c) and (d), we showtrains of pulses I k (black) and I ⊥ (red) obtained from specificHHG spectral region of H23 to H31 for N (c) and the modelatom (d). The laser parameters used are I= 1 . × /cm and λ = 1400 nm with ε = 0 . Z = 0 . R = 2 . 079 a.u., ζ = 0 . ρ = 1 . comparedwith H +2 , here, we use a weaker driving laser intensity of I = 1 . × /cm and a longer laser wavelength of λ = 1400 nm. Relevant results at different angles θ with ε = 0 . I p to N are also presentedhere. We focus on lower harmonic orders near the thresh-old. One can observe from Fig. 6(a) that the parallelspectra of N at θ = 0 ◦ show the larger intensities thancases of other angles and the model atom. Around H25,somewhat higher than the threshold harmonic of H19,harmonics of θ = 0 ◦ also show a ellipticity hump, moreremarkable than other cases, as seen in Fig. 6(b). Wemention that in our extended simulations with chang-ing the laser parameters, this hump holds. The trains ofpulses obtained from the spectral region of H23 to H31for N at the parallel alignment and for the model atomare presented in Figs. 6(c) and 6(d). When the results ofmodel atom for parallel and perpendicular pulses of I k ( t )and I ⊥ ( t ) do not show a obvious time delay in Fig. 6(d),the time delay for results of N can be clearly identifiedin Fig. 6(c). In addition, the relative intensity of theperpendicular pulse in comparison with the parallel oneis also remarkably larger for N than for the model atom.Different from H +2 , whose ground-state wavefunctionis dominated by the 1 s orbital, the valence-orbital wave-function of real N is composed of the 2 p z orbital (about70%) and 1 s and 2 s orbitals (about 30%) [31]. Thebound-continuum transition dipole d ( p ) calculated with2 p z -combination molecular orbital has a sin-type interfer-ence term, in contrast to the cos-type one observed in Eq.(8). For cos-type interference term, the minimum of Eq.(10) appears at the momentum of p sr = π/ ( R cos θ R ),and for sin-type one, that appears at p sr = 2 π/ ( R cos θ R )which is larger than the cos one. Accordingly, theinterference minima in the HHG spectra for cos-typemolecules generally appear at lower harmonic energythan sin-type ones. This situation is somewhat morecomplex for molecules with a mix of cos-type and sin-type interference terms such as N . However, in bothcases, the minima appear at lower harmonic energy forsmaller angles θ R (which is near to the alignment angle θ for a small laser ellipticity) and harmonics at lower en-ergy near the threshold usually have larger amplitudes.It is the reason that the parallel alignment is preferredfor generating a strong elliptically-polarized EUV pulse,as explored in the paper. IV. SUMMARY In conclusion, we have studied the polarization proper-ties of HHG from aligned molecules in strong elliptically-polarized laser fields with small laser ellipticity. We haveshown that the addition of the vertical component ofthe laser field with a small intensity has important in-fluences on not only the yields but also the polarizationof harmonics. In particular, for the parallel alignmentof the molecule, for which the perpendicular harmonicsdisappear in a linearly-polarized laser field, the additionof the small vertical component induces a strong emis- sion of perpendicular harmonics and an accompanyinghigh ellipticity of harmonics near or below the thresh-old with forming a ellipticity hump. We show that thephenomenon arises from the effects of two-center inter-ference which influence differently on parallel and per-pendicular harmonics, resulting in a remarkable phasedifference between these two harmonic components. Asthe intensities of parallel and perpendicular harmonicsare usually comparable near the threshold, the inherentphase difference and comparable intensities of these twocomponents lead to the appearance of the HHG ellip-ticity hump around the threshold. By comparison, theHHG ellipticity for larger alignment angles appears athigher harmonic orders located in the plateau region ofthe HHG spectra. Because harmonics near or below thethreshold usually have remarkably higher intensities thanthose in the HHG plateau, the ellipticity hump for theparallel alignment suggests a manner for generating abright elliptically-polarized EUV pulses. Because the in-tramolecular interference generally occurs at lower har-monic orders for the parallel alignment, we expect thatthe ellipticity hump discussed here will also appear formore species of molecules. 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