Isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field

doi:10.1364/OE.20.009713

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Isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field

Hongchuan Du, Laoyong Luo, Xiaoshan Wang, and Bitao Hu »View Author Affiliations

Optics Express, Vol. 20, Issue 9, pp. 9713-9725 (2012)
http://dx.doi.org/10.1364/OE.20.009713

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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical methods
3. Results and discussions
4. Conclusion
– References and links

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Abstract

We theoretically investigate the isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field. It is found that the large initial population of the excited state can lead to the high density of the free electrons in the medium and the large distortion of the driving laser field after propagation, though it benefits large enhancement of harmonic intensity in single atom response. These effects can weaken the phase-match of the macroscopic supercontinuum. On the contrary, the small initial population of 4% can generate well phase-match intense supercontinuum. We also investigate an isolated attosecond pulse generation by using a filter centered on axis to select the harmonics in the far field. Our results reveal that the radius of the spatial filter should be chosen to be small enough to reduce the duration of the isolated attosecond pulse due to the curvature effect of spatiotemporal profiles of the generated attosecond pulses in the far field.

1. Introduction

Isolated attosecond pulses have offered a robust tool for opening a new field of physics and chemistry. Electronic motion in atoms [1

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002). [CrossRef] [PubMed]

], molecules, nanoparticles [2

T. Pfeifer, M. Abel, P. Nagel, A. Jullien, Z. Loh, M. Bell, D. Neumark, and S. Leone, “Time-resolved spectroscopy of attosecond quantum dynamics,” Chem. Phys. Lett. 463, 11–24 (2008). [CrossRef]

], and solids [3

A. Cavalieri, N. Müller, T. Uphues, V. Yakovlev, A. Balthuška, B. Hovrath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007). [CrossRef] [PubMed]

] can be probed in real time by using the phase-locked visible pump plus isolated attosecond probe technique. These applications, however, are still limited by the low photon flux and long duration of the isolated attosecond pulse. Many efforts have been made on the spectral and temporal characteristics of HHG in order to broaden the bandwidth of the generated isolated attosecond pulses and enhance the pulse efficiency. Nowadays, isolated attosecond pulses have been experimentally generated mainly using two techniques: a few-cycle driving pulse [4

E. Goulielmakis, M. Schultze, M. Hofstetter, V. Yakovlev, J. Gagnon, M. Uiberacker, A. Aquila, E. Gullikson, D. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008). [CrossRef] [PubMed]

–6

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006). [CrossRef] [PubMed]

] or polarization gating [6

–10

H. Mashiko, S. Gilbertson, M. Chini, X. Feng, C. Yun, H. Wang, S. Khan, S. Chen, and Z. Chang, “Extreme ultraviolet supercontinua supporting pulse durations of less than one atomic unit of time,” Opt. Lett. 34, 3337–3339 (2009). [CrossRef] [PubMed]

]. The 100-as barrier has been first brought through by Goulielmarkis et al. [4

] using the few-cycle driving pulse. In their experiment, a sub-4-fs near-single-cycle driving pulse has been employed to generate a 40-eV supercontinuum and an 80-as pulse with the pulse energy of 0.5 pJ has been filtered out. Very recently, Ferrari et al. generated an isolated attosecond pulse with the pulse energy that reaches nanojoule-level by above-saturation few-cycle fields [5

F. Ferrari, F. Calegari, M. Lucchini, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “High-energy isolated attosecond pulses generated by above-saturation few-cycle fields,” Nat. Photonics 4, 875–879 (2010). [CrossRef]

]. Chang et al. obtained a very broadband xuv continuous spectrum with double-optical-gating technique (DOG), which supports 16-as isolated pulse generation [10

]. Theoretically, Gaarde et al. presented a method to generate an isolated attosecond pulse by using the reshaping of the fundamental laser field and a spatial filter in the far field [11

M. Gaarde and K. Schafer, “Generating single attosecond pulses via spatial filtering,” Opt. Lett. 31, 3188–3190 (2006). [CrossRef] [PubMed]

,12

M. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A 74, 053401 (2006). [CrossRef]

]. Jin et al. extended this method to the few-cycle mid-infrared driving pulse [13

C. Jin, A-T Le, C. Trallero-Herrero, and C. Lin, “Generation of isolated attosecond pulses in the far field by spatial filtering with an intense few-cycle mid-infrared laser,” Phys. Rev. A 84, 043411 (2011). [CrossRef]

The three-step model [14

P. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993). [CrossRef] [PubMed]

] indicates that the HHG process can be controlled via manipulated the ionization, acceleration, and recombination steps. Altucci et al. investigated isolated attosecond pulse generation in a few-cycle [15

C. Altucci, V. Tosa, and R. Velotta, “Beyond the single-atom response in isolated attosecond-pulse generation,” Phys. Rev. A 75, 061401(R) (2007). [CrossRef]

] and multi-cycle [16

C. Altucci, R. Velotta, V. Tosa, P. Villoresi, F. Frassetto, L. Poletto, C. Vozzi, F. Calegari, M. Negro, S. De Silvestri, and S. Stagira, “Interplay between group-delay-dispersion-induced polarization gating and ionization to generate isolated attosecond pulses from multicycle lasers,” Opt. Lett. 35, 2798–2880 (2010). [CrossRef] [PubMed]

] polarization gating regime by controlling the recombination step. The two-color and multi-color schemes can control electron dynamics or confine the ionization process within a short time to generate broadband isolated attosecond pulses [17

E. Takahashi, P. Lan, O. Mücke, Y. Nabekawa, and K. Midorikawa, “Infrared two-color multicycle laser field synthesis for generating an intense attosecond pulse,” Phys. Rev. Lett. 104, 233901 (2010). [CrossRef] [PubMed]

–26

G. Orlando, P. Corso, E. Fiordilino, and F. Persico, “A three-colour scheme to generate isolated attosecond pulses,” J. Phys. B 43, 025602 (2010). [CrossRef]

]. It is demonstrated recently that near-continuum spectra can be generated with the multicycle two-color field by carefully adjusting the wavelength of the control pulse [17

]. Since the harmonic efficiency is mainly decided by the ionization rate when the intensity of the driving field is far below the saturation intensity of the target, it has been firstly proposed by Gaarde et al. that the harmonic efficiency can be significantly enhanced by adding a xuv field [27

M. Gaarde and K. Schafer, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005). [CrossRef]

,28

J. Biegert, A. Heinrich, C. Hauri, W. Kornelis, P. Schlup, M. Anscombe, K. Schafer, M. Gaarde, and U. Keller, “Enhancement of high-order harmonic emission using attosecond pulse trains,” Laser Phys. 15, 899–902 (2005).

]. The coherent superposition state of the target atom is another alternative way to enhance the harmonic efficiency [29

J. Watson, A. Sanpera, X. Chen, and K. Burnett, “Harmonic generation from a coherent superposition of states,” Phys. Rev. A 53, R1962–R1965 (1996). [CrossRef] [PubMed]

,30

D. Milos̆ević, “Theoretical analysis of high-order harmonic generation from a coherent superposition of states,” J. Opt. Soc. Am. B 23, 308–317 (2006). [CrossRef]

]. But the large population of the excited state will lead to the high density of the free electrons in the medium. These high-density free electrons will weaken the phase-match of the xuv supercontinuum and distort the electric field of the driving pulse. These may lead to the low efficiencies of the macroscopic supercontinuum. Very recently, Hong et al. investigated the isolated attosecond pulse generation in pre-excited medium driven by a intense 4-fs few-cycle laser [31

W. Hong, Q. Zhang, X. Zhu, and P. Lu, “Intense isolated attosecond pulse generation in pre-excited medium,” Opt. Express 19, 4728–4739 (2011). [CrossRef] [PubMed]

]. Unfortunately, only very few laboratories can routinely produce such short driving pulse, which limits the spreading of their scheme.

In this work, we extend our scheme in ref [21

H. Du and B. Hu, “Propagation effects of isolated attosecond pulse generation with a multicycle chirped and chirped-free two-color field,” Phys. Rev. A 84,023817 (2011). [CrossRef]

] to the pre-excited system. We investigate the propagation effects of isolated attosecond pulse generation in pre-excited medium with a multicycle chirped and chirped-free two-color field, which relaxes the required pump pulse duration of the scheme in Ref [31

W. Hong, Q. Zhang, X. Zhu, and P. Lu, “Intense isolated attosecond pulse generation in pre-excited medium,” Opt. Express 19, 4728–4739 (2011). [CrossRef] [PubMed]

]. Our results reveal that though the large initial population of the excited state benefits large enhancement of harmonic intensity in single atom response, it can lead to the high density of the free electrons in the media and the large distortion of the electric field after propagation. Both the effects can weaken the phase-match of the macroscopic supercontinuum. Instead, small initial population of 4% can generate well phase-match intense supercontinuum. We also investigate an isolated attosecond pulse generation by using a filter centered on axis to select the harmonics in the far field. It is found that spatiotemporal profiles of the attosecond pulses in the far field present a curved distribution. The radius of the spatial filter should be chosen to be small enough to reduce the duration of the isolated attosecond pulse in order to avoid this curvature effect in the far field.

2. Theoretical methods

The simulation is carried out by taking into account both the single-atom response to the laser pulse and the collective response of macroscopic gas to the laser and high-order harmonic field. The single-atom response is calculated with the time-dependent Schrödinger equation. In our scheme, the initial coherent superposition state can be given by

ψ (x, 0) = α | g 〉 + β | e 〉

(1)

where the ground state |g〉 and excited state |e〉 are chosen to be the states with the binding energies of 54.4eV and 14.6eV corresponding to the 1s and 2s states of the He⁺ ion, respectively. α and β are the amplitudes of the ground and excited states, and α² + β² = 1. The initial population of the excited state can be expressed as p = β². The one-dimensional TDSE is solved accurately and efficiently by means of the split-operator method [32

M. Feit, J. Fleck Jr., and A. Steiger, “Solution of the schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412–433 (1982). [CrossRef]

]. Once the time evolution of the wave function ψ(x,t) is determined, the time-dependent induced dipole acceleration can be given by means of Ehrenfest’s theorem.

a (t) = - 〈 ψ (x, t) | \frac{\partial V (x)}{\partial x} - E (t) | ψ (x, t) 〉

(2)

The ionization probability can be given

1 - g (t) = 1 - \sum_{n = 1}^{N} {| < φ_{n} (x, 0) | ψ (x, t) > |}^{2} .

