Tailoring OAM spectrum of high-order harmonic generation driven by two mixed Laguerre-Gaussian beams with nonzero radial nodes

Beiyu Wang(汪倍羽), Jiaxin Han(韩嘉鑫), and Cheng Jin(金成),2,†

1Department of Applied Physics,Nanjing University of Science and Technology,Nanjing 210094,China

2MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing,Nanjing University of Science and Technology,Nanjing 210094,China

Keywords: orbital angular momentum(OAM)spectrum,high-order harmonic generation,Laguerre-Gaussian beam,radial nodes,phase-matching

1.Introduction

An optical vortex refers to a light beam characterized by a helical phase structure, resulting in a spiral wavefront and the formation of a doughnut-shaped intensity distribution.[1-4]The presence of a phase singularity at the center of the beam defines its unique properties.Each photon in such a beam carries an orbital angular momentum (OAM) ofl¯h, wherelis the topological charge,and ¯hdenotes the reduced Planck constant.The OAM associated with a vortex beam, also known as an OAM beam, can be transferred to atoms or molecules through the interaction between light and matter,[5-8]involving the exchange of momentum.The concept of the OAM beam was first demonstrated by Allenet al.in 1992 using a Laguerre-Gaussian (LG) beam.[9]The OAM beam provides an additional degree of freedom.With a unique OAM value(or single OAM mode), the vortex beam has been diversely applied in large-capacity communications,[10-13]gravitational wave detection,[14,15]optical tweezers and manipulation,[16,17]quantum information processing,[18,19]and other fields.[20-24]

It is important to note that a single vortex beam can also contain multiple OAM states, with the distribution of different OAM channels, known as the OAM spectrum, which greatly broadens its application areas.With a wide OAM spectrum, the vortex beam can be used in the ultrasensitive angular measurement,[25]the generation of arbitrary superpositions of atomic rotational states in a Bose-Einstein condensate,[26]the measurement of object parameters,[27]the performance of rotations and reflections on images,[28]and so on.To generate OAM spectra in the infrared (IR) and visible spectral ranges, there are several methods available including computer-generated holograms on the spatial light modulators,[29]specially designed metasurfaces,[30]integrated angular momentum gratings,[31,32]and coherent mixing of pre-generated vortex beams.[33]However, using optical elements is difficult to generate and tailor the OAM spectrum of light in the extreme ultraviolet(XUV)or soft x-rays.

One alternative approach is to utilize the high-order harmonic generation (HHG) resulting from the nonlinear interaction between an intense femtosecond IR laser and a gas medium.[34-41]This approach has been successfully employed to generate the vortex beam with high-value OAM in the XUV.[42,43]For example,Gairepyet al.[44]and Konget al.[45]employed a noncollimated scheme by mixing two-color laser beams to control the OAM of HHG.Sansonet al.[46,47]conducted a measurement of HHG beams using an XUV Hartmann wave-front sensor, revealing the presence of multiple vortex modes.Regoet al.[48]theoretically investigated the generation of vortex HHG driven by non-pure LG modes and identified the nonperturbative characteristics of the OAM spectrum.Williet al.[49]utilized two-color anti-spiral LG beams to generate HHG with specific OAM values, and they also coherently superimposed LG beams with different polarization states to generate the HHG and to control the OAM value.[50]Jinet al.[51]simulated the phase-matching conditions of HHG driven by a single LG beam with the nonzero OAM.Recently, Guanet al.[52]analyzed the macroscopic phase-matching mechanism for the OAM spectra of HHG by mixing two LG vortex modes and demonstrated that the OAM spectra can be readily tailored by varying the harmonic order,the position of the gas medium, and the energy ratio of two LG beams.Other aspects of OAM in the XUV through the HHG have also been intensively studied theoretically.[53-57]In these studies,the LG beam with zero radial node has been often used as the fundamental optical vortex.It is interesting to see how the vortex HHG can be changed by considering the nonzero radial nodes in the driving LG beam, especially the modification of the OAM spectrum.

