## Abstract

We theoretically investigate the broadband water-window supercontinuum generation from pre-excited medium with a mid-infrared pulse. We find that the wavelength scaling of the harmonic yield from near-visible (0.8 *μ*m) to mid-infrared (1.8 *μ*m) in single-atom level is *λ*^{−2.7}. Using an intense phase-stabilized few-cycle 1.6 *μ*m laser pulse, a broadband water window supercontinuum with bandwidth of approximately 140 eV is obtained. We also investigate the macroscopic effects and find that large initial population of the excited state leads to the high-density of free electrons, which shift the carrier-envelop phase of the driving pulse and further diminish the water-window supercontinuum generation. The highly-ionized medium also results in poor temporal and spatial properties of the attosecond pulse. Instead, small initial population of the excited state can produce well phase-matched xuv supercontinuum in water-window region and an 100-as pulse with central wavelength of 2.8 nm and pulse energy of 0.15 nJ can be filtered out.

© 2011 Optical Society of America

## 1. Introduction

High-order harmonic generation (HHG) is one of the major topics in the interaction of intense laser fields with atoms. It has provided an attractive way towards attosecond extreme ultraviolet (XUV) light source, making a breakthrough in attosecond science [1] and nonlinear optics in the extreme ultraviolet region [2]. One of the major applications of HHG is to produce isolated attosecond pulses. In the past few years, how to broaden the bandwidth of the supercontinuum and shorten the pulse duration of the isolated attosecond pulse have attracted great interests, because the generation of shorter attosecond pulses enables a higher time resolution for ultrafast measurements [3, 4]. Recently, Cavalieri and co-workers [2] have broke through the 100-as-barriers. In their experiment, a sub-4-fs near one optical cycle driving pulse has been employed to generated a 40 eV supercontinuum and an 80-as pulse has been filtered out, with the pulse energy of 0.5 pJ.

The HHG process can be well explained by the quasi-classical three-step model [5, 6]. First, the electron is ionized by tunnelling through the potential barrier, then it is driven by the laser field treated as a free electron, finally, it may return to the atomic core and emit a harmonic photon in the transition back to the ground state. The maximum kinetic energy acquired by a free electron from the laser field when it recombines with the atomic core is 3.17*U _{p}*(where

*U*=

_{p}*I*/4

*ω*

^{2}is the ponderomotive energy of the laser field with frequency

*ω*and intensity

*I*), which coincides with the cutoff energy of the emitted harmonic photons. Manipulating the three steps in HHG process (ionization, acceleration and recombination) offers a possibility to control the electron trajectories for the generation of a broadband supercontinuum. For instance, two-color schemes have been proposed to control electron dynamics to generate broadband attosecond pulses [7]. An alternative way to broaden the bandwidth of the supercontinuum is applying polarization gating technique (PG) [8, 9] and double-optical-gating technique (DOG) [10]. It has been demonstrated that the electron only returns efficiently in a short time period when PG is adopted and then an isolated attosecond pulse can be produced. Recently, a very broadband XUV continuum, which supports a 16-as isolated pulse generation, has been produced with double-optical-gating technique (DOG).

The well-known *I _{p}* + 3.17

*U*law for the position of the HHG cutoff (where

_{p}*I*is the atomic binding energy) shows that using mid-infrared lasers allows the generation of higher order harmonics [11] and offers the possibility for the generation of broader supercontinuum in water-window [the spectra range between the K-absorption edges of carbon (284 eV) and oxygen (543 eV)] region. Hong

_{p}*et al.*have found that the two-color schemes in the mid-IR regime are much more effective in electron trajectory control than those in the near-IR regime [12]. The DOG and PG techniques have also been applied in the mid-IR region to produce broadband supercontinuum [13, 14]. However, a theoretical investigation about the wavelength scaling from near-visible (0.8

*μ*m) to mid-infrared (1.8

*μ*m) of high order harmonics shows that the integration of the harmonic yield follows a

*λ*

^{−(5–6)}scaling at constant intensity [15]. The spreading of the electron wave packet leads to a decrease of the HHG yield in mid-infrared regime [15], which leads to low harmonic efficiency thus low energy of attosecond pulse.

