Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds

Qingbin Zhang; Lixin He; Pengfei Lan; Peixiang Lu

doi:10.1364/OE.22.013213

1. Introduction

High-order harmonic generation (HHG) driving by femtosecond laser pulses has attracted a lot of attention in the past years for the emission emerges as a broad spectrum with attosecond temporal structure, which may sever as an important tool for accessing the fastest time scales relevant to electron dynamics in atoms, molecules and materials [1]. The unique properties of HHG have uncovered new knowledge of fundamental physics in unexplored precision, including the atomic photoexcition and photoionization [2, 3], electron-electron or electron-nuclei interaction and correlation [4–8], molecular dissociation and electron localization [9–13], and electron dynamics in nanosystems [14–16]. The HHG process can be well described in terms of three-step model [17]. In detail, an electron first tunnels through Coulomb-like potential barrier lowered by the laser field. Once free, it oscillates and gains kinetic energy in the continuum in response to the laser field. Finally, the ionized electron may recombine with the parent ions and emit a harmonic photon with energy up to I_p +3.17U_p, where I_p is the ionization potential and U_p is the ponderomotive energy. Following this process, it turns out that the harmonic radiation is usually repeated each half cycle of the multi-cycle Ti:sapphire driving field, resulting in an attosecond pulse train (APT). However, for a straightforward attosecond metrology, it is necessary to isolate one clearly defined attosecond pulse, namely IAP.

To generate an IAP, the emission time of the harmonics should be confined in one-half-cycle, which has been achieved in experiment by directly using a 5-fs laser field. In this scheme, a single 250-as pulse carrying a bandwidth limited to 10 eV is obtained [18]. One is expected to extend the applications of IAP by further reducing its pulse duration, because an improved temporal resolution is essentially required for investigating the processes, such as intra-atomic energy transfer [19], charge transfer [20] and the rearrangement of electrons [21]. G. Sansone et al. demonstrated the compression of the generated IAP for as short as 130 as by polarization control of 5-fs laser field [22]. Later, Goulielmakis et al. have employed a near-single-cycle pulse and successfully produced an isolated 80-as pulse, which first broke through the 100-as-barrier [23]. Recently, by using a 7-fs, 750-nm laser field, Zhao et al. have obtained a 67-as pulse, which is known as the shortest IAP at present [24]. Even though a broader spectrum enables the generation of shorter IAP, which can be achieved in principle by increasing the intensity of the commonly used 800-nm Ti:sapphire laser source, a well-know challenge is that a higher intensity of driving laser leads to a poor efficiency due to the ground-state depletion. Alternatively, a promising route for generating harmonics of higher photon energy is to use a driving laser with longer wavelength. The ponderomotive energy U_p scales as λ², therefore the laser wavelength becomes an effective knob to extend harmonic spectrum for short IAP generation while keeps a moderate laser intensity [25]. This attractive prospect has motivated research on the wavelength scaling of the HHG yield. However, both the theoretical predictions [26–28] and experimental results [29,30] demonstrate a rapid drop in the single-atom HHG yield as the wavelength increases, which scales as λ⁻⁵–λ⁻⁶. This effect hinders the development of intense IAP source by directly using driving field with longer wavelength.

The low intensity of IAP will set limits for many applications, such as ultrafast coherent imaging [31], nonlinear processes in the XUV range [32–34] as well as straightforward attosecond-pump/attosecond-probe spectroscopy [35, 36]. To improve the HH yield, not only the microscopic single-atom response but also the macroscopic phase-matching condition should be optimized. As the coherent length in the gas medium is, however, limited by the dephasing of the atomic dipole oscillators and the plasma defocusing of the laser pulse, the ionization must maintain at low level, typically a few percent, above which the conversion efficiency rapidly decreases. In this case, the optimized phase matching (PM) using loosely focused 800-nm high-energy laser field has been proposed to constructively build up the harmonic emission from different atoms in the gas medium, led to μJ level HH with improved conversion efficiency [37–42]. On the other hand, the energy of the IAP barely reaches the nJ level. Although the energy of IAP is also promising to be scaled up by using high-energy driving laser, the requirements including few-cycle pulse duration and stabilized CEP become the main obstacles. To overcome these limitations, the interferometric polarization gating (IPG) and generalized double optical gating (GDOG) techniques have been developed [43, 44]. Since the available pulse duration of driving laser falls into multi-cycle regime, it holds the significant promise for the generation of high-flux IAP. Tzallas et al. have realized 20 nJ sub-femotosecond IAP at 50 eV with IPG technology and Feng et al. have demonstrated an 100 nJ extreme ultraviolet (XUV) super-continuum supporting 230-as IAP at 35 eV using GDOG technology. Alternatively, the two-color field has also been extensively investigated for IAP generation because of its potential to break the symmetry of electric field and extend the cutoff energy as the pioneering works have demonstrated [45–48]. For instance, an optimized mixing field was proposed by Kim et al. for extending the continuum length through analyzing the behavior of the difference amplitude of the strongest and next strongest field peaks [49]. However, it is still restricted to the CEP stabilized case. Moreover, the temporal delay and intensity ratio variation of the two-color field during propagation have not been included, which are very important for evaluating the performance of generating energetic IAP practically. Very recently, by combining a two-color field and an energy-scaling method of HHG, a 500-as IAP with the pulse energy as high as 1.3 μJ has been experimentally achieved, which is the highest pulse energy of an IAP, to the best of our knowledge [50].

