Enhancing circularly polarized XUV vortices from bicircular Laguerre-Gaussian fields

Zhiyang Lin; Zhiyang Lin; You Chen; Jixiong Pu; Xuanyang Lai; Xuanyang Lai

doi:10.1364/OE.450049

1. Introduction

Laguerre-Gaussian (LG) fields, which have spiral phase fronts, can carry orbital angular momentum (OAM) of topological charge. These fields are called the optical vortices [1] and have witnessed an extraordinary progress in multidisciplinary applications [2], such as super-resolution imaging [3], optical communication [4], kinematic micromanipulation [5] and strong-field physics [6]. Later on, due to the great progress of the strong-field physics, the optical vortices have been extended from the near-infrared region to the extreme ultraviolet (XUV) region, by using the high-order harmonic generation (HHG) [7]. The HHG can be well understood in terms of a simple semiclassical three-step model [8,9]: When an atom interacts with strong laser field, the outmost electron, liberated and subsequently driven by the laser field, may revisit and then recombine with its parent ion, releasing an XUV photon. Through HHG driven by the infrared vortices, the XUV vortices with OAM equal to the harmonic order have been generated [10–13]. Furthermore, in order to reduce the OAM of the generated XUV vortices, a non-collinear scheme has been employed for the harmonic generation [11,14,15].

Recently, vortices with spin angular momentum (SAM) have received great attention. For light, the SAM is associated with their polarization. The circularly polarized vortices with the SAM of ${\pm} 1$ have opened new prospects for controlling, e.g., chiral structures [16,17]. Similarly, through HHG, it is found that the circularly polarized XUV vortices can be also generated with two coplanar counter-rotating circularly polarized LG fields (i.e., $r\omega {\bf \circlearrowright } \oplus s\omega {\bf \circlearrowleft}$ with frequency $\omega $ and integers r and s) [18–22]. The OAM and SAM of the XUV vortices satisfy the following the selection rules [18]

(1)$$LG_{{l_r},0}^{r\omega ,{\bf \circlearrowright }} \oplus LG_{{l_s},0}^{s\omega ,{\bf \circlearrowleft}} \to \left\{ \begin{array}{l} {\omega_q} = q\omega = mr\omega + ns\omega \\ m - n ={\pm} 1\\ {l_{Hq}} = m{l_r} + n{l_s} \end{array} \right.$$

where ${l_r}$(${l_s}$) and ${l_{Hq}}$ denote the OAM of the driving LG fields with frequency $r\omega$($s\omega$) and the ${q^{th}}$ harmonic, respectively, ${\omega _q}$ is the photon energy of the ${q^{th}}$ harmonic, $m$ and $n$ are the number of absorbed $r\omega$ and $s\omega$ photons, respectively, and $m - n$ is the corresponding SAM. These selection rules can be also understood by using the conservation law of the linear combination of the OAM and SAM (coined as torus-knot angular momentum, TKAM) [21,22]. According to these selection rules, the SAM and OAM of the XUV vortices can be flexibly tuned by tailoring the component-OAM of the bicircular LG fields. For example, the circularly polarized XUV vortices with a low OAM of 1 or −1, which equals to the component-OAM of the driving fields $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{ - 1,0}^{2\omega ,{\bf \circlearrowleft}},$ have been experimentally obtained [19].

Currently, great progress has been achieved on the generation of the circularly polarized XUV vortices with different OAM, by using the HHG driven by the bicircular LG fields. On the other hand, for the practical application, brilliant circularly polarized XUV vortex beam is critically required [23–25], especially for the study of the nonlinear processes. Therefore, when we control the OAM of the circularly polarized XUV vortices with the different bicircular LG fields, its influence on the amplitude of the generated vortices should be also considered accordingly. However, how its amplitude is affected by the bicircular LG fields has not yet been well studied.

In this work, we theoretically study the generation of the circularly polarized XUV vortices from HHG driven by bicircular LG fields with a frequency ratio of, e.g., 1:2 ($r = 1$, $s = 2$) and 1:3 ($r = 1$, $s = 3$), respectively. Our simulation with strong-field approximation (SFA) theory shows that the amplitude of the circularly polarized XUV vortices from the ω-3ω bicircular LG field is about one order of magnitude larger than that from the ω-2ω field. Moreover, we find that such significant enhancement can be also observed for the ω-3ω bicircular LG fields with different component OAMs. Our analysis shows that the amplitude enhancement for the ω-3ω field origins from the single-atom response of the HHG. Furthermore, in terms of quantum-orbit theory within SFA, the enhancement is ascribed to that, compared with the ω-2ω field, the ω-3ω field has more segments within one cycle and the corresponding electron orbit has the shorter travel time and the higher tunneling ionization rate.