An effective way to increase the harmonic efficiency of harmonics driven by mid-infrared pulses is to populate part of the electron wave packet at the excited state. It has been demonstrated that the ionization rate can be significantly enhanced when atomic systems are prepared in a coherent superposition of ground and excited states [16], since part of the electron wave packet on the excited state can be much more easily projected to the continuum due to its relatively small ionization potential. Thereby the harmonic efficiency can be significantly increased compared with HHG generated from the ground state alone. In this paper, we theoretical investigate the generation of a broadband water-window supercontinuum in pre-excited medium. A better wavelength scaling of the harmonic yield is demonstrated in infrared region. Using a few-cycle phase-stabilized 1.6 *μ*m driving pulse when the target is prepared in a coherent superposition of two electron states, a supercontinuum in water-window region with bandwidth of approximately 140 eV is straightforwardly obtained. The macroscopic effects are also investigated associated with the initial population of the excited state. The results show that in large initial population case the harmonic spectra is deeply modulated and the temporal and spatial properties of the generated pulses are poor. Instead, small initial population of the excited state can produce well phase-matched water-window xuv supercontinuum and a 100 as attosecond pulse can be filtered out with pulse energy of 0.15 nJ and central wavelength of 2.8 nm.

## 2. Theoretical model

#### 2.1. Single-atom response

At single-atom level, we perform a quantum simulation by solving the time-dependent Schrödinger equation. In the single-active-electron (SAE) and dipole approximations, the Hamiltonian is

(in atomic units), where*V*(

*x*) is the atomic Coulomb potential and

*E*(

*t*) is the electric field. We choose the soft-core model within SAE approximation where

*a*and

*b*are parameters describing different ions. For a He

^{+}ion, we choose

*a*= 2 and

*b*= 0.5 to retrieve the ionization energy of the ground state (1

*s*), i.e. 54.4 eV, and the first excited state (2

*s*) ionization energy is 13.6 eV. In our simulation, a 8-fs 1600-nm few-cycle laser pulse with intensity of 6 × 10

^{14}W/cm

^{2}is used. The electric field of the driving pulse can be written as

*E*

_{0},

*ϕ*and

*ω*are the amplitude, carrier envelope phase (CEP) and laser frequency of the driving pulse, respectively.

The initial state is prepared as a coherent superposition of the 1*s* and 2*s* states

*α*and

*β*are the amplitudes of the ground and first excited states and

*φ*is the initial relative phase between two states.

*φ*is set to be zero because it has no influence on the results. The initial population of the excited state is described as

*p*=

*β*

^{2}.

The instantaneous wave function can be expressed as

where*ψ*(

_{g}*r,t*) and

*ψ*(

_{e}*r,t*) are the instantaneous wave functions originating from the ground state and the excited state, respectively.

In order to obtain the harmonic spectrum, we should calculate the time-dependent dipole acceleration

The harmonic spectrum is then obtained by Fourier transforming of the time-dependent dipole accelerationIn order to describe the features of harmonic spectra, we split up different contributions to the dipole acceleration for the coherent superposition as [16]

*c*> is the continuum state.

*γ*and

_{e}*γ*, are the time-dependent amplitudes of the excited and continuum states. Here we assume the ground state is not depleted, so the term $<{\psi}_{g}\left(r,t\right)\left|\widehat{\ddot{x}}\right|{\psi}_{g}\left(r,t\right)>$ doesn’t contribute to the harmonic generation. Then we can rewrite the relevant contributions to the acceleration as

_{c}^{14}W/cm

^{2}(as shown in Fig. 1), the ionization rate can be highly increased and so are the conversion efficiency. As a result, this part of dipole acceleration would play an irreplaceable role on the harmonic spectra.

#### 2.2. Macroscopic propagation response

The macroscopic response can be described by the co-propagation of the laser and harmonics beams which is simulated by numerically solving Maxwell wave equations for the fields of the laser pulse *E _{f}* and the harmonics

*E*in cylindrical coordinates separately.

_{h}*ω*is the plasma frequency and is given by

_{p}*P*(

_{nl}*ρ*,

*z*,

*t*) = [

*n*

_{0}–

*n*(

_{e}*ρ*,

*z*,

*t*)]

*d*(

_{nl}*ρ*,

*z*,

*t*), where

*n*

_{0}is the gas density and

*n*is the free-electron density.

_{e}This model takes into account both temporal plasma-induced phase modulation and spatial plasma-lensing effects on the laser pulse.The procedures of solution of above equations have been clearly described in [17].

## 3. Result and discussion

Firstly, we investigate the population variations of ground and excited states. Figure 1 shows the normalized populations of the excited state (red dashed line) and the ground state (blue dashed line) as functions of time when the initial state is a coherent superposition of the ground and excited states with equally weighted initial populations. The electric field (green line) of the driving pulse is also shown in this figure. The excited state ionizes completely within the first 1.5 optical cycles while the ground state is hardly ionized. Since harmonic generation originates from the coherent dipole transition between the continuum and the bound states, only those states that remain populated during the pulse will contribute to the harmonic generation. The interference term $<{\psi}_{g}\left(r,t\right)\left|\widehat{\ddot{x}}\right|{\psi}_{e}\left(r,t\right)>$, where the excited state is responsible for the ionization and the dipole transition is between the continuum and the ground state, plays a more important role than other two terms in HHG process. Therefore, conversion efficiency can be highly increased in the coherent superposition case.

