## Abstract

The ionization of hydrogen by a chirped XUV pulse in the presence of a few cycle infrared laser pulse has been investigated. The electron momentum distribution has been obtained by treating the interaction of the atom with the XUV radiation at the first order of the time-dependent perturbation theory and describing the emitted electron through the Coulomb-Volkov wavefunction. The results of the calculations agree with the ones found by solving numerically the time-dependent Schrödinger equation. It has been found that depending on the delay between the pulses the combined effect of the XUV chirp and of the steering action on the infrared field brings about asymmetries in the electron momentum distribution. These asymmetries may give information on both the chirp and the XUV pulse duration.

© 2012 OSA

## 1. Introduction

The progress achieved in laser technology has made it possible to produce attosecond extreme ultraviolet (XUV) and soft X-ray pulses, that in the last years have become important tools for investigating the electronic dynamics of atoms and molecules occurring on the attosecond time scale [1, 2]. The application of these sources requires the knowledge of their characteristics and in particular their duration. XUV pulse durations are currently measured by cross correlation techniques, based on photoionization of a target atom by the XUV pulse in the presence of the infrared (IR) pulse [3, 4]. For XUV pulse encompassing several infrared radiation periods, the cross correlation exploits the appearance of sidebands in photoelectron energy spectrum or the ponderomotive shift of these peaks. These methods cannot be directly extended for measuring attosecond XUV pulse durations, as the energy resolution is limited by the uncertainty relation Δ*E*Δ*t* > *h*̄. In fact, the sidebands disappear and, consequently, the ponderomotive shift is not longer observable, when the XUV duration is shorter than the period of the infrared radiation field. Different cross correlation methods have been proposed to measure attosecond pulse duration [5–7]. In Ref. [5] the attosecond XUV duration has been determined by using a classical model that relates the XUV duration to the shift and the broadening of the energy spectrum of the emitted electrons, whose motion has been described classically. The validity range of this classical method was investigated in ref. [6] by using a quantum mechanical analysis, in the framework of the strong field approximation. Based on the possibility of resolving the emitted electron signal in energy and angle, Itatani et al. [7] derived a method founded on the streak camera principle for determining the duration of attosecond pulses. The duration of the XUV pulse is determined by measuring the width of the photoelectron energy spectrum at a given observation angle, when the laser field is linearly polarized, and by the angular spread of the photoelectron moving with a given energy on the plane perpendicular to the propagation direction of the collinear pulses, when the IR radiation is circularly polarized. We remark that the streak camera principle was also used for retrieving the electric field of a linearly polarized laser pulse [8]. In this experiment the XUV pulse was approximately ten times shorter than the optical period, so that it was possible to correlate the recorded electron energy to the laser field vector potential at the time of ionization. By varying the delay time of the XUV pulse with respect to the laser pulse and analyzing the recorded electron spectra, the electric field of the laser pulse could be reconstructed. A method providing the complete evolution of the streaking electric field as well as the complex amplitude of the XUV pulse (FROG-CRAB) was proposed by Mairesse and Quéré [9]. It consists in generating an electron wave packet in the continuum by photoionizing atoms with the attosecond XUV pulse (including train of pulses), and in using a low frequency dressing laser field (including multi cycle fields) as a phase gate for FROG-like measurements on this wavepacket [2, 10]. It is the aim of the present work to analyze and discuss the properties of the momentum distribution of the electron emitted from the ground state of the H atom by a relatively weak single attosecond XUV pulse in the presence of an IR laser pulse under particular conditions that will be specified below. We note that the effect of an additional IR laser pulse on the momentum distributions of photoelectrons produced by a few-cycle attosecond XUV pulse with well-defined carrier envelope phase has been addressed in ref. [11]. Here our focus is on the effect produced by a linearly chirped attosecond XUV pulse on the photoelectron spectra. In fact, it will be shown that a chirped attosecond pulse in the presence of IR laser pulse may bring about asymmetries in the photoelectron momentum distributions that can be used for measuring both pulse duration and chirp.

## 2. Theory

The case study is the hydrogen atom ionization by the simultaneous action of a XUV pulse and an IR radiation. An approximate analytical form of the differential ionization probability giving the electron momentum distribution (EMD) produced by the XUV ionization in the presence of the IR radiation may be derived by treating the interaction of the atom with the XUV pulse at the first order of the time-dependent perturbation theory and describing the freed electron by the Coulomb-Volkov wavefunction that is assumed, though approximately, to account for the electron interaction with both the Coulomb and the IR fields [12, 13]. With the above approximation,by taking both the pulses linearly polarized along the z-axis and in dipole approximation, the differential transition probability from the atomic ground state to a continuum state characterized by the electron canonical momentum **q** ≡ (*q _{x}*,

*q*,

_{y}*q*) may be written, in atomic units, as

_{z}**q**(

*t*) =

**q**+

**A**

*(*

_{L}*t*)/

*c*is the instantaneous mechanical momentum,

**A**

*(*

_{L}*t*) the vector potential associated to the IR pulse,

**û**

*a unit vector directed along the z-axis,*

_{z}*E*(

_{H}*t*) the XUV electric field,

**d**

_{q}_{(t)}the field free dipole transition matrix element between the atomic ground state and the state describing the photoelectron emitted with mechanical momentum

**q**(

*t*),

*I*= −0.5

_{p}*a.u.*the ground state energy and

*τ*the total IR pulse duration. We remark that Eq. (1) does not account for the channel pertaining to the ionization due to the infrared action through the above threshold ionization (ATI). Therefore, the validity of

_{L}*P*(

**q**) is confined to photoelectron energy ranges well separated from the energies characterizing the photoelectrons generated by ATI. In our calculations an attosecond, linearly chirped Gaussian, XUV pulse will be assumed with the electric field given by

*E*

_{0}

*is the field amplitude,*

_{H}*t*the instant at which the pulse reaches its maximum,

_{H}*τ*the pulse duration, taken as full width at half maximum (FWHM), for the transform-limited pulse, and

_{H}*ω*the central photon energy at

_{H}*t*=

*t*.

_{H}*β*stands for the dimensionless linear chirp rate: positive (negative) chirp corresponds to the instantaneous frequency increasing (decreasing) with time. The duration (FWHM) of the chirped pulse is ${\tau}_{CH}={\tau}_{H}\sqrt{1+{\beta}^{2}}$. The IR laser electric field, with frequency

*ω*and field amplitude

_{L}*E*

_{0}

*, is taken as*

_{L}*f*(

*t*) = cos

^{2}

*πt*/

*τ*for −

_{L}*τ*/2 ≤

_{L}*t*≤

*τ*/2 and zero elsewhere. In order to have an integer number of cycles we assume

_{L}*τ*=

_{L}*n*, with

_{L}T_{L}*T*= 2

_{L}*π*/

*ω*the period of the carrier. The time lag between the maxima of the two pulses is given by

_{L}*t*. The vector potential associated to IR pulse, taken in Gaussian units as ${\mathbf{A}}_{L}(t)=-c{\int}_{-{\tau}_{L}/2}^{t}d{t}^{\prime}{\mathbf{E}}_{L}({t}^{\prime})$, turns out to be zero for

_{H}*t*≤ −

*τ*/2 and

_{L}*t*≤

*τ*/2. We note that for

_{L}*t*>

*τ*/2,

_{L}**A**

*(*

_{L}*t*) = 0 and

**q**(

*t*) =

**q**.

Owing to the cylindrical symmetry with respect to the z-axis, the differential transition probability *P*(*q _{x}*,

*q*,

_{y}*q*) will be shown in the (

_{z}*q*,

_{x}*q*) plane, having put

_{z}*q*= 0 without loss of generality.

_{y}## 3. Results and discussion

Figure 1(a) shows the momentum distribution of electrons stripped from the ground state of hydrogen atoms by a linearly chirped Gaussian XUV pulse (*τ _{H}* = 150

*asec*FWHM, $\beta =\sqrt{3}$) with the central photon energy

*ω*= 90

_{H}*eV*and peak intensity

*I*= 1011

_{H}*W*/

*cm*

^{2}in the presence of a 6 cycle IR pulse with wavelength 750 nm and peak intensity

*I*= 2·10

_{L}^{13}

*W*/

*cm*

^{2}. The center of the attosecond pulse is assumed to be positioned at the peak of the laser field (

*t*= 0). The results of Fig. 1(a) show the breakdown of the photoelectron momentum distribution invariance under the reflection through the plane

_{H}*q*= 0. This invariance is commonly observed when the atomic ionization is caused by a sole very long XUV pulse. In particular, for the electron emitted along the z-direction (

_{z}*q*= 0), the two peaks centered at about the kinetic momenta ${q}_{z}=\pm \sqrt{2\left({\omega}_{X}+{I}_{p}\right)}$ turn out to be quite different in that the peak located at

_{x}*q*> 0 is lower and broader than the other one centered at

_{z}*q*< 0. Calculations here not reported show that the peaks positions invert when

_{z}*β*changes sign, i.e. the photoelectron momentum distributions are invariant under both the transformations

*q*→ −

_{z}*q*and

_{z}*β*→ −

*β*. Moreover, it may be shown that by keeping fixed

*ω*,

_{L}*ω*and

_{H}*τ*,

_{H}*P*(

*q*,

_{x}*q*,

_{y}*q*) results to be very sensitive to the variations of both the linear chirp and the IR pulse intensity

_{z}*I*.

_{L}Figure 1(b) shows the electron momentum distribution evaluated by choosing the same parameters as those of Fig. 1(a), but with the delay time *t _{H}* =

*T*/4. In this case the main effect of the presence of IR pulse is the shifting of the EMD, in the momentum space, by Δ

_{L}*q*= −

_{z}*A*(

_{L}*t*)/

_{H}*c*, where

*A*(

_{L}*t*) is the value of the vector potential of the IR pulse at the instant of birth of the electron assumed to occur at the peak of the XUV pulse. This shift has been experimentally observed in the ionization of electrons ejected from the 4p state of krypton atoms under simultaneous irradiation of a 90

_{H}*eV*transform limited X-ray pulse and a femtosecond IR pulse (

*λ*= 750

*nm*) [6]. We remark that the results reported in Fig. 1(b) show that the asymmetries in the two peaks for electron ejection along z-axis, found for

*t*= 0, become vanishingly small for

_{H}*t*=

_{H}*T*/4. The main features of the EMD shown in the Fig. 1 may be conveniently illustrated by considering that, for a sufficiently short XUV pulse (

_{L}*τ*<

_{CH}*T*/4), Eq. (1) may be evaluated by expanding the integrand in power series of (

_{L}*t*−

*t*) and by keeping, in the exponent in curly brackets, terms up to the second order in (

_{H}*t*−

*t*). Then, the differential transition probability, taking the XUV radiation in the rotating wave approximation, assumes the simple form

_{H}Equation (4) allows us to discuss the main features of the EMD evaluated at *t _{H}* = 0 and

*t*=

_{H}*T*/4, already shown in Fig. 1. More generally, from Eq. (4), it turns out that the peaks of the photoelectron momentum distributions, for given values of

_{L}*q*, are located about at ${q}_{z}=-A\left({t}_{H}\right)/c\pm \sqrt{{q}_{x}^{2}+2\left({\omega}_{X}+{I}_{P}\right)}$ and that the breadth of the latter depends on the sign of

_{x}*q*. For

_{z}*β*> 0 and fixed value of

*q*, the momentum distribution of the electron ejected with

_{x}**û**

*parallel to*

_{z}q_{z}**E**

*(*

_{L}*t*) are found to be broader than the ones pertaining to electron emission with

_{H}**û**

*opposite to*

_{z}q_{z}**E**

*(*

_{L}*t*), as already shown in Fig. 1(a) for the particular case

_{H}*t*= 0. These asymmetries originate from the combined effect of the XUV chirp and of the steering action of the laser pulse on the freed electron. They tend to disappear for electron ejection along the direction perpendicular to the laser electric field, or, as shown in Fig. 1(b), when the time delay is

_{H}*t*=

_{H}*T*/4, as the steering effect extinguishes being

_{L}*E*(

_{L}*T*/4) = 0. Moreover, we note that the EMD are invariant under the simultaneous transformations

_{L}**q →**−

**q**+ 2

**A**

*(*

_{L}*t*)/

*c*and

*β*→ −

*β*. From the above considerations it follows that the XUV chirp influence on the EMD characterized by opposite

*q*becomes more effective when

_{z}*t*= 0. This circumstance may be exploited to get information on both the chirp and XUV pulse duration. In fact, for electron emission along the z-direction and for time delay

_{H}*t*= 0, Eq. (4) predicts that the height of the peaks of the EMD for forward emission decreases monotonically by increasing

_{H}*I*, while for backward electron emission the height of the peaks first increases by increasing

_{L}*I*and, after reaching its maximum, decreases monotonically. In Fig. 3 the peaks height for, respectively, forward and backward electron emission, evaluated for two different values of

_{L}*β*by means of Eq. (4), is shown as a function of

*I*and compared with the results obtained by using Eq. (1) and with the ones found by performing the numerical integration of the TDSE. We note that (see Eq. (4)) for

_{L}*β*> 0 (

*β*< 0) the highest peak in the EMD occurs for backward (forward) electron emission. These results suggest a way for determining the value of

*β*and

*τ*. These parameters may be obtained by simultaneously detecting, as a function of

_{CH}*I*, the momentum distribution of the electrons emitted, respectively, in the forward and backward directions. This task can be accomplished by using the same stereodetector arrangement as the one used in Ref. [16] for recording electrons emitted in opposite directions. By remembering that

_{L}*A*(

_{L}*t*) = 0 when

_{H}*t*= 0, the highest peaks in EMD, by assuming

_{H}*β*> 0, occurs when

*b*given by Eq. (7) is zero, i.e. for such laser field strength

*Ē*

_{0}

*that*

_{L}*Ē*

_{0}

*̄ = −2*

_{L}q*βa*with $\overline{q}=\sqrt{2\left({\omega}_{X}+{I}_{P}\right)}$. By denoting by

*R*the ratio between the values of the peaks of the EMD recorded respectively in the backward and forward direction at the field strength

*Ē*

_{0}

*, it is easily found that $\beta =\sqrt{\frac{{R}^{2}-1}{4}}$ and ${\tau}_{CH}=\sqrt{\frac{8\beta \text{ln}2}{\overline{q}{\overline{E}}_{0L}}}$.*

_{L}Before concluding, we observe that the results here reported on, found for hydrogen, may be extended to any atomic system. According to Eq. (4), the features of electron momentum distribution are given by Π(*β*, *τ _{H}*) · ℱ, that is independent of the atomic system taken under consideration, the characteristics of the atom being incorporated into the dipole transition matrix element evaluated at the instantaneous mechanical electron momentum

**q**(

*t*).

_{H}This work is supported in part by the Italian Ministry of University and Scientific Research.

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