1. Introduction
Vortices are prevalent phenomena that exist in various natural systems, ranging from quantum vortices in liquid nitrogen to ocean circulation, typhoon vortices, and spiral galaxies in the Milky Way. These vortices are observed not only in macroscopic materials, but also in electromagnetic and optical fields [1]. The interaction between light and matter constitutes a fundamental physical process in the natural world. The evolution of modern ultrafast laser technology has empowered researchers to observe and manipulate physical, chemical, and biological processes at exceedingly small temporal and spatial scales. These technological developments have facilitated the study of the interaction between light and matter at the atomic and molecular levels, leading to the discovery of new physical effects and laws [2–9]. Among these discoveries, Ramsey interference [10] in laser-produced electron wave packets has been investigated in both Rydberg states [11] and continuum for linearly polarized lasers.
The development of ultrafast laser technology has opened the door to precise control and monitoring of atomic-scale electron dynamics. Notably, the advent of single attosecond pulses of extremely ultraviolet (XUV) light has pioneered a transformative era in this field, enabling systematic measurements and real-time observations of electron behavior at the atomic level [9]. In recent years, considerable attention has been directed towards the generation and manipulation of vortex states in light beams [1, 12] and the electron vortex [13–17]. These vortex states possess inherent orbital angular momentum, akin to twisted light and electron beams [18–20].
The presence of vortex quasiparticles in atomic systems has been demonstrated through the interaction of atomic hydrogen with a short, rectangular, half-cycle electric field pulse [21]. Furthermore, many theoretical investigations have significantly enriched our understanding of electron vortex phenomena. These studies have delved into various aspects, including photoionization, double ionization [22], and molecular photoionization [17, 23–25] under the influence of intense [16, 26, 27] or bichromatic [13, 17, 23, 26] laser fields. The generation of vortex-shaped photoelectron wave packets in the photoionization of helium atoms [13, 28] and hydrogen molecular ions [29] has been proposed by employing time-delayed counter-rotating circularly polarized attosecond pulses. In addition to the time delay, researchers have also explored the control of other laser parameters, such as the carrier-envelope phase (CEP) [17, 27], relative optical phase [25, 30], polarization state [13, 23, 28, 31, 32], orbital symmetry [33, 34], and chirp [35]. Moreover, vortices have not only been observed in photoionization but also in photodissociation [29] and photodetachment [36, 37]. The presence of an electron vortex has been experimentally verified [32, 38–40] in femtosecond multiphoton ionization of potassium and sodium atoms using counter-rotating circularly polarized and co-rotating circularly polarized laser fields [41, 42].
In this work, we investigate the electron vortex in the photoionization of hydrogen atoms under irradiation from a pair of counter-rotating circularly polarized laser fields by numerically solving the two-dimensional time-dependent Schrödinger equation (2D TDSE). Our findings demonstrate that the application of two attosecond pulses with a time delay results in a vortex structure in the momentum distribution. This highlights the sensitivity of the electron vortex to the angular frequency, time delay, and rotation of the laser pulse, as previously shown in [17, 28]. Specifically, our focus lies in scrutinizing the influence of chirping on the structural characteristics of the electron vortex. By introducing positive and negative chirp parameters, we explore how the vortex structure is affected by different chirp parameters in the momentum distribution. This provides an effective and new method for controlling the momentum distribution of electrons.
The paper is organized as follows. In section 2 , we briefly introduce the numerical method for solving the 2D TDSE and extracting physical variables. Then in section 3 , we show and discuss the photoelectron momentum distributions (PMDs) under different chirp parameters and time delays. A summary and conclusion are given in section 4 . Unless specifically stated otherwise, the atomic units (a.u.) ℏ = me = e = 1 are used throughout this work.
2. Theory and models
2.1. Solving the TDSE
Accurately modeling the ionization process in the presence of an intense laser field typically necessitates the numerical solution of the TDSE. This numerical approach simulates the interaction between an intense electromagnetic field and an atomic target, enabling the derivation of the final electron wave function during the ionization process. Consequently, this methodology enables an in-depth exploration of the physical phenomena and underlying mechanisms associated with the interaction between the intense field pulse and matter. Furthermore, these discoveries offer crucial theoretical support for experimental investigations.
