Carrier-envelope-phase and helicity control of electron vortices and spirals in photodetachment

M. M. Majczak; F. Cajiao Vélez; J. Z. Kamiński; K. Krajewska; K. Krajewska

doi:10.1364/OE.473929

1. Introduction

During the last decade there has been an increasing interest towards the electron momentum spirals (often referred as ‘vortices’) in laser-induced photoionization [1–14] or photodetachment [15]. They manifest themselves, in the probability distribution of photoelectrons, as zones of large probability which follow concentric Fermat spirals with well-defined number of arms [1]. However, other type of structures, known as electron vortices, can also be found in the momentum distribution of photoelectrons; they appear as continuous lines in the three-dimensional momentum space where the probability amplitude of ionization (or detachment) vanishes, and its phase changes from zero to integer multiples of $2\pi$ around them [15–19]. Those two types of structures are fundamentally different as demonstrated by their physical properties; for instance, electron vortices carry a nonvanishing and quantized orbital-angular momentum (OAM) [20,21], while spirals are characterized by a null OAM.

While electron vortices and spirals are formed due to the same physical phenomena, i.e., by subtile interference effects in the probability amplitude of ionization, whether one or the other are observed depends on the light field configuration and the target atom (or ion) [9,18]. Several theoretical studies have predicted the formation of momentum spirals in photoionization from a variety of atomic and molecular targets [1,3,5,12] and diverse laser field configurations [2,4]. For instance, it has been shown that a sequence of two counterrotating circularly-polarized and ultrashort laser pulses leads to momentum spirals, whereas single pulses or trains of co-rotating pulses lead to the formation of electron vortices [15]. However, the spiral formation has not been theoretically explained. Note that co-rotating trains of bichromatic laser pulses may also lead to the spiral formation [3,5].

According to the analysis presented in Ref. [1], which is performed in the perturbation regime of laser-matter interactions, the energy spectra of photoelectrons obtained in the photoionization of He atoms by two co-rotating laser pulses is circularly-symmetric; i.e., annular zones of zero probability, similar to Newton’s rings, are observed. Moreover, a change of the relative carrier-envelope phase (CEP) between the driving pulses leaves the angular symmetry unchanged. Hence, the effect of the CEP on the electron distribution is, at most, detected as a variation of the rings radial locations or their intensities. Note that in the nonperturbative regime, the probability distribution of photoelectrons stimulated by co-rotating pulses is not circularly symmetric, and vortical structures are observed together with the annular zones of zero probability [15].

In this paper, we further advance the theoretical understanding of vortical and spiral structures in photodetachment driven by circularly-polarized laser pulses. Various pulse configurations are considered, including isolated pulses and pairs of pulses in co-rotating and counter-rotating schemes. The calculations presented here are based on the strong-field approximation (SFA) [22–24] in a nonperturbative regime, which provides a remarkable agreement with ab initio methods of solving numerically the Schrödinger equation [15,17,19]. Our analysis focuses on the CEP effects over the formation of vortex and spiral structures. We demonstrate that the vortical pattern in the probability amplitude of photoelectrons rotates in the polarization plane by the CEP of the driving pulse. The same is observed in the co-rotating configuration of two identical pulses. For the latter, a new type of vortex structures is also discovered. For pulses with opposite helicities, on the other hand, momentum spirals are observed. More specifically, we show that the formation of spirals is closely related to annihilation of vortex-antivortex pairs, which occurs for laser fields with time-reversal symmetry. In that case, the spirals can also be rotated when varying the CEP. Since the manipulation of electron vortical and spiral structures in photodetachment is sensitive to the handedness and the CEP of the laser field, it might provide additional means of field characterization.

We use atomic units (a.u.) along this paper. For our theoretical derivations we set $\hbar =1$ but show the electron charge, $e=-|e|$, and mass, $m_\mathrm {e}$, explicitly. Our numerical illustrations are presented in terms of the atomic units of momentum $p_{\rm at}={\alpha }m_\mathrm {e} c$ and energy $E_{\rm at}=\alpha ^2m_\mathrm {e} c^2$, where $\alpha$ is the fine-structure constant and $c$ is the speed of light. Furthermore, the atomic unit of length corresponds to the Bohr radius, $a_0=\hbar /p_{\rm at}$, whereas the atomic unit of electric field strength equals ${\cal E}_{\rm at}=\alpha ^2m_{\rm e}c^2/(|e|a_0)$.

2. Theoretical formulation

The theoretical derivations presented here are based on the SFA. As it was shown in Refs. [15,17,19], the SFA is an excellent analytical tool for the treatment of photodetachment from negative ions. A direct comparison between the results obtained within this framework and the numerical solution of the time-dependent Schrödinger equation has shown remarkable quantitative similarities. This is actually expected, as the SFA neglects the Coulomb interaction between the freed electron and the residue. In laser-induced photodetachment the residue is a neutral atom, hence the Coulomb interaction during the electron evolution in the continuum is absent. In contrast, in photoionization, the parent ion has a positive charge and the Coulomb potential modifies the electron dynamics, particularly in the low-energy regime. However, the SFA presents several advantages as compared to other more sophisticated ab initio calculations, including a faster numerical computation and simpler analytical expressions, which can be used to understand, in a deeper way, the physical phenomena under consideration.

