Frequency tunable single attosecond pulse production from aligned diatomic molecules ionized by intense laser field

Mikhail Yu. Emelin; Mikhail Yu. Ryabikin; Alexander M. Sergeev

doi:10.1364/OE.18.002269

1. Introduction

High-order harmonic generation (HHG) in intense laser-gas interactions provides a very useful tool for the attosecond pulse production [1, 2]. The generation of isolated pulses as short as 90 attoseconds in the HHG process with extremely short laser pulses has been reported recently [3]. First proof-of-principle experiments have demonstrated the feasibility of precise time-resolved measurements and control of atomic and molecular processes using attosecond pulses (see, e.g., review [4]).

Future applications of HHG to ultrafast x-ray spectroscopy and quantum control will greatly benefit from the ability to spectrally shape the harmonic field [5]. Of great importance for time resolved x-ray absorption spectroscopy [6] is the availability of the radiation with the photon energies reaching or even exceeding 1 keV, whose feasibility using the HHG source has been shown recently [7]. The key point on the way of putting such extremely short-pulse HHG-based soft x-ray sources into practice is to enhance the brilliance of high-harmonic generation.

Molecular systems have attracted considerable recent interest as the media for HHG due to the fact that, because of their higher complexity as compared with the atomic ones, they provide additional means for optimization of the HHG process [8–13], which are beyond the possibilities of their simple atomic counterparts. A variety of new opportunities is provided by using excited molecular transients to enhance the efficiency of high-harmonic and attosecond pulse generation and to control their spectral and temporal characteristics. These enhancement and control, which are often based on various interference phenomena, can be implemented by producing molecular rotational [14, 15], vibrational [16, 17], and dissociative [18] wave packets.

In this paper, we present the results of our theoretical study of single attosecond pulse generation during ionization of molecules by an intense few-optical-cycle laser pulse. We discuss the possibilities for highly efficient production of light pulses as short as about 100 as in the process of Coulomb-barrier suppression ionization (BSI) of molecules. We demonstrate the possibility for continuously tunable x-ray production due to the quantum interference of the electron wave packets emanating from different nuclei of a stretched aligned molecule in the process of ionization.

2. Numerical results

There have been many theoretical works in the past on HHG from molecular systems and, specifically, from hydrogen molecular ions (see, e.g., [9–12, 16–23]). The new point in our study is that we concentrate on high harmonic generation in the BSI regime. The details of the x-ray spectral shapes generated in this regime are not yet fully understood.

If a rapidly increasing electric field in a few-cycle driving pulse or at the leading edge of a high-power laser pulse exceeds the critical value, atoms are ionized during a short time within one optical cycle (see, e.g., [24]). The free-electron wave packet is then strongly localized. Upon returning to a parent ion, this compact wave packet gives rise to a short burst of radiation due to the classical bremsstrahlung mechanism. Unlike in the commonly treated high-harmonic regime, in which only a part of the electron wave packet is set free each half-cycle of the driving field, in the case of quick bound-state depletion high-harmonic radiation is not produced via continuum–bound transitions (leading to the well-known cut-off law [25, 26]) but rather via continuum–continuum transitions. To show more explicitly the difference between these two regimes, we will discuss more particularly the roles of different types of transitions in HHG.

The atomic or molecular nonlinear radiative response can be calculated via the time-dependent dipole acceleration expectation value [27]

R (t) = ∭ {| Ψ (x, y, z, t) |}^{2} \frac{\partial V}{\partial z} d x d y d z

using Ehrenfest’s theorem. Here ψ is the electron wave function and

V (r)

is the electron-nuclei interaction potential. The wave function can be represented as the sum of parts corresponding to the bound (

ψ_{b}

) and free (

ψ_{f}

) electron:

ψ = c_{b} ψ_{b} + c_{f} ψ_{f}

. The squared modulus of the wave function is then written as

{| ψ |}^{2} = {| c_{b} |}^{2} {| ψ_{b} |}^{2} + c_{b}^{*} c_{f} ψ_{b}^{*} ψ_{f} + c_{b} c_{f}^{*} ψ_{b} ψ_{f}^{*} + {| c_{f} |}^{2} {| ψ_{f} |}^{2} .