Within the dipole approximation, the 2D TDSE can be expressed as
$\begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial }{\partial t}\psi (t)=\left[\displaystyle \frac{{p}^{2}}{2}+V(x,y)+{V}_{\mathrm{int}}\right]\psi (t),\end{eqnarray}$
where V(x, y) represents the Coulomb potential of hydrogen atoms. In order to avoid the singularity at the origin, we introduce a soft-core parameter a, so the potential reads $\begin{eqnarray}V(x,y)=\displaystyle \frac{-1}{\sqrt{{x}^{2}+{y}^{2}+a}},\end{eqnarray}$
where a = 0.64 is chosen to obtain proper ground state energy Eg = − 0.5. The potential term Vint denotes the interaction between the electron and the external laser field. In the length gauge, it is given by ${V}_{\mathrm{int}}=\vec{r}\cdot \vec{E}(t)$. The electric field $\vec{E}(t)$ is composed of two components along the x- and y-directions, denoted as $\vec{E}(t)={E}_{x}(t)\hat{x}+{E}_{y}(t)\hat{y}$. The expressions for Ex(t) and Ey(t) are given by $\begin{eqnarray}\begin{array}{l}{E}_{x}(t)=\displaystyle \frac{1}{\sqrt{2}}{E}_{0}f(t)\cos (\omega t+\phi ),\\ {E}_{y}(t)=\displaystyle \frac{\eta }{\sqrt{2}}{E}_{0}f(t)\sin (\omega t+\phi ).\end{array}\end{eqnarray}$
Here, E0 and ω represent the electric field amplitude and central frequency of the laser field, respectively, and f(t) denotes the envelope of each laser pulse. In our simulations, we use a sine-squared profile envelope. The parameter η corresponds to the helicity of a circularly polarized pulse. Specifically, the circularly polarized pulse rotates clockwise with η = 1, while rotating counterclockwise with η = –1. The carrier-envelope phase φ is set to 0 in our following simulations. We apply two circularly polarized laser pulses with a time delay of td, and the total electric field is expressed as $\begin{eqnarray}\vec{E}(t)=[{E}_{x}(t)+{E}_{x}(t-{t}_{d})]\hat{x}+[{E}_{y}(t)+{E}_{y}(t-{t}_{d})]\hat{y}.\end{eqnarray}$
To determine the time evolution of wave function and obtain the final state wave function after the electron interaction with the external laser field, we adopt the splitting-operator method in which the time propagation of the electron wave function is given by
$\begin{eqnarray}\psi (t+{\rm{\Delta }}t)\approx {{\rm{e}}}^{-{\rm{i}}{V}{\rm{\Delta }}t/2}{{\rm{e}}}^{-{\rm{i}}{T}{\rm{\Delta }}t}{{\rm{e}}}^{-{\rm{i}}{V}{\rm{\Delta }}t/2}\psi (t).\end{eqnarray}$
In our simulations, the box size is increased up to 800 for each dimension, and the step size of time is set to Δt = 0.01. In addition, a mask function is employed to avoid spurious reflections from the boundaries. When the final step of time propagation of the wave function is complete, we record the ionization part as [1 − M(r)]ψ(x, y; tf), where ψ(x, y; tf) denotes the wave function at the last time step. The mask function for absorption reads
$\begin{eqnarray}M(r)=\left\{\begin{array}{ll}1, & r\leqslant {r}_{b},\\ {{\rm{e}}}^{-\alpha (r-{r}_{b})}, & r\gt {r}_{b},\end{array}\right.\end{eqnarray}$
with α = 1, $r=\sqrt{{x}^{2}+{y}^{2}}$, and rb = 15 a.u., which corresponds to the boundary of the ionized wave function. Then, the PMDs are obtained by performing Fourier transform on the final wave function of the photoelectron. In this work, the propagation in the free field allows 10 optical cycles (o.c.) to relax. The initial ground-state wave function of hydrogen atoms can also be obtained by using the imaginary-time propagation method.2.2. Chirped attosecond pulses
In this study, we employ laser pulses with a time-dependent frequency characterized by the chirp parameters. Specifically, we utilize a linear chirp, denoted as [43]
$\begin{eqnarray}\omega (t)={\omega }_{0}+\beta t,\end{eqnarray}$
where ω0 is the central frequency of the laser pulse and β is the linear chirp parameter.In figure 1, we present schematics of the power spectrum and the time-dependent frequency of the laser pulses. The solid lines denote the frequencies of the first pulse, while the dashed lines denote those for the second pulse. Different colors (black, blue, and green) represent different chirp parameters (0, 0.01, and 0.02). The top panel displays the power spectra of laser pulses with different chirp parameters. For a pulsed laser beam, a linear chirp generally broadens the power spectrum but does not change the temporal profile of the power spectrum [44]. The larger the chirp parameter, the broader its power spectrum. In the middle and bottom panels, we display examples of the instantaneous frequencies of two pulses with a time delay of td = 2 o.c. for different chirp parameters. The time delay of pulses is defined by the time delay of the pulse central frequencies.