Even if the two-color field brings us the hope for intense IAP generation, we still face some raised questions such as: (i) What is the best shape of a two-color field to generate a shorter IAP using commercial available multi-cycle driving laser sources without/with CEP stabilization? (ii) What is the future potential for scaling up this short IAP, especially considering the influence of the employment of two-color field on phase match? Now, several types of high-energy femtosecond Ti-sapphire laser systems centered at 800 nm are commercial available for fundamental driving field, for example ALPHA 10 (25 fs/10 Hz/1.2 J) and ALPHA KHz (25 fs/1 KHz/20 mJ) with CEP stabilization option from Thales Group. Moreover, the development of high-energy OPA/OPCPA [51–53] allows us to employ even sub-30 fs/100 mJ control field centered at 1200 nm to 2000 nm in near future. Can we take advantage of these newly developed multi-cycle high-energy sources for generating an intense short IAP? In this work, we first devote to generate a short IAP by seeking the best shape of 25-fs multi-cycle two-color driving field. To avoid the influence of the CEP-instability of the laser pulses on the IAP generation, we present a shaped two-color field with optimized wavelength, by means of the peak amplitude-wavelength analysis. With this shaped two-color laser field, a broadband continua can be generated in a broader CEP range and IAPs with durations of 250 as can be created even without CEP stabilization. In addition, with the possibility of stabilizing the CEP of both high-energy fundamental and control laser fields, we also systematically optimize the shape of the two-color field in the carrier-envelope phase (CEP), wavelength and intensity ratio simultaneously to further reduce the duration of an IAP toward 100 as. Using this optimized two-color field, we directly produce a 65 eV broadband supercontinuum, which supports a 81-as IAP. To improve the energy of IAP with two-color field, we then discuss the influence of intensity ratio and group delay of the two-color field on the macroscopic build-up of HH. It is found that the method of phase-matching still works for the continual harmonic radiation as long as the ionization level and pressure of the gas medium are low enough. Therefore the energy scaling method for HHG can be employed by using loosely focused multi-cycle high-energy two-color field, which will pave the way for the realization of IAP with high focused intensity toward 100-as pulse duration.

The rest of this paper is organized as follows. The section 2 is dedicated to the modeling tools utilized to carry on our analysis. The section 3 has a first part in which the two-color field is optimized toward the goal of obtaining IAP with short and stable pulse duration against CEP. In the second part, the optimum CEP, intensity ratio and wavelength of two-color driving field are designed for producing an IAP with even shorter pulse duration. The third part analyzes the possible influence on achieving phase-matched harmonic introduced by two-color field. The forth and fifth parts do simulation with macroscopic effects, to examine the performance of the designed two-color fields in first and second parts on obtaining phase-matched IAPs. Finally, we present a summary of our conclusions and discuss the prospects for intense IAP based on the shaped two-color scheme.

2. Theoretical model

In our calculation, the Lewenstein model is applied to calculate the harmonic radiation with single-atom response [54]. In this model, the time-dependent dipole momentum is described as

\begin{array}{l} d_{n l} (t) & = & i \int_{- \infty}^{t} d t^{'} {[\frac{π}{ε + i (t - t^{'}) / 2}]}^{3 / 2} \\ \times d_{rec} [p_{s t} (t^{'}, t) - A (t)] d_{ion} [p_{s t} (t^{'}, t) - A (t^{'})] \\ \times exp [- i S_{s t} (t^{'}, t)] E (t^{'}) g (t^{'}) + c . c . \end{array}

In this equation, E(t) is the laser field, A(t) is the corresponding vector potential, ε is a positive regularization constant. p_st and S_st are the stationary momentum and quasiclassical action, respectively. g(t) represents the ground-state amplitude, which is given by ADK tunneling model. Then the harmonic spectrum can be obtained by the Fourier transforming the time-dependent dipole acceleration a⃗(t),

a_{q} = {| \frac{1}{T} \int_{0}^{T} \vec{a} (t) exp (- i q ω t) |}^{2},

where

\vec{a} (t) = {\vec{\ddot{d}}}_{n l} (t)

, T and ω are the duration and angular frequency of the driving pulse, respectively. q corresponds to the harmonic order. Here, to qualitatively evaluate the continuous length of the generated harmonic spectrum, we define the electric field ratio as [49]:

R = E_{1 m} / E_{2 m},

where E_1m denotes the strongest amplitude of the synthesized two-color field, and E_2m is the second strongest one. Generally, the larger the ratio R is, the broader the continuous spectrum is. This is because the maximum energy of the recombining electron is proportional to the square of the electric field, and then the bandwidth of the continuous spectrum is approximately proportional to the difference between the square of E_1m and E_2m.

Macroscopically, the collective response of the gas medium is described by the copropagation of the laser (E_l) and harmonics (E_h) fields, which can be simulated by numerically solving the Maxwell wave equations in the cylindrical coordinate system and assuming radial symmetry. The pulse evolution in a medium can be written as

\nabla^{2} E_{l} (r, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E_{l} (r, z, t)}{\partial t^{2}} = \frac{ω^{2}}{c^{2}} (1 - n_{eff}^{2}) E_{l} (r, z, t),

where η_eff is the effective refractive index of the medium. The η_eff can be written in a form of

η_{eff} = η_{0} (r, z, t) + η_{2} I (r, z, t) - \frac{ω_{p}^{2} (r, z, t)}{2 ω} .

The effective refractive index accounts for the refraction, absorption, kerr nonlinearity and plasma defocusing.

ω_{p} = e \sqrt{4 π n_{e} (r, z, t) / m_{e}}

is the plasma frequency for the presence of a density n_e of free electrons. The time dependent n_e(t) can be expressed as

n_{0} [1 - exp (- \int_{- \infty}^{t} w (t^{'}) d t^{'})]

with gas density n₀ and electron ionization rate w(t). On the other hand, the propagation equation of harmonic field can be expanded as

\nabla^{2} E_{h} (r, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E_{h} (r, z, t)}{\partial t^{2}} = \frac{ω_{p}^{2} (r, z, t)}{c^{2}} E_{h} (r, z, t) + μ_{0} \frac{\partial^{2} P_{n l} (r, z, t)}{\partial t^{2}},

where P_nl(r, z, t) = [n₀ − n_e(r, z, t)]d_nl(r, z, t) is the nonlinear polarization of the medium. The equations here take into account both temporal plasma induced phase modulation and the spatial plasma lensing effects on the driving field. Equations (4) and (6) can be numerically solved with the Crank-Nicholson method.