To conform the harmonic efficiency increasing, we calculate the harmonic spectra at single-atom level. Figure 2 presents the harmonic spectra of the He^{+} ion in coherent superposition state case (red thick line) and ground state case (blue dotted line), respectively. The CEP of the driving pulse is set to be 0.6 *π*. The harmonic efficiency of the superposition state case is about 6 orders of magnitude higher than that of the ground case with the same cutoff harmonic order of approximately 620th. A water-window supercontinuum with the bandwidth of approximately 140 eV is straightforwardly obtained. The inset is the harmonic spectra when the wavelength of the driving pulse is 0.8 *μ*m (near-visible) which is shown for comparison. We can see that the bandwidth of the supercontinuum is further broadened and more energetic harmonic photons in water-window region are generated using a 1.6 *μ*m driving pulse than that of a 0.8 *μ*m driving pulse with equal intensity.

A deeper insight is obtained by investigating the temporal and spectral structures of HHG using wavelet time-frequency analysis method. Figure 3 presents the time profiles of the harmonics when the initial state is superposition state. The maximum harmonic order of the highest peak is about 620th, agreeing well with harmonic spectra cutoff harmonic order. We can also see that multi-recombinations of the electron wave packet are remarkable. This is due to the high ionization rate of the excited state.

To further demonstrate the advantage of using coherent superposition state, we investigate the wavelength scaling of HHG yield from near-visible (0.8 *μ*m) to mid-infrared (1.8 *μ*m). The wavelength scaling is evaluated by integrating the HHG yield (defined as radiated energy per unit time) from 50 to 80 eV, shown in Fig. 4. As pointed out by J.Tate and co-workers [15], the harmonic yield follows a *λ*^{−(5–6)} scaling at constant intensity when the initial state is only ground state. However, in our simulation, when using coherent superposition state, the harmonic yield scales as *λ*^{−2.7}, which falls more slowly as the increasing of the driving laser wavelength. Although the physics underlying the scaling law is not yet fully understood: the scaling law of the HHG yield (*λ*^{−(5–6)}) at constant intensity has discrepancy on the theory predicted *λ*^{−3} and high-order returns of the electron wave packet have not been paid enough attention to in previous analysis [18, 19]. It is expected that the superposition state case may have potential to help us understand this owing to its noteworthy multi-recombinations of the electron wave packet.

As for few-cycle driving pulses, the CEP plays an important role [20]. Figure 5(a) shows the population variations of the excited state as functions of CEP. Since the ionization rate of excited state has a significant influence on HHG process in superposition state case, we can predict that the different variations of the population of the excited state will change the structure of the harmonic spectra. As shown in Fig. 5(b), the harmonic efficiency and bandwidth of the generated xuv supercontinuum sensitively depend on the CEP of driving pulse. This is quite different from the ground state case, of which the CEP only influences the modulation of the supercontinuum and slightly changes its bandwidth and the harmonic cutoff. A qualitative explanation to the results is that the ionization time of the electron trajectory coincides with the time when the population of the excited state decreases fastest at some CEPs, which forms a broadband supercontinuum with high conversion efficiency While at other CEPs this condition can’t be satisfied and the bandwidth are relatively narrow and the harmonic efficiencies are low. Next, we investigate the macroscopic effects. As described above, high ionization rate of the excited state is responsible for the high harmonic efficiency in superposition state case. Since the population of the excited state is almost totally depleted during the laser pulse while the ground state is hardly ionized, more populations of the excited state mean higher conversion efficiency of the harmonics at single-atom level. However, considering macroscopic effects, more populations of the excited state bring high density of free electrons into the medium, which may deteriorate the phase-matching condition and change the spatiotemporal propagation dynamics of the laser field including the spatial defocusing of the beam and temporal distortion of the electric field [21]. To demonstrate this issue, we first perform three-dimensional propagation simulations for driving and harmonic fields in the gas medium. The parameters are as below: the beam waist of the focused laser beam is 50 *μ*m and the gas target is 500 *μ*m long with the density of 2.6 × 10^{18}/cm^{2}. The gas jet is placed 1 mm after the laser focus. Figure 6 shows the macroscopic harmonic spectra generated from He^{+} in coherent superposition state with different initial excited state populations of *p* = 0.05 (red line), *p* = 0.1 (blue line) and *p* = 0.3 (green line). We can see that the modulation in the water-widow supercontinuum region is weaken in the case of *p* = 0.05 after propagation, which implies the phase-matching conditions of one electron trajectory is satisfied. While in the cases of *p* = 0.1 and *p* = 0.3, the supercontinua are more deeply modulated than that at single-atom level. Moreover, in these two cases, the harmonic cutoffs are changed in comparison with the single-atom response, which may be due to the distortion of the electric field after propagation. Figure 7 shows the electric field of the driving pulses after propagation with the excited state populations of *p* = 0.05 (red line), *p* = 0.1 (blue line) and *p* = 0.3 (green line), respectively. The dashed line shown in this figure is the initial driving field at the entrance of the gas jet. It is found that the electric field distortion is unconspicuous for the case of *p* = 0.05. On the contrary, it is considerable for the cases of *p* = 0.1 and *p* = 0.3. This distortion changes the temporal profiles of the electric field, which can be regarded as CEP shifts. As shown in Fig. 5, the efficiencies and cutoffs of the supercontinua strongly depend on the CEP, so the CEP shifts caused by the distortion of the electric field after propagation change the macroscopic xuv supercontinua.

