Attosecond ionization control for broadband supercontinuum generation using a weak 400-nm few-cycle controlling pulse

doi:10.1364/OE.20.027226

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Attosecond ionization control for broadband supercontinuum generation using a weak 400-nm few-cycle controlling pulse

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

Optics Express, Vol. 20, Issue 24, pp. 27226-27241 (2012)
http://dx.doi.org/10.1364/OE.20.027226

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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 demonstrate a method for generating the broadband supercontinuum. It is found that a weak 400-nm few-cycle pulse can be used to replace the ultraviolet attosecond pulse for controlling the ionization dynamics of the electron wave packets when a long-wavelength driving pulse is adopted. By adding a 400-nm few-cycle laser pulse to a 2000-nm driving pulse at proper time, only a quantum path can be selected to effectively contribute to the harmonics, leading to the efficient generation of a broadband supercontinuum. Moreover, our scheme is stable against nearly all the small parameter shift of the driving pulse and the controlling pulse. The macroscopic investigation reveals that the macroscopic supercontinuum with the bandwidth of about 165eV can be obtained. Then isolated sub-110-as pulses can be directly generated. Moreover, the generated attosecond pulse has a divergence angle of about 0.1mrad in the far field, which indicates its beam quality is good. Besides, it is also found that a near-field spatial filter can be used to select the different quantum paths (short or long) in the far field.

1. Introduction

High-order harmonic generation (HHG) by laser-atom (or molecule) interaction is a well-known method for generating coherent soft X-rays and extreme ultraviolet (XUV) attosecond sources [1

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009). [CrossRef]

A. Rundruist, C. Durfee III, Z. Chang, C. Herne, S. Backus, M. Murnane, and H. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science 280, 1412–1415 (1998). [CrossRef]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002). [PubMed]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature (London) 414, 509–513 (2001). [CrossRef]

–5

P. Paul, E. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001). [CrossRef] [PubMed]

]. So far, high-order harmonic generation is also the unique way to produce isolated attosecond pulses and attosecond pulse trains. In the last decade, attosecond pulse trains and isolated attosecond pulses have been successfully produced experimentally [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]

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]

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]

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]

–10

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]

]. Recently, the 100-as barrier has been first brought through by Goulielmakis et al. [6

] using a 3.3-fs driving pulse. However, it is still an extreme challenge to generate a broadband supercontinuum that supports the isolated attosecond pulse generation with shorter duration, since the duration of the driving pulse can hardly be further compressed. Moreover, the low intensity of isolated attosecond pulses also limits their applications. In addition, Salières et al. investigated the spatial structure of harmonics in order to generate the high-order harmonics with good spatial coherence property [11

P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of the high-order harmonics,” Phys. Rev. Lett. 74, 3376–3379 (1995). [CrossRef]

P. Salières, T. Ditmire, M. Perry, A. L’Huillier, and M. Lewenstein, “Angular distributions of the high-order harmonics generated by a femtosecond laser,” J. Phys. B 29, 4771–4786 (1996). [CrossRef]

–13

F. Schapper, M. Holler, T. Auguste, A. Zair, M. Weger, P. Salières, L. Gallmann, and U. Keller, “Spatial finger-print of quantum path interferences in high order harmonic generation,” Opt. Express 18, 2987–2994 (2010). [CrossRef] [PubMed]

The HHG process can be well understood by the three-step model [14

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

]. In detail, the electron firstly tunnels through the barrier formed by the Coulomb potential and the laser field, then it oscillates almost freely in the laser field and gains kinetic energy, and finally it returns to the ground state by recombining with the parent ion. During the recombination process, a photon with energy equal to ionization potential plus the kinetic energy is released. This classical picture indicates that the HHG process can be controlled by manipulating different steps for the broadband supercontinuum generation. It has been proposed that a two-color or multi-color field can control the acceleration process or confine the ionization process within half a cycle to produce a broadband supercontinuum [15

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]

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]

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]

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]

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]

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]

H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A 81, 063813 (2010). [CrossRef]

H. Du, L. Luo, X. Wang, and B. Hu, “Isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field,” Opt. Express 20, 9713–9725 (2012). [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]

–24

P. Lan, E. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A 82, 053413 (2010). [CrossRef]

]. Another effective way is the polarization gating technique [25

P. Corkum, N. Burnett, and M. Ivanov, “Subfemtosecond pulses,” Opt. Lett. 19, 1870–1872 (1994). [CrossRef] [PubMed]

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

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]

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]

–29

H. Du and B. Hu, “Broadband supercontinuum generation method combining mid-infrared chirped-pulse modulation and generalized polarization gating,” Opt. Express 18, 25958–25966 (2010). [CrossRef] [PubMed]

] that can confine the recombination of the electrons into one half-cycle, then a broadband supercontinuum in the plateau can be obtained. Besides, since an ultraviolet (uv) attosecond pulse can enhance significantly the ionization rate of the target [30

K. Schafer, M. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004). [CrossRef] [PubMed]

K. Ishikawa, “Photoemission and ionization of He⁺ under simultaneous irradiation of fundamental laser and high-order harmonic pulses,” Phys. Rev. Lett. 91, 043002 (2003). [CrossRef] [PubMed]

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

–33

A. Bandrauk and N. Shon, “Attosecond control of ionization and high-order harmonic generation in molecules,” Phys. Rev. A 66, 031401(R) (2002). [CrossRef]

], the uv-assisted ionization enhancement has been also suggested to gate the ionization times to generate the efficient broadband supercontinuum [34

P. Lan, P. Lu, W. Cao, and X. Wang, “Efficient generation of an isolated single-cycle attosecond pulse,” Phys. Rev. A 76, 043808 (2007). [CrossRef]

]. Unfortunately, only few laboratories can routinely produce such an ultraviolet (uv) attosecond pulse, which limits the spreading of their scheme. However, can the ultraviolet (uv) attosecond pulse be replaced by a realizable laser pulse to popularize this scheme? In this work, we give an affirmative answer. It is found that a weak 400-nm few-cycle pulse can be used to replace the ultraviolet attosecond pulse for controlling the ionization dynamics of the electron wave packets when a 2000-nm driving pulse is adopted. Only a quantum path can be selected to effectively contribute to the harmonics, leading to the efficient generation of a broadband supercontinuum by adding a 400-nm few-cycle laser pulse to a 2000-nm driving pulse at proper time. In addition, the 3D propagation is also carried out to investigate the spectral, temporal and spacial characteristics of the generated attosecond pulse.

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 the macroscopic gas to the laser and high-harmonic field. The single-atom response is calculated with the Lewenstein model [35

M. Lewenstein, Ph. Balcou, M. Ivanov, A. L’Huillier, and P. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef] [PubMed]

]. In this model, the instantaneous dipole moment of an atom is described as (in atom units)

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

(1)

In this equation, E(t) is the electric field of the laser pulse, A(t) is its associated vector potential, and ε is a positive regularization constant. p_st and S_st are the stationary momentum and quasiclassical action, which are given by

p_{s t} (t^{'}, t) = \frac{1}{t - t^{'}} \int_{t^{'}}^{t} A (t^{″}) d t^{″},

(2)

S_{s t} (t^{'}, t) = (t - t^{'}) I_{p} - \frac{1}{2} p_{s t}^{2} (t^{'}, t) (t - t^{'}) + \frac{1}{2} \int_{t^{'}}^{t} A^{2} (t^{″}) d t^{″},

(3)

where I_p is the ionization energy of the atom, and d(p) is the dipole matrix element for transitions from the ground state to the continuum state. For hydrogenlike atoms, it can be approximated as [36

H. Bethe and E. Salpeter, Quantum mechanics of one and two electron atoms (Academic, New York, 1957).

]

d (p) = i \frac{2^{7 / 2}}{π} {(2 I_{p})}^{5 / 4} \frac{p}{{(p^{2} + 2 I_{p})}^{3}} .