3. Results and discussion

In our investigation, The electric field of the synthesized two-color laser pulse can be expressed as:

E (t) = E_{0} exp (- 2 ln 2 t^{2} / {τ_{0}}^{2}) cos (ω_{0} t + ϕ_{0}) + E_{1} exp (- 2 ln 2 t^{2} / {τ_{1}}^{2}) cos (ω_{1} t + ϕ_{1}),

where E₀ and E₁ are the electric field amplitudes, ω₀ and ω₁ are the corresponding angular frequencies of the driving and control pulses. τ₀ and τ₁ correspond to pulse durations (FWHM) of the two pulses. ϕ₀ and ϕ₁ are the CEPs of the fundamental and control fields, and can be adjusted independently. Typically, the infrared control field is generated through OPA/OPCPA technology, in which the control field is resulted from the ampliation of a white light continuum (WLC) produced by focusing a portion of fundamental energy on a nonlinear medium. Thus the control field inherits the phase fluctuation of the fundamental field. The fine-tuning of the CEPs can be achieved by correcting the optical path of one of fundamental and control fields by means of inserting a wedge prism-pair independently. We choose neon as the gas medium, of which the ionization energy is 21.6 eV. Although both helium and neon are advantageous compared with argon due to their lower absorption at high photon energy and higher ionization potential. However, since helium has exceptionally small effective nonlinearity, neon is expected to be the most practical gas for efficiently generating HH above 100 eV.

In general, the IAP generation can be decomposed into the single-atom response and the coherent addition of XUV radiation during pulse propagation, often called macroscopic effects. The single-atom response determines the characteristics of obtained continues harmonics, namely, the emission time and pulse duration of an IAP. The propagation effect, such as the phase matching between harmonic radiation and driving pulses, changes the ultimate efficiency.

3.1. Single-atom response: IAP generation with non-CEP-stabilized two-color field

We first discuss the IAP generation using shaped two-color field within the frame of single-atome response. For the practical application of IAPs, the pulse durations are required to be not only short but also stable. At present, for a large number of existing high-energy femtosecond laser installations, the laser pulse durations are longer than 20 fs and the CEPs can not be stabilized. In this case, the pulse duration of a generated IAP may drastically change with the fluctuation of CEPs from shot to shot. To seek the best shape for generating IAPs with stable pulse duration, we perform the peak amplitude-wavelength analysis by calculating the R values with the control field of different wavelengths. In our calculations, a 25-fs [full width at half maximum (FWHM)], 800-nm multi-cycle fundamental laser field, together with a control field in the wavelength region of 1200–2400 nm is used as the driving two-color field. As shown in Fig. 1, for different intensity ratios, there are two valleys in the curves, which are marked as A and B, respectively. Among them, the valley A appears at 1330 nm and valley B is around 1600 nm. The R values of valleys A and B almost keep constant as the intensity ratio is increased, which results in the gradients of R values at A and B are equal to 0. Consequently, the R value is relatively stabilized with the wavelengths of control field at valley A or B. Since the R value of valley A is larger than that of B, it is reasonable to choose 1330 nm (A) as the optimum control field wavelength to generate a broad continuous harmonic with a stable bandwidth.

Fig. 1 The variation of the ratio R as a function of the control field wavelength using 25-fs two-color fields with different intensity ratios: (E₁/E₀)²=0.05 (blue solid line), (E₁/E₀)²=0.1 (green dashed line), and (E₁/E₀)²=0.15 (red chain line). Here, ϕ₀ and ϕ₁ are both set to π.

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We next investigate the dependence of HHG on the CEPs of the driving and control fields. Figure 2(a) shows the distribution of R values with respect to ϕ₀ and ϕ₁ for the optimum 1330-nm control field. For comparison, the result calculated with a 1250-nm control field (which corresponds to the nearest peak around valley A in Fig. 1) is also presented in Fig. 2(d). One can clearly see that, in these two cases, the variation of R values follows a periodic strip-like distribution. When the fundamental and control fields are simultaneously reversed (with a π difference in ϕ₀ and ϕ₁), the observed R value (or the harmonic spectrum) remains almost the same. By scanning the R values in Figs. 2(a) and 2(d), we find that the maximum values of R both appear at the CEP combination (ϕ₀, ϕ₁) of (π, π). To obtain a deeper insight, we further present the ϕ₁-dependent ratio R with ϕ₀=π in Fig. 2(b) and the ϕ₀-dependent ratio R with ϕ₁=π in Fig. 2(c). Figures 2(e) and 2(f) are the corresponding results for the 1250-nm control field. One can see that, with ϕ₀ (or ϕ₁) deviating from the optimizing CEP of π, the values of R decrease much more rapidly in the case of 1250-nm control field [see Figs. 2(e) and 2(f)]. Nevertheless, with the 1330-nm control field [see Figs. 2(b) and 2(c)], values of R can maintain almost stable with ϕ₀ varying from 0.8π to 1.2π and ϕ₁ from 0.9π to 1.1π.

Fig. 2 (a) The distribution of R values as a function of ϕ₀ and ϕ₁, (b) ϕ₁-dependent R values with ϕ₀=π and (c) ϕ₀-dependent R values with ϕ₁=π in the mixed field with a control field of 1330-nm. (d)–(f) Same to (a)–(c), but for a 1250-nm control field. Here, the intensity ratios (E₁/E₀)² of the driving and control fields are chosen as 0.1.

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To examine the analysis above, Fig. 3 displays the harmonic spectra calculated with the 1330-nm and 1250-nm control fields. It is noted that, to generate efficient harmonic emission, the ground-state population depletion must be effectively reduced and the ionization probability should be kept below 1% for Ne atom, with 800-nm Ti:sapphire driving laser alone. As the control field is much weaker than the fundamental field, here, it is reasonable to comply with this rule. Thus, the intensity of fundamental field is set to be 2.5×10¹⁴ W/cm² with an intensity ratio $E_{1}^{2} / E_{0}^{2}$ of 0.1. For different CEP combinations, the continuous ranges of the spectra can be well maintained at ∼ 20 eV with the 1330-nm control fields [see Fig. 3(a)], whereas in the case of 1250-nm control fields [see Fig. 3(b)], the bandwidths of the continua can be effectively extended but change rapidly as the CEP is changed. Therefore the 1330-nm control field is preferable to generating broadband harmonic spectra with stable continuous bandwidths in a broader range of CEPs. On the other hand, with the specific CEP combination and control field wavelength, even broader continuum of 32 eV is produced [see green line in Fig. 3(b)]. Since the shortest IAP we can obtain is of course limited by the bandwidth of the generated continuous harmonics, this result will motive us to look for the best CEP and control wavelength for a shorter IAP. This issue will be discussed in subsection 3.2.

Fig. 3 (a) Calculated harmonic spectra for different CEP combinations with the 1330-nm control field. (b) Same to (a), but for the 1250-nm control field. Here, the intensity ratios $E_{1}^{2} / E_{0}^{2}$ are fixed to 0.1.

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We then consider the IAPs generated with the optimized 25-fs synthesized laser field, of which the wavelength of the control field is 1330 nm. Since the control field is typically phase-locked to the fundamental field due to the OPA/OPCPA process, we assume the control field shares the same CEP with the fundamental field. Figure 4(a) shows the harmonic spectra generated in the synthesized field with the CEP changing from 0.5π to 1.5π. For most CEPs, the continuous harmonics are generated in the range from 130 eV (85^th) to 150 eV (98^th). Due to the existence of the short and long quantum paths, obvious interference fringes can be found in the continuous parts of the harmonic spectra. By superposing the harmonics in the continuous part, the temporal profiles of attosecond pulses are obtained as presented in Fig. 4(b). One can clearly see that only one main attosecond pulse is generated at each CEP value, and the obtained pulse durations are rather stable. Note that in the range of 0.75π to 1.35π, there are two close branches in the temporal profiles of the attosecond pulses, which correspond to the short and long quantum paths of the harmonic emission, respectively.

Fig. 4 (a) The CEP dependence of HHG with single-atom response in the optimized 25-fs two-color field. (b) Temporal profiles of attosecond pulses generated by selecting continuous harmonics in (a).

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3.2. Single atom response: IAP generation with CEP-stabilized two-color field

With the recent progresses on the stabilization of CEP of high-energy laser pulses for 800-nm fundamental field and 1200-nm to 2000-nm control field. The constraint on using CEP stabilized driving laser pulses has been eased. As has been discussed in subsection 3.1, a broader supercontinuum can be produced at a specific CEP. Here, we further systematically optimize a multi-cycle two-color field with stabilized CEP toward the goal of obtaining a shorter IAP. As we all know, the role of CEP becomes increasingly important with the reduction of laser pulse duration. Therefore, we choose 15-fs two-color laser field to enhance the dependence of IAP generation on CEP, and then pick out the best CEP combination. The intensity of the fundamental field is set to be 2.8 × 10¹⁴ W/cm², and the intensity ratio of control field is 4% to keep the ionization probability below 1%. We first calculate the values of the ratio R with ϕ₀ and ϕ₁ varying from 0–2π. Figure 5 shows the variation of R values as a function of ϕ₀ and ϕ₁ for different wavelengths of the control fields [Figs. 5(a)–5(d) are for 1200 nm, 1600 nm, 2000 nm and 2400 nm, respectively]. We find that the values of the ratio R still follow a strip-like distribution (similar to that in Fig. 2). With the increase of the control field wavelength, the width of the strip area is gradually enlarged. This result means that for a shorter wavelength of the control field, the ratio R is more sensitive to the variation of ϕ₀ and ϕ₁. Besides, we also find that in the cases of Figs. 5(a)–5(d), the maximum values of R all appear at the CEP combination (ϕ₀, ϕ₁) of (π, π), from which we can conclude that the optimum CEP combination for a broadband supercontinuum generation is (π, π). To confirm the above prediction, we next calculate the harmonic spectra with different CEP combinations of the fundamental and control fields. Corresponding results are presented in Fig. 6. In Fig. 6(a), the wavelength of the control field is 1600 nm and the intensity is 1.1× 10¹³ W/cm². For the CEP combination of (π, π) (R=1.213), the continuum bandwidth of the generated harmonic spectrum (blue line) is demonstrated to be 22 eV [Here, we call the smooth part (100 eV −122 eV) of the spectrum as continuum]. When the CEPs of the fundamental and control fields are varied, the continuum range is obviously reduced. For comparison, we also calculate the harmonic spectra with (π, 1.1π) (R=1.134, green line) and (π, 1.2π) (R=1.054, red line), corresponding continuum ranges are 14 eV and 6 eV, which are both smaller than that with (π, π). In addition, similar results can also be found in the case of 2400-nm control field, as shown in Fig. 6(b). For the combinations of (π, π) (R=1.140, blue line), (π, 1.1π) (R=1.138, green line) and (π, 1.2π) (R=1.135, red line), the bandwidths of the continuum spectra are 12 eV, 12 eV and 6 eV, respectively. Comparing Fig. 6(a) and Fig. 6(b), in the case of 1600-nm control field, the continuum bandwidths (values of R) decrease more rapidly as the CEPs are changed, which denotes a stronger dependence on the CEPs of the fundamental and control pulses.

Fig. 5 The dependence of the R value on both the CEPs of the fundamental and control laser fields. The wavelength of control field is chosen as (a) 1200 nm, (b) 1600 nm, (c) 2000 nm, and (d) 2400 nm, respectively

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Fig. 6 (a) The harmonic spectra calculated with different CEP combinations (ϕ₀, ϕ₁) of the fundamental and control fields: (π, π) (blue line), (π, 1.1π) (green line), and (π, 1.2π) (red line). The wavelength and intensity of the control field are 1600 nm and 1.1 × 10¹³ W/cm², respectively. (b) Same to (a), but for 2400-nm control field.

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On the other hand, as can be seen in Fig. 6, when a 1600-nm control field is mixed, the maximal continuum bandwidth is demonstrated to be 22 eV, while it is only 12 eV for 2400-nm control field. The wavelength of the control field apparently also plays an important role in the extension of the continuum. To find an optimum wavelength of the control field, we have calculated the R ratios with the wavelength of control field changing from 1200 nm to 2400 nm. In our calculations, the CEPs of the two-color fields are both set to be π and the intensity of the control field is still maintained at 1.1 × 10¹³ W/cm². We find that the highest R ratio of the two-color field is 1.251 as shown in Fig. 7(a), and the corresponding wavelength of control field is 1776 nm. To verify the results in Fig. 7(a), we also present the harmonic spectra in Fig. 7(b). For the case of 1776 nm (blue line), the bandwidth of the continuous spectrum is 35 eV, which is indeed much broader than cases of 1500 nm (20 eV, green line) and 2400 nm (12 eV, red line). To sum up, with the control field intensity of 1.1 × 10¹³ W/cm², we obtain a 35 eV supercontinuum at the optimium control field wavelength of 1776 nm.