For the reasons stated above, we shall limit our analysis of photodetachment from negative ions to the framework of the SFA. Even though the probability amplitude of detachment was initially calculated by Gribakin and Kuchiev in Ref. [25], and further explored elsewhere (see, e.g., [18,19]), here we shall present the most important results along its derivation.

2.1 Probability amplitude of photodetachment

It is assumed that, in the remote past, the electron is found in the ground state of the H$^-$ anion ($s$-electron) of energy $E_0$, which we denote as $\Phi _{0}(\boldsymbol {r},t)=\mathrm {e}^{-\mathrm {i} E_0 t}\Phi _{0}(\boldsymbol {r})$. By the action of the laser field, which lasts for a time $T_p$, the electron is promoted to the continuum. The probability amplitude of photodetachment, under the scope of the SFA, is given by [15,17–19,25]

(1)$${\mathcal{A}}(\boldsymbol{p})={-}\mathrm{i} \int_{0}^{T_p} \textrm{d} t\int\textrm{d}^3 r\,\psi^*_{\boldsymbol{p}}(\boldsymbol{r},t)H_{\rm I}(\boldsymbol{r},t)\Phi_0(\boldsymbol{r},t),$$

where $\psi _{\boldsymbol {p}}(\boldsymbol {r},t)$ is the Volkov state of the electron in the laser field [26] with an asymptotic momentum $\boldsymbol {p}$. In the following, we shall consider only the length gauge for our calculations, as it was suggested in Ref. [25]. The Volkov solution, in this gauge, reads

(2)$$\psi_{\boldsymbol{p}}(\boldsymbol{r},t)=\exp\left[\mathrm{i} (\boldsymbol{p}-e\boldsymbol{A}(t))\cdot\boldsymbol{r}-\frac{\mathrm{i}}{2m_\mathrm{e}}\int_0^t\textrm{d} t'(\boldsymbol{p}-e\boldsymbol{A}(t'))^2\right],$$

with $\boldsymbol {A}(t)$ defining the vector potential of the laser field in the dipole approximation. The interaction Hamiltonian, $H_{\rm I}(\boldsymbol {r},t)$, takes the form

(3)$${H}_{\rm I}(\boldsymbol{r},t)={-}e{\boldsymbol{\mathcal{E}}}(t)\cdot\boldsymbol{r},$$

where ${\boldsymbol {\mathcal {E}}}(t)=-\partial _t\boldsymbol {A}(t)$ is the oscillating electric field corresponding to the vector potential $\boldsymbol {A}(t)$.

The unperturbed ground-state wave function of the electron in the H$^-$ anion, $\Phi _0(\boldsymbol {r})$, is determined according to the zero-range potential model [25,27]

(4)$$\Phi_{0}({\boldsymbol{r}})= \frac{\cal N}{\sqrt{4\pi}a_0}\frac{\mathrm{e}^{-\kappa r/a_0}}{r},$$

where ${\cal N}$ is a scaling factor chosen such that the properties of the anion coincide with more advanced ab initio calculations or experimental data [27], and $\kappa$ relates to the ground state energy, $E_{0}=-\alpha ^2m_{e}c^2\kappa ^2/2$. For our numerical illustrations we use the values ${\cal N}=0.75$ and $\kappa =0.2354$ [25], which correspond to an ionization potential for H$^-$ equal to $|E_0|\approx 0.754$ eV.

From Eqs. (1) to (3), and after some algebraic manipulations (for details see, e.g., Refs. [18,19]), the probability amplitude of detachment takes the form

(5)$${\mathcal{A}}(\boldsymbol{p})=\mathrm{i} e\int_0^{T_p}\textrm{d} t\,{\boldsymbol{\mathcal{E}}}(t)\cdot \tilde{\boldsymbol{\Phi}}_0(\boldsymbol{p}-e\boldsymbol{A}(t))\mathrm{e}^{\mathrm{i} G_{\boldsymbol{p}}(t)}.$$

Here, we have introduced the function,

(6)$$\tilde{\boldsymbol{\Phi}}_{0}(\boldsymbol{p})=\mathrm{i}{\boldsymbol\nabla}_{\boldsymbol{p}}\tilde{\Phi}_{0}(\boldsymbol{p})={-}\mathrm{i}\frac{4\sqrt{\pi}\,{\cal N}}{(\kappa^{2}+\boldsymbol{p}^2)^{2}}{\boldsymbol{p}},$$

where $\tilde {\Phi }_{0}(\boldsymbol {p})$ is the Fourier transform of the ground-state wave function of the anion $\Phi _0(\boldsymbol {r})$. Furthermore, in Eq. (5) we have also introduced the function $G_{\boldsymbol {p}}(t)$, which is defined as

(7)$$G_{\boldsymbol{p}}(t)=\frac{1}{2m_\mathrm{e}}\int_0^t\textrm{d} t'\Big[\boldsymbol{p}-e\boldsymbol{A}(t')\Big]^2-E_0t.$$

Hence, by integrating numerically Eq. (5) and taking into account Eqs. (6) and (7), we calculate the probability amplitude of photodetachment. It is worth noting that, while the above formulation concerns the length gauge, the velocity gauge leads to the same numerical results [28]. The driving laser field used in our illustrations is described in Sec. 2.3, after we comment on the properties of ${\mathcal {A}}(\boldsymbol {p})$ in the three-dimensional (3D) momentum space.