Figure 1. Power spectra and instantaneous frequencies of chirped pulses. Top panel: power spectra of chirped pulses with different chirp parameters. Middle panel: instantaneous frequencies of two equal chirped pulses in sequence with time delay. Bottom panel: instantaneous frequencies of two opposite chirped pulses in sequence with time delay. Solid lines are for the first pulses, and dashed lines indicate the second chirped pulses. Chirp parameters are β = 0, 0.01 and 0.02, respectively. |
However, in contrast to chirp-free pulses, there exists an additional frequency-dependent time delay denoted as τω between the first and second chirped pulses. Let us consider this time delay for a specified frequency. In the middle panel, we present the instantaneous frequencies of two sequent pulses with equal chirp parameters, i.e. β1 = β2. In the bottom panel, we choose β1 = –β2. For a specified pulse frequency ω, the time delay τω can be determined by
$\begin{eqnarray}{\tau }_{\omega }=\left(\omega -{\omega }_{0}\right)\left(\displaystyle \frac{1}{{\beta }_{2}}-\displaystyle \frac{1}{{\beta }_{1}}\right)+{t}_{d}.\end{eqnarray}$
This expression indicates that the time delay for a given frequency can be divided into two parts: the time delay induced by chirp (the first term) and the time delay of the pulses with regard to the central frequency (the second term, td).In the scenario where β1 = β2 = β, the time delay τω is expressed as
$\begin{eqnarray}{\tau }_{\omega }={t}_{d}.\end{eqnarray}$
In other words, when two pulses have the same chirp parameters, the time delay for all frequencies equals the central frequency's time delay td.In the scenario where β1 = − β2 = β, the time delay τω is described by
$\begin{eqnarray}{\tau }_{\omega }=-\displaystyle \frac{2}{\beta }(\omega -{\omega }_{0})+{t}_{d}.\end{eqnarray}$
It is evident that, even when the time delay for the central frequency of the pulses becomes zero (td = 0), the time delay for a specific frequency ω still exists, unless chirp-free pulses are considered. This chirp-induced time delay can give rise to a special vortex in PMDs.2.3. Electron vortex in PMDs of single-photon ionization
When hydrogen atoms are exposed to a pair of counter-rotating, chirp-free circularly polarized pulses, a vortex structure can be observed in the PMDs. This distinctive distribution is an indicator of the interference among partial waves characterized by different magnetic quantum numbers. Extensive predictions and investigations have been devoted to this phenomenon. The time delay td between two circularly polarized pulses plays a prominent role in generating an electron momentum vortex during the photoionization process.
In figure 2, we show the PMDs obtained from two counter-rotating circularly polarized laser pulses with time delays of 0 (left) and 2 o.c. (right). When there is no time delay (td = 0), the two counter-rotating laser pulses add up along the x-direction and cancel each other out along the y-direction, resulting in a linearly polarized pulse along the x-axis. In this case, the observed PMD exhibits a dipolar double-lobe structure, which is identical to that of single-photon ionization by a linearly polarized pulse [45, 46]. No vortex structure is observed in this case. The vortex structure becomes visible as the time delay is increased to 2 o.c. Such a phenomenon has been extensively studied and discussed in previous works [13, 17, 25, 47, 48]. The generation of the vortex is contingent upon the phase difference between the photoionization processes induced by the first and the second laser pulses with time delay.