(4)

The g(t′) in Eq. (1) represents the ground state amplitude:

g (t) = exp [- \int_{- \infty}^{t} ω (t^{″}) d t^{″}],

(5)

where ω(t″) is the ionization rate which is calculated by ADK tunneling model [37

M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms and of atoms ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

ω (t) = ω_{p} {| C_{n^{*}} |}^{2} {(\frac{4 ω_{p}}{ω_{t}})}^{2 n^{*} - 1} exp (- \frac{4 ω_{p}}{3 ω_{t}}),

(6)

where

ω_{p} = \frac{I_{p}}{\bar{h}}, ω_{t} = \frac{e | E_{f} (t) |}{\sqrt{2 m_{e} I_{p}}}, n^{*} = Z {(\frac{I_{p h}}{I_{p}})}^{1 / 2}, {| C_{n^{*}} |}^{2} = \frac{2^{2 n^{*}}}{n^{*} Γ (n^{*} + 1) Γ (n^{*})},

(7)

where Z is the net resulting charge of the atom, and I_ph is the ionization potential of the hydrogen atom, and e and m_e are electron charge and mass, respectively.

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

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

(8)

where a⃗(t) = d̈_nl(t), T and ω are the duration and frequency of the driving pulse, respectively. q corresponds to the harmonic order.

The collective response of the macroscopic medium is described by the propagation of the laser and the high harmonic field, which can be written separately [38

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

(9)

\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}} .

(10)

Where E 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} ε_{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, which is due to the low gas density [38

]. 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 [38

The far-field harmonic emissions can be obtained from near-field harmonic emissions at the exit face of a gas medium through a Hankel transformation [39

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

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

–41

C. Jin, A.-T. Le, and C. Lin, “Medium propagation effects in high-order harmonic generation of Ar and N₂,” Phys. Rev. A 83, 023411 (2011). [CrossRef]

E_{h}^{f} (r_{f}, z_{f}, ω) = - i k \int_{0}^{r_{0}} \frac{E_{h} (r_{f}, z^{'}, ω)}{z_{f} - z^{'}} J_{0} (\frac{k r r_{f}}{z_{f} - z^{'}}) exp [\frac{i k (r^{2} + r_{f}^{2})}{2 (z_{f} - z^{'})}] r d r,

(11)

where J₀ is the zero-order Bessel function, z_f is the far-field position from the laser focus, r_f is the transverse coordinate in the far field, and the wave vector k is given by k = ω/c. r₀ is the radius of a near-field spatial filter.

3. Results and discussions

In order to demonstrate our scheme, we first analyze the HHG process from He atom driven by the 2000-nm driving pulse alone in terms of the three-step model [14

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

], which presents a clear physical picture. Figure 1 shows the classical sketch of the HHG process in the 2000-nm driving pulse alone. We present the electric field of the 2000-nm driving pulse in Fig. 1(a). Figure 1(b) presents the dependence of harmonic order on the ionization (blue dots) and recombination times (red circles) in this driving pulse. As shown in Fig. 1(b), the 2000-nm driving pulse yields two couples of ionization and recombination times per half-cycle (known as short and long trajectories). The electron ionized around t = −0.4T₀ (T₀ is the optical cycle of the 2000-nm driving pulse) is driven by the driving field at the highest amplitude and returns to the parent ion near t = 0.25T₀, leading to the emissions of the harmonics with the maximal order of 420. The electron is ionized around t = −1.0T₀ and returns to the parent ion near t = −0.3T₀, leading to the emissions of the harmonics with the second maximal order of 365. Therefore, more than one return contribute to the lower harmonics and the isolated attosecond pulse can be only obtained by filtering the harmonics in the cutoff, which prevents the generation of the broadband isolated attosecond pulse with a shorter duration. If the ionization rate around t = −0.4T₀ is enhanced significantly, the recombination quantum path near t = 0.25T₀ would mainly contribute to the harmonic emissions and a broadband supercontinuum would be obtained. According to this idea, Lan et al. [34

P. Lan, P. Lu, W. Cao, and X. Wang, “Efficient generation of an isolated single-cycle attosecond pulse,” Phys. Rev. A 76, 043808 (2007). [CrossRef]