2.2 Vortices vs Nodes

The differences between nodal surfaces and vortex lines have been thoroughly analyzed in Ref. [15]. However, for the sake of clarity, we present here their main properties. As presented in Ref. [29], vortex lines appear in the 3D momentum space as closed loops or continuous curves (starting and ending at infinity). While the complex probability amplitude of detachment ${\cal A}(\boldsymbol {p})$ vanishes along those lines, around them the amplitude’s phase changes from $0$ to $2\pi m$, where $m=\pm 1,\pm 2,\ldots$ is an integer number called the topological charge (see, e.g., Refs. [20,21]). In contrast, nodes appear as surfaces of vanishing probability, accross which the phase of ${\cal A}(\boldsymbol {p})$ jumps by $\pm \pi$. Note that in a two-dimensional plane vortex lines are typically visualized as single points, while nodal surfaces appear as curves (either closed or open) of zero probability. In this paper, however, we show some exceptional topological vortex structures as well.

As was elaborated in Ref. [15], the appearance of vortices in photodetachment can be quantified by the quantization condition,

(8)$$m=\frac{1}{2\pi}\oint_{\cal C} \boldsymbol{\pi}(\boldsymbol{p})\cdot \textrm{d}\boldsymbol{p},$$

where $\boldsymbol {\pi }(\boldsymbol {p})$ is related to the probability amplitude of photodetachment ${\mathcal {A}}(\boldsymbol {p})$,

(9)$$\boldsymbol{\pi}(\boldsymbol{p})=\frac{1}{|{\mathcal{A}}(\boldsymbol{p})|^2} {\rm Re}\,[{\mathcal{A}}^*(\boldsymbol{p})(-\mathrm{i}\boldsymbol{\nabla}_{\boldsymbol{p}}){\mathcal{A}}(\boldsymbol{p})].$$

This quantity can be treated as the Berry connection in the parameter space of momenta $\boldsymbol {p}$ [30]. While $\boldsymbol {\pi }(\boldsymbol {p})$ is gauge dependent, the topological charge (8) is not. Also, the vortices and nodal curves are gauge-independent [15]. In the following, we assume for simplicity that the closed path of integration ${\cal C}$ in Eq. (8) lies in the $p_xp_y$-plane (i.e., we set $p_z=0$). More precisely, we choose ${\cal C}$ to be a circular contour of radius $p_r$ centered at momentum $\boldsymbol {p}_0=(p_{0x},p_{0y},0)$ and counterclockwise oriented. Hence, it can be parametrized by an angle $\varphi _{\boldsymbol {p}}$ such that

(10)$$p_x=p_{0x}+p_r\cos{\varphi_{\boldsymbol{p}}},\; p_y=p_{0y}+p_r\sin{\varphi_{\boldsymbol{p}}}, \; p_z=0.$$

It is straighforward to show in this case that the quantization condition (8) becomes

(11)$$m(p_r)=\frac{1}{2\pi}\int_0^{2\pi}\textrm{d}\varphi_{\boldsymbol{p}}\frac{{\rm Im}\big[{\mathcal{A}}^*(\boldsymbol{p})\partial_{\varphi_{\boldsymbol{p}}}{\mathcal{A}}(\boldsymbol{p})\big]}{|{\mathcal{A}}(\boldsymbol{p})|^2},$$

where $m=m(p_r)$. Actually, the contour ${\cal C}$ may contain multiple vortices. In such case, $m(p_r)$ represents a total topological charge enclosed by the contour. If, however, ${\cal C}$ encircles an isolated vortex point, which for instance can be realized if an individual vortex stays enclosed by the contour in the limit $p_r\rightarrow 0$, Eq. (11) represents its individual topological charge. This definition will be used in Sec. 3. to interpret our numerical results.

2.3 Laser field

As we plan to demonstrate in this paper, the properties of the laser field are imprinted onto vortical and spiral structures in photodetachment. For this reason, a careful definition of the driving laser field is important. We define an isolated $N_{\rm osc}$-cycle laser pulse, lasting for time $\tau _p$, by the following time-dependent electric field,

(12)$${\boldsymbol{\mathcal{E}}}_{\chi\sigma}(t)=\begin{cases} {\cal E}_0\sin^2\bigl(\frac{\omega t}{2N_{\rm osc}}\bigr)\boldsymbol{F}(t,\chi,\sigma), & 0\leqslant t\leqslant \tau_p , \cr 0, & \textrm{otherwise}, \end{cases}$$

with the circularly-polarized carrier wave,

(13)$$\boldsymbol{F}(t,\chi,\sigma)=\sin(\omega t+\chi)\boldsymbol{e}_x-\sigma\cos(\omega t+\chi)\boldsymbol{e}_y,$$

and ${\cal E}_0={\cal E}_{\rm at}\sqrt {\frac {I}{2I_{\rm at}}}$. Here, $I$ denotes the peak intensity of the pulse expressed in units of $I_{\rm at}=3.51\times 10^{16}$ W/cm$^2$, $\omega =2\pi N_{\rm osc}/\tau _p$ is the carrier frequency of the field, whereas $\chi$ is the carrier envelope phase of the pulse. The polarization properties of the field are controlled by the helicity, $\sigma$, and in our further analysis we shall choose circularly-polarized pulses with either counterclockwise ($\sigma =+1$) or clockwise ($\sigma =-1$) orientations. We also define a train of $N_{\rm rep}$ such pulses,

(14)$${\boldsymbol{\mathcal{E}}}_{\chi_1\sigma_1,\chi_2\sigma_2,\ldots}(t)=\sum_{\ell=1}^{N_{\rm rep}} {\boldsymbol{\mathcal{E}}}_{\chi_\ell\sigma_\ell}\Bigl(t-(\ell-1)\tau_p\Bigr),$$

with, in principle, different $\chi _\ell$ and $\sigma _\ell$. (Note that, under such conditions, the total duration of the train of pulses is $T_p=N_{\rm rep}\tau _p$.) Hence, the electromagnetic vector potential describing the laser field can be defined,

(15)$$\boldsymbol{A}_{\chi_1\sigma_1,\chi_2\sigma_2,\ldots}(t)={-}\int_{-\infty}^t{\boldsymbol{\mathcal{E}}}_{\chi_1\sigma_1,\chi_2\sigma_2,\ldots}(t')\textrm{d} t',$$

where it holds that

(16)$$\lim_{t\rightarrow \pm\infty} \boldsymbol{A}_{\chi_1\sigma_1,\chi_2\sigma_2,\ldots}(t) =\boldsymbol{0}.$$

Hence, we use the notation $(\chi _1,\sigma _1;\cdots ;\chi _\ell,\sigma _\ell ;\cdots ;\chi _{N_{\rm rep}},\sigma _{N_{\rm rep}})$ to identify each field configuration. From $\sigma _\ell$ we only retain its sign, i.e., we write $\sigma _\ell =\pm$.

While the above definitions are very general, in the following we shall consider either an isolated pulse ($N_{\rm rep}=1$) or a sequence of two pulses ($N_{\rm rep}=2$) with co-rotating or counter-rotating circular polarizations. For our numerical illustrations we shall keep the peak intensity, wavelength, and number of field oscillations within the $\sin ^2$ envelope fixed ($I=2.5\times 10^{11}$W/cm$^2$, $\lambda =4000$ nm, and $N_{\rm osc}=3$, respectively), while allowing the CEP and helicity to vary independently for each pulse in the train.

3. Laser-field control of vortical and spiral structures

We start by analyzing the creation of electron vortices in photodetachment driven by a single laser pulse [$N_{\rm rep}=1$ in Eq. (14)]. In Fig. 1, we present the color mappings of detachment probability amplitude ${\mathcal {A}}(\boldsymbol {p})$ [Eqs. (5) to (7)] calculated in the $p_xp_y$-plane ($p_z=0)$. While in the upper panels we show the amplitude’s modulus, i.e., $|{\mathcal {A}}(\boldsymbol {p})|^\nu$, where $\nu =0.5$ has been chosen for visual purposes, in the lower panels we present its phase, ${\rm arg}[{\mathcal {A}}(\boldsymbol {p})]/\pi$. Each column corresponds to a different field configuration. In Fig. 1(a), which is for the configuration $(0,+)$, we observe the appearance of four vortices; they manifest themselves as points of vanishing probability for which the amplitude’s phase changes continuously from $-\pi$ to $\pi$ around them. Hence, we assign to each of them the topological charge $m=1$. Two of such vortices are located along the line $p_y=0$ while the other two appear at $p_x=0.025p_{\rm at}$ [31]. In Fig. 1(b), which corresponds to a driving field of the type $(\pi /3,+)$, we observe the exact same configuration of vortices, but rotated by an angle $\theta =\pi /3$ with respect to the center of coordinates. Finally, in Fig. 1(c) we present the results corresponding to the driving pulse configuration $(2\pi /3,-)$, such that the CEP takes now the value $2\pi /3$ and the helicity is inverted, $\sigma =-1$ (clockwise field evolution). This time, we deal with antivortices ($m=-1$) and the rotation angle $\theta =4\pi /3=-2\pi /3$ mod $2\pi$. Hence, we conclude that the effect of the CEP and helicity of the driving pulse ($N_{\rm rep}=1$) over the vortex pattern (or, in general, over the probability distribution of photoelectrons in momentum space) is an overall rotation by an angle $\theta =\chi \sigma$ with respect to the origin of coordinates.

