An Effective Method for Generating Isolated Attosecond Pulses from a Solid Crystal Film

Abstract

Solids subjected to strong-field laser excitation can produce high harmonics, making high-order harmonic generation (HHG) one of the most effective methods for creating ultrafast coherent light sources, such as isolated attosecond pulses (IAPs). While extensive research has been conducted on generating IAPs through HHG in gaseous media, studies focusing on solid media are relatively limited. In crystals, the presence of numerous ionization and recombination sites, combined with high density and periodic structure, results in more complex interference dynamics. This complexity paves the way for unique applications in generating IAPs. Using an argon (Ar) crystal as a specific example, we have proposed and theoretically demonstrated an innovative approach for generating IAPs from a solid crystal film using a multi-cycle conventional driving laser pulse.

1. Introduction

The ultrafast motion of electrons in atoms, molecules, and condensed matter can generally involve attosecond timescales. The attosecond light pulse can provide unusual functionality for probing, initiating, driving, and controlling the ultrafast electronic dynamics with unprecedented high temporal and spatial resolutions simultaneously. The progress of attosecond science is closely linked to the improvement of attosecond light sources in terms of shorter and more intense attosecond pulses. Indeed, following its first synthesis and characterization [1,2], with a tendency towards reducing the pulse durations and increasing the pulse intensities [3,4,5,6,7,8,9,10,11,12,13,14,15,16], attosecond light pulses have been and will continue to open up new venues for studying both fundamental and applied sciences, enabling a number of exciting possibilities [5].

Solids driven by a strong-field laser can produce high-order harmonics. The signal intensity of high-order harmonic spectrum decreases sharply after the fundamental frequency, followed by a plateau region of harmonics with similar intensities until reaching a cutoff point for a stable decrease in signal intensity. Generally speaking, there is a supercontinuum spectrum near the cut-off point. The generation of attosecond pulses can be obtained by combining a spectral filtering of supercontinuum spectrum, and a far-field spatial filtering [17]. It is well known that the driving laser pulse used for HHG can be categorized into two classes: few-cycle pulses and multi-cycle pulses. When using few-cycle laser pulses, the supercontinuum spectrum is emitted exclusively at the peak of the laser envelope. By selecting this supercontinuum spectrum, one can generate an isolated attosecond pulse (IAP). In contrast, when employing multi-cycle laser pulses, the supercontinuum spectrum bursts every half cycle period. By selecting this supercontinuum spectrum, it becomes possible to create an attosecond pulse train (APT).

The ultrashort temporal resolution of IAPs makes them a powerful tool for studying and manipulating the ultrafast dynamic processes in atoms, molecules, and solid materials, driving significant advances in both fundamental scientific research and technological applications, such as: (1) IAPs can capture transient processes of electron movement in atoms and molecules, which is crucial for understanding electron transfer and bond formation/breakage in chemical reactions [18]. (2) In the presence of strong laser fields, IAPs are used to study the nonlinear responses and tunneling ionization of electrons, which holds potential for the development of next-generation electronic devices [11]. (3) IAPs can be used in time-resolved photoelectron spectroscopy and X-ray absorption spectroscopy, allowing for the analysis of changes in electronic structures and excited-state dynamics in materials [4]. (4) By precisely controlling and measuring a material’s response to attosecond pulses, researchers can explore ultrafast changes in optical properties, which has potential applications in photonics and nanophotonics [12]. (5) IAPs are used to investigate ultrafast electron transfer and energy transfer processes in complex biomolecules, helping to reveal the fundamental mechanisms of life processes [19].

Recently, high-order harmonic generation (HHG) is recognized as one of the best methods for generating ultrafast coherent light sources of IAPs. Over the past decade, significant progress has been made in generating IAPs through both experimental and theoretical explorations [11,20,21,22]. While there is a wealth of research on creating IAPs by HHG in gas media, studies on creating IAPs by HHG in solid media are scarce. From a technical standpoint, producing multi-cycle conventional driving laser pulses is more accessible for most ultrafast laser laboratories than generating few-cycle driving laser pulses. To the best of our knowledge, in order to create IAP with multi-cycle driving laser pulses, it is necessary to incorporate various additional technologies, such as polarization gating [11], employing a synthetic electric field consisting of two-color laser pulses [23,24], double optical gating [25], and others [26,27,28,29], unlike the use of few-cycle driving laser pulses [7,22,30,31].