The first term in right-hand member of Eq. (2) corresponds to the bound-bound transitions, the second and third terms correspond to the transitions between bound and continuum states, and the last term describes the continuum-continuum transitions. The first term, which describes intraatomic transitions, is responsible for the fast-falling low-frequency part of the harmonic spectrum and is unimportant for attosecond pulse production. In the commonly treated case of weak ionization, the bound-state population remains significant. In this regime, the second and third terms of Eq. (2) are dominant for the response

R (t)

, while the contribution of the continuum-continuum term is relatively small and is not usually taken into account. Since the high-energy photon emission in this case is predominantly due to the electron recombination with the parent ion, the theory of HHG concentrates usually on the continuum-bound transitions.

In contrast to that case, in the BSI regime examined in this paper, all bound states become depopulated almost entirely; therefore, the response $R (t)$ is dominated by the free-free part, i.e., the last term of Eq. (2). This mechanism is also of potential use for a single attosecond pulse production [28]. However, HHG in this regime is reputed to be inefficient because of the spreading of the free-electron wave packet (the so-called “wave-packet-spreading regime” [29]). Smooth free-electron probability distribution ${| ψ_{f} |}^{2}$ would indeed lead to a weak dipole response. Nevertheless, an efficiency of the frequency conversion in this regime can be enhanced dramatically if the properly prepared initial states are used. For example, for the case of a hydrogen atom prepared in an excited s-state, an efficiency of the visible to XUV frequency conversion can be enhanced by orders of magnitude for an optimally chosen value of the principal quantum number n of the initial state [30, 31]. This is due to the slowed spreading of the laser-driven electron wave packet because of higher degree of delocalization of the initial state wave function. In this paper, we will discuss another possibility, which is based not on the electron wave-packet spreading slowdown but rather on its strong space modulation. We will demonstrate that manipulations of vibrational and rotational degrees of freedom of a molecule can lead to a greatly enhanced efficiency of attosecond pulse production with the added bonus of a wide-range frequency tuning of the generated radiation.

In molecules, because of the presence of several nuclei, there are several sources of the free-electron de Broglie waves arising from ionization. These waves interfere in the course of further propagation. For a diatomic molecule, this two-center interference, in a sense, is analogous to the Young’s two-slit interference of the light beams. In the case of ionization of aligned molecules in a rapidly increasing laser field, the resulting wave packet gains a regular structure with the characteristic spatial scale dependent on the geometry of the system. This interference modulation induces characteristic patterns in the emission spectra.

Our study of the time evolution of the electron wave packet released by an intense few-cycle pulse from the H₂ ⁺ molecular ion is based on the 3D numerical solution of the time-dependent Schrödinger equation (TDSE). To numerically integrate the TDSE, we employed the split-operator fast Fourier transform technique [32]. For H₂ ⁺ ion, we considered the two-center potential $V (x, y, z)$ with Coulomb singularities. Calculations were made for different fixed values of the internuclear distance D and the molecular orientation angle θ with respect to the laser electric field, which was considered linearly polarized along the z axis. The harmonic emission spectra were obtained by Fourier transformation of $R (t)$ into the frequency domain. Atomic units are used throughout the paper.

The numerical results presented below were obtained for the H₂ ⁺ ion driven by an optical pulse with electric field of super-Gaussian shape

E (t) = 2 \exp {[- 5 {(ω_{0} t / 2 π - 1)}^{4}]}^{} \sin ω_{0} t

with

ω_{0}

=0.114 corresponding to λ=400 nm; laser pulse peak intensity and duration are, respectively, 1.4×10¹⁷ Wcm⁻² and 1.4 fs. It should be mentioned that the particular waveform we chose is only an example and all the physics discussed here is not closely tied to this particular choice. The requirements we impose on the waveform are rather general: the pulse has to be intense enough to provide rapid ionization (during one half-cycle of the field) and, in addition, steeper rising edge of the pulse is desirable to more strongly accelerate the wave packet. State-of-the-art laser tools for attosecond technology are already capable of meeting these requirements [33, 34].

Figure 1 shows the electron wave-packet time evolution during ionization of H₂ ⁺ by the laser pulse Eq. (3) for three different orientations of the molecular axis: θ=90° (Fig. 1(a), Media 1), θ=45° (Fig. 1(b), Media 2), and θ=0° (Fig. 1(c), Media 3). The molecule is stretched to the internuclear distance D = 20. Snapshots shown in Fig. 2 correspond to three different internuclear distances in H₂ ⁺ aligned parallel to the electric field: (a) D=14, (b) D=20, and (c) D=28.

Fig. 1 Snapshots of the electron wave packet recolliding with the molecular core in the process of H₂ ⁺ ionization by a laser pulse for different orientations of the molecular axis: (a) θ=90° (Media 1), (b) θ=45° (Media 2), and (c) θ=0° (Media 3). Internuclear distance is D=20.

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Fig. 2 Snapshots of the electron wave packet recolliding with the molecular core in the process of H₂ ⁺ ionization by a laser pulse for different internuclear distances: (a) D=14, (b) D=20, and (c) D=28. Molecular axis is parallel to the laser field (θ=0°).

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As movies show, the electron wave-packet evolution begins from the detachment of two separate wave packets emanating from the vicinity of two different nuclei. Later on, when these sub-packets start to overlap, the quantum interference fringes arise which are tilted at angle 90°–θ with respect to z axis. The fringe separation depends on the internuclear distance - see Fig. 2. More specifically, it is inversely proportional to D.

In each case shown here, a spatial envelope of the whole wave packet is roughly the same, replicating in general terms the shape of the wave packet for the case of atomic BSI (cf. Figure 2 of Ref [16].). However, the distinctive feature of the wave packet for the case of a stretched molecule is that the electron probability distribution is sliced, because of the two-center interference, and both the arrangement of slices and fineness of slicing are of great consequences to both the dipole response and the harmonic emission spectrum.

In accordance with Eq. (1), space modulation of a free-electron wave packet (Figs. 1 and 2) is mapped onto the temporal modulation of the induced dipole acceleration (Fig. 3 ). This temporal modulation is the key point of our scheme, since it provides the translation of a significant part of the molecular response to higher frequency range where the harmonic yield is otherwise very low.

Fig. 3 Time profile of the attosecond burst in the molecular dipole response for the cases of parallel (red line) and perpendicular (green line) orientation of the molecular axis. Internuclear distance is (a) D=14 and (b) D=29.

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Figure 3 plots the time profile of the molecular dipole response within the time interval corresponding to the excitation of an attosecond burst during the electron recollision with the parent ion.

For perpendicular orientation of the molecular axis, the longitudinal profile of the wave packet is as smooth as in the atomic case. Hense, the dipole response is smooth as well (Fig. 3, green line), being also atomic-like (cf. Figure 4(a) of Ref [31].). As a result of this smoothness, the emission spectrum steadily decreases with the harmonic order (Fig. 4(a), black line), in good agreement with the calculation for the hydrogen atom (cf. Figure 5 of Ref [29].).

Fig. 4 Harmonic spectra of H₂ ⁺ ion driven by the laser pulse Eq. (3) for (a) various orientations of the molecular axis (internuclear distance D=29) and (b) various internuclear distances (orientation angle θ=0°).

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Fig. 5 Harmonic intensities for a stretched H₂ ⁺ ion plotted (a) at the (ω, θ) plane (internuclear distance D=29) and (b) at the (ω, D) plane (orientation angle θ=0°).

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For parallel orientation of the axis, the longitudinal spatial structure of the electron wave packet is quasi-periodical that is reflected in a regular temporal modulation of the dipole response (Fig. 3, red line). For a more strongly stretched molecule (Fig. 3(b)), this modulation is faster, because of a finer slicing of the wave packet. As a result of this quasi-periodical temporal modulation of the dipole response, a pronounced maximum appears in the emission spectrum. This maximum shifts towards shorter wavelengths with decreasing molecular orientation angle [Fig. 4(a)]. A similar shift is observed with increasing nuclear separation [Fig. 4(b)].

It should be mentioned that the blue shifting of the maximum of the emission spectrum with decreasing angle θ [Fig. 4(a)] is accompanied by a significant increase of the harmonic yield. This can be explained by the fact that, for orientation angle closer to θ=0°, the parts of the wave packet aimed directly at the nuclei are of higher density, see Fig. 1.