] proposed an efficient method for producing a broadband supercontinuum by enhancing the contribution from one return with an UV pulse. Unfortunately, so far, such an ultraviolet (uv) attosecond pulse can be routinely produced in only few laboratories. Moreover, the intensity of the ultraviolet (uv) attosecond pulse is also very low. These limit the spreading of their scheme. Because an optical cycle of the 2000-nm driving pulse is 5 times long as that of the 400-nm laser pulse, a 400-nm laser pulse lasts five times less than the 2000-nm driving pulse when the same optical cycles are contained. Hence, a weak 400-nm few-cycle pulse can be introduced to replace the ultraviolet attosecond pulse for controlling the ionization dynamics of the electron wave packets when a 2000-nm driving pulse is adopted. The schematic illustration of our scheme is shown in Fig. 2. A weak 400-nm few-cycle pulse is synthesized to the 2000-nm driving pulse. Experimentally, this scheme can be carried out with a Ti: sapphire laser system. The laser beam is split into a stronger beam and a weaker one. The stronger one is used to produce the 2000-nm driving pulse via an optical parametric amplifier, and the weaker one is used to generate the 400-nm controlling pulse. In our calculation, the electric field of the combined field can be expressed as

E (t) = E_{0} f_{0} (t) cos (ω_{0}) t + \sqrt{a} E_{0} f_{1} (t - t_{delay}) cos [ω_{1} (t - t_{delay}) + ϕ] .

(12)

Here, E₀ is the electric amplitude of the driving pulse, and ω₀ and ω₁ are the frequencies of the driving and controlling pulses, respectively.

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

and

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

present the profiles of the two pulses, and τ₀ = 2. T₀ and τ₁ = 2. T₁ are the pulse durations of the driving and controlling pulse, where T₀ and T₁ are the optical periods of the driving and controlling pulses. ϕ is the relative phase and is chosen to be π. τ_delay is the time-delay of the controlling and driving pulses and is chosen to be −0.425T₀. a is the rate of the intensities of the controlling and driving pulses. E₀ and a are chosen to be 0.075a.u.and 0.4, respectively. As shown in Fig. 2, the electron is only effectively ionized around t = −0.4T₀ in the combined field. Consequently, only the return near t = 0.25T₀ is selected to effectively contribute to the harmonics, which will lead to the efficient generation of the broadband super-continuum.

Fig. 1 (a) Electric field of the 2000-nm driving pulse, and (b) classical sketch of the electron dynamics in the driving pulse.

Fig. 2 Electric fields of the 2000-nm driving pulse (bold black curve) and the 400-nm few-cycle controlling pulse (thin red curve), and the evolution of the ionization probability (dashed blue curve) in the combined field.

To verify the above scheme, we calculate the harmonic spectrum using the Lewenstein model [35

M. Lewenstein, Ph. Balcou, M. Ivanov, A. L’Huillier, and P. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994). [CrossRef] [PubMed]

]. Here the neutral species depletion is considered using the ADK model. Figure 3(a) shows the harmonic spectrum in the combined field (solid red curve). For comparison, the harmonic spectrum in the 2000-nm driving pulse alone is also presented (dotted black curve). One can see that the overall spectral structure is irregular for the harmonics below 300ω₀, and only the harmonics near the cutoff are continuous in the 2000-nm driving pulse alone. For this case, the isolated attosecond pulse can be only obtained by synthesizing the harmonics near the cutoff. However, in the combined field, the harmonic spectrum shows a different structure in contrast to that in the 2000-nm driving pulse alone. It can be seen that the harmonic spectrum is supercontinuous above 160ω₀ and a supercontinuum with the bandwidth of 160eV is generated. Moreover, the harmonic intensity is 2 or 3 orders of magnitude higher than that in the 2000-nm driving pulse alone. In order to further understand the emission times of the harmonics, the time-frequency distribution in the combined field is shown in Fig. 3(b). One can clearly see that only the quantum path P1 around t = 0.25T₀ mainly contributes to the harmonics above 160ω₀. Thereby only the return P1 is selected to effectively contribute to the generation of the supercontinuum, which is in good agreement with the above classical results. Moreover, the yields of the harmonics generated from the short trajectory are more efficient than that of the harmonics generated from the long trajectory, which leads to the generation of the smooth supercontinuum. In this scheme, the structure of the supercontinuum in the microscopic level is insensitive to the small changes in the intensity and duration of the controlling field. Figure 4 presents the harmonic spectra generated at the different intensities of the controlling pulse. Compared with a = 0.4, the structure of the supercontinuum hardly changes though the intensity decreases for a = 0.2. However, when the intensity of the controlling pulse decreases to a = 0.1, the structure of the harmonic spectrum becomes irregular. This implies that the continuous harmonics in the plateau can not be obtained when the intensity of the controlling pulse is too low. Then, the intensity of the controlling field is set as a = 0.4 and the pulse duration is scanned. Figure 5 shows the harmonic spectra generated at the different durations of the controlling pulse. For clarity, the harmonic spectrum with τ₁ = 4.5T₁ (solid glaucous curve) has been shifted 2 units down. As shown in this figure, the supercontinuum can be still obtained for τ₁ = 1.5T₁ and τ₁ = 2.5T₁. Moreover, the structure and intensity of the supercontinuum hardly change. However, when the duration increases to τ₁ = 4.5T₁, the harmonic spectrum below 360ω₀ becomes irregular and only the harmonics above 360ω₀ are continuous. For this case, the bandwidth of the supercontinuum is about 40eV, which is much less than that of the supercontinuum with τ₁ = 2.0T₁. According to our calculation, the duration of the controlling pulse doesn’t exceed τ₁ = 3.5T₁ to generate the broadband supercontinuum.