Here our main goal is to identify how the radial node affects the OAM spectrum and the phase-matching condition of HHG driven by two mixed LG beams.Two LG modes are with equal wavelength and with topological charges of 1 and 2,and their radial nodes are simultaneously set as 0,1,and 2,respectively.The article is arranged in the following.Section 2 will give a brief review of theoretical methods for simulating the vortex HHG, including the propagation equations of harmonic field, the formulation of far-field harmonic emissions,and the quantitative rescattering(QRS)model,and it will also present the comparison of spatial profiles and phase distributions of three mixed LG beams.In Section 3, we will show the spatial profiles of harmonics in the plateau and the cutoff,and will give the OAM spectra of far-field harmonics.We will reveal the phase matching of harmonics by employing two approaches.One is to calculate the coherence length through the phase mismatch between the driving laser and the high harmonic, and the other is to simulate the harmonic field in the gas medium by solving the three-dimensional Maxwell’s wave equation.The conclusions will be presented in Section 4.

2.Theoretical methods

To simulate the vortex HHG driven by the mixed LG beams, we employ the general propagation theory in a gas medium, which has been established in Refs.[51,52].Here we will only adopt some key equations,and will focus on the variation of spatial features of the mixed LG beams with the radial nodes.

2.1.Propagation equations of high-harmonic field in the gas medium

We assume that the fundamental LG beams carrying orbital angular momentum (OAM) remain unchanged while propagating through an ionized gas medium,resembling their behaviors in free space.The propagation characteristics of the vortex high-harmonic field in the ionizing medium can be described by(in a Cartesian coordinate)[51,52,58]

HerePnl(x,y,z,t) is the nonlinear polarization,n0is the neutral atom density,ne(x,y,z,t) is the free electron density, andw(τ)is the tunnel ionization rate.The induced dipole momentD(x,y,z,t)is calculated in the local laser field.Effects of absorption and free electron dispersion are neglected.

2.2.Far-field harmonic emissions

Near-field harmonics are obtained from the exit face of a gas medium (zout).After a long distance propagation in vacuum,they reach the detector and are called far-field harmonics,which can be obtained in the frequency domain by using Huygens’ integral under the paraxial and Fresnel approximations(in a Cartesian coordinate)as[51,52,59]

HereL=zf-zout,zfis the far-field position from laser focus,xfandyfare the transverse coordinates in the far field,and the wave vectorkis given byk=ω/c.Note that Eq.(4)is used to calculate the harmonic field on a plane atzfperpendicular to the propagation axis, while far-field harmonics are computed in a spherical surface with a radiusL.

2.3.Quantitative rescattering model for single-atom response

Single-atom induced dipole moment, denoted asD(t) in Eq.(3), can be accurately calculated by using the quantitative rescattering(QRS)model.[60,61]In this model,D(t)is expressed in the frequency domain asD(ω),which can be written as[58]

HereNrepresents the ionization probability taken at the end of the laser pulse,d(ω) corresponds to the complex photorecombination (PR) transition dipole matrix element, andW(ω) is the complex microscopic wave packet.The QRS model improves the strong-field approximation (SFA) by using a more accurate scattering wave in the calculation of the PR transition dipole, instead of using a simple plane wave in the SFA.[62]However, the returning wave packet remains the same as that used in the SFA.In practice,the QRS obtains the accurate induced dipole moment as[60,61]

where bothDsfa(ω)anddqrs(ω)are complex numbers,whiledsfa(ω)is either a pure real or pure imaginary number.Nqrsis calculated by the ADK theory.[63]

2.4.Fundamental Laguerre-Gaussian beams

Under the paraxial and slowly varying transverse amplitude approximations, the electric field of the Laguerre-Gaussian (LG) beam can be expressed in a cylindrical coordinates as[51,52,59]

For the driving laser,we consider coherently mixing two LG vortex modes with topological charges ofl1=1 andl2=2,and their numbers of radial node are chosen asp1=p2=0,1, and 2 in three sets, respectively.Laser wavelength is fixed at 800 nm.To ensure a significant overlap between two beams in each set,we choose beam waists at the focus of 25µm forl1=1 and 17.7µm forl2=2 such that the maximum intensity appears in the same radial distance forp1=p2=0.These parameters are maintained forp1=p2/=0 even though it is not possible to achieve complete spatial overlap of two beams.To balance the energy distribution, we adjust the amplitudes ofE0so that the energies of two beams are always equal in three sets.By coherently combining the two beams,we obtain a maximum laser intensity of 2.5×1014W/cm2at the focus.This intensity is sufficiently high to drive the high-order harmonic generation(HHG)process with argon gas.