As a point of reference, the presence of multiple ionization and recombination sites in solid crystals, combined with their high density and periodic structure, leads to richer interference dynamics, enabling the possibility of unique applications for generation of IAPs [32,33]. Interesting enough, Ndabashimiye et al. [34] presented the experimental observation of HHG from argon (Ar) and krypton (Kr) in both gas and crystal phases irradiated by laser pulses of 50 fs, 1333 nm, and 1500 nm, finding that the HHG spectra of crystal-phase show multiple plateaus that extend far beyond the atomic limit of the corresponding gas-phase harmonics measured under similar conditions. You et al. [35] reported the experimental observation of HHG from single-crystal MgO irradiated by laser pulse of 50 fs, 1333 nm, demonstrating that the conversion efficiency of HHG in solids is sensitive to laser orientation with respect to the crystal. Here, of particular interest for us was to explore the exciting possibilities of IAP generation opened up by the availability of HHG from crystals. Inspired by the above experiments, taking the Ar crystal as a concrete example, we are motivated to numerically investigate how IAP can be generated from an solid crystal film by the multi-cycle laser pulse utilized in above experiments.

The rest of this paper is organized as follows. First, we describe in brief the theoretical framework and the simulation details. Second, the calculated results and discussion are presented. Finally, we summarize the conclusion of this paper.

2. Materials and Methods

2.1. Light-Matter Interaction Model

We consider an irradiation of a free-standing thin film, composed of valence electrons and ionic cores, in a vacuum by an ultrashort light pulse of a linearly polarized plane wave at normal incidence. In order to obtain time evolution of the electromagnetic fields and the electronic excitations inside the thin film, our calculation is based on a coupled microscopic first-principles scheme, in which both dynamics of electron motion and light propagation are treated consistently and simultaneously using a common spatial grid. It should be noted that atomic units (a.u.) are used, unless stated otherwise.

The microscopic Maxwell equations are solved to describe the propagation of electromagnetic fields using the Gaussian unit system and Weyl gauge. In this gauge, the scalar potential $\varphi (\mathit{r},t)$ is set to zero [36], and the vector potential ${\mathit{A}}_{\mathrm{EM}}(\mathit{r},t)$ satisfies the following equation:

$$\left({\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{▽}^2\right){\mathit{A}}_{\mathrm{EM}}(\mathit{r},t)={\displaystyle \frac{4\pi }{c}}\mathit{j}(\mathit{r},t),$$

(1)

where c is the scalar speed of light in vacuum, $\mathit{j}(\mathit{r},t)$ represents the electric current density.

The electron dynamics is described by first-principles time-dependent Kohn–Sham (TDKS) equation, which is the basic equation of time-dependent density-functional theory (TDDFT), expressed as

$$\begin{array}{c}\hfill \mathbf{i}\hslash {\displaystyle \frac{\partial {\psi }_i(\mathit{r},t)}{\partial t}}=\left\{{\displaystyle \frac{1}{2m_{\mathrm{e}}}}{\left(-\mathbf{i}\hslash ▽+{\displaystyle \frac{e}{c}}{\mathit{A}}_{\mathrm{eff}}(\mathit{r},t)\right)}^2+V_{\mathrm{eff}}\right\}\times {\psi }_i(\mathit{r},t),\end{array}$$

(2)

where e ($e>0$), $m_{\mathrm{e}}$, ℏ are the elementary charge, electron mass, and reduced Planck constant, respectively. The index i runs over the occupied Kohn–Sham orbitals $\left\{{\psi }_i(\mathit{r},t)\right\}$. ${\mathit{A}}_{\mathrm{eff}}$ and $V_{\mathrm{eff}}$ can be decomposed into

$${\mathit{A}}_{\mathrm{eff}}={\mathit{A}}_{\mathrm{ext}}+{\mathit{A}}_{\mathrm{EM}}+{\mathit{A}}_{\mathrm{xc}},$$

(3)

$$V_{\mathrm{eff}}=V_{\mathrm{H}}+V_{\mathrm{ion}}+V_{\mathrm{xc}},$$

(4)

with the incident light pulse as an external field ${\mathit{A}}_{\mathrm{ext}}$. The exchange-correlation (XC) contribution to the vector potential and scalar potential are denoted as ${\mathit{A}}_{\mathrm{xc}}$ and $V_{\mathrm{xc}}$, respectively. $V_{\mathrm{ion}}$ is the ionic (pseudo-)potential. The Hartree potential energy, describing the classical electrostatic interaction between electrons, $V_{\mathrm{H}}$ is defined as

$$V_{\mathrm{H}}(\mathit{r},t)=e^2\int d{\mathit{r}}^{\prime }{\displaystyle \frac{n_{\mathrm{e}}({\mathit{r}}^{\prime },t)}{|{\mathit{r}}^{\prime }-\mathit{r}|}},$$

(5)

in which $n_{\mathrm{e}}(\mathit{r},t)$ is the electron number density, given by

$$n_{\mathrm{e}}(\mathit{r},t)=\sum _i{f}_i{\left|{\psi }_i(\mathit{r},t)\right|}^2,$$

(6)

with occupations $f_i$.

From the Kohn–Sham orbitals, the electron number current density ${\mathit{j}}_{\mathrm{e}}(\mathit{r},t)$ is obtained as below:

$$\begin{array}{c}\hfill {\mathit{j}}_{\mathrm{e}}(\mathit{r},t)=\sum _i{\displaystyle \frac{1}{2m_e}}{f}_i\left[{\psi }_i^{*}(\mathit{r},t)\times \left(-\mathbf{i}\hslash ▽+{\displaystyle \frac{e}{c}}{\mathit{A}}_{\mathrm{eff}}(\mathit{r},t)\right){\psi }_i(\mathit{r},t)+c.c.\right].\end{array}$$

(7)

The electric current density is given by $\mathit{j}(\mathit{r},t)=-e{\mathit{j}}_{\mathrm{e}}(\mathit{r},t)$. Here, the positions of the ions have been frozen, so there is no electric current density coming from the ionic motion.

2.2. Computational Details

To make the problem manageable, we perform several (controlled) approximations. The transverse geometry [37] is used to treat the bulk polarization response of the infinitely extended system along the polarization orientation. In order to reduce computational cost, a pseudopotential model for the Ar atom is constructed with 8 valence electrons being explicitly considered. The interaction between valence electrons and the ionic cores [Ar(1s²2s²2p⁶)] is described by an optimized norm-conserving Vanderbilt pseudopotential [38] for the Ar thin film. In the present calculations, an adiabatic generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof (PBE) parameterization for the scalar XC potential ${\mathrm{V}}_{\mathrm{xc}}$ is used. In view of the numerical complexity of the present simulation, ${\mathit{A}}_{\mathrm{xc}}$ is [39] neglected. We set a coordinate system as shown in Figure 1: a thin film of Ar exists within the interval z ∈ [0, L], with (001) surfaces located at z = 0 and z = L planes. Namely, we set an Ar thin film of thickness L = 9a, where a = 10.93 a.u. is the lattice constant of Ar in the face centered cubic unit cell containing four atoms. The thin film is placed in a rectangular computational box, with three side lengths of ${\mathrm{L}}_{\mathrm{x},\mathrm{y}}=\mathrm{a}$ in the x and y directions and ${\mathrm{L}}_{\mathrm{z}}=\mathrm{L}+2\mathrm{d}$ for the z direction, where d = 9.45 a.u. is the thickness of vacuum layers attached to both sides of the film. The linearly polarized incident laser pulse propagates along the z-axis, with its polarization aligned at an angle $\theta$ with the x-axis. The time/frequency characteristics of the transmitted pulses is analyzed.

Periodic boundary conditions are imposed to the boundaries in the x and y directions of the computational box. At the boundaries in the x and y directions of the computational box, the periodic boundary conditions for $\left\{{\psi }_i(\mathit{r},t)\right\}$ and $V_{\mathrm{eff}}$ are imposed. At the boundaries in the z directions of the computational box, absorbing boundary condition for the vector potential $\mathit{A}$ is applied. The thin film is infinitely periodic in the x and y directions.

Before beginning the time evolution calculation, we first calculate the ground state, which will serve as the initial state for the time evolution calculations. At the beginning of the time evolution calculation, the vector potential ${\mathit{A}}_{\mathrm{ext}}$ of incident wave is assumed to be linearly polarized along the $\widehat{\mathit{e}}$ unit vector and propagating along the z-axis. The time profile of the incident light pulse is given by

$${\mathit{A}}_{\mathrm{ext}}=\widehat{\mathit{e}}{A}^{\left(i\right)}(t-{\displaystyle \frac{z}{c}})$$

(8)

with $A^{\left(i\right)}\left(t\right)=-{\displaystyle \frac{E_0}{\omega }}co{s}^2\left[{\displaystyle \frac{\pi }{T}}(t-{\displaystyle \frac{z}{c}}-{\displaystyle \frac{T}{2}})\right]sin\left[\omega (t-{\displaystyle \frac{z}{c}})+\varphi \right],(0<t-z/c<T)$, where $E_0$ is the maximum electric field strength, which is related to the laser peak intensity ($I_0$) as $I_0=c{E_0}^2/8\pi$, $\omega$ is the carrier frequency, $\varphi$ is the carrier-envelope phase (CEP), and T is the pulse duration related to the usual measure ${\tau }_p$ (full width at half-maximum intensity (FWHM)) by ${\tau }_p=0.364T$.

We use a multi-cycle driving laser pulse with T = 2067 a.u. (50 fs), $\omega$ = 0.03417 a.u. (0.93 eV), and $I_0$ = 2.6 × 10¹³ W/cm², which are the same as those used in the experiments by Ndabashimiye et al. [34] and You et al. [35]. To achieve an energy convergence of less than 4.7 × 10⁻⁷ in the ground state calculation, a grid spacing of 0.273 a.u. is chosen for the spatial finite difference calculation. The time step is set to 0.01 a.u. (1 a.u. of time corresponds to 0.024189 fs) to ensure the stability of time evolution calculations. In order to sample the Brillouin zone effectively while balancing computational efficiency and result convergence, we have chosen to set the number of k points in the 2D reciprocal space as 1 × 1. Numerical simulations of light-matter interaction process were performed with the open-source software SALMON code (version 2.2.0) [42].

As shown in Figure 1, in the present simulation, when a linearly polarized laser pulse at an angle $\theta$ to the x-axis propagates through an Ar crystal film along the z-axis, the polarization vector of the transmitted light field lies within the xy-plane. To facilitate the analysis of the time/frequency characteristics of the emitted harmonics, the HHG spectrum is calculated by the Fourier transformation of the transmitted electric filed given by

$$\begin{array}{cc}\hfill H\left(\omega \right)& =H_x\left(\omega \right)+H_y\left(\omega \right),\hfill \\ \hfill H_x\left(\omega \right)& \propto |\int E_x\left(t\right)e^{\mathbf{i}\omega t}dt{|}^2,\hfill \\ \hfill H_y\left(\omega \right)& \propto |\int E_y\left(t\right)e^{\mathbf{i}\omega t}dt{|}^2,\hfill \end{array}$$

(9)

where $E_x\left(t\right),E_y\left(t\right)$ are the x and y components of the transmitted electric field in xy-plane, respectively. In the Gaussian unit system, the relationship between the electric field and vector potential is expressed as $\mathit{E}\left(t\right)=-{\displaystyle \frac{1}{c}}{\displaystyle \frac{d\mathit{A}\left(t\right)}{dt}}$. The time intensity profile synthesized from the spectral range between frequencies ${\omega }_1$ and ${\omega }_2$ can be given by inverse Fourier transformation of the harmonic spectrum [43],

$$\begin{array}{cc}\hfill I\left(t\right)& =I_x\left(t\right)+I_y\left(t\right),\hfill \\ \hfill I_x\left(t\right)\propto |{\displaystyle \frac{1}{2\pi }}{\int }_{{\omega }_2}^{{\omega }_1}& e^{-\mathbf{i}\omega t}\int E_x\left(t^{\prime }\right)e^{\mathbf{i}\omega t^{\prime }}d{t}^{\prime }d\omega {|}^2,\hfill \\ \hfill I_y\left(t\right)\propto |{\displaystyle \frac{1}{2\pi }}{\int }_{{\omega }_2}^{{\omega }_1}& e^{-\mathbf{i}\omega t}\int E_y\left(t^{\prime }\right)e^{\mathbf{i}\omega t^{\prime }}d{t}^{\prime }d\omega {|}^2.\hfill \end{array}$$

(10)

From the theoretical point of view, by extracting the cutoff components of the high-order harmonics spectrum with ${\omega }_1$ and ${\omega }_2$, it is possible to obtain an IAP.