Fig. 3 (a) Harmonic spectra generated in the combined field (solid red curve) and in the 2000-nm driving pulse alone (dotted black curve), and (b) time-frequency distribution in the combined field.

Fig. 4 Harmonic spectra generated at the different intensities of the controlling pulse. Other parameters are the same as in Fig. 2.

Fig. 5 Harmonic spectra generated at the different durations of the controlling pulse. Other parameters are the same as in Fig. 2.

Next, we further investigate the influence of the driving pulse duration. Figure 6 shows the harmonic spectra generated at the different durations of the driving pulse. Other parameters are the same as in Fig. 2. As shown in this figure, when the pulse duration increases to τ₀ = 4.0T₀, the bandwidth and the intensity of the supercontinuum decrease. However, when the pulse duration further increases to τ₀ = 8.0T₀, the bandwidth of the supercontinuum hardly changes except for a slight reduction of the intensity compared to the case of τ₀ = 4.0T₀. Thus our scheme is also available for the driving pulse with the longer pulse duration to generate the broadband supercontinuum.

Fig. 6 Harmonic spectra generated at the different durations of the driving pulse. Other parameters are the same as in Fig. 2.

Because the laser field interacts with a macroscopic number of atoms in experiment, the full description of the HHG process requires solving not only the strong-field laser-atom interaction at the microscopic level, but also the Maxwell wave equation which describes the propagation of the radiation through the nonlinear medium at the macroscopic level [42

M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B 41, 132001 (2008). [CrossRef]

]. Moreover, the macroscopic effect often alters the results of the single-atom scheme. In order to fully demonstrate our scheme, we perform the nonadiabatic three-dimensional (3D) propagation simulations [38

] for fundamental and harmonic field in the gas target. We consider a focused Gaussian laser beam with waist of 50μm and a 0.4-mm long gas jet with the gas density of 2.5 × 10¹⁸/cm³. The gas jet is placed 2.0mm after the laser focus. Other parameters are the same as in Fig. 2. Figure 7 shows the macroscopic spectrum in the combined field (bold red curve). For comparison, the single-atom result is also presented (thin blue curve). One can clearly see that the interference fringes through the plateau to the cutoff are all removed after propagation, which implies that the continuous harmonics from a single path are perfectly phase-matched. Since the relative time-delay between the controlling pulse and the driving pulse can affect the generation of the supercontinuum, we further discuss the influences of the relative time-delay. Figure 8 presents the macroscopic harmonic spectra produced at the different time-delays between the 2000-nm and 400-nm fields. As shown in this figure, for the three different time-delays, the bandwidth of the macroscopic supercontinuous spectra hardly changes. Namely, the macroscopic broadband supercontinuum can be obtained when the time-delay is within the range from 0.4T₀ to 0.45T₀. In the current laser technology, the time-delay can be controlled in dozens of attosecond by a piezoelectric translator [43

H. Mashiko, S. Gilbertson, C. Li, S. Khan, M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008). [CrossRef] [PubMed]

]. Thus our scheme is stable against small changes in the relative time-delay.

Fig. 7 The harmonic spectrum after 3D propagation in the combined field.

Fig. 8 The macroscopic harmonic spectra generated at the different time-delays between the 2000-nm and 400-nm fields.