We further investigate the temporal profiles of the supercontinuum after propagation by means of applying a square window with width of 50 harmonics to the supercontinuum from 550th to 600th harmonic orders for each case. The results are shown in Figs. 8(a)–(c). The bandwidth is 39 eV, approximately 100 as to the Fourier-transform limit. For the case of *p* = 0.05, a pure attosecond pulse is filtered out from the double-peak structure, which indicates that the harmonics from short quantum path are well phase-matched while that from the other electron trajectories are suppressed. On the contrary, the satellite pulses become noticeable for the case of *p* = 0.1 and multi-peaks attosecond pulses with equal weighted intensities are appeared for the case of *p* = 0.3 because of the deep modulation of the supercontinua. The spatial distribution is also an important characteristic of the generated attosecond pulse. Figures 8(d)–(f) show the spatiotemporal profiles of the attosecond pulse generation in the cases of *p* = 0.05, *p* = 0.1 and *p* = 0.3. This characteristic is related to the high-density of free electrons. In the spatial domain, the refractive index introduced by the free electrons causes the laser phase front to advance faster on axis, which leads to defousing of the beam. The higher density of free electrons in the cases of *p* = 0.1 and *p* = 0.3 make the spatiotemporal distribution much more complicated than that for the case of *p* = 0.05. We further calculate the emitted power of the attosecond pulses and find it is estimated to be approximately 0.15 nJ in the case of *p* = 0.05.

## 4. Conclusion

In summary, we theoretically investigate the generation of a broadband water-window supercontinuum with high efficiency. When the medium (He^{+}) is prepared in a coherent superposition of ground and excited states, a better wavelength scaling of the harmonic yield is demonstrated: the harmonic yield scales as *λ*^{−2.7} at constant intensity, which falls more slowly as the increase of the driving laser wavelength yet the physical mechanism of this scaling law should be further discussed. It should be emphasize that the *λ*^{−2.7} scaling keeps unchanged when the initial population of the excited is higher than 2%. Using a phase-stabilized few-cycle 1.6 *μ*m laser pulse to irradiate the prepared excited medium, the enhancement of the harmonic efficiency is achieved due to the high ionization rate of the excited state and a water-window supercontinuum of about 140 eV is obtained straightforwardly. We also investigate the macroscopic effects. The results show that large initial population of the excited state causes the medium to be high-ionized and introduces high density of the free electrons, which lead to the distortion of the electric field and make the supercontinuum deeply modulated. The highly-ionized medium also results in poor temporal and spatial properties of the attosecond pulse. On the other hand, small initial population of the excited state can produce well phase-matched xuv supercontinuum and an isolated 100-as attosecond pulse with the pulse energy of 0.15 nJ and central wavelength of 2.8 nm can be filtered out. We also calculated different initial populations of the excited state and find that 2% – 7% initial population of the excited state can produce an isolated attosecond pulse. When it increases to 8%–15%, the supercontinuum is modulated and a satellite pulse is emerged. When the initial population of the excited state becomes more, for instance, 30%, the supercontinuum becomes deeply modulated and multi-peaks attosecond pulses with equal weighted intensities are appeared.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants No. 60925021, 10904045, 10734080 and the National Basic Research Program of China under Grant No. 2011CB808103. This work was partially supported by the State Key Laboratory of Precision Spectroscopy of East China Normal University.

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