In Fig.1, we illustrate the intensity and phase distributions of three mixed LG beams.Note that near-field results are shown using the real size of the driving laser beam around the laser focus while far-field ones are displayed in terms of the divergence angle to eliminate the dependence on the propagation distance.In the near field,forp1=p2=0,the intensity distribution exhibits a single revolving and half-ring structure in Fig.1(a).Unlike the single-valued OAM beam,the cylindrical symmetry in the intensity distribution is no longer exited.The phase is not evenly changed with the azimuthal angle in Fig.1(d).With the increase of radial nodes,additional revolving ring structures appear in the intensity distribution beyond the half-ring with maximum intensity, resulting in a total ofp+1 rings,see Figs.1(b)and 1(c).Aspincreases,the radius of the half-ring with the maximum intensity decreases and the radius of outmost intensity ring increases.The phase changes in multiple incomplete rings,and there exhibits the phase shift between two neighboring rings, see Figs.1(e) and 1(f).In the far field, three mixed beams display the same number of intensity rings as that in the near field,and the ring with maximum intensity remains the most inner one in Figs.1(g)-1(i).The phase distributions of the three cases are quite different as shown in Figs.1(j)-1(l).The shape of phase distribution varies dramatically fromp=0 top=2 due to the Gouy phaseζ(z′).Note that fromz′=0 toz′=+∞,the phase shift is-π(-3/2π),-2π(-5/2π),or-3π(-7/2π)for LG1,0(LG2,0),LG1,1(LG2,1), or LG1,2(LG2,2) mode, respectively.Thus,mixing two LG beams results in the complicated Gouy phase shift in the far field.

3.Results and discussion

3.1.Comparison of spatial profiles and OAM spectra of H13 generated by mixed two LG beams

In the simulations, we employ a Gaussian envelope to model the temporal profile of the laser pulse.The pulse has a full-width-at-half-maximum (FWHM) duration of 10 optical cycles (or 26.7 fs) at 800 nm and the carrier-envelope-phase(CEP)of 0.The temporal waveform and laser wavelength are kept the same for all LG beams.The spatial part is obtained by coherently combining two LG beams as shown in Fig.1.A 1 mm long Ar gas jet with uniform density distribution is applied and its center is located at 1.5 mm after laser focus.

For three sets of mixed LG beams,we show the intensity distributions of 13th-order harmonic (H13) after propagation in the medium(near field,at exit face of gas jet)and in vacuum(far field, far away from gas jet) in Fig.2.In the near field,first, harmonic emissions are mostly distributed on the right half plane for LG1,0+LG2,0in Fig.2(a), and on the left half plane for LG1,1+LG2,1and LG1,2+LG2,2in Figs.2(b) and 2(c).Second,the distribution of harmonic intensity is centered between about 20 µm to 40 µm and its radius gets smaller and smaller whenpis increased from 0 to 2.These areas of harmonic emissions are consistent with ring regions having the maximum intensity of driving laser in Figs.1(a)-1(c).In the far field, intensity distribution of H13 by LG1,0+LG2,0remains a single half-ring structure in Fig.2(d), but multiple rings appear in the emission distribution in Fig.2(e) whenpis increased to 1.In Fig.2(f), whenp=2, the multiple ring structure is quite different from that in Fig.2(e).

Fig.1.Transverse intensity and phase profiles of mixed LG beams in the near field(z′=1.5 mm,left)and in the far field(z′=+∞,right).First row: LG1,0+LG2,0;second row: LG1,1+LG2,1;third row: LG1,2+LG2,2.The phase is defined from-π to π,and I0=1014 W/cm2.