3. Results and Discussion

Figure 2 gives a typical result of the time/frequency characteristics of the transmitted light for $\theta =0^{\circ }$, which is the direction with the highest intensity of harmonics as reported by You et al. [35]. The HHG spectrum is calculated by the Fourier transformation of the transmitted pulse using Equation (9). The time profile synthesized from the marked regions ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3$ and ${\mathrm{C}}_4$ of plateaus in HHG spectrum can be given by inverse Fourier transformation of the harmonic spectrum using Equation (10), respectively. In the left panel of Figure 2, the appearance of multiple plateaus is in good agreement with experimental observations [34], proving that the model parameters used in our calculations are reasonable. It is important to mention that the experiment observed multiple plateaus structures up to the second order, whereas our calculations indicated the presence of multiple plateaus structures up to the 4th order. Note that there is a discrepancy between experimental and calculated spectra. The experimental spectra show clear peaks in the plateau region, while the calculated spectra exhibit strong fluctuations without a clear pattern due to the absence of some more complex mechanisms, such as dephasing [44,45] or the spatiotemporal distribution of the emission events on the mesoscopic scale [46]. However, the mystery surrounding the clean peak formation observed in the experiment, which is challenging to replicate in theoretical models, has not been fully resolved [46,47]. For the sake of clarity in our discussion, we will assign labels to the four plateaus as follows: ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, ${\mathrm{P}}_3$ and ${\mathrm{P}}_4$, along with their respective marked regions near the cut-off point: ${\mathrm{C}}_1$ (0.807∼0.987 a.u.), ${\mathrm{C}}_2$ (1.392∼1.485 a.u.), ${\mathrm{C}}_3$ (1.863∼1.986 a.u.) and ${\mathrm{C}}_4$ (2.454∼2.55 a.u.). The right panel of Figure 2 shows the time intensity profile of APTs obtained from the marked regions ${\mathrm{C}}_1,{\mathrm{C}}_2,{\mathrm{C}}_3$ and ${\mathrm{C}}_4$, respectively. Just like in the case of gas, the APT can also be generated in crystal solids by superimposing the supercontinuum spectrum near the cut-off point. Interestingly enough, in an APT generated from a crystal solid, the isolation of a single attosecond pulse can occur due to the effective suppression of other pulses by the richer interference dynamics in the crystal solid as compared with the case of gas. Obviously, by applying a saturable absorber [40,41], which can absorb low intensity light and let it pass through when the light intensity is high enough. The APT with the isolation of a single attosecond pulse can become IAP shown in Figure 1.

As is well known, the conversion efficiency of HHG in solids is sensitive to laser orientation with respect to the crystal [35]. Furthermore, we are motivated to investigate how the isolation of a single attosecond pulse in APT generated from a crystal solid depends on the polarization direction of multi-cycle driving laser pulse with respect to the crystal.

Figure 3 shows the HHG spectra driven by laser pulses with different polarization directions $\theta$ (upper panels) and the temporal intensity distribution of APTs synthesized from the HHG spectral marked regions (middle panels). The marked ranges used to generate APT has been optimized to achieve maximum isolation of a single attosecond pulse for a specific laser polarization direction. As shown in the middle panels of Figure 3, several interesting findings are as follows: (1) The isolation of a single attosecond pulse in APT is sensitive to the polarization directions $\theta$. (2) The isolation of a single attosecond pulse in APT is most remarkable when $\theta =0^{\circ }$, leading to the generation of an IAP with the highest peak intensity and narrowest FWHM compared to other polarization directions. To understand the origin of anisotropy of HHG in bulk crystals, You et al. [35] perform semi-classical calculations of real-space electron trajectories inside MgO crystal, leading to a simple real-space picture of HHG in solids as arising from coherent collisions of the delocalized charge carriers with the periodic potential. In this real-space picture, the harmonic signal is strongly enhanced or reduced depending on whether the semi-classical electron trajectories are directed towards or away from neighboring atomic sites. However, Lakhotia et al. [48] experimentally studied the irregular crystal orientation dependence of extreme ultraviolet high harmonics in MgF₂, observing that not all harmonics follow the same angular dependence, i.e., they do not maximize their yield at the same crystal direction. In addition, Gertsvolf et al. [49] experimentally studied orientation-dependent multiphoton ionization in wide band gap crystals of $\alpha$ quartz ($\alpha$-SiO₂), sapphire ($\alpha$-Al₂O₃) and lithium fluoride (LiF) in terms of linearly polarized 45 fs, 800 nm laser pulses, implying that the anisotropy of HHG may be due to the possible dependence of tunneling rate on crystallographic directions. Overall, from previous publications one conclusion that can be drawn at this stage is that the efficiency of HHG depends not only on the crystal’s orientation, but also on the materials used.

As shown in the lower panels of Figure 3, several interesting findings are as follows: (1) In the case of $\theta ={15}^{\circ }$ and ${30}^{\circ }$, the semi-classical electron trajectories that avoid neighboring atomic sites in the crystal should lead to a significant reduction in harmonic efficiency, as expected from the real-space picture. (2) In the case of $\theta =0^{\circ }$ and ${45}^{\circ }$, the semi-classical electron trajectories that connect neighboring atomic sites in the crystal should lead to a significant enhance in harmonic efficiency according to the real-space picture. Intuitively, the closer the distance between electron trajectories, the stronger the interference effect between them will be. Compared with $\theta ={45}^{\circ }$, when $\theta =0^{\circ }$, the closer distance between semi-classical electron trajectories lead to the stronger the interference effect, and then result in higher harmonic efficiency and more remarkable isolation, and finally enabling a novel IAP generation method.

In solids, the mechanism of HHG can generally be classified into intra-band and inter-band mechanisms within the band picture. However, during strong field interactions, the energy band of a crystal will undergo dynamic and violent distortion. Consequently, the debate surrounding the contribution of these mechanisms to HHG remains ongoing, with no consensus reached thus far [50]. This indicates that the energy band model may not be the most suitable framework for explaining the HHG process that occurs when intense laser pulses interact with solids. On the one hand, by using the same laser parameters and crystalline solid as in the experiment [34], the calculated HHG spectrum can provide an initial benchmark for the model and simulation parameters used in this study, and future work is still required to analyze the results from the model in order to make it a benchmark in the true sense. Despite this, the utilization of this model will establish an initial basis for studying HHG in crystalline solids at the accurate level of ab initio calculations, which play a crucial role in predicting realistic physical mechanisms. On the other hand, selecting the same laser parameters and crystalline solid used in the experiment ensures that our research results can be most directly verified through experimentation.

4. Conclusions

HHG is acknowledged as one of the most effective methods for producing ultrafast coherent light sources of IAPs. While there is a significant amount of research on generating IAPs by HHG in gas media, studies on generating IAPs by HHG in solid media are limited. The presence of multiple ionization and recombination sites in crystals, along with their high density and periodic structure, results in more richer interference dynamics, allowing for the potential of generating IAP. Taking the same laser parameters and crystalline solid used in the experiment as a concrete example, we have proposed and theoretically demonstrated an innovative idea for generation of IAPs from a crystalline solid film by a multi-cycle conventional driving laser pulse, that are easily accessible for most ultrafast laser laboratories, combined with a saturable absorber, based on a coupled microscopic first-principles model, in which both dynamics of electron motion and light propagation are treated consistently and simultaneously.

This study has proposed and theoretically demonstrated the IAP generation in Ar crystal with a multi-cycle driving laser pulse. Our aim is not to comprehensively consider all factors that affect the IAP generation in crystalline solid, as this is beyond the available computational cost. Rather, we attempt to highlight how IAP can be generated from a crystal film by intense multi-cycle laser pulse. Thus, this deserves to be investigated in future research both theoretically and experimentally.