In the following, we investigate the attosecond pulse generation making use of the macroscopic supercontinuum (bold red curve) in Fig. 7. Figure 9 presents the isolated attosecond pulses by synthesizing the harmonics of 160–220, 220–280, and 280–340 order, respectively. As shown in this figure, by applying a square window to the supercontinuum, the isolated sub-110-as pulses with extremely high signal-to-noise ratio are directly obtained without any chirp compensation. Moreover, this supercontinuum with the bandwidth of 165eV can support the attosecond pulse duration below 24as with proper chirp compensation. Since the spatiotemporal properties of the attosecond pulses are very important for practical applications, we further investigate the spatiotemporal profiles of the isolated attosecond pulses in the near field. Figure 10(a) and (b) present the spatiotemporal profiles of the attosecond pulses at the end of the medium generated by superposing the harmonics of (a) 160th–220th and (b) 280th–340th, respectively. For the attosecond pulse generated by superposing the harmonics of 160th–220th, as shown in Fig. 10(a), the near-field spatiotemporal distribution shows a crescent-like structure. And the attosecond pulse has a second maximal intensity around the radius of 27μm. For the attosecond pulse generated by superposing the harmonics of 280th–340th, as shown in Fig. 10(b), the crescent-like distribution is obviously degenerate. And the attosecond pulse has a maximal intensity around the radius of 23μm. Moreover, the isolated attosecond pulse has a strong spatial chirp and is generated later off-axis than on-axis. In order to clearly understand the near-field spatiotemporal characteristics of the generated attosecond pulses, we investigate the spatial distribution of macroscopic HHG spectrum at the exit face. The result is shown in Fig. 11. One can clearly see that there is a broadband macroscopic supercontinuum in the region (radius: −10μm—10μm and harmonic order: 150th–380th). And the intensity of the harmonics in the cutoff is stronger than that of the harmonics in the plateau for each value of radius, which leads to the appearance of the ring structure in the radial direction in the range from 150th to 380th. Moreover, the intensity of the harmonics in the cutoff decreases as the radius increases. The ring structure of the macroscopic harmonic spectrum leads to the appearance of a second maximal intensity of the isolated attosecond pulse generated by superposing the harmonics of 160th–220th around the radius of 27μm and the appearance of a maximal intensity of the isolated attosecond pulse generated by superposing the harmonics of 280th–340th around the radius of 23μm.

Fig. 9 Isolated attosecond pulses centered at different frequencies.

Fig. 10 Spatiotemporal profiles of the attosecond pulses at the end of the medium generated by superposing the harmonics of (a) 160th–220th and (b) 280th–340th, respectively.

Fig. 11 Spatial distribution of macroscopic HHG spectrum at the exit face.

The far-field spatiotemporal characteristics of the attosecond pulse are very important for its practical applications. We further investigate the spatial profile and spatiotemporal characteristic of the isolated attosecond pulse in the far field. In the calculation, the far-field position is located 1.0m from the laser focus. The spatial profile of isolated attosecond pulse generated by superposing the harmonics from 220 to 280 is shown in Fig. 12(a). It can be seen from this figure that the divergence angle of the generated isolated attosecond pulse is about 0.1mrad according to our calculation. This indicates that a large amount of the energy is radiated on axis and the isolated attosecond pulse processes a good beam quality. In order to fully show the far-field characteristic of the isolated attosecond pulse, figure 12(b) presents the spatiotemporal profile of the isolated attosecond pulse in the far field. As shown in this figure, the attosecond pulse is radiated at t = 0.25T₀ on axis and the generated attosecond pulse presents a curved spatial distribution. Fortunately, a large energy of the attosecond pulse is radiated on axis. Thus, a strong isolated attosecond pulse can be directly obtained in the far field.

Fig. 12 The far-field spatial profile and far-field spatiotemporal profile of isolated attosecond pulse generated by superposing the harmonics from 220 to 280.

Next,we investigate the influence of the near-field filter to the far-field characteristic of the generated attosecond pulse. Figure 13(a) presents the far-field profile of the generated attosecond pulse by using the near-field filter with a radius of 5μm. Other parameters are the same as in Fig. 12. As shown in Fig. 13(a), the far-field spatial distribution is a Gaussian-like stucture. Moreover, the divergence angle is larger than that without the near-field spatial filter as shown in Fig. 12(a). This implies that the better beam quality can be obtained without the near-field filter. In order to clearly show the far-field spatiotemporal characteristic of the generated attosecond pulse, figure 13(b) presents the spatiotemporal profile of the isolated attosecond pulse with the near-field filter. As shown in this figure, the generated attosecond pulse also presents a curved spatial distribution. It is quite surprising that the attosecond pulse is radiated at t = 0.05T₀ on axis, which is obviously earlier than that without the near-field filter. In order to further understand the different radiation times of the attosecond pulses without and with the near-field filter, figure 14 shows the time-frequency distribution of on-axis harmonic spectra in the far field without and with the near-field filter. As shown in Fig. 14(a), when the total near-field harmonics are ”projected” into the far field, the intensity of the long quantum path is stronger than that of the short one. In other words, the long quantum path is successfully selected in the far field for this case. However, when the near-field filter is performed, only the harmonics in the window from 0 to 5μm in the near field are ”projected” in to the far field. As shown in Fig. 14(b), the intensity of the short quantum path is stronger than that of the long one. Namely, the short quantum path is successfully selected in the far field. Therefore, the different radiation times of the far-field attosecond pulses without and with the near-field filter come from the different selection (short or long) of the quantum path. Moreover, our results also imply that a near-field spatial filter can be also used to select different quantum paths (short or long) in the far field. In addition, we also calculate the far-field profiles of isolated attosecond pulses with the near-field filter at different radiuses, which is shown in Fig. 15. It is clear that the divergence angle of the attosecond pulse can be significantly affected by the radius of the near-field filter. For comparison, the far-field profile of isolated attosecond pulse without the near-field filter is also presented (solid glaucous curve). At first glance, the far-field profile without the near-field filter is much closer to that using the near-field filter with a radius of 5μm. This is because that the normalized intensity is used in this figure. In fact, they are quite different. The intensity of the attosecond pulse without the near-field filter is far larger than that using the near-field filter with a radius of 5μm. Moreover, as shown in Fig. 12(b), when the near-field filter is not adopted, the attosecond pulse is radiated at t = 0.25T₀ on axis, which corresponds to the long quantum path. However, as shown in Fig. 13(b), when the near-field filter with a radius of 5μm is adopted, the attosecond pulse is radiated at t = 0.05T₀ on axis, which corresponds to the short quantum path.

Fig. 13 The far-field profile and far-field spatiotemporal profile of isolated attosecond pulse generated by superposing the harmonics from 220 to 280 using the near-field filter with a radius of 5μm.

Fig. 14 Time-frequency distribution of on-axis harmonic spectra in the far field without (a) and with (b) the near-field filter.

Fig. 15 The far-field profiles of isolated attosecond pulses with the near-field filter at different radiuses. The solid glaucous curve is the far-field profile of isolated attosecond pulse without the near-field filter.

4. Conclusion

In summary, we investigate the broadband supercontinuum generation by adding a 400-nm few-cycle laser pulse to a 2000-nm driving pulse. It is found that a weak 400-nm few-cycle pulse can be used to replace the ultraviolet attosecond pulse for controlling the ionization dynamics of the electron wave packets when a long-wavelength driving pulse is adopted. By properly selecting the time-delay between the two pulses, only a quantum path can be selected to effectively contribute to the harmonics, leading to the efficient generation of a broadband supercontinuum. Moreover, the microscopic broadband supercontinuum can be produced for nearly all the small parameter shift of the driving pulse and the controlling pulse. The macroscopic propagation is also carried out and the macroscopic supercontinuum with the bandwidth of about 165eV can be generated. Then isolated sub-110-as pulses can be directly obtained. Moreover, the generated attosecond pulse has a good beam quality in the far field. In addition, we also find that a near-field spatial filter can be used to select the different quantum paths (short or long) in the far field.