Fig. 7 (a) The variation of the R value as a function of the wavelength of control field. The R values at 1500 nm, 1776 nm and 2400 nm are indicated by green, blue and red lines, respectively. (b) the harmonic spectra calculated with different wavelengths of control field. Here, the CEP combination (ϕ₀, ϕ₁) is set to be (π, π) and the laser intensity of the control field is 1.1 × 10¹³ W/cm².

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For a two-color laser field, when the intensity of control field is changed, it will make a difference in the strongest amplitude E_1m, and also in the second strongest one E_2m, which will directly influence the values of the ratio R (or the bandwidth of continuous spectrum). Therefore, to obtain an IAP with broader bandwidth, it is essential to know how the ratio R depends on the intensity of the control field. Since it is still a technical challenge to stabilize the CEP of high-energy laser source with pulse duration shorter than 25 fs, we set the pulse duration of both the fundamental and control fields to 25 fs. The fundamental intensity is 1.45 × 10¹⁴ W/cm². In this case, the ionization probability is lower than 1% even if the intensity ratio of control field is increased to 0.9. Figure 8 shows the variation of ratio R with respect to the wavelength of the control field and the intensity ratio $E_{1}^{2} / E_{0}^{2}$ of the two-color field. The bright branch (red color) corresponds to a broader bandwidth of continues spectrum. With a weak control field [ $E_{1}^{2} / E_{0}^{2} < 0.06$ ], only one bright branch exists (marked as C in the illustration), which means that there is only one optimum wavelength for the extended continuum generation. Corresponding optimum wavelengths of branch C are around 1750 nm. When the intensity of the control field is further increased ( $E_{1}^{2} / E_{0}^{2} > 0.06$ ), two bright branches emerge, specifically, branch A is around 1400 nm and branch B is around 1800 nm. In this range, for a given $E_{1}^{2} / E_{0}^{2}$ , two optimum wavelengths can be adopted to produce extended continua. The closer observation is that branch A is somewhat brighter than B, which is more appropriate for the generation of broadband continua. By scanning the values of R in Fig. 8, we find the maximum (R=1.228) appears at 1400 nm with $E_{1}^{2} / E_{0}^{2} = 0.866$ (the corresponding intensity of control field is 1.25 × 10¹⁴ W/cm²).

Fig. 8 The dependence of the ratio R on both the laser intensity ratio $E_{1}^{2} / E_{0}^{2}$ and wavelength of the control field with the optimum CEP combination (π, π).

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Consequently, we can obtain the optimum two-color field, of which the CEP combination is (π, π), the wavelength and intensity of the control field are 1400 nm and 1.25 × 10¹⁴W/cm², respectively. Figure 9(a) displays the temporal envelops of the synthesized field (green solid line) and the 800-nm fundamental field (black dashed line). It is obvious that in the synthesized field, the maximum amplitude E_1m of the synthesized field is increased and the difference between E_1m and E_2m is enlarged. With such a two-color field interacting with neon, we have calculated the harmonic spectrum by using Lewenstein model. The generated harmonic spectra is presented as the green line in Fig. 9(b). One can clearly see that, due to the increase of E_1m, the cutoff of the spectrum is extremely extend to 200 eV and a smoothed supercontinuum with a bandwidth of 65 eV (from 160 eV to 225 eV) is successfully produced, which supports a transform-limited (TL) IAP with pulse duration below 100 as. By superposing the 110^th –140^th harmonics, an IAP with duration of 110 as is obtained directly as shown in Fig. 9(d). A deeper insight can be obtained by investigating the harmonic emission times in terms of the time-frequency analysis. Calculation details can be found in Ref. [55] and the references therein. The result is shown in Fig. 9(c). It is clear that there are two branches, contributed by the short and long quantum paths. These two quantum paths give rise to the modulation in the continuous part of the spectrum shown in Fig. 9(b) and the sub pulses shown in Fig. 9(d).

Fig. 9 (a) The electric field of the optimum two-color field, (b) calculated harmonic spectra for single-atom response, (c) time-frequency diagram of the spectrum, and (d) the temporal profiles of the attosecond pulses by superposing the continuous harmonics. Inset: the TL pulse of the IAP shown in (d).

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3.3. Influence of two-color field on phase-matched harmonic generation

We then turn to the macroscopic effect on HHG, which is the most significant factor to produce an efficient IAP. As with any nonlinear process, the main technical obstacle to a high conversion efficiency for HH/IAP is the difference in the phase velocities between propagation of the fundamental and harmonic frequencies in the nonlinear medium. Because of this phase mismatch, the phase of the driving field will shift by π rad relative to the harmonic field as it propagates. This means that harmonic field generated early in the nonlinear medium will be exactly out of phase with harmonic field generated a certain propagation distance later. This dephasing distance is called the coherence length L_c = π/Δk, where Δk is the phase mismatch. Therefore phase match of IAP can be achieved by balancing the various dispersion terms or a zero net phase mismatch. If a monochromatic driving field is used for generating IAP, the total phase mismatch Δk can simply be written by

Δ k = Δ k_{n} + Δ k_{p} + Δ k_{g} + Δ k_{p l} .

The phase mismatch contributions from neutral gas dispersion Δk_n, plasma dispersion Δk_p, geometrical phase Δk_g and atomic dipole polarization Δk_pl, given by

\begin{array}{l} Δ k_{i} & = q \frac{\partial ϕ_{f, i} (z)}{\partial z} - \frac{\partial ϕ_{q, i} (z)}{\partial z} \\ = q k_{f, i} - k_{q, i}, (i = n, p, g, p l), \end{array}

where ϕ_f,i(z), ϕ_q,i(z) represent the phase of driving field and q^th harmonic field at propagation position z, and k_f,i, k_q,i are the corresponding wavevector. This simple relationship is valid because any phase shift δϕ of the monochromatic driving field during propagation maps a phase change qδϕ to the generated q^th harmonic. However, in the case of two-color field used to drive HHG, there are not only the phase shift of each field but also the relative phase slip between the fundamental and control fields themselves, induced by the dispersion in gas medium. It has been demonstrated that this relative phase slip will lead to an additional phase to the generated q^th harmonic [56–58]. Moreover, the intensity ratio of control field may also vary with respect to the propagation position z for different beam focusing conditions, which will also alter the phase of q^th harmonic. Therefore we rewrite a robust phase mismatch by

Δ k = \frac{\partial Φ_{q} (z)}{\partial z} - \frac{\partial Φ_{q, pro} (z)}{\partial z},

where Φ_q(z) is the phase of newly generated q^th harmonic at position z, while Φ_q,pro(z) is the phase of q^th harmonic generated early in the nonlinear medium and propagated to position z. The phase of q^th harmonic field with wavelength λ_q propagating in a gas medium is given by [59]

Φ_{q, pro} (z) = Φ_{q} (z_{0}) + [\frac{2 π}{λ} + \frac{2 π N_{a} n (λ_{q})}{λ} - N_{e} r_{e} λ_{q}] (z - z_{0}),

where Φ_q(z₀) is the phase at start position z₀, N_a is the atom density, N_e is the free electron density in the medium, n(λ_q) is the refractive index per unit neutral atom density at wavelength λ_q, and r_e is the classical electron radius. In Eq. (11), the contribution from the Guoy term of the harmonic light is much smaller than that for the fundamental and control field by a factor of ≈ 1/q² and is therefore not included for high harmonic orders. In order to estimate the phase Φ(z), we neglect the envelop of both fundamental and control fields and only take the term of carrier waves into account for simplicity, then the harmonic phase can be written approximately in the term of

\begin{array}{l} ϕ (S_{0}, S_{1}) = & E_{0}^{2} {\frac{1}{2 ω_{0}^{2}} S_{0}^{2} (t_{r} - t_{i}) + \frac{E_{1}^{2}}{E_{0}^{2}} \frac{1}{2 ω_{1}^{2}} S_{1}^{2} (t_{r} - t_{i}) + \frac{E_{1}}{E_{0}} \frac{1}{ω_{0} ω_{1}} S_{0} S_{1} \\ + \frac{1}{8 ω_{0}^{3}} [sin (2 ω_{0} t_{r}) - sin (2 ω_{0} t_{i})] + \frac{E_{1}^{2}}{E_{0}^{2}} \frac{1}{8 ω_{1}^{3}} [sin (2 ω_{1} t_{r} + 2 Δ ϕ) - sin (2 ω_{1} t_{i} + 2 Δ ϕ)] \\ + \frac{E_{1}}{E_{0}} \frac{1}{ω_{0} ω_{1}} {\frac{sin [(ω_{0} + ω_{1}) t_{r} + Δ ϕ]}{ω_{0} + ω_{1}} - \frac{sin [(ω_{0} - ω_{1}) t_{r} - Δ ϕ]}{ω_{0} - ω_{1}}} \\ - \frac{1}{4 ω_{0}^{2}} (t_{r} - t_{i}) - \frac{E_{1}^{2}}{E_{0}^{2}} \frac{1}{4 ω_{1}^{2}} (t_{r} - t_{i})} \\ + q ω_{0} t_{r}, \end{array}

where

S_{0} = \frac{cos (ω_{0} t_{r}) - cos (ω_{0} t_{i})}{ω_{0} (t_{i} - t_{r})},

S_{1} = \frac{cos (ω_{1} t_{r} + Δ ϕ) - cos (ω_{1} t_{i} + Δ ϕ)}{ω_{1} (t_{i} - t_{r})},

Δϕ describes the relative phase slip of the two fields during propagation, t_i and t_r are the birth moment and recollision moment of freed electron contributed to the q^th harmonic. The t_i and t_r are supposed to be chosen such that

P_{s} = \frac{1}{t_{i} - t_{r}} \int_{t_{r}}^{t_{i}} d t^{″} A (t^{″}),

\frac{{[P_{s} + A (t_{i})]}^{2}}{2} + I_{p},

{\frac{[P_{s} + A (t_{r})]}{2}}^{2} + I_{p} = q ω .

Here A and P_s are the vector potential of driving field and the stationary canonical momentum of electron, respectively. P_s, t_i, and t_r can be solved from these saddle-point equations [54,60].

As one can see from Eq. (12), the changes of intensity ratio $E_{1}^{2} / E_{0}^{2}$ and relative phase Δϕ between the fundamental and control fields alters the phase of generated harmonic at position z. Even if the intensity of control field is low, the term of “E₀E₁” will amplify the influence on harmonic phase. Furthermore, the shaped driving laser field will lead to the changes of t_i and t_r according the Eqs. (15)–(17), which also contributes to the harmonic phase. Therefore it is necessary to investigate the dependence of harmonic phase on the intensity ratio and relative phase slip before we go into the macroscopic build-up of IAP.

In Figs. 10(a)–10(e), we calculate the birth moment t_i and recollision moment t_r contributed to the 30^th–120^th harmonic with the intensity ratio of control field varied from 0 to 0.8, the marked symbols represent t_i and t_r of the 80^th harmonic for short and long trajectories. It is found that both the t_i and t_b gradually change with the increasing of intensity ratio. Since the emission resulted from the short trajectory usually presents a smaller beam divergence than the long trajectory, we then focus on the short one. Figures 10(f)–10(j) show the calculated phases of 75^th–85^th harmonics, an obvious phase slip can be observed for each harmonic. For this reason, the intensity ratio induced phase variation should be taken into account for achieving phase-matched harmonics with two-color driving field.

Fig. 10 (a)–(e) The birth moment t_i and recollision moment t_r contributed to the 30^th–120^th harmonic with intensity ratios of 0, 0.2, 0.4, 0.6 and 0.8, respectively. The blue solid and red solid line denote the long and short trajectories, respectively. The marked blue squares and diamonds represent the t_i and t_r of 80^th harmonic for long trajectory, the marked red circles and triangles correspond the t_i and t_r of 80^th harmonic for short trajectory. (f)–(j) The calculated phases of 75^th–85^th harmonics for short trajectory, the intensity ratios are the same as (a)–(e). The marked stars represent the phases of 80^th harmonic.

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As we know, the beam radius as well as the intensity of a gaussian laser is not constant along the propagation direction. Therefore a free focused two-color field through a gas medium may suffer from a variation of intensity ratio across the focal region. In Fig. 11(a), we show the intensity ratios within the focal region in different focal geometries with the f-number f_# = f/D= 600, f_#= 300 and f_#= 150, respectively. A more dramatic change of the intensity ratio is observed for the tightly focused geometry, leading to a narrower parabolic phase curves of the 80^th harmonic shown in Fig. 11(b) and a steeper dispersion curve in Fig. 11(c). For comparison, we also calculate the dispersion with f_#= 300 for a constant intensity ratio along propagation direction as shown in Fig. 11(d). The difference of the dispersion curves shown in Fig. 11(d) indicate the evolution of intensity ratio indeed change the phase of a specific harmonic, and consequently the phase-matching condition. Moreover, the variation of intensity ratio may also significantly change the cutoff of harmonic radiation, which hampers the selection of continuous harmonics near cutoff region for IAP generation with our two-color driving scheme.

Fig. 11 (a) The dependence of intensity ratio on different focal geometries across the focal region of the two-color field. (b) The phase Φ_q(z) of the generated 80^th harmonic (contributed by short trajectory) for these three focal geometries along propagation direction. (c) The dispersion ∂Φ_q(z)/∂z. In (a), (b) and (c) the intensity ratio is set to be 0.86 at the waist of the focused gaussian laser beam. (d). The comparison of calculated dispersion with f_#= 300 for a constant intensity ratio along propagation direction (brown line) and varied intensity ratio same as that in (c) (green line).

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To avoid the possible drawbacks introduced by the evolution of intensity ratio, it is necessary to correct the focal geometries of both fundamental and control field for a constant intensity ratio during propagation. Therefore, $f_{0 #} / f_{1 #} = \sqrt{λ_{1} / λ_{0}}$ is required, where f_0# and f_1# are the f-numbers of fundamental and control field, λ₀ and λ₀ are the corresponding wavelengths. In experiment, this can be achieved by using two separate focusing lenses for the fundamental and control fields.

It is also noted that the pressure-dependent group delay between fundamental and control fields should also be taken into account. In direct analogy to the influence of intensity ratio on harmonic phase, the temporal delay introduces an additional relative phase Δϕ = ω₁δt between the driving fields, which will also change the shape of the combined fields during propagation, and consequently the harmonic phase. Here, the group delay between a 800-nm fundamental field and a 1400-nm control field in neon is estimated to be about 1 as/cm with a gas pressure of 1 torr. As the refractive index is proportional to the gas pressure, the group delay δt increases linearly with gas pressure. Since the gas medium may be extended from several millimeter to centimeters for efficiently extracting harmonic energy, the gas pressure must be low enough to maintain the waveform of this synthesized field during propagation.

3.4. Phase-matched IAP generation using non-CEP-stabilized two-color field

In this subsection, we present the phase-matching effects on efficient generation of IAP with the designed two-color field, which has been demonstrated for single-atom response in subsection 3.1. In general, the focused geometries for IAP generation fall into two categories: tightly focused and loosely focused. We first consider the tightly focused one. To demonstrate this issue, we perform the simulation of the co-propagation of the laser and harmonic fields by focusing fundamental and control fields with f_0# = 137 and f_1# = 106 to a 0.5-mm neon gas jet, corresponding to laser beams with waists of 35 μm and 45 μm. Other laser parameters are the same as those in Fig. 4. The density of the gas medium is 3.5 × 10¹⁷/cm³ (10 Torr). In practice, the phase-matching condition can be achieved for only very low levels of ionization, at most a few percent (about 1%) for neon [42]. At the intensities used in this situation, the variation of the intensity dependent phase of the harmonic emission during propagation is stronger than the dispersion of the free electrons. Typically, the phase mismatch induced by neutral and plasma dispersion are the magnitudes of 10⁰ cm⁻¹, while 10² cm⁻¹ for Guoy phase and the intensity dependent atomic phase. Thus the brightest emission from a tight focal geometry is often achieved through balancing the Guoy phase and the intensity dependent phase. The phase-matching condition can be achieved by adjusting the gas pressure and the position of the laser focus. We place the gas jet 1.5 mm after the laser focus, therefore the short quantum path can be effectively enhanced. The macroscopic harmonic spectrum with different CEPs are shown in Fig. 12(a). For most CEP values, the continuous harmonics are generated in the range from 125 eV (80^th) to 145 eV (92^th). We further investigate the temporal characteristics of the smoothed supercontinuum by applying a square window with a width of 20 eV to the supercontinuum at different CEPs, it is clearly shown that only one branch in Fig. 12(b), which indicates we obtain IAP at a broad range of CEP.

Fig. 12 (a) The CEP dependence of on-axis harmonic spectrum with collective response in the optimized 25 fs two-color field. (b)Temporal profiles of attosecond pulses generated by selecting continuous harmonics in (a). The focused beam waists are 35 μm and 45 μm for the fundamental and control fields, respectively. Other laser parameters are the same as Fig. 4.

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In Fig. 13(a), we plot the temporal profiles of the obtained IAPs. It is shown that the IAPs centered at 135 eV with pulse duration ∼250 as are directly obtained. Although the birth of the IAPs with different CEPs are not synchronized, it is still very useful for pump-probe experiment. Usually, the pump pulse can be spatially separated into two pulses on a beam splitter. One will be used for preparing the driving fields for IAP generation, and the other one will be used as pump or probe pulse in experiment. In this case the generated IAP is locked to the infrared laser field shot to shot because they come from a common source. For a pump-probe experiment, the other important parameter is the stability of pulse duration. In Fig. 13(b), we present the durations of IAPs against different CEPs. It is found that the fluctuation of IAP duration is confined within 30 as.

Fig. 13 (a) The produced IAPs with different CEPs. (b) The relationship between obtained pulse duration and CEP. The parameters are the same as Fig. 12.

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For a two-color scheme, the synchronization between the fundamental and control field is usually achieved using Mach-Zehnder non-collinear interferometer setup. This technique suffers from instability in the optical path lengths of the two arms due to mechanical vibration and fluctuating environmental conditions. The fluctuation of time delay for a free-running interferometer is typically around 250 as RMS. Of course, the longer (shorter) arm of the interferometer will result in a larger (smaller) fluctuation. We then calculate the obtained pulse durations of IAPs with time delay ranging from −250 as to 250 as. The results show that the fluctuation of IAP duration is still stabilized to within 45 as. Even though a more exact and stable control of time delay can be achieved technically, it usually requires a lot of efforts. For this reason, our two-color scheme with optimum wavelength provides a simple way for the realization of IAPs with stable duration.

To further increase the harmonic yield toward intense IAP, what we can do is to broaden the cross section of laser field in gas medium [37, 41]. We therefore discuss the collective effect for a loosely focused two-color field. In a loosely focused geometry, the transverse component of the wave vectors, the on-axis intensity and the gouy phase of the laser field do not notably change. In this case, the major sources of phase mismatch are due to a pressure-dependent (neutral and free-electron plasma dispersion) term. Since the gas medium is extended to several centimeters in the loosely focused geometry, the group delay between the fundamental control fields will play a more important role in alerting the phase of the produced harmonic. For this reason, the gas pressure is kept bellow 1 torr to maintain the waveform of the synthesized field, and consequently the phase of harmonic. Then the generation of IAPs in a 2-cm long neon gas cell is simulated. We focus the two-color field 1.4 cm before this gas cell optimally, with f_0# = 600 and f_1# = 464. We obtain similar results to those shown in Fig. 12, while the conversion efficiency is significantly improved.

3.5. Phase-matched IAP generation using CEP-stabilized two-color field

To obtain an intense IAP with shorter pulse duration, the performance of the optimum two-color field designed in subsection 3.2 is examined by including macroscopic effects. For the tightly focused case, the two-color field is focused at 2 mm before a 0.5-mm neon gas jet. In Fig. 14, the contour plot shows the spatial distribution of harmonic spectrum at the exit of the gas medium and the thin lines denote the on-axis harmonic spectra. Compared with the result of single-atom response (cyan line), the intensity of the spectrum include macroscopic effect (white line) is markedly enhanced while the modulation in spectrum is suppressed. This result indicates that the selection of the short quantum path is achieved. By synthesizing 30 orders of harmonics, an IAP with duration of 108 as is produced as shown in the inset plot of Fig. 14. The loosely focused case is also checked with a 2-cm neon gas cell. The focus of the two-color field is optimized to 1.2 cm before this cell, according to the phase-matching condition. The resulted macroscopic response of harmonic spectrum and its spatial distribution are shown in Fig. 15. The temporal profile of the produced IAP is also calculated. An IAP with a pulse duration of 105 as with 65-ev bandwidth is obtained, which approaches the TL of 81 as. Since a loosely focused geometry can be applied to produce phase-matched IAP with short pulse duration, high-energy laser systems can be used for further scaling up the energy of the IAP.

Fig. 14 The spatial distribution of harmonic radiation at the exit of gas medium. The thin white line denote the on-axis harmonic spectrum, and the single-atom response is plotted with cyan line for comparison. Inset: the obtained IAP by superposing the 105^th to 135^th harmonics in the continuous part. The focused beam waists are 35 μm and 45 μm for the fundamental and control fields, respectively. Other parameters are the same as Fig. 9.

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Fig. 15 Same as Fig. 14, but for loosely focused geometry. f_0# = 600 and f_1# = 464 correspond to the beam waists of 140 μm and 180 μm for the fundamental and control fields.

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4. Conclusion

In conclusion, we systematically investigate the IAP generation with the multi-cycle two-color laser scheme. In this work, the two-color field is synthesized by a 25-fs, 800-nm driving pulse and a 25-fs control pulse. By performing peak amplitude-wavelength analysis, the wavelength of the control field is optimized to 1330 nm. In this case, the broadband continua with stable bandwidth can be produced in a broad range of CEPs. With an intensity ratio of 0.1, IAPs with duration of ∼ 250 as have been successfully generated in the designed two-color field without CEP stabilization. Therefore, this optimized shape of two-color field is helpful to relax the requirement on both laser pulse duration and CEP control for short IAP generation in experiment. Moreover, with the utilization of the CEP-stabilized driving laser source, we also propose an optimization of two-color field to generate a shorter IAP. It is found that the best CEP combination of the driving and control fields for an extended continuum generation appears at (π, π) and the optimum wavelength of the control field is demonstrated to be 1400 nm with an increased intensity ratio of 0.866. With such a 25-fs synthesized field, an broadband supercontinuum with the bandwidth of 65 eV is successfully produced. By discussing the influence of group delay and relative intensity of the two-color field on the macroscopic build-up of IAP, we conclude that the method of phase-matching still works as long as the ionization level and gas pressure is low enough. Under proper phase-matching condition, an efficient IAP down to 100 as can be obtained with loosely focused geometry.

The shaped two-color field, is efficient and scalable in output energy of the IAP, which provides us with the design parameters of an efficient IAP toward 100 as. In neon gas, according to the HHG conversion efficiency of ∼5 × 10⁻⁷ for the loosely focused 800-nm driving laser obtained in ref. [42], together with the theoretically predicted wavelength scaling law of λ⁻² for the 800-nm/1400-nm two-color field [61], the conversion efficiency for our two-color field reaches 10⁻⁷. Meanwhile, 40-TW class (e.g., 25 fs, 1 J) Ti:sapphire laser systems are commercially available. For example we estimate the energy of IAP driving by 200-mJ fundamental pulse and 200-mJ control pulse (obtained from OPCPA with the conversion efficiency of 25%), the IAP with energy as high as 0.5 μJ could be reached by selecting more than 30 harmonic orders in the continuous spectrum. As demonstrated in Ref. [62], coherent soft-x-ray from harmonic source can be focused to μm spot size. With the reported optical losses from the IR/XUV beam splitter and multilayer concave mirror [37, 62], the intensity of the produced IAP is estimated to readily exceed 10¹⁴ W/cm².

We believe that the shaped multi-cycle two-color field has great potential to markedly increase the energy while reduce the pulse duration of IAP toward 100 as, which will pave the way for ultrafast XUV nonlinearity but also ultrafast intra-atomic dynamics. Especially, an intense IAP can open the door to demonstrate the attosecond pump/attosecond probe experiment.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants No. 11204095, 11234004, 61275126 and the 973 Program of China under Grant No. 2011CB808103.

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Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

3.1. Single-atom response: IAP generation with non-CEP-stabilized two-color field

3.2. Single atom response: IAP generation with CEP-stabilized two-color field

3.3. Influence of two-color field on phase-matched harmonic generation

3.4. Phase-matched IAP generation using non-CEP-stabilized two-color field

3.5. Phase-matched IAP generation using CEP-stabilized two-color field

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (15)

Equations (17)

Optics Express