It is important to emphasize that, according to the data presented in Fig. 4, the harmonic yield in the range around 3 nm is enhanced by at least eight orders of magnitude as compared to the atomic case. It follows from Fig. 3 that this short-wavelength radiation is temporally confined to the time interval about 100 as. It agrees well with the estimation of the duration of the harmonic burst produced via the BSI mechanism as the duration of the transit of the electron wave-packet central part across the core: $τ_{h} \approx a (t_{r}) / V (t_{r})$ [29], where $a (t_{r})$ and $V (t_{r})$ are, respectively, the characteristic size of the electron wave packet and its mean velocity at the recollision time $t_{r}$ .

3. Theory

The positions of the interference maxima in the harmonic spectra discussed above can be estimated analytically as follows.

If the internuclear distance is large enough as compared with the Bohr radius, the molecular ground state $Ψ_{0}^{M}$ can be represented as a linear combination of atomic orbitals in its simplest form:

Ψ_{0}^{M} (r, D) = \frac{1}{\sqrt{2}} [Ψ_{0}^{a} (r + D / 2) + Ψ_{0}^{a} (r - D / 2)],

where

Ψ_{0}^{a} (r)

is the hydrogen ground state and the overlap integral between the two atomic orbitals is neglected;

D = R_{1} - R_{2}

(

R_{i}

is the radius-vector of the i-th nucleus). Further, we adopt the simplifying assumption that the electron wave packet remains undistorted until the instant of barrier-suppression ionization, after which time the electron becomes completely free from the atomic potential [29–31]. Under this assumption, the time evolution of the atomic electron wave packet can be described as a free evolution of a radially symmetric function (in the center-of-mass system of the wave packet) as follows [35, 31]:

Ψ^{a} (r, t) = - \frac{2 i}{r \sqrt{2 π i t}} \exp (\frac{i r^{2}}{2 t}) \int_{0}^{\infty} x Ψ^{a} (x, 0) \exp (\frac{i x^{2}}{2 t}) \sin (\frac{r x}{t}) d x .

Taking into account strong localization of the atomic ground state, for large t (much larger than atomic unit) one can use the stationary phase approximation by setting

\exp (i x^{2} / 2 t) \approx 1

in the integrand of Eq. (5). Equation (5) can be then represented as

Ψ^{a} (r, t) = i^{5 / 2} \exp (\frac{i r^{2}}{2 t}) | Ψ^{a} (r, t) | .

By analogy with Eq. (4), the molecular free-electron wave function using Eq. (6) can be written as

Ψ^{M} (r, t) = \frac{i^{5 / 2}}{\sqrt{2}} [\exp (i r_{1}^{2} / 2 t) | Ψ^{a} (r_{1}, t) | + \exp (i r_{2}^{2} / 2 t) | Ψ^{a} (r_{2}, t) |],

where

r_{i}

and r are the radius-vectors of the electron relative to the center of mass of i-th sub-packet and of the whole wave packet, respectively. Since at large t the two sub-packets get overlapped (see movies), we have

| Ψ^{a} (r_{1}, t) | \approx | Ψ^{a} (r_{2}, t) | \approx | Ψ^{a} (r, t) | .

From (7) and (8) we obtain

{| Ψ^{M} (r, t) |}^{2} = \frac{{| Ψ^{a} (r, t) |}^{2}}{2} {| \exp (i r_{1}^{2} / 2 t) + \exp (i r_{2}^{2} / 2 t) |}^{2} = \frac{{| Ψ^{a} (r, t) |}^{2}}{2} [1 + \cos (D r / t)] .

As follows from Eq. (9), the molecular free-electron wave packet is spatially modulated with the modulation period

Λ = 2 π τ / D,

where τ is the electron excursion time. This spatial modulation results in the temporal modulation of the attosecond burst (see Fig. 3) at a frequency

Ω = 2 π V \cos θ / Λ = V D \cos θ / τ .

As a result, the interference maximum in the harmonic spectrum appears whose position in determined by Eq. (11). Therefore, changing θ or D values by producing, for example, molecular rotational, vibrational or dissociative wave packet enables one to control the spectral content of the attosecond pulse. Since V and τ values are field-dependent, the emission spectrum can also be controlled by changing the laser pulse shape.