Fig.2.Spatial intensity profiles of H13 in the near field(left column)and in the far field(middle column)driven by three sets of mixed LG beams.OAM spectra are calculated by using far-field harmonic fields and given by the perturbative theory (right column).First row: LG1,0+LG2,0;second row: LG1,1+LG2,1;third row: LG1,2+LG2,2.

We next decompose the far-field harmonic to calculate the contribution of different OAM modes.The complex electric field of far-field harmonic is expressed asEfar(ω,rfar,φ),in which thezcoordinate is omitted.We perform the azimuthal Fourier transform along the azimuthal coordinateφin the following:[52,64,65]

wherelis the topological charge of OAM mode,and the normalized weight of each mode is obtained by the integration alongrfaras

For comparison, we also evaluate the OAM spectrum of HHG in the perturbative regime.To simplify the analysis,we focus solely on the azimuthal-angle dependence in the LG beam.The electric field of LG beam can be expressed asE1,p=E1eil1φ(orE2,p=E1eil2φ) with the complex amplitudeEi=1,2and the topological chargeli=1,2.For the perturbative harmonic,its electric fieldEh(ω)is proportional to theq-th power of the driving laser field as[52]

and the normalized weight for the OAM mode with topological chargel=(q-n)l1+nl2is obtained as

whereω=qω0with the fundamental frequencyω0.

The calculated OAM spectra with far-field harmonic fields(including the phase,not shown)and perturbative OAM spectrum for H13 are both plotted in Figs.2(g)-2(i).One can see that the calculated OAM spectrum is greatly tailored by varying the radial node in the driving laser.In comparison with the perturbative OAM spectrum, the calculated one is similar to it in both the spectral range and the peak OAM mode whenp=0, see Fig.2(g).But the calculated OAM spectra have broader ranges in Figs.2(h) and 2(i) when the radial node is increased to 1 or 2.They have the same OAM peak atl=19.In addition, the OAM spectrum ofp=2 has an extra peak structure aroundl=8.This indicates that the radial node in the driving laser can not only broaden the OAM spectrum of HHG but also change its shape.

3.2.Analysis of phase-matching mechanism of H13 driven by different mixed LG beams

We then employ two approaches to uncover the origins of the change of high harmonics with the radial node in the driving laser, i.e., coherence length analysis and numerical simulations based on Maxwell’s wave equations.The second approach can be performed by solving the propagation equations of the high-harmonic field in Eq.(1).Two methods allow us to examine the phase-matching conditions and the accumulation of harmonic strength within the gas medium.For a given azimuthal angle of 90◦,intensity distributions of mixed LG laser beams,maps of the coherence length of H13,and the evolution of harmonic field in the medium are plotted in thez-rplane in Fig.3.

Let us introduce the first approach in detail.The total wave vectork1for the fundamental mixed LG beam can be expressed as[51]

whereezis the unit vector along thezdirection,k0=ω0/cwith the laser frequencyω0, andϕ(r,φ,z) is the geometric phase of the mixed LG beam.The effective wave vectorKto describe the spatial dependence of atomic phase in the singleatom response is

HereΦq,dip(r,φ,z)is the intrinsic dipole phase, which can be written as

Fig.3.Laser intensity distributions of three mixed LG beams (left column), maps of the coherence length of H13 due to short and long trajectories,respectively(two middle columns),and the evolution of harmonic field in the medium(right column).Results are shown for a fixed azimuthal angle of 90◦.First row: LG1,0+LG2,0;second row: LG1,1+LG2,1;third row: LG1,2+LG2,2.Note that the white color indicates that the coherence length is longer than 1 mm.The area between two dashed lines specifies the position of the gas medium and I0=1014 W/cm2.