(3)

The ionization rate can be given [33

M. Vafaee, H. Sabzyan, Z. Vafaee, and A. Katanforoush, “Detailed instantaneous ionization rate of $H_{2}^{+}$ in an intense laser field,” Phys. Rev. A 74, 043416 (2006). [CrossRef]

]

w (t) = - \frac{d \ln (\sum_{n = 1}^{N} {| < φ_{n} (x, 0) | ψ (x, t) > |}^{2})}{d t} .

(4)

φ_n(x, 0) denotes the wave function of ground state and (n–1)th excited state.

The harmonic spectrum is then obtained by Fourier transforming the time-dependent dipole acceleration a⃗(t):

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

(5)

T and ω are the duration and frequency of the driving pulse, respectively. q corresponds to the harmonic order.

The collective response of the macroscope medium is described by the propagation of the laser and the high harmonic field, which can be written separately [34

E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimentional model of high-order harmonic generation in the few-optical cycle regime,” Phys. Rev. A 61, 063801 (2000). [CrossRef]

]

\nabla^{2} E_{f} (ρ, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E_{f} (ρ, z, t)}{\partial t^{2}} = \frac{ω_{p}^{2} (ρ, z, t)}{c^{2}} E_{f} (ρ, z, t),

(6)

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

(7)

Where E_f and E_h are the laser and high harmonic field; ω_p is the plasma frequency and is given by

ω_{p} = e \sqrt{n_{e} (ρ, z, t) / m_{e} ɛ_{0}}

and P_nl = [n₀ – n_e(ρ,z,t)]d_nl(ρ,z,t) is the nonlinear polarization generated by the medium. n₀ is the gas density and

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

is the free-electron density in the gas. This propagation model takes into account both the temporal plasma-induced phase modulation and the spatial plasma lensing effects, but does not consider the linear gas dispersion, the depletion of the fundamental beam during the HHG process and absorption of high harmonics. The absorption length L_abs for the given harmonic corresponds to λ₀/4πqβ [35

E. Takahashi, T. Kanai, K. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window x ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. 101, 253901 (2008). [CrossRef] [PubMed]

]. β is the absorptive parameter of the complex refractive index [36

B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50 – 30,000eV, Z = 1 – 92,” At. Data Nucl. Data Tables 54, 181–342 (1993). [CrossRef]

], given by

β = N_{0} r_{e} λ_{0}^{2} f_{2} (ω) / 2 π q^{2}

. Then the absorption length L_abs can be given by L_abs = q/2N₀r_eλ₀ f₂(ω), where N₀ is the neutral atom density, λ₀ is the wavelength of the driving pulse, r_e is the classical electron radius, and f₂ are atomic scattering factors which can be obtained from Refs. [36

]. In our calculation, N₀ = 2.6×10¹⁸cm⁻³, λ₀ = 800nm. For the 80th harmonic of helium atom, the absorption length L_abs is about 1.6cm. The length of the gas medium we adopted is much shorter than the absorption length L_abs. Hence, the re-absorption of the HHG beam can be negligible. Then the induced refractive index n can be approximately described by the refractive index in vacuum (n = 1). These equations can be solved with Crank-Nicholson method. The calculation details can be found in [34

,37

P. Lan, P. Lu, Q. Li, F. Li, W. Hong, and Q. Zhang, “Macroscopic effects for quantum control of broadband isolated attosecond pulse generation with a two-color field,” Phys. Rev. A 79, 043413 (2009). [CrossRef]

–39

Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A 70, 043802 (2004). [CrossRef]

In the near field, the total intensity of an API or an IAP can be calculated by [40

Ph. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using high-order harmonics,” Phys. Rev. Lett. 77, 1234–1237 (1996). [CrossRef] [PubMed]

]

I_{near} (t) = \int_{0}^{\infty} 2 π r d r {| \int_{ω_{1}}^{ω_{2}} E_{h} (r, ω) e^{(i ω t)} d ω |}^{2} .

(8)

In the far field, the total intensity of an API or an IAP can be calculated by

I_{near} (t) = \int_{0}^{r_{0}} 2 π r d r {| \int_{ω_{1}}^{ω_{2}} E_{h}^{f} (r, ω) e^{(i ω t)} d ω |}^{2} .

(9)

3. Results and discussions

In order to clearly demonstrate our scheme, we first investigate the HHG process according to the classical three-step model. In our calculation, the electric field of the laser pulse is expressed as

E (t) = E_{0} f_{0} (t) cos (ω_{0} t + δ (t)) + E_{1} f_{1} (t) cos (ω_{1} t + ϕ_{C E P}),

(10)

where E₀ and E₁ are the amplitudes of the driving and controlling fields, and ω₀ and ω₁ are the frequencies of the driving and controlling fields, respectively; ϕ_CEP is the relative phase and is set as 0.15π;

f_{0} (t) = exp (- 2 \ln 2 t^{2} / τ_{0}^{2})

and

f_{1} (t) = exp (- 2 \ln 2 t^{2} / τ_{1}^{2})

present the pulse envelopes of the driving and controlling fields, respectively. τ₀ and τ₁ are the pulse durations (full width at half maximum) of the driving and controlling fields. The time-dependent phase of the driving pulse is given by δ(t) = −β tanh(t/τ) [41

J. Carrera and S. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A 75, 033807 (2007). [CrossRef]

,42

Y. Xiang, Y. Niu, and S. Gong, “Control of the high-order harmonics cutoff through the combination of a chirped laser and static electric field,” Phys. Rev. A 79, 053419 (2009). [CrossRef]

]. β and τ are set as 6.25 and 800a.u., respectively. In our simulation, we choose ω₀ = 0.057a.u. and ω₁ = 0.0285a.u. corresponding to λ₀ = 800nm and λ₁ = 1600nm, respectively. E₀ and E₁ are set as 0.07a.u. and

\sqrt{0.1} E_{0}

, respectively. τ₀ = 3T₀ and τ₁ = 2T₁, where T₀ and T₁ are the periods of the driving and controlling fields. In our scheme, the excited state is responsible for ionization during the laser-matter interaction, while the ionization of the ground state is negligible. Figure 1(a) shows the ionization probability (dashed dotted pink curve)) in the chirped two-color field and the electric fields of the chirped driving pulse (dotted blue curve), controlling pulse (dashed red curve), and the chirped two-color field (solid black curve). In Fig. 1(b), we present the dependence of the electron kinetic energy on the ionization (blue dots) and recombination times (red circles) in the chirped two-color field. In our calculation, we only consider the electrons return to the ground state. As shown in Fig. 1(b), the harmonics above the 78th are emitted from the quantum path P1. Moreover, according to the three-step model, the harmonic efficiency is mainly decided by the ionization rate when the intensity of the driving field is far below the saturation intensity of the target. As shown in Fig. 1(a), the ionization of the electron path P1 takes place very rapidly. Hence, the supercontinuum with high efficiency can be emitted by adopting the coherent superposition state.

Fig. 1 (a) Electric fields of the chirped driving pulse (dotted blue curve), the control pulse (dashed red curve), the chirped two-color field (solid black curve), and the evolution of the ionization probability (dashed dotted pink curve). (b) Classical sketch of the electron dynamics in the chirped two-color pulse.

In the following, we further calculate the harmonic spectrum to confirm the classical sketch above. The harmonic spectrum is shown in Fig. 2(a). The harmonic spectrum from the pre-excited system with p=0.04 is shown bold red curve. For comparison, the harmonic spectrum from the pre-excited system with p=0.0 (the initial state is prepared as the ground state) is also presented (thin black curve). As shown in this figure, the harmonic intensity from the superposition state is 3–4 orders higher than that from the ground state alone. A similar enhancement of the harmonic yield by combining an attosecond pulse train with an IR driving laser has also been observed [27

M. Gaarde and K. Schafer, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005). [CrossRef]

,28

]. The spectrum cutoff is approximately the 92th harmonic and the spectrum above the 80th is continuous. The modulations on the supercontinuum are attributed to the interference of the short and the long quantum paths. To deeply understand the harmonic spectrum structure, Fig. 2(b) shows the time-frequency distribution of the HHG from the pre-excited system with p=0.04 in terms of the time-frequency analysis method [43

P. Antoine, B. Piraux, and A. Maquet, “Time profile of harmonics generated by a single atom in a strong electromagnetic field,” Phys. Rev. A 51, R1750–R1753 (1995). [CrossRef] [PubMed]

]. As shown in this figure, there are two main peaks contributing to the harmonics with the maximum orders of approximately the 92th and 78th, marked as P1, and P2, respectively. The quantum path P1 contributes mainly to the harmonics above the 78th and forms a supercontinuum with the bandwidth of 22 eV. These results are well in agreement with the classical results shown in Fig. 1.

Fig. 2 (a)The harmonic spectrum from the pre-excited system with p=0.04 (bold red curve) and p=0.0 (thin black curve) and (b) its time-frequency distribution from the pre-excited system with p=0.04.

In our scheme, the population of the exited state is totally depleted after the laser pulse while that of the ground state is hardly ionized as shown in Fig. 1(a), then the population of the excited state directly determines the ionization probability of the system and the density of the free electrons in the medium. However, high-density free electrons can lead to the spatial defocusing of the beam and temporal distortion of the driving laser field, which will influence the phase matching of HHG and the isolated attosecond pulse generation. Hence, there exists an optimal population of the excited state to realize the enhancement of the harmonic intensity and good phase matching of HHG. In addition, the coexistence of the short and long trajectories as shown in Fig. 2 prevents from the isolated attosecond pulse generation. It is well known that the collective response of the macroscopic gas allows one adjust the phase matching condition, such that eliminates the one quantum trajectories [44

P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. Paulus, H. Walther, R. Kopold, W. Becker, D. Milosević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001). [CrossRef] [PubMed]

]. To confirm the optimal population of the excited state and generate an isolated attosecond pulse, we perform the nonadiabatic three-dimensional (3D) propagation simulations [34

] for the fundamental and harmonic field in the gas target. We consider a tightly focused Gaussian laser beam with a beam waist of 30μm and a 0.75mm long gas jet with a density of 2.6 × 10¹⁸cm⁻³. The gas jet is placed 2mm after the laser focus. Other parameters are same as in Fig. 1. Figure 3(a) presents the on-axis harmonic spectra in the pre-excited medium with different initial population of the excited state of p = 0.02 (dashed black curve), p =0.04 (solid red curve) and p = 0.06 (dotted blue curve). One can see that the efficiency of the supercontinua for the cases of p=0.02 and p=0.04 is significantly enhanced after propagation, and the interference fringes are largely removed. Particularly, for the case of p=0.04, the harmonics above 70th are well phase matched and become smooth, indicating the elimination of one trajectory. On the contrary, for the case of p=0.06, there are the interference fringes. Moreover, the modulation of the supercontinua becomes deeper as the initial population of the excited state unceasingly increases. This can be attributed to two reasons. The first one is the high-density free electrons originating from the large population of the excited state, which weaken the phase-match of the supercontinuum. The anther one is distortion of the electric field after propagation. Figure 3(b) presents the on-axis electric field of the driving pulse at the end of the jet for the three cases. As shown in this figure, the distortions of the driving pulse become noticeable as the initial population of the excited state increases.