Figure 5 demonstrates wide-range tunability of attosecond radiation via manipulations of rotational and vibrational degrees of freedom of a molecule. This figure plots the harmonic intensities at the (ω, θ) and (ω, D) planes for the case of ionization of H₂ ⁺ by the laser pulse Eq. (3).

From the comparison of Eq. (11) with Fig. 5 we conclude that the cosine dependence on θ and the linear dependence on D of the central frequency of the interference maximum in the harmonic spectrum, which follow from our analytical approach, agree perfectly with the numerical results shown in Figs. 5(a) and 5(b), respectively.

All means mentioned above can be used not only for the tuning of the carrier frequency of the attosecond pulse but also for a more general spectral shaping of the harmonic field. For example, Fig. 6 illustrates the possibility to efficiently generate x-ray radiation with a continuous spectrum covering the full water window.

Fig. 6 Generation of the x-ray radiation matching the water window. Laser pulse is given by Eq. (3); internuclear distance D=30, orientation angle θ=0°.

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It should be emphasized that strong bond-length and orientation dependence of the emission spectrum of a stretched molecule due to the quantum interference effect discussed above can be used in reverse, i.e. for probing molecular vibration-rotational dynamics. Particularly, Fig. 4(b) suggests that using an intense ultrashort laser pulse as a probe, one can monitor the molecular vibration or dissociation by analyzing the pump-probe time-delay dependence of the high harmonic spectra of aligned molecules. This possibility can provide a new item for the toolbox of the molecular dynamic imaging [36]. Because of the 100-as duration of the underlying electron recollision process (see Fig. 3), the changes of the molecular configuration during the probing are negligible. Therefore, the resulting harmonic signal, in fact, bears the imprint of the instantaneous molecular structure.

We notice that the harmonic spectra shown above are sometimes influenced by the other kind of interference. This interference arises not in the initial (detachment) but rather in the late (recollision) stage of the attosecond pulse production process and is due to the presence of several scattering centers in a molecule. If the internuclear distance in a diatomic molecule is equal to an odd number of half-periods of a spatial modulation of the free-electron wave packet [Fig. 2(c)]

D = (2 n + 1) Λ /2 = \sqrt{(2 n + 1) π τ}, n = 0, 1, \dots,

a rather narrow and deep minimum is observed at the center of the peak in the emission spectrum, see curves for D=28 and D=32 on Fig. 4 (b). This minimum is due to the destructive interference of contributions to the emission from the vicinity of two neighboring ions. Using Eqs. (11) and (12), the positions of such minima can be written as

Ω_{n}^{*} = V \cos θ \sqrt{(2 n + 1) π / τ}, n = 0, 1, \dots

The interference minima determined by Eq. (13) can be used as reference points providing additional information useful for the molecular dynamic imaging.

4. Conclusion

Our study shows that quantum interference in ionization of excited molecules by intense laser field can be exploited efficiently for both the x-ray emission control and molecular dynamic imaging. For BSI of a stretched molecule, the shape of the ionized electron wave packet is extremely sensitive to molecular alignment and internuclear separation. By the example of H₂ ⁺ ion, we have shown that manipulations of molecular vibrational and rotational degrees of freedom can result in tuning harmonic spectra over a wide range within the soft x-ray band with the maximum photon energies approaching 1 keV. Furthermore, high sensitivity of HHG spectra to the molecular configuration offers the opportunities for ultrafast dynamic imaging of excited molecular transients with extremely high time resolution.

Acknowledgments

We acknowledge financial support from the Presidium of RAS, RFBR (grants Nº 08-02-01173 and Nº 09-02-97078-r-povolzh’ye-a), and the Russian President’s Grant Nº 1931.2008.2. M.Y.E. also acknowledges support from the Dynasty Foundation.

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Frequency tunable single attosecond pulse production from aligned diatomic molecules ionized by intense laser field

Abstract

1. Introduction

2. Numerical results

3. Theory

4. Conclusion

Acknowledgments

References and links

Supplementary Material (3)

Cited By

Figures (6)

Equations (13)

Optics Express