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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9.	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]
10.	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]
11.	P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of the high-order harmonics,” Phys. Rev. Lett. 74, 3376–3379 (1995). [CrossRef]
12.	P. Salières, T. Ditmire, M. Perry, A. L’Huillier, and M. Lewenstein, “Angular distributions of the high-order harmonics generated by a femtosecond laser,” J. Phys. B 29, 4771–4786 (1996). [CrossRef]
13.	F. Schapper, M. Holler, T. Auguste, A. Zair, M. Weger, P. Salières, L. Gallmann, and U. Keller, “Spatial finger-print of quantum path interferences in high order harmonic generation,” Opt. Express 18, 2987–2994 (2010). [CrossRef] [PubMed]
14.	P. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993). [CrossRef]
15.	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]
16.	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]
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18.	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]
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22.	H. Du, L. Luo, X. Wang, and B. Hu, “Isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field,” Opt. Express 20, 9713–9725 (2012). [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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29.	H. Du and B. Hu, “Broadband supercontinuum generation method combining mid-infrared chirped-pulse modulation and generalized polarization gating,” Opt. Express 18, 25958–25966 (2010). [CrossRef] [PubMed]
30.	K. Schafer, M. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004). [CrossRef] [PubMed]
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42.	M. Gaarde, J. Tate, and K. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. B 41, 132001 (2008). [CrossRef]
43.	H. Mashiko, S. Gilbertson, C. Li, S. Khan, M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008). [CrossRef] [PubMed]

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

ToC Category:
Supercontinuum Generation and Ultra-Short Pulses

History
Original Manuscript: September 14, 2012
Revised Manuscript: November 5, 2012
Manuscript Accepted: November 5, 2012
Published: November 19, 2012

Virtual Issues
Nonlinear Photonics (2012) Optics Express

Citation
Hongchuan Du, Laoyong Luo, Xiaoshan Wang, and Bitao Hu, "Attosecond ionization control for broadband supercontinuum generation using a weak 400-nm few-cycle controlling pulse," Opt. Express 20, 27226-27241 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27226

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References

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys.81, 163–234 (2009). [CrossRef]
A. Rundruist, C. Durfee, Z. Chang, C. Herne, S. Backus, M. Murnane, and H. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science280, 1412–1415 (1998). [CrossRef]
R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science297, 376–378 (2002). [PubMed]
M. Hentschel, R. Kienberger, Ch. Spielmann, G. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature (London)414, 509–513 (2001). [CrossRef]
P. Paul, E. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science292, 1689–1692 (2001). [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]
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]
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]
P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of the high-order harmonics,” Phys. Rev. Lett.74, 3376–3379 (1995). [CrossRef]
P. Salières, T. Ditmire, M. Perry, A. L’Huillier, and M. Lewenstein, “Angular distributions of the high-order harmonics generated by a femtosecond laser,” J. Phys. B29, 4771–4786 (1996). [CrossRef]
F. Schapper, M. Holler, T. Auguste, A. Zair, M. Weger, P. Salières, L. Gallmann, and U. Keller, “Spatial finger-print of quantum path interferences in high order harmonic generation,” Opt. Express18, 2987–2994 (2010). [CrossRef] [PubMed]
P. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett.71, 1994–1997 (1993). [CrossRef]
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]
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]
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]
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]
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]
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]
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, L. Luo, X. Wang, and B. Hu, “Isolated attosecond pulse generation from pre-excited medium with a chirped and chirped-free two-color field,” Opt. Express20, 9713–9725 (2012). [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]
P. Lan, E. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A82, 053413 (2010). [CrossRef]
P. Corkum, N. Burnett, and M. Ivanov, “Subfemtosecond pulses,” Opt. Lett.19, 1870–1872 (1994). [CrossRef] [PubMed]
Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A70, 043802 (2004). [CrossRef]
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]
H. Du and B. Hu, “Broadband supercontinuum generation method combining mid-infrared chirped-pulse modulation and generalized polarization gating,” Opt. Express18, 25958–25966 (2010). [CrossRef] [PubMed]
K. Schafer, M. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett.92, 023003 (2004). [CrossRef] [PubMed]
K. Ishikawa, “Photoemission and ionization of He+ under simultaneous irradiation of fundamental laser and high-order harmonic pulses,” Phys. Rev. Lett.91, 043002 (2003). [CrossRef] [PubMed]
M. Gaarde, K. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A72, 013411 (2005). [CrossRef]
A. Bandrauk and N. Shon, “Attosecond control of ionization and high-order harmonic generation in molecules,” Phys. Rev. A66, 031401(R) (2002). [CrossRef]
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