Fig. 1. Modulus (upper panels) and phase (lower panels) of the probability amplitude of photodetachment ${\mathcal {A}}(\boldsymbol {p})$ [Eqs. (5) to (7)] in the $p_xp_y$-plane ($p_z=0$). The modulus has been raised to the power $\nu =0.5$ and the phase is presented in units of $\pi$, which is done only for visual purposes. A circularly-polarized driving pulse ($N_{\rm rep}=1$) comprising $N_{\rm osc}=3$ oscillations within a $\sin ^2$ envelope [see, Eqs. (12) to (14)] has been used. While we keep its peak intensity ($I=2.5\times 10^{11}$W/cm$^2$) and wavelength ($\lambda =4000$ nm) fixed, each column corresponds to a different CEP and helicity of the pulse. Specifically, the column (a) is for the laser field configuration $(\chi,\sigma )=(0,+)$, meaning that the CEP is zero and the laser pulse rotates counterclockwise. The column (b) corresponds to the configuration $(\pi /3,+)$, whereas the column (c) is for $(2\pi /3,-)$.

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Next, we analyze the case when a pair of identical pulses [$N_{\rm rep}=2$ in Eq. (14)] interacts with the H$^-$ ion. In Fig. 2 we show the same as in Fig. 1 but this time we vary the CEP, which is common for both co-rotating pulses. According to our convention, the driving fields considered here are labelled as $(\chi,\sigma ;\chi,\sigma )$. In Fig. 2(a), which corresponds to a configuration $(0,+;0,+)$, we observe the formation of the same four vortices as in Fig. 1(a) with additional nodal surfaces. The latter appear as annular zones of zero probability located at different radii (see, Ref. [15]). Note that the regions of high probability resemble the Newton’s rings reported in the perturbative photoionization of He atoms [1]. In Fig. 2(b), which is for the configuration $(\pi /3,+;\pi /3,+)$ we see that the very same system of vortices and nodal surfaces is formed, but it is rotated by an angle $\theta =\pi /3$ with respect to the origin of coordinates. For the train configuration $(2\pi /3,-;2\pi /3,-)$, in Fig. 2(c), the rotation is by an angle $\theta =4\pi /3=-2\pi /3$ mod $2\pi$, while vortices are transformed into antivortices. The same has been seen in Fig. 1 for a single pulse. (We would like to mention that similar rotations of the distribution of photoelectrons with the CEP has been observed in photoionization of H$_2^+$ by trains of bichromatic laser pulses [3].) Hence, the only difference imposed by the second identical pulse on ${\cal A}(\boldsymbol {p})$ is the appearance of annular nodal surfaces. This is confirmed by theoretical analysis. Namely, it follows from Eq. (46) in Ref. [19] that the probability amplitude of detachment driven by a sequence of two identical pulses can be expressed in terms of the probability amplitude of detachment by a single pulse ${\cal A}(\boldsymbol {p};\chi,\sigma )$ such that

(17)$${\cal A}(\boldsymbol{p})=2\mathrm{e}^{\mathrm{i} \langle G_{\boldsymbol{p}}\rangle \tau_p/2}{\cal A}(\boldsymbol{p};\chi,\sigma)\cos\left(\langle G_{\boldsymbol{p}}\rangle \tau_p/2\right).$$

Here,

(18)$$\langle G_{\boldsymbol{p}}\rangle=\frac{(\boldsymbol{p}-e\langle\boldsymbol{A_{\chi\sigma}}\rangle)^2}{2m_{\rm e}}+\frac{e^2(\langle \boldsymbol{A}_{\chi\sigma}^2\rangle-\langle \boldsymbol{A}_{\chi\sigma}\rangle^2)}{2m_{\rm e}}-E_0,$$

where $\langle \ldots \rangle$ denotes the time-averaged value over a single $(\chi,\sigma )$-pulse duration from the train,

(19)$$\langle f\rangle=\frac{1}{\tau_p}\int_0^{\tau_p}\textrm{d} t\,f(t).$$

This means that, in addition to the vortex lines that originate from ${\cal A}(\boldsymbol {p};\chi,\sigma )$, additional zeroes of ${\cal A}(\boldsymbol {p})$ will be manifested. These are the zeroes of the cosine function in Eq. (17), $\boldsymbol {p}_\ell$, satisfying the following equation $(\boldsymbol {p}_\ell -e\langle \boldsymbol {A}_{\chi \sigma }\rangle )^2=P_\ell ^2$, where $P_\ell ^2=2m_{\rm e}\pi (2\ell +1)/\tau _p-e^2(\langle \boldsymbol {A}_{\chi \sigma }^2\rangle -\langle \boldsymbol {A}_{\chi \sigma }\rangle ^2)+2m_{\rm e}E_0$. While in principle $\ell \in \mathbb {Z}$, we have $P_\ell ^2>0$ which imposes the lower limit on $\ell$. It is clear that the additional nodes of the probability amplitude (17) form circles centered at $e\langle \boldsymbol {A}_{\chi \sigma }\rangle$ and having radius $P_\ell$. While we pass across those zeroes, the cosine function changes its sign, contributing to the phase of ${\cal A}(\boldsymbol {p})$ which jumps by $\pm \pi$. This agrees with our numerical results, showing the appearance of additional nodal rings in Fig. 2.