Fig. 3 (a) On-axis harmonic spectra in the pre-excited medium with different initial population of the excited state of p = 0.02 (dashed black curve), p =0.04 (solid red curve) and p = 0.06 (dotted blue curve). (b) The electric fields of the driving pulses after propagation with the excited-state populations of p = 0.02 (dashed black curve), p =0.04 (solid red curve) and p = 0.06 (dotted blue curve).

For the tight focusing laser pulses, the efficiency of the HHG is limited by the geometrical phase shift [45

Ph. Balcou and A. L’Huillier, “ Phase-matching effects in strong-field harmonic generation,” Phys. Rev. A 47, 1447–1459 (1993). [CrossRef] [PubMed]

] experienced by the fundamental beam (Gouy shift) and by defocusing [46

C. Altucci, T. Starczewski, E. Mevel, C. Wahlström, B. Carré, and A. L’Huillier, “ Influence of atomic density in high-order harmonic generation,” J. Opt. Soc. Am. B 13, 148–156 (1996). [CrossRef]

] due to the large density of free electrons. Next, we further investigate the broadband super-continuum generation at different propagation distances. Figure 4 presents on-axis harmonic spectra after propagating different distances with the population of the excited state of p=0.04. As shown in this figure, when the propagation distance increases from 0.5mm to 0.75mm, the intensity of supercontinuum increases gradually. However, when the propagation distance increases from 0.75mm to 1.0mm, the intensity of supercontinuum decreases obviously. Hence, we can conclude that the optimum thickness of the target is about 0.75mm.

Fig. 4 On-axis harmonic spectra after propagating different distances with the population of the excited state of p=0.04.

Next, we will take p=0.04 as example to investigate the isolated attosecond pulse generation. To unambiguously show that one of the trajectories has been eliminated, Fig. 5(a) shows that the normalized temporal profiles of the attosecond pulses generated by filtering the 85th–95th harmonics on the on-axis macroscopic high-harmonic spectrum (solid red curve) and on the single atom high harmonic spectrum (dashed black curve). As shown in this figure, the long trajectory indeed is eliminated after propagation.

Fig. 5 (a) Normalized temporal profiles of the on-axis attosecond pulses generated by filtering the 85th–95th harmonics in the macroscopic high-harmonic spectrum (solid red curve) and in the single atom high-harmonic spectrum (dashed black curve). (b) Temporal profiles of the on-axis attosecond pulses generated by filtering the 75th–95th harmonics in the macroscopic high-harmonic spectrum.

In the following, we further investigate the attosecond pulse generation. Figure 5(b) shows the temporal profile of the on-axis attosecond pulse by superposing the 75th–95th harmonics in the macroscopic high-harmonic spectrum. As shown in this figure, a pure isolated 133as pulse can be directly obtained without any chirp compensation. The characteristics of the macroscopic attosecond pulses also include the spatial properties. We further investigate the spatiotemporal profiles of the isolated attosecond pulse in the near field, which is shown in Fig. 6(a). As shown in this figure, isolated attosecond pulse can be obtained in the near field. The near-field spatial profile of isolated attosecond pulse is an annular-like distribution. In order to distinctly show the annular-like distribution, Fig. 6(b) presents the normalized intensity of the attosecond pulses along the line t = 0.144T₀ in Fig. 6(a). It is obvious that the isolated pulse shows an annular-like distribution in the near field. Figure 6(c) shows the time profile of the isolated attosecond pulse generated in the near field. As shown in this figure, a pure isolated attosecond pulse with the duration of 226as can be directly obtained in the near field. The duration of the radially integrated attosecond pulse is longer than that of the on-axis attosecond pulse.

Fig. 6 (a) Spatiotemporal profiles of the attosecond pulses at the exit of gas jet. (b) Normalized intensity of the attosecond pulses along the line t = 0.144T₀ in Fig. 6(a). (c) Time profile of radially integrated attosecond pulse in the near field.

The far-field spatiotemporal characteristics of the attosecond pulse are essential for its applications. We further investigate the spatiotemporal profile of the isolated attosecond pulse in the far field through a Hankel transformation [47

A. L’Huillier, Ph. Balcou, S. Candel, K. Schafer, and K. Kulander, “Calculations of high-order harmonic-generation processes in xenon at 1064 nm,” Phys. Rev. A 46, 2778–2790 (1992). [CrossRef]

,48

V. Tosa, K. T. Kim, and C. H. Nam, “Macroscopic generation of attosecond-pulse trains in strongly ionized media,” Phys. Rev. A 79, 043828 (2009). [CrossRef]

]. The results are shown in Fig. 7. Figure 7(a) shows that the spatiotemporal profiles of the attosecond pulses in the far field (z=0.5m). We can see that the generated isolated attosecond pulse presents a curved spatial distribution, which is attributed to different travelling distance of on-axis and off-axis harmonics. This effect can lead to the long duration of attosecond pulses when a spatial filter centered on axis with a large radius is applied. Figure 7(c) presents the time profile of radially integrated attosecond pulse in the far field with a radius r₀=0.3mm. It can be seen that a pure isolated attosecond pulse with the duration of 330as can be obtained in this case. Here, the throughput of the aperture is estimated to be 2% of the initially generated light. In order to avoid this curvature effect, the radius of the spatial filter should be chosen to be small enough. Moreover, when the radius of the spatial filter is small, the far-field spatial profile of isolated attosecond pulse is a Gaussian profile distribution, as shown in Fig. 7(b). It is believed that the spatial quality of isolated attosecond pulse with a Gaussian profile is better than that with an annular-like profile. Figure 8(a) shows the time profiles of radially integrated attosecond pulses in the far field (z=0.5m) using a spatial filter with different radius. We can see that the duration and intensity of the isolated attosecond pulse reduces with the increasing spatial filter radius. When a spatial filter with a radius of 0.1mm is applied, a pure isolated attosecond pulse with the duration of 137as which is very close to that of the on-axis attosecond pulse can be obtained. However, the throughput of the aperture is estimated to be 0.2% of the initially generated light for this case. For z=1.0m, the same conclusion can be obtained as shown in Fig. 8(b). The position of the spatial filter in the far field can be easily adjusted in an experiment. From Fig. 8, we can also see the change of attosecond pulses with the far-field position. As shown in this figure, using the same filter, when the radius of the spatial filter is small enough (r₀=0.1mm), the duration of the isolated attosecond pulse hardly changes with z, and the intensity of the isolated attosecond pulse reduces as the far field position z increases. However, for the biggish radius of the spatial filter (r₀=0.2mm and 0.3mm), both the duration and intensity of the isolated attosecond pulse reduce as the far field position z increases.

Fig. 7 (a) Spatiotemporal profiles of the attosecond pulses in the far field (z=0.5m). (b) Normalized intensity of the attosecond pulses along the line t = 0.144T₀ in Fig. 7(a). (c) Time profile of radially integrated attosecond pulse in the far field (z=0.5m) with a radius r₀ = 0.3mm.

Fig. 8 Time profiles of radially integrated attosecond pulses in the far field (a) z=0.5m and (b) z=1.0m is obtained using a spatial filter with different radius.

Finally, we discuss the generality of this scheme. Firstly, the initial population p of the excited state is estimated to be less than 8% to generate an isolated attosecond pulse. Secondly, the relative phase can be varied between 0 and 0.3π to generate broadband supercontinuum. Thirdly, the laser intensity should be moderate to ensure that the excited state is rapidly depleted to enhance the ionization and to also ensure that the ground state of the He⁺ isn’t significantly ionized. According to our calculation, the scheme can work well when the laser intensity is within 8.5 × 10¹³W/cm² and 2.2 × 10¹⁴W/cm².

4. Conclusion

In summary, we have investigated isolated attosecond pulse generation in pre-excited medium with a chirped and chirped-free two-color field. We found that the initial population of the excited state is responsible of the density of the free electrons in the medium. The large initial population of the excited state can lead to the high density of the free electrons in the media and the large distortion of the electric field after propagation. These can weaken the phase-match of the macroscopic supercontinuum. However, small initial population of 4% can generate well phase-match intense supercontinuum. In addition, we also investigated an isolated attosecond pulse generation by using a filter centered on axis to select the harmonics in the far field. It was shown that spatiotemporal profiles of the attosecond pulses in the far field present a curved spatial distribution due to different travelling distance of on-axis and off-axis harmonics. The radius of the spatial filter should be chosen to be small enough to reduce the duration of the isolated attosecond pulse to avoid this curvature effect. In this case, the far-field spatial profile of isolated attosecond pulse presents a Gaussian profile distribution.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 91026021, 11075068, 10875054, 11175076 and 10975065), the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2010-k08) and Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.

References and links

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6.	G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006). [CrossRef] [PubMed]
7.	P. Tzallas, E. Skantzakis, C. Kalpouzos, E. Benis, G. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys. 3, 846–850 (2007). [CrossRef]
8.	S. Gilbertson, S. Khan, Y. Wu, M. Chini, and Z. Chang, “Isolated attosecond pulse generation without the need to stabilize the carrier-envelope phase of driving lasers,” Phys. Rev. Lett. 105, 093902 (2010). [CrossRef] [PubMed]
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18.	W. Cao, P. Lu, P. Lan, X. Wang, and G. Yang, “Efficient isolated attosecond pulse generation from a multi-cycle two-color laser field,” Opt. Express 15, 530–535 (2007). [CrossRef] [PubMed]
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22.	T. Pfeifer, L. Gallmann, M. Abel, D. Neumark, and S. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett. 31, 975–977 (2006). [CrossRef] [PubMed]
23.	F. Calegari, C. Vozzi, M. Negro, G. Sansone, F. Frassetto, L. Poletto, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Efficient continuum generation exceeding 200 eV by intense ultrashort two-color driver,” Opt. Lett. 34, 3125–3127 (2009). [CrossRef] [PubMed]
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25.	C. Vozzi, F. Calegari, F. Frassetto, L. Poletto, G. Sansone, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Coherent continuum generation above 100 eV driven by an ir parametric source in a two-color scheme,” Phys. Rev. A 79, 033842 (2009). [CrossRef]
26.	G. Orlando, P. Corso, E. Fiordilino, and F. Persico, “A three-colour scheme to generate isolated attosecond pulses,” J. Phys. B 43, 025602 (2010). [CrossRef]
27.	M. Gaarde and K. Schafer, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005). [CrossRef]
28.	J. Biegert, A. Heinrich, C. Hauri, W. Kornelis, P. Schlup, M. Anscombe, K. Schafer, M. Gaarde, and U. Keller, “Enhancement of high-order harmonic emission using attosecond pulse trains,” Laser Phys. 15, 899–902 (2005).
29.	J. Watson, A. Sanpera, X. Chen, and K. Burnett, “Harmonic generation from a coherent superposition of states,” Phys. Rev. A 53, R1962–R1965 (1996). [CrossRef] [PubMed]
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31.	W. Hong, Q. Zhang, X. Zhu, and P. Lu, “Intense isolated attosecond pulse generation in pre-excited medium,” Opt. Express 19, 4728–4739 (2011). [CrossRef] [PubMed]
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38.	M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B. 41, 132001 (2008). [CrossRef]
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40.	Ph. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using high-order harmonics,” Phys. Rev. Lett. 77, 1234–1237 (1996). [CrossRef] [PubMed]
41.	J. Carrera and S. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A 75, 033807 (2007). [CrossRef]
42.	Y. Xiang, Y. Niu, and S. Gong, “Control of the high-order harmonics cutoff through the combination of a chirped laser and static electric field,” Phys. Rev. A 79, 053419 (2009). [CrossRef]
43.	P. Antoine, B. Piraux, and A. Maquet, “Time profile of harmonics generated by a single atom in a strong electromagnetic field,” Phys. Rev. A 51, R1750–R1753 (1995). [CrossRef] [PubMed]
44.	P. Salières, B. Carré, L. Le Déroff, F. Grasbon, G. Paulus, H. Walther, R. Kopold, W. Becker, D. Milosević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292, 902–905 (2001). [CrossRef] [PubMed]
45.	Ph. Balcou and A. L’Huillier, “ Phase-matching effects in strong-field harmonic generation,” Phys. Rev. A 47, 1447–1459 (1993). [CrossRef] [PubMed]
46.	C. Altucci, T. Starczewski, E. Mevel, C. Wahlström, B. Carré, and A. L’Huillier, “ Influence of atomic density in high-order harmonic generation,” J. Opt. Soc. Am. B 13, 148–156 (1996). [CrossRef]
47.	A. L’Huillier, Ph. Balcou, S. Candel, K. Schafer, and K. Kulander, “Calculations of high-order harmonic-generation processes in xenon at 1064 nm,” Phys. Rev. A 46, 2778–2790 (1992). [CrossRef]
48.	V. Tosa, K. T. Kim, and C. H. Nam, “Macroscopic generation of attosecond-pulse trains in strongly ionized media,” Phys. Rev. A 79, 043828 (2009). [CrossRef]

OCIS Codes
(190.4160) Nonlinear optics : Multiharmonic generation
(300.6560) Spectroscopy : Spectroscopy, x-ray
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Ultrafast Optics

History
Original Manuscript: February 7, 2012
Revised Manuscript: April 2, 2012
Manuscript Accepted: April 9, 2012
Published: April 13, 2012

Citation
Hongchuan Du, Laoyong Luo, Xiaoshan Wang, and Bitao Hu, "Isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field," Opt. Express 20, 9713-9725 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9713