We show the calculated spatial coherence length in thezrplane for H13 in Figs.3(d)-3(f)and 3(g)-3(i)by only considering harmonic emissions from either short or long trajectories.The intrinsic phase accumulated by an electron following the short or long trajectory is quite different, resulting in two distinct maps of harmonic coherence length.Spatial intensity profiles of mixed LG beams are also plotted in Figs.3(a)-3(c)for easily identifying effective phase-matching regions.These figures can help understand the spatial distributions of near-field harmonics in Fig.2.For example, in Fig.3(a), the laser intensity aroundr=20 µm is decreased within the gas medium along the propagation axis(labeled by two dashed lines).In the same spatial areas in Figs.3(d)and 3(g), the phase-matching regions for the short trajectory, indicated by the white color, are much better compared to that for the long trajectory,which is predominantly represented by blue color.The favorable phase-matching condition for the short trajectory leads to the gradual growth of the harmonic field, as depicted in Fig.3(j), especially afterz= 1.5 mm.By increasing the radial node, one can observe the significant difference in the map of coherence length.Forp=1,considering the spatial area of effective intensity distribution of driving laser,the phase-matching region of short trajectory aroundr=12µm starts from a narrow one atz=0 mm and becomes broader along the propagation axis in Fig.3(e),while the long trajectory exhibits very poor phase-matching in the same area in Fig.3(h).Due to the interference of harmonic emissions from both trajectories, the harmonic field cannot continuously grow up within the 1-mm-long gas medium in Fig.3(k).Furthermore,even though the phase-matching condition of short trajectory forp=2 appears similar to that forp=1,see the comparison of Figs.3(e)and 3(f),the harmonic field only constantly grows beforez=1.5 mm in Fig.3(l).This is because the white area in Fig.3(f)is narrower than that in Fig.3(e),meanwhile the length of white area of long trajectory in Fig.3(i)is much shorter than the length of gas medium,which is not enough for accumulating harmonic yields.Thus,the phase-matching condition of the harmonic is greatly modified by varying the radial node in the mixed LG beam,leading to a significant difference in the spatial harmonic profile in the near field in Fig.2.

3.3.Comparison of emission properties of H29 in the cutoff generated by different mixed LG beams

For the harmonics in the cutoff,we choose H29 as a representative for presentation and analysis.Its spatial profiles in both near and far fields are plotted in Fig.4 together with its far-field OAM spectra.It is evident that intensity structures are much cleaner compared to results for H13,particularly forp=1 and 2 in Figs.4(b) and 4(c).Furthermore, regardless of thepvalues,all the distributions of harmonic emissions exhibit a single half-ring structure in both near and far fields.For the OAM spectra,the calculated ones remain broader than the perturbative ones for three cases.And they also show a symmetrical distribution, unlike that of H13.Whenp=0,the peak of the calculated OAM spectrum is the same as that of perturbative one, see Fig.4(g).And its peak is shifted by increasing the radial node as shown in Figs.4(h) and 4(i).Thus in the cutoff, the clean spatial profile (including the phase) of harmonic leads to the broad and symmetrical distribution of the OAM spectrum regardless of radial node in the driving laser.

Fig.4.Same as Fig.2 but for H29.Dashed lines indicate different azimuthal angles used in Fig.5.

Fig.5.Similar figure to Fig.3 except for H29.Note that maps of coherence length(middle column)for cutoff harmonic are plotted where short and long trajectories are merged.Azimuthal angle is 45◦,135◦,and 225◦in the first,second,and third rows,respectively.

To understand the presence of the single caritive-ring structure in the spatial profile in the near field, we analyze the phase-matching conditions of cutoff harmonic H29,where the short and long trajectories are coming together.In Fig.5 we choose different azimuthal angles for three mixed LG beams, which are plotted in Figs.4(a)-4(c) by dashed lines,thus forp=2, the peak intensity of driving laser appears atz<0 mm.Considering the area of effective intensity distribution,a quite broad white color region occurs off axis whenp=0 in Fig.5(d),explaining the growth of the harmonic field in a wide radial range in Fig.5(g).Whenpincreases, the phase-matching area aroundr=10µm becomes narrower in Figs.5(e)and 5(f), implying that the harmonic field can only be accumulated within a limited region.In Figs.5(h)and 5(i),the evolution of the harmonic field develops in a long and slender shape, which is consistent with the phase-matching picture.Therefore,the single caritive-ring structure exhibiting in the spatial profile of near-field harmonic is due to the good(or limited) phase-matching achieved off axis for cutoff harmonic.Since high harmonics in the plateau and cutoff regions are generated from different electron trajectories,their phasematching conditions are quite different as shown in Figs.3 and 5.Thus the behavior of the OAM spectrum in the far field exhibits a strong dependence on the harmonic order.

3.4.Illustration of harmonic vortex features for selected high harmonics in the plateau

We finally show the spatial profiles in the far field and the corresponding OAM spectra in Fig.6 for another two harmonics in the plateau, i.e., H15 and H21.In the case ofp=0,the spatial distribution of harmonic intensity keeps the single half-ring structure,see Figs.6(a)and 6(b).Thus the calculated OAM spectrum resembles the perturbative one as shown in Figs.6(c)and 6(d).In addition, the OAM spectrum becomes broader as the harmonic order increases.Forp=1,as shown in Figs.6(e) and 6(f), the multiple-ring structure appears in the spatial profile, and the number of ring decreases with the increase of harmonic order.Taking into account of results for H13 and H29,we can conclude that the multiple-ring structure occurring in the presence of radial node in the driving laser can be gradually eliminated by increasing the harmonic order.The OAM spectrum can maintain a reasonably well symmetric distribution,see Figs.6(g)and 6(h).Whenp=2,the harmonic intensity has a similar multiple-ring structure to that forp=1,but its distribution is more irregular,as shown in Figs.6(i)and 6(j),leading to the asymmetrical OAM spectral distribution in Figs.6(k)and 6(l).For example,in Fig.6(k),there is a noticeable increase in the weight of OAM mode,which is developed on the left side of peak one, determined by the perturbative theory.So the OAM spectrum can also be readily tailored with the harmonic order.This is valid even though considering the radial node in the driving laser.

Fig.6.Intensity profiles of H15 (first column) and H21 (second column) in the far field and corresponding OAM spectra (third and fourth columns).First row: LG1,0+LG2,0;second row: LG1,1+LG2,1;third row: LG1,2+LG2,2.

Note that we have performed simulations for vortex high harmonics driven by the mixed LG beams with higher radial nodes, specifically withp=3 and 4.We have checked that for both plateau and cutoff harmonics, the spanning range of the OAM spectrum and its shape do not change much when the radial node is increased from 2 to 4.This is because when the radial node is set as 3 or 4 the mixed LG beams show a similar ring with the maximum intensity around the laser focus to that when the radial node is 2.This intensity ring contributes mostly to the generation of vortex HHG.This also indicates that effective tailoring of the OAM spectrum of HHG by varying the radial node is limited in the range of 0 to 2.

4.Conclusion

We proposed to effectively modify the orbital angular momentum (OAM) spectrum of vortex HHG by varying the radial node in the two mixed LG beams.Our investigation was based on the well-established and rigorous approach for simulating the vortex HHG from the gas medium,i.e.,solving the Maxwell’s wave equation of harmonic field in the medium,computing the single-atom response with the QRS model,and projecting the near-field harmonic to the far field using Huygens’ integral.We found that by increasing the radial node(from 0 to 2) in the driving laser the single half-ring structure in the spatial profile switches to the multiple caritive-ring structure for high harmonics at the plateau.For high harmonics at the cutoff,the single half-ring structure appearing in the spatial profile forp=0 does not change with the radial node.We showed that by varying the radial node,the OAM spectrum of far-field harmonic can change its shape and broaden its distribution region.For a given radial node of 1 or 2, with the increase of harmonic order,the mode range of the OAM spectrum becomes wider and wider, and its distribution tends to be more symmetric.We adopted two methods to uncover the phase-matching mechanism of high harmonics in the plateau and the cutoff, i.e., analyzing the spatial map of coherence length and the evolution of harmonic field in the medium.By varying the radial node, the spatial intensity and phase distributions of mixed two LG beams are greatly changed,leading to the difference in the favorable phase-matching regions off axis,which is consistent with the growth of harmonic field in the medium.

This work offers a practical way to tailor the OAM spectra of high-order harmonics in the XUV by modifying the properties of driving LG beams.Meanwhile it further strengthens our standing of the phase-matching nature of vortex HHG driven by nonpure OAM modes with radial nodes.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant Nos.12274230, 91950102, and 11834004)and the Funding of Nanjing University of Science and Technology(Grant No.TSXK2022D005).