Fig. 2. The same as in Fig. 1 but for a pair of two identical laser pulses ($N_{\rm rep}=2$) [configuration $(\chi,\sigma ;\chi,\sigma )$]. The column (a) is for the configuration $(0,+;0,+)$ (see, Ref. [15]), (b) is for $(\pi /3,+;\pi /3,+)$, whereas (c) is for $(2\pi /3,-;2\pi /3,-)$.

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A more general situation is met when one of the CEPs of co-rotating pulses is kept constant (say $\chi _1$) and we vary the other one ($\chi$). In Fig. 3 we present the probability amplitude of photodetachment driven by a pair of pulses in the configurations $(\pi /3,+;\chi,+)$. We set for column (a) $\chi =0$, (b) $\chi =8\pi /50$, and (c) $\chi =14\pi /50$. This time, four vortices with the same topological charge ($m=1$) and one antivortex ($m=-1$) are present at $\chi =0$. Also, an open ring of seemingly low probability is visible there at larger photoelectron momenta. By calculating the total topological charge of vortices encicled by contours $C_1$ and $C_2$ in panel (a), we obtain $m=3$ and 4, respectively. This means that the open ring is in fact a vortex carrying the topological charge $m=1$. Thus, we deal with a degenerate case when the cut of the vortex line through the plane $p_z=0$ has topological structure. To the best of our knowledge, this is the first time that such a degenerate structure, which carries nonvanishing OAM and extends along a well-defined region in a two-dimensional momentum space, is reported in the literature. By changing the CEP there is a continuous expansion of the structured vortex, together with antivortex migration towards larger $|\boldsymbol {p}|$ [column (b)]. When both antivortex and degenerated vortex merge, a full nodal surface (annular region of zero probability) is formed [column (c)]. Figure 3(b) also captures a new structred vortex at smaller momenta, which expands with increasing $\chi$ [Fig. 3(c)]. In addition, in Fig. 3(c) we observe a new vortex-antivortex pair, which has emerged out of a simple node at smaller $\chi$ (not shown in the figure). Hence, vortex-antivortex pair creation, their migration, and further recombination lead to a cyclic formation of nodal lines propagating outwards in the $p_xp_y$-plane while changing $\chi$. As expected, when $\chi =2\pi$, we end up with the same momentum pattern as for $\chi =0$. Importantly, only when $\chi _1=\chi$ the momentum distributions exhibit exclusively nodal lines with annular shape, as presented in Fig. 2. Note that, in contrast to the results presented in Ref. [1] (perturbative regime), the probability amplitude of photodetachment in Fig. 3 is not circularly symmetric. Hence, the formation of vortex patterns and their migration can be observed with increasing the relative CEP.

Fig. 3. The same as in Fig. 2 but for a driving pair of pulses in the configuration $(\chi _1,\sigma ;\chi,\sigma )$, with $\chi _1=\pi /3$ and $\sigma =+1$. Here, the column (a) is for $\chi =0$, (b) for $\chi =8\pi /50$, and (c) for $\chi =14\pi /50$. Except of nodal surfaces and typical vortex points, we observe structured vortices in a shape of open rings. It follows from the total topological charge balance in relation to contours $C_1$ and $C_2$ that the open ring presented in column (a) is a vortex of the topological charge $m=1$. Similar calculation can be performed for other structured vortices, like a new open ring visible in column (b).

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Next, we analyze the electron photodetachment driven by a pair of counterrotating pulses ($\sigma _1=1$ and $\sigma _2=-1$). Here, the most interesting case relates to a pair of pulses in configuration $(\chi,+;-\chi,-)$ [or $(-\chi,-;\chi,+)$]. In Fig. 4 we show the respective modulus of the probability amplitude of photodetachment (upper panels) and its phase (lower panels) for (a) $\chi =0$, (b) $\chi =\pi /3$, and (c) $\chi =2\pi /3$. We observe momentum spirals with no vortices (see, also Ref. [15]). Interestingly, in columns (b) and (c) the original momentum spirals presented in column (a) are recovered, but they are rotated by an angle $\theta =\chi$ in the polarization plane. As we have checked for other pulse configurations, $(\chi _1,+;\chi,-)$ with $\chi _1\neq -\chi$, electron spirals are not formed. To illustrate that, we present in Fig. 5 a typical momentum distributon for a counterrotating pulse configuration, where this time we consider $(\pi /3,+;\pi /3,-)$. Instead of spirals like the ones shown in Fig. 4, we observe spiral branches that do not meet, as well as vortex and antivortex points. If we change now the CEP of the second pulse, the vortex-antivortex pairs will be created and annihilated for as long as $\chi _1\neq -\chi$. However, when $\chi _1=-\chi$ the complete dissapearance of vortex-antivortex pairs is observed, as illustrated in Fig. 4. As we argue next, this is directly related to the time-reversal property of the laser field.

Fig. 4. The same as in Fig. 2 but for a driving pair of pulses in the counterrotating configuration. Specifically, the column (a) corresponds to $(0,+;0,-)$ (see, Ref. [15]), (b) is obtained for $(\pi /3,+;-\pi /3,-)$, whereas (c) is for $(2\pi /3,+;-2\pi /3,-)$. The angle $\theta$ indicates that the entire distribution rotates while changing the CEP and it occurs that $\theta =\chi$.

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Fig. 5. The same as in Fig. 2 but for a driving pair of pulses in the counterrotating configuration $(\pi /3,+;\pi /3,-)$. One can see the spiral branches and vortex-antivortex pairs.

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Since now on, we focus on the case when detachment is driven by a pair of counterrotating circularly-polarized laser pulses with opposite CEPs [i.e., the laser fields are in the configurations $(\chi,\sigma )$ and $(-\chi,-\sigma )$ for single pulses or $(\chi,\sigma ;-\chi,-\sigma )$ for trains of two pulses]. As an example, in Fig. 6 we plot the color mappings of probability amplitude of detachment ${\cal A}(\boldsymbol {p})$ for: (a) an isolated pulse with $\sigma =1$ and $\chi =\pi /3$, (b) the same with $\sigma =-1$ and $\chi =-\pi /3$, and (c) the sequence of those pulses $(\pi /3,+;-\pi /3,-)$. As before, we keep $p_z=0$, and we present separately the modulus and the phase of ${\cal A}(\boldsymbol {p})$ in each case. When comparing the results for individual pulses [Figs. 6(a) and 6(b)], the absolute values of probability amplitudes turn out to be identical. Specifically, the zeroes of ${\cal A}(\boldsymbol {p})$ in both cases are located at the exactly same momenta. As it follows from the mappings of the phase of probability amplitudes, those points have a vortex character. More specifically, in both cases we observe two pairs of vortex points, with $m=1$ in Fig. 6(a) and $m=-1$ in Fig. 6(b). Those properties can be explained if one realizes that the respective probability amplitudes are related by the time-reversal symmetry. Namely, it follows from Eq. (12) that ${\boldsymbol {\mathcal {E}}}_{\chi \sigma }(t)=-{\boldsymbol {\mathcal {E}}}_{-\chi -\sigma }(2\tau _p-t)$ and, hence, $\boldsymbol {A}_{\chi \sigma }(t)=\boldsymbol {A}_{-\chi -\sigma }(2\tau _p-t)$ for $0\leqslant t\leqslant \tau _p$. Taking this into account, one concludes from Eqs. (5) to (7) that the probability amplitudes of photodetachment driven by pulses $(\chi,\sigma )$ and $(-\chi,-\sigma )$ from the train are related such that

(20)$${\cal A}(\boldsymbol{p};-\chi,-\sigma)={-}[{\cal A}(\boldsymbol{p};\chi,\sigma)]^*\,\mathrm{e}^{2\mathrm{i}\langle G_{\boldsymbol{p}}\rangle\tau_p},$$

with $\langle G_{\boldsymbol {p}}\rangle$ given by Eq. (18). Here, ${\cal A}(\boldsymbol {p};\chi,\sigma )$ is calculated for $0\leqslant t\leqslant \tau _p$ whereas ${\cal A}(\boldsymbol {p};-\chi,-\sigma )$ for $\tau _p\leqslant t\leqslant 2\tau _p$. As we have checked, the additional phase factor in Eq. (20) does not contribute to the contour integral in Eq. (11). Therefore, the vortex-like points for pulses $(\chi,\sigma )$ and $(-\chi,-\sigma )$, while located at the same momenta, will carry opposite topological charges. A very distinct spiral pattern is observed when detachment occurs in a sequence of both pulses [Fig. 6(c)]. In this case, there is no vortices. Instead, we observe the nodal lines of the probability amplitude, with its phase jumping by $\pm \pi$ accross those lines. These features can be explained by the formula,

(21)$$\begin{aligned} {\cal A}(\boldsymbol{p}) &= {\cal A}(\boldsymbol{p};\chi,\sigma)+{\cal A}(\boldsymbol{p};-\chi,-\sigma) \\ &= 2\mathrm{i}|{\cal A}(\boldsymbol{p};\chi,\sigma)|\mathrm{e}^{\mathrm{i}\langle G_{\boldsymbol{p}}\rangle\tau_p}\sin\Bigl[{\rm arg}[{\cal A}(\boldsymbol{p};\chi,\sigma)]-\langle G_{\boldsymbol{p}}\rangle\tau_p\Bigr], \end{aligned}$$