This paper is organized as follows. In Sec. 2, we briefly introduce the SFA theory for the high-harmonic vortex and the quantum-orbit theory within SFA. In Sec. 3, we present the SFA simulations of the circularly polarized XUV vortices from HHG driven by the different bicircular LG fields. Subsequently, we employ the quantum-orbit theory to understand our simulation results. Finally, in Sec. 4 our conclusions are given. Atomic units (a.u.) are used unless stated otherwise.

2. Theoretical methods

2.1 SFA theory for the high-harmonic vortex

In this work, the XUV vortices from HHG driven by a bicircular LG field are simulated with SFA theory [26,27]. Specifically, we firstly calculate the HHG spectra of a single atom. In the SFA theory, the time-dependent electric dipole moment of a single atom is given by [28] :

(2)$$\begin{aligned} \textbf{d}(t) =&{-} i{(\frac{{2\pi }}{i})^{3/2}}\mathop \int \nolimits_0^\infty \frac{{\textrm{d}\tau }}{{{\tau ^{3/2}}}}\left\langle {{\psi_{il^{\prime}m^{\prime}}}} \right|\textbf{r}|{{\textbf{k}_{st}} + \textbf{A}(t)} \rangle \\ &\times \left\langle {{\textbf{k}_{st}} + \textbf{A}(t - \tau )} \right|\textbf{r} \cdot \textbf{E}(t - \tau )|{{\psi_{il^{\prime}m^{\prime}}}} \rangle \exp ( - i{S_{st}}) \end{aligned}$$

where $\textbf{E}(t)$ is the electric field of the driving laser pulse, $t$ is the recombination time of the electron, $\tau = t - {t_0}$ is the travel time with the ionization time ${t_0}$, $\textbf{A}(t) ={-} \int_{}^t {\textbf{E}(t^{\prime})dt^{\prime} }$ is its associated vector potential, $\textbf{K}_{st} = -\int_{t_0}^{t} dt^{\prime}{\textbf{A}}(t^{\prime})/\tau$ is the stationary momentum and ${S_{st}} = \int_{{t_0}}^t {{{dt^{\prime}{{[{\textbf{k}_{\textrm{st}}} + \textbf{A}(t^{\prime})]}^2}} / 2} + {I_p}\tau }$ denotes the quasiclassical action with the ionization potential ${I_p}$. The ground state wave function ${\psi _{il^{\prime}m^{\prime}}}$ is a hydrogen-like wave function with $m^{\prime}$ he magnetic quantum number and $l^{\prime}$ the orbital quantum number. After obtaining the electric dipole moment, the single-atom harmonic spectrum is obtained by performing the Fourier transform of the dipole moment, $\textbf{A}_q^{(near)} = \int {\exp (i\omega t)\textbf{d}(t)dt}$. Note that the SFA theory requires that the electron during its excursion between the ionization and the recombination experiences a spatially uniform field, unlike the LG field. However, the maximum excursion distance of the ionized electron is usually much smaller than the spatial size of the LG field [11,14] and thus, it is still feasible to use the SFA theory in our simulations.

Secondly, we calculate the harmonic spectra of atoms at the different spatial positions in the gas jet. In this work, the spatial positions of the gas jet are assumed in a two-dimensional (2D) plane placed at 2 mm after the focus position [10–12], which is chosen for compensating the intrinsic phase of the harmonics and the Gouy phase to optimize longitudinal phase-matching conditions. By using the same procedure as the single-atom response, we obtain the complex amplitude of the ${q^{th}}$ harmonic in the near-field, i.e., $\textbf{A}_q^{\textrm{(near)}}(\rho ^{\prime},\varphi ^{\prime})$ with the radius $\rho ^{\prime}$ and the polar angle $\varphi ^{\prime}$ of the near-field plane.

Thirdly, according to the Fraunhofer diffraction [29], we express the far-field complex amplitude of the ${q^{th}}$ harmonic as

(3)$$\begin{aligned} A_q^{\textrm{(far)}}(\beta ,\varphi ) = &\mathop \int \nolimits_0^\infty \mathop \int \nolimits_0^{2\pi } \rho ^{\prime}d\rho ^{\prime}d\varphi ^{\prime}A_q^{\textrm{(near)}}(\rho ^{\prime},\varphi ^{\prime})\\ &\times exp( - i\frac{{2\pi }}{{{\lambda _q}}}\rho ^{\prime}\textrm{tan(}\beta \textrm{)cos(}\varphi - \varphi ^{\prime}\textrm{)}) \end{aligned}$$

where ${\lambda _q}$ is the wavelength of the ${q^{th}}$ harmonic, $\beta$ is the angle of divergence and $\varphi$ is the polar angle in the far field. In practice, it has been shown that the far-field Fraunhofer diffraction image can be equally obtained by performing a 2D spatial Fourier transