Efficient generation of high beam-quality attosecond pulse with polarization-gating Bessel-Gauss beam from highly-ionized media

Single attosecond pulse generation with polarization gating Bessel-Gauss beam in relatively strongly-ionized media is investigated. The results show that Bessel-Gauss beam has the ability to suppress the spatial plasma dispersion effects caused by high density of free electrons, thus the laser field can maintain its spatial profile through highly-ionized medium. This indicates the use of Bessel-Gauss beam has advantages over Gaussian beam in high harmonic generation under high ionization conditions. In our scheme, significant improvement of spatiotemporal properties of harmonics is achieved and an isolated attosecond pulse with high beam quality is filtered out using polarization gating. © 2012 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.4160) Multiharmonic generation; (300.6560) Spectroscope, x-ray. References and links 1. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009). 2. E. Goulielmakis, M. Schultze, M. 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Introduction
High-order harmonic generation (HHG) is a highly nonlinear process occurring in the interaction of intense laser fields with atoms and molecules.It has been widely studied for the past two decades for its potential of producing ultrashort attosecond pulses [1][2][3], and as shortwavelength light sources [4,5].The physical mechanism of HHG process in single-atom level can be well understood by the three-step model [6,7], i.e., an electron is ionized by tunnelling through the potential barrier suppressed by the laser field, then it is driven by the laser field treated as a free electron, and finally it may return to the atomic core and emit a harmonic photon in the transition back to the ground state.The recombination occurs every half of an optical cycle, resulting in attosecond pulse train (APT) [8].Experimentally, an isolated attosecond pulse is preferred in straightforward attosecond metrology [9][10][11].Generation of isolated attosecond pulse can be achieved in two different approaches: the selection of the highestenergy part of the harmonic spectrum generated by linearly polarized pulses [12][13][14][15] and the control of the polarization state of the pulse to allow HHG only in a short time window while the lase pulse can be treated as linearly polarized, i.e., polarization gating (PG) [16,17].The PG technique has been successfully implemented for the generation of xuv continua in argon and neon using few-cycle infrared carrier-envelope phase (CEP) stabilized pulse [18].Recently the PG technique has been extended to generate single attosecond pulse in multicycle pulse regime referred to as double optical gating (DOG) [19] and to generate more energetic harmonic bursts as modulated polarization gating (MPG) [20,21].These schemes open new perspectives for the generation of intense attosecond pulses.
Since the laser field interacts with a macroscopic number of atoms, the full description of HHG process requires solving not only the strong-field laser-atom interaction at single-atom level, but also the propagation of the fundamental laser and harmonic fields through the gas medium at macroscopic level [5].The macroscopic HHG is influenced by propagation conditions, such as the laser intensity, the focusing geometry, the gas pressure, the gas-jet position with respect to the laser focus, the free electrons density in the gas, etc. Particularly, phase matching in the propagation plays an important role in HHG [22][23][24].When both the ionization probability and the atomic density are low, the fundamental laser field is not modified through the medium, which implies that macroscopic propagation effects on the driving field can be neglected.The plasma dispersion and absorbtion of the harmonic field can also be neglected.However, under low-ionization conditions the intensity of the harmonics is very low, simply because single-atom harmonic yield depends on the ionization probability according to three-step model.In order to increase the harmonic yield, an approach with laser pulse focused in high-pressure strongly-ionized medium is feasible, just because it allows one to have a larger and denser interaction volume.However, in highly-ionized medium, large density of free electrons may distort the laser field, changing the spatiotemporal properties of the laser field [25].Temporal and spatial modifications of the laser pulse can influence spectral and spatial characteristics of the harmonic field.This effect becomes a more serious issue in PG scheme, because HHG sensitively depends on the polarization properties of the laser field.The distortion of the laser pulse may change the spatial characteristics of HHG and lead to poor beam quality of the generated attosecond pulse.In order to reduce the distortion of the laser pulse and improve the spatial beam quality of HHG, the use of Bessel-Gauss (BG) beam for generation of high-order harmonics has been proposed [26][27][28].BG beam offers realization of a slowly varying geometry both for the laser field and for HHG at low and intermediate orders in plateau region, which has the potential to improve the spatial quality of HHG [27].
In this paper, we study high harmonic generation using polarization-gating Bessel-Gauss beam, in relatively highly-ionized media.Numerical simulations of macroscopic 3-dimensional (3D) propagations of both fundamental and harmonic fields are carried out to investigate the spectral, temporal and spacial characteristics of the generated attosecond pulse.The results show that the distortion of the fundamental laser field caused by free electrons is greatly suppressed with Bessel-Gauss beam by controlling the focusing half-angles.Significant improvement of spatiotemporal properties of harmonics in plateau region is achieved in comparison with Gaussian beam and an isolated attosecond pulse with high beam quality can be filtered out with harmonics in plateau region.
The rest of this paper is arranged as follows: in Sec. 2, we briefly summarize the theoretical method and the essential equations for describing the macroscopic propagation and the calculation of single-atom dipoles in the generalized Lewenstein model [16].We also show the spatial properties of Bessel-Gauss beam in this section.In Sec. 3, the results are shown and analyzed.First, we show the electric field components used in PG technique and the time dependent ellipticity of the laser field.Then we present the HHG spectrum in single-atom level and the time-frequency distribution.Next, spatiotemporal profiles of the two electric field components before and after propagation are shown and compared.Finally , the characteristics of harmonics in plateau region, including on-axis spectra, spatiotemporal profile, near-field and far-field distribution are discussed.A short summary in Sec. 4 concludes this paper and potential applications of attosecond pulse with high beam quality are discussed.

Theoretical model
We employ xenon as the target atom and 20 fs, 1.8 μm, 5.5 × 10 13 W/cm 2 , driving laser pulse with time dependent ellipticity.The polarization gating is created by two counterrotating,circularly polarized Gaussian pulses with a proper delay between them [15].The laser pulse propagates in the z direction.The electric fields of the combined pulses polarized along x and y directions respectively are the peak amplitude E 0 , frequency ω, pulse duration τ p , and CEP φ are the same for the left and right circularly polarized fundamental pulse and t d is the delay between the two pulses.We set φ = 0, τ p = 2T and t d = τ p , where T is the optical cycle of the field.
Numerical simulations of HHG include both the response at single-atom level induced by the laser field and the co-propagation of the laser pulse and harmonics.The single-atom response is calculated within the well-known Lewenstein model and its generalized form [7,16].The nonlinear dipole moment is [in atom units(a.u.)] where E(t) is the electric field of the laser pulse, A(t) is the corresponding vector potential, ε is a positive regularization constant, p st and S st are the stationary momentum and quasi-classical action, d(p) is the field free dipole transition matrix element between the ground state and the continuum state, and g(t) is the depletion of ground state, which can be calculated with the ADK theory [29].The nonlinear dipole moments along the x and y directions are calculated respectively by The propagations of the fundamental laser and harmonic fields in an ionizing medium have been described in detail in [5], so it is only briefly discussed here.The propagations of the fundamental field and harmonic field are solved separately by 3D Maxwell wave equations in cylindrical coordinate, which are described by In the above two equations, E f and E h are laser and harmonic fields, respectively.
is the plasma frequency, where m e and e are the mass and charge of an electron, respectively, and n e (t) is the density of free electrons.n e (t) = n 0 1 − exp − t −∞ γ(t )dt , which can be calculated by ADK theory [29].n 0 is the atomic density in the gas.The absorption term J abs (t) due to the ionization of the medium is given by where γ(t) is the ionization rate and I p is the ionization potential.
is the nonlinear polarization generated by the medium, where d nl (t) is the single-atom-induced dipole moment.The propagation equations here take into account the temporal plasma induced phase modulation, the spatial plasma dispersion effects on the driving field and the loss of energy of the driving field due to the ionization of the medium.Equation ( 6) and ( 7) can be numerically solved by using the Crank-Nicholson routine in the frequency domain [30,31].
A Bessel-Gauss beam can be obtained by focusing an initial Gaussian beam with a conical lens, namely, an axicon [32], placed in the focal plane (waist) of the Gaussian beam.The characteristics of a BG beam have been described in [26].In our notation, the spatial distribution of the electric field for a BG beam is written as where E f ocus = E BG (0, 0) is the field amplitude at the focus, J 0 is the zeroth-order Bessel function , b BG the confocal parameter, k is the wave number, and γ is the focusing half-angle of the Bessel-Gauss beam.In the present paper, we vary γ from 0 • to 1 • .For γ = 0 • , we recover a Gaussian beam with the same confocal parameter.We first give a brief discussion about the single-atom response using PG technique.In Fig. 1, the two electric field components E x (t), E y (t) and the time dependent ellipticity are presented.As we can see, only in a short time window near the middle of the pulse can the laser pulse be approximately treated as linear polarized.So the harmonics can be generated efficiently in this time region.The corresponding harmonic spectrum and the time-frequency analysis are shown in Fig. 2. One can clearly see from Fig. 2(a) that the harmonics from 90th to 170th become continuous.The modulation in the supercontinuum is originated from the interference between the long and short quantum paths with nearly equivalent intensities.This modulation can be cancelled in the process of propagation because of the different phase-matching conditions of the short and long quantum paths [23,36].As shown in Fig. 2 is only generated in a small time range with width of 0.5 optical cycle around the linear part of the driving pulse, which agrees well with our analysis.Next, we focus on the propagation effects of the fundamental and harmonic fields in macroscopic medium with different spatial distributions.In Fig. 3, we present the spatiotemporal profiles of the laser pulses at the entrance of a gas jet.Three cases with different focusing half-angles of 0 • , 0.5 • and 1.0 • are given.For γ = 0 • , a Gaussian beam can be reproduced.We can see that the radius of the central lobe decreases with γ, whereas the number and the amplitude of the secondary maxima increase.Figure 4 shows the spatiotemporal profiles of the laser pulses corresponding to Fig. 3 at the exit of the gas cell.A 1-mm-long gas jet with an atomic density of 2.5 × 10 24 m −3 is placed 1 mm after the waist of the initial Gaussian beam.In our simulations, we employ xenon as the target atom.The peak intensity of each circularly polarized pulse is 5.5 × 10 13 W/cm 2 , which would give an ionization probability of 30% at the end of the laser pulse, according to the empirical ADK formula.Under this relatively high ionization conditions, the fundamental field is perturbed both temporally and spatially as it propagates through the medium for Gaussian beam (γ = 0 • ).Spatially, the peak intensity of the electric field in different half optical cycle shifts off the axis, which means that the peak intensity occurs at different times for different radial positions.The beam becomes broader and more divergent and the duration of the pulse varies with radius.While as the angle γ increases, the spatial reshaping is suppressed.For γ = 1.0 • , the reshaping effect caused by free electrons is unconspicuous.This may due to that the change of the intensity distribution as γ increases affects the plasma intensity.For γ = 0.0 • spatial distribution is Gaussian shape and the laser energy is concentrated in the central lobe.For γ = 0.5 • about 13% of the total laser energy is concentrated in the second maxima.For γ = 1.0 • this amount is substantially higher.Due to the high intensity of the central lobe, the ionization of electrons is mostly from this part.Although the second maxima contains 13% of the laser energy for γ = 0.5 • and more for γ = 1.0 • , the maximum intensity is much lower than the central lobe.Our calculations show that the contribution of the second maxima to the plasma density is small.As γ increases, the radius of the axicon caustic decreases, so dose the plasma density in this region.Therefore, the spatial reshaping is suppressed.Temporally, a large dynamical blueshift [33] is observed.We present E x on-axis as a function of time, as illustrated in Fig. 5.As we can see, it is strongly depend on γ.Around the central part of the pulse, we can see the amount of blueshift of the on-axis electric field is drop with increasing γ.Under axicon focusing different segments of the caustic are formed from different concentric rings in the laser beam.Thus, the portion of laser energy from the certain radius forms a segment of the caustic and then diverges in the ring structure.Therefore, the blueshift is substantially less than for the focused Gaussian beam and its amount drops with increasing γ.This dynamical blueshift has greater influence in Gaussian beam case with PG technique.Because the spatial reshaping of Gaussian beam leads the blueshift to varying with radius, the time window in which harmonics can emit efficiently also varies with radius, leading to complicated spatiotemporal characteristics of the harmonics.We then investigate the on-axis macroscopic HHG spectra with different angles of 0 • (Gaussian beam), 0.5 • and 1.0 • , respectively, as plotted in Fig. 6.One can clearly see that the interference fringes are reduced for all angles compared with single-atom HHG spectrum through the plateau to cutoff region, which indicates that the on-axis phase-matching of one quantum path is better than the other one.In off-axis region, the phase-matching conditions can be differ-ent: the on-axis well phase-matched path may not realize phase matching off-axis whereas the other path can be phase-matched [23].Under axicon focusing, it is possible to control the phase velocity by simply changing γ.The axial wavevector k z will be shortened with respect to the plane wave k 0 by an amount of k 0 (1 − cosγ).Therefore, the phase mismatch can be expressed as Δk = (atomic term) − (plasma term) − qα, where q is the harmonic order [26].The atomic phase term and −qα has the same sign in our condition.It is possible to choose a suitable γ to compensate the phase mismatch.For the half-focusing angles of 0 • and 0.5 • , the supercontinua are well phase-matched from about 90th to the cutoff, while the phased-matched harmonics are only 40 harmonic orders from 110th to 150th for the angles of 1.0 • .Besides, there are some irregular modulations in the supercontinuum for the angles of 1.0 • .The results indicate that the phase-matching conditions for γ = 1.0 • is not as well as that for γ = 0 • and γ = 0.5 • .However, the phase-matching of the harmonics is sensitive to the propagation conditions, such as the position of the gas cell, the atomic density, etc.We can improve the phase-matching for the angles of 1.0 • by more carefully choosing the parameters.

Result and discussion
In order to examine the influence of the distortion of the fundamental field during propagation on the properties of HHG, we show the spatiotemporal characteristics of harmonics in plateau region with different angles in Fig. 7. Harmonics from 110th to 150th are selected and five different angles 0 • , 0.3 • , 0.5 • , 0.7 • and 1.0 • are chosen.As can be seen in Fig. 7(a), with Gaussian beam driving, the radiation of the harmonics is more intense off axis than that on axis, owing to the spatial reshaping of the driving field during propagation.Since the peak intensity shifts off the axis, the harmonics emerging from the nonlinear medium which generated by the strongly reshaped laser pulse also have the off-axis peak intensity.Moreover, the spatiotemporal distribution has some fine structures: the most intense emission of harmonics occurs at different times for different radial positions, which may be related to that the reshaping of the laser pulse during propagation changes the polarization properties of the laser pulse and the time window in which the pulse can be treated as linear polarized varies with radius.In Fig. 4, as the angle increases, the reshaping of the laser pulse is reduced.As a result, the spatiotemporal distributions gradually becomes regular.In Fig. 7 that the short path is well phase matched during propagation and an isolated attosecond pulse can be obtained.
To give a more explicitly description, we respectively calculate the spatial and temporal profiles.The spatial profile is obtained by integrating the spatiotemporal profile with respect to time and the temporal profile is obtained by integrals of radius.Spatial profiles are presented in Fig. 8(a) and 8(b).As we can see, the spatial profile with Gaussian beam shows an annular-like distribution.While for the BG beam with γ = 1.0 • , the profile becomes a nearly rectangular-like structure with a dip at R = 3.5 μm.The results illustrate the point that BG beam can improve the spatial quality of the generated harmonics.Temporal profiles are plotted in Fig. 8(c) and 8(d).
For the Gaussian beam case, there are two attosecond pulses corresponding to short and long quantum paths,respectively.Their emission times is so close that the neighboring parts overlap.On the contrary, with BG beam an isolated pulse can be straightforwardly obtained.Notice that there is a satellite pulse near the end of the main pulse with intensity less than 10% of the main pulse.This may related to the crescent-like spatiotemporal profile as shown in Fig. 7(e).The emission times of the harmonics off-axis and on-axis have a little time delay while the off-axis radiation is much weaker, forming a satellite pulse at the end of the main pulse.
In the experiment, harmonics are not measured at the exit of the medium, the near-field harmonics are refocused on the Xe target for the photoelectron generation.To further investigate the beam quality of the attosecond pulses, we calculated the far-field in this second focus using a Hankel transform of the near field [34,35].The far-field distributions for Gaussian and BG beam cases are presented in Fig. 9(a) and 9(b), respectively.The corresponding light spots are shown in Fig. 9(c) and 9(d).For Gaussian beam case, the profiles are still annular-like and there are more annuluses in the outer area of the main spot, due to large divergences of the harmonics.Instead, For BG beam case with γ = 1.0 • , the profiles becomes Gaussian-like.The results further demonstrate that using BG beam can highly improve the beam quality of the generated attosecond pulse.
In addition, at ionization level of 30% in our scheme, we find that axicon focusing with γ = 1.0 • allows sustaining phase-matching for harmonic generation.We then investigate ionization level and half-focusing angle dependence of harmonic efficiency.we do it by changing the intensity of laser pulse and the axicon focusing angle γ while keeping neutral gas density unchanged.The results show that for an ionization probability of 30%, γ = 1.0 • can sustain phase-matching while high harmonic efficiency can still be preserved.For a higher ionization probability, say 40%, γ should increase to 1.2 • in order to sustain phase-matching and preserve high harmonic efficiency.When reaching the highest useful ionization rate 50%, γ should be 1.5 • .However, from a practical point of view, only axicon with small base angle, i.e., of small enough thickness, can be used to focus short pulses, without altering the spatial, spectral, and temporal profiles of the laser pulse.In [26], γ varies between 0.0 • and 1.2 • .γ = 1.5 • seems too large.So we decrease the neutral gas density.Our calculations show that the optimal axicon focusing angle γ and neutral gas density are 1.2 • and 1.6 × 10 24 m −3 , respectively, under the highest useful ionization rate 50%.

Conclusion
In conclusion, we investigate high-order harmonic generation with polarization-gating Bessel-Gauss beam, in a relatively high-ionized media of xenon.The results show that Bessel-Gauss beam can suppress the distortion of the fundamental laser field caused by large density of free electrons.By controlling the focusing half-angle of the Bessel-Gauss beam, the fundamental field can maintain its spatial profile through the medium, in contrast with the Gaussian beam case where the laser field is severely reshaped during propagation.This indicate that use of Bessel-Gauss beam has advantages over the traditional Gaussian beam in high harmonic generation under high ionization condition.In our scheme spatiotemporal characteristics of the harmonics can be significantly improved and an isolated attosecond pulse with Gaussian-like spatial distribution can be filtered out with harmonics from 110th to 150th in plateau region.Such a light source, with low divergence and high beam quality, has many potential applications in various fields, including high-precision metrology, and microscopy and holography with nanometer resolution [37].Moreover, the short time duration of the short pulse (a few hundred attoseconds) will enable Extreme ultraviolet (EUV) microscopy and holography to be performed with ultrahigh time resolution.

Fig. 1 .
Fig.1.Electric field components E x (red solid line) and E y (blue dash-dotted line).CEP is chosen to be 0. The inset is the time dependent Ellipticity.
HHG spectra with different half-focusing angles: γ = 0 • (blue solid line), γ = 0.5 • (green solid line) and γ = 1.0 • (red solid line).The results for the different angles have been vertically shifted for visual convenience.The intensities for all the angles are almost equal in reality.

Fig. 9 .
Fig. 9. Far-field spatial distributions and the light spots of the attosecond pulse with Gaussian beam (γ = 0 • ) and BG beam (γ = 1.0 • ).(a) and (b) spatial distributions with Gaussian beam and BG beam, respectively, (c) and (d) light spots with Gaussian and BG beam, respectively.The position is 50 cm away from the exit of the medium for both two cases.