where we have used Eq. (20). Note that the probability amplitude of detachment by a sequence of time-reversed pulses $(\chi,\sigma ;-\chi,-\sigma )$ [Eq. (21)] is entirely defined by the properties of the first pulse from the train, lasting from 0 to $\tau _p$. Moreover, Eq. (21) shows that the vortices and antivortices observed in Figs. 6(a) and 6(b), respectively, annihilate each other into simple nodes. At the same time, new nodes appear as indicated by the sine function in Eq. (21). More specifically, they occur at momenta such that ${\rm arg}[{\cal A}(\boldsymbol {p};\chi,\sigma )]-\langle G_{\boldsymbol {p}}\rangle \tau _p=k\pi$, where $k\in \mathbb {Z}$. Let us assume now that ${\cal A}(\boldsymbol {p};\chi,\sigma )$ has a vortex (antivortex) with topological charge $m=1$ ($m=-1$) at the momentum $\boldsymbol {p}=\boldsymbol {p}_0$. This means that around that point ${\rm arg}[{\cal A}(\boldsymbol {p};\chi,\sigma )]\approx \varphi _{\boldsymbol {p}}$, where $\varphi _{\boldsymbol {p}}$ changes by $2\pi$ ($-2\pi$). It follows from Eq. (21) that in the close vicinity of $\boldsymbol {p}_0$ one observes additional two zeros at $\varphi _{\boldsymbol {p}}^+\approx \langle G_{\boldsymbol {p}_0}\rangle \tau _p$ (mod $2\pi$) and $\varphi _{\boldsymbol {p}}^-\approx \pi +\langle G_{\boldsymbol {p}_0}\rangle \tau _p$ (mod $2\pi$). Thus, the nodal curve is formed. Note that the sine function also contributes to the phase of the total probability amplitude in Eq. (21). Namely, it changes the sign while passing across the node, which results in a disconuity of ${\rm arg}[{\cal A}(\boldsymbol {p})]$ by $\pm \pi$ across the nodal line.

Fig. 6. The same as in Figs. 1 and 2 but for the following laser field configurations: (a) is for $(\pi /3,+)$, (b) for $(-\pi /3,-)$, and (c) for $(\pi /3,+;-\pi /3,-)$. While vortices [column (a)] and antivortices [column (b)] that appear at the same momenta and carry the opposite topological charges annihilate each other, the spiral pattern is formed [column (c)].

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4. Conclusions

We have presented an in-depth analysis of the formation of electron vortices and spirals in photodetachment driven by either an isolated circularly polarized laser pulse or a pair of such pulses. We have seen that, by changing the CEP and helicity of individual pulses, it is possible to control the vortical structures and the appearance of spirals in momentum distributions of photoelectrons. Thus, generalizing our previous results presented in Refs. [15,19].

Specifically, in the case of photodetachment driven by a single pulse, we have seen that the respective momentum distributions of photoelectrons (and so, the vortical patterns as well) are rotated by the pulse CEP. The rotation direction, on the other hand, depends on the pulse helicity. The same features have been also observed in photodetachment driven by two identical pulses where, in addition to vortex structures, circular nodal lines appear.

Here, a novel type of vortex structure has been identified. Typically, a vortex and an antivortex in a two-dimensional momentum plane are represented as isolated points. Such vortices and antivortices have been observed in our previous papers. This time, however, for two co-rotating pulses of different CEPs, we have discovered vortices (antivortices) with nontrivial topological properties. Those structured vortices are represented in Fig. 3, forming open rings of zero probability.

Clear understanding of a mechanism behind the spiral formation has been provided in this paper. So far, it has been shown that spiral structures differ from vortices, as they do not carry a topological charge [15]. Also in Ref. [15], it has been realized that spiral structures are formed when the sequence of two counterrotating pulses with zero CEPs drive photodetachment. Our current analysis goes far beyond that. First of all, we have identified the conditions imposed on the laser field such that the momentum spirals are observed. We have demonstrated, both analytically and numerically, that this happens when detachment is induced by two pulses of time-reversal symmetry. This largely generalizes our previous results. Second of all, we have shown that the spiral formation is accompanied by the absolute disappearance of vortices and antivortices. Namely, we have attributed the spiral formation to annihilation of vortices with antivortices which are generated by time-reversed pulses comprising the train. Finally, we have observed spiral rotation which is controlled by the CEP and helicity of time-reversed pulses.

Note that our findings can, in principle, have applications in ultrashort laser pulse diagnostics, as the CEP can be directly inferred from the vortex pattern or spiral orientation in the two-dimensional momentum distribution of photoelectrons. Hence, it is expected that the CEP can be experimentally determined by a careful measurement of the rotation angle of those structures.

Funding

Narodowe Centrum Nauki (2018/30/Q/ST2/00236, 2018/31/B/ST2/01251).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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31. The results shown in Fig. 1(a) were already presented in Fig. 4 of Ref. [15]. However, we show them here for two main reasons: firstly, by comparing our results with those presented in Ref. [15], we corroborate that the computational code used in our calculations leads to accurate results. Secondly, the case (0, + ) is a benchmark to determine the influence of both CEP and helicity over vortex formation.

Carrier-envelope-phase and helicity control of electron vortices and spirals in photodetachment

Abstract

1. Introduction

2. Theoretical formulation

2.1 Probability amplitude of photodetachment

2.2 Vortices vs Nodes

2.3 Laser field

3. Laser-field control of vortical and spiral structures

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (21)

Optics Express