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References

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature419, 803–807 (2002). [CrossRef] [PubMed]
T. Pfeifer, M. Abel, P. Nagel, A. Jullien, Z. Loh, M. Bell, D. Neumark, and S. Leone, “Time-resolved spectroscopy of attosecond quantum dynamics,” Chem. Phys. Lett.463, 11–24 (2008). [CrossRef]
A. Cavalieri, N. Müller, T. Uphues, V. Yakovlev, A. Balthuška, B. Hovrath, B. Schmidt, L. Blumel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature449, 1029–1032 (2007). [CrossRef] [PubMed]
E. Goulielmakis, M. Schultze, M. Hofstetter, V. Yakovlev, J. Gagnon, M. Uiberacker, A. Aquila, E. Gullikson, D. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science320, 1614–1617 (2008). [CrossRef] [PubMed]
F. Ferrari, F. Calegari, M. Lucchini, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “High-energy isolated attosecond pulses generated by above-saturation few-cycle fields,” Nat. Photonics4, 875–879 (2010). [CrossRef]
G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science314, 443–446 (2006). [CrossRef] [PubMed]
P. Tzallas, E. Skantzakis, C. Kalpouzos, E. Benis, G. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007). [CrossRef]
S. Gilbertson, S. Khan, Y. Wu, M. Chini, and Z. Chang, “Isolated attosecond pulse generation without the need to stabilize the carrier-envelope phase of driving lasers,” Phys. Rev. Lett.105, 093902 (2010). [CrossRef] [PubMed]
X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett.103, 183901 (2009). [CrossRef] [PubMed]
H. Mashiko, S. Gilbertson, M. Chini, X. Feng, C. Yun, H. Wang, S. Khan, S. Chen, and Z. Chang, “Extreme ultraviolet supercontinua supporting pulse durations of less than one atomic unit of time,” Opt. Lett.34, 3337–3339 (2009). [CrossRef] [PubMed]
M. Gaarde and K. Schafer, “Generating single attosecond pulses via spatial filtering,” Opt. Lett.31, 3188–3190 (2006). [CrossRef] [PubMed]
M. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A74, 053401 (2006). [CrossRef]
C. Jin, A-T Le, C. Trallero-Herrero, and C. Lin, “Generation of isolated attosecond pulses in the far field by spatial filtering with an intense few-cycle mid-infrared laser,” Phys. Rev. A84, 043411 (2011). [CrossRef]
P. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett.71, 1994–1997 (1993). [CrossRef] [PubMed]
C. Altucci, V. Tosa, and R. Velotta, “Beyond the single-atom response in isolated attosecond-pulse generation,” Phys. Rev. A75, 061401(R) (2007). [CrossRef]
C. Altucci, R. Velotta, V. Tosa, P. Villoresi, F. Frassetto, L. Poletto, C. Vozzi, F. Calegari, M. Negro, S. De Silvestri, and S. Stagira, “Interplay between group-delay-dispersion-induced polarization gating and ionization to generate isolated attosecond pulses from multicycle lasers,” Opt. Lett.35, 2798–2880 (2010). [CrossRef] [PubMed]
E. Takahashi, P. Lan, O. Mücke, Y. Nabekawa, and K. Midorikawa, “Infrared two-color multicycle laser field synthesis for generating an intense attosecond pulse,” Phys. Rev. Lett.104, 233901 (2010). [CrossRef] [PubMed]
W. Cao, P. Lu, P. Lan, X. Wang, and G. Yang, “Efficient isolated attosecond pulse generation from a multi-cycle two-color laser field,” Opt. Express15, 530–535 (2007). [CrossRef] [PubMed]
W. Hong, P. Lu, Q. Li, and Q. Zhang, “Broadband water window supercontinuum generation with a tailored mid-IR pulse in neutral media,” Opt. Lett.34, 2102–2104 (2009). [CrossRef] [PubMed]
H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A81, 063813 (2010). [CrossRef]
H. Du and B. Hu, “Propagation effects of isolated attosecond pulse generation with a multicycle chirped and chirped-free two-color field,” Phys. Rev. A84,023817 (2011). [CrossRef]
T. Pfeifer, L. Gallmann, M. Abel, D. Neumark, and S. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett.31, 975–977 (2006). [CrossRef] [PubMed]
F. Calegari, C. Vozzi, M. Negro, G. Sansone, F. Frassetto, L. Poletto, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Efficient continuum generation exceeding 200 eV by intense ultrashort two-color driver,” Opt. Lett.34, 3125–3127 (2009). [CrossRef] [PubMed]
Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett.98, 203901 (2007). [CrossRef] [PubMed]
C. Vozzi, F. Calegari, F. Frassetto, L. Poletto, G. Sansone, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Coherent continuum generation above 100 eV driven by an ir parametric source in a two-color scheme,” Phys. Rev. A79, 033842 (2009). [CrossRef]
G. Orlando, P. Corso, E. Fiordilino, and F. Persico, “A three-colour scheme to generate isolated attosecond pulses,” J. Phys. B43, 025602 (2010). [CrossRef]
M. Gaarde and K. Schafer, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A72, 013411 (2005). [CrossRef]
J. Biegert, A. Heinrich, C. Hauri, W. Kornelis, P. Schlup, M. Anscombe, K. Schafer, M. Gaarde, and U. Keller, “Enhancement of high-order harmonic emission using attosecond pulse trains,” Laser Phys.15, 899–902 (2005).
J. Watson, A. Sanpera, X. Chen, and K. Burnett, “Harmonic generation from a coherent superposition of states,” Phys. Rev. A53, R1962–R1965 (1996). [CrossRef] [PubMed]
D. Milos̆ević, “Theoretical analysis of high-order harmonic generation from a coherent superposition of states,” J. Opt. Soc. Am. B23, 308–317 (2006). [CrossRef]
W. Hong, Q. Zhang, X. Zhu, and P. Lu, “Intense isolated attosecond pulse generation in pre-excited medium,” Opt. Express19, 4728–4739 (2011). [CrossRef] [PubMed]
M. Feit, J. Fleck, and A. Steiger, “Solution of the schrödinger equation by a spectral method,” J. Comput. Phys.47, 412–433 (1982). [CrossRef]
M. Vafaee, H. Sabzyan, Z. Vafaee, and A. Katanforoush, “Detailed instantaneous ionization rate of H2+ in an intense laser field,” Phys. Rev. A74, 043416 (2006). [CrossRef]
E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimentional model of high-order harmonic generation in the few-optical cycle regime,” Phys. Rev. A61, 063801 (2000). [CrossRef]
E. Takahashi, T. Kanai, K. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window x ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett.101, 253901 (2008). [CrossRef] [PubMed]
B. Henke, E. Gullikson, and J. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50 – 30,000eV, Z = 1 – 92,” At. Data Nucl. Data Tables54, 181–342 (1993). [CrossRef]
P. Lan, P. Lu, Q. Li, F. Li, W. Hong, and Q. Zhang, “Macroscopic effects for quantum control of broadband isolated attosecond pulse generation with a two-color field,” Phys. Rev. A79, 043413 (2009). [CrossRef]
M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B.41, 132001 (2008). [CrossRef]
Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A70, 043802 (2004). [CrossRef]
Ph. Antoine, A. L’Huillier, and M. Lewenstein, “Attosecond pulse trains using high-order harmonics,” Phys. Rev. Lett.77, 1234–1237 (1996). [CrossRef] [PubMed]
J. Carrera and S. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A75, 033807 (2007). [CrossRef]
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