Figure 2. Photoelectron momentum distributions of hydrogen atoms by a pair of counter-rotating circularly polarized attosecond pulses without chirp. The time delay is 0 (left) and 2 o.c. (right). The laser parameters are as follow: central frequency ω0 = 1.52, peak intensity I0 = 1014 W cm−2, pulse duration τ = 8 o.c. |
For the single-photon ionization process, the probability density distribution of the final photoelectron can be described by the squared modulus of the wave function [47]:
$\begin{eqnarray}| {\rm{\Psi }}(E,\theta ,\phi ){| }^{2}=2\rho (E,\theta )\left[1-\cos (2\phi -{{Et}}_{d})\right],\end{eqnarray}$
where ρ(E, θ) = ∣ψ1,±1(E, θ, φ)∣2. E = ω0 − Ip is the final kinetic energy of the photoelectron and Ip is the ground energy of hydrogen atoms. The azimuthal angle θ is set to π/2 for the current 2D simulation. Two Archimedean spirals are observed in the isophase contour maps within the polarization plane, to be specific in the φ − E plane or the px − py plane. For chirp-free fixed-frequency laser pulses, the manipulation of the vortex pattern can be achieved by solely adjusting the time delay. As shown in the left panel of figure 2, when there is no time delay (td = 0), the maximum photoelectron energy Emax is independent of φ. However, in the presence of a time delay between the two pulses, ${E}_{\max }=\left[2\phi -(2n+1)\pi \right]/{t}_{d}$, which becomes dependent on φ.Furthermore, when chirp is introduced, the modulation of the laser pulse frequency will also exert an influence on Emax. Consequently, chirp can be regarded as a parameter to control vortex dynamics, which is similar to time delay. Therefore, an additional parameter can be utilized to manipulate the electron vortex by adjusting the chirp parameters. In this case, equation (11 ) should be rewritten as:
$\begin{eqnarray}| {\rm{\Psi }}(E,\phi ){| }^{2}=2\rho (E)[1-\cos (2\phi -E{\tau }_{\omega })],\end{eqnarray}$
where E = ω(t) − Ip, and τω represents the time delay between the two pulses at a instantaneous frequency ω.3. Results and discussion
In this section we discuss the effect of chirp on the generation of the vortex in the PMDs of hydrogen atoms. In the figure 3, we illustrate the PMDs induced by a pair of counter-rotating circularly polarized attosecond pulses with identical chirp parameters (β1 = β2 = β). The time delays are set to td = 0 and 2 o.c. for the left and right panels, respectively. From top to bottom, the chirp parameters are β = 0.01 and 0.02. Results with negative chirps are not presented because they are similar to their positive chirp counterparts. In the left panels, the PMDs give exactly the same results as those obtained with the linearly polarized field [45, 46]. As a result, there no vortex structures exist. The only difference from top to bottom is that the PMDs are broadened due to the fact that chirped pulses have wider power spectra, as shown in the top panel of figure 1. However, when the time delay is introduced, e.g. td = 2 o.c., the vortex structure becomes visible. As we have discussed above, two pulses with equal chirp parameters (β1 = β2 = β) have a constant time delay for all frequencies, that is τω = td. Therefore, the PMDs are similar to the results described by equation (11 ). In addition, since the chirped pulses have broadened spectra, we observe longer spiral arms for larger chirp parameters.
Figure 3. Photoelectron momentum distributions of hydrogen atoms after photoionization by a pair of counter-rotating circularly polarized attosecond pulses. The time delay is 0 (left panels) and 2 o.c. (right panels). The laser parameters are as follow: central frequency ω0 = 1.52, peak intensity I0 = 1014 W cm−2, pulse duration τ = 8 o.c. From top to bottom, the chirp parameters are β = 0.01 and 0.02, respectively. |
In the following discussion, we focus on the photoionization of hydrogen atoms by a pair of counter-rotating circularly polarized pulses with opposite chirp parameters (β1 = –β2 = β), where the instantaneous frequencies are shown in the bottom panel of figure 1. We examine the no-delay scenario first. In figure 4, we present the PMDs of hydrogen atoms at different chirp parameters. From the top-left panel to the bottom-right panel, the chirp parameters are ±0.01, ∓ 0.01, ± 0.02 and ∓0.02, respectively. It is interesting to note that, despite the absence of a time delay between two pulses, a pronounced vortex structure can be clearly observed in all situations. Unlike the vortex in figure 3, the spiral arms are reversible, and split into two tails ultimately. Similar phenomena were also observed by Strandquist et al [35] in a study of the photoionization of helium atoms. According to equation (10 ), it is evident that even though the time delay of the counter-rotating pulses is zero, the time delay for a given frequency is nonzero, i.e. τω = –2(ω − ω0)/β. As the instantaneous frequency ω increases, the sign of τω value is reversed at ω = ω0, which leads to splitting into two tails. Furthermore, this opposite chirp parameter β is the essential reason for the reversible spiral arms
Figure 4. Photoelectron momentum distributions of hydrogen atoms after photoionization by a pair of counter-rotating circularly polarized attosecond pulses with td = 0. The laser parameters are as follow: central frequency ω0 = 1.52, peak intensity I0 = 1014 W cm−2, pulse duration of each pulse τ = 8 o.c. The chirp parameters for the first and second pulses from top left panel to bottom right panel are ±0.01, ∓ 0.01, ± 0.02 and ∓0.02, respectively. |
Previous research has shown that time delay plays an important role in vortex creation [13, 17, 23, 25, 28]. We now examine a scenario with both chirp and time delay. In figure 5, we increase the time delay from td = 0 to 2 o.c. and present the corresponding PMDs in comparison to figure 4. On the two left panels, the positive chirped pulse comes first, followed by the negative chirped pulse with 2 o.c. time delay, and vice versa for the panels on the right. If the positive chirped pulse comes first, it produces a larger τω at low frequencies than that at high frequencies of laser pulse. This leads to narrower fringes in the PMD of the lower-frequency regime than in the higher-frequency regime. On the other hand, when the negative chirped pulse comes first, it yields a large τω for higher frequency but small τω for lower frequency. Therefore, we can observe slimmer fringes in the higher-frequency regime and broader fringes in the lower-frequency part in the panels on the right.
Figure 5. Photoelectron momentum distributions of hydrogen atoms after photoionization by a pair of counter-rotating circularly polarized attosecond pulses with time delay td = 2 o.c. The laser parameters are as follow: central frequency ω0 = 1.52, peak intensity I0 = 1014 W cm−2, pulse duration of each pulse τ = 8 o.c. The chirp parameters for the first and second pulses from top left panel to bottom right panel are ±0.01, ∓ 0.01, ± 0.02 and ∓0.02, respectively. |
It is worth mentioning that switching the order of a pair of counter-rotating pulses without chirp should give their counterpart of PMDs [13]. But it is evident that this approach does not work for chirped pulses, as shown in figure 5 above. In figures 6(a) and (b), we plot the instantaneous frequencies of two pulses that were applied to produce the bottom panels of figure 5. We notice that when switching the order of left- and right-rotating pulses, the curves of instantaneous frequency also switch the crossing point (tc) from ω(tc) < ω0 to ω(tc) > ω0, as shown in figures 6(c) and (d). To obtain the same crossing point, we need to switch the order of pulses and their chirp parameters simultaneously.
Figure 6. Relationship between the instantaneous frequency and time. The green solid lines are for the first pulses, and green dashed lines indicate the second pulses. Laser parameters: (a) td = 2 o. c. , β = ±0.02, (b) td = 2 o. c. , β = ∓0.02, (c) td = − 2 o. c. , β = ±0.02, (d) td = − 2 o. c. , β = ∓0.02. |
When we apply the laser parameters of figures 6(c) and (d), the PMDs shown in Figure 7 are obtained. In this figure, we provide both numerical simulation results (top panels) obtained by the TDSE and analytical results (bottom panels) obtained by equation (12 ). As expected, the case of figure 6(c) produces the counterpart PMD of figure 6(b), while the case of figure 6(d) produces the counterpart PMD of figure 6(a). The numerical simulation results are in good agreement with the analytical model predictions.
Figure 7. Photoelectron momentum distributions of hydrogen atoms after photoionization by a pair of counter-rotating circularly polarized attosecond pulses. Top panels: results from TDSE simulations; bottom panels: analytical results from the equation ( |
Finally, in figure 8, we depict the PMDs for a pair of counter-rotating circularly polarized attosecond pulses with a longer time delay of td = 6 o.c. Once again, in the low-frequency regime we observe a narrower interference pattern in the left panels and thicker fringes in the right panels. Although all graphs have equal values of td, the corresponding chirp-induced time delay τω is different. As the td increases to a long enough duration, the total frequency-dependent time delay will remain positive and not change its sign. In this case, the PMDs display more fringes and do not exhibit any visible splitting structure in the vortex spirals. On the other hand, the larger chirp parameters in the pulses result in more fringes in the PMDs and wider vortex structures for both cases due to their broader spectra.
Figure 8. Photoelectron momentum distributions of hydrogen atoms after photoionization by a pair of counter-rotating circularly polarized attosecond pulses with time delay td = 6 o.c. Other laser parameters are the same as figure 5. |
4. Summary and conclusions
In summary, we have investigated vortex generation in the PMDs of hydrogen atoms after photoionization by a pair of counter-rotating circularly-polarized chirped attosecond pulses by numerically solving the TDSE and employing the analytical perturbative model. We found that using a pair of chirped pulses allows us to observe vortex structures in the PMDs even when the two laser pulses have no time delay but with opposite chirp parameters.
We show that vortices produced by chirped pulses are quite different to those from chirp-free pulses. A frequency-dependent time delay τω is introduced to understand the splitting of vortex spirals. According to our analytical model, we demonstrate that it is triggered by the sign change of the chirp-induced frequency-dependent time delay. To produce the counterpart of the PMD from a pair of counter-rotating circularly polarized chirped pulses, both the chirp parameters and the pulse ordering must be inverted. Our findings in this work provide an additional method for manipulating the electron vortex using attosecond laser pulses. It is also expected that the present approach can be extended to the generation of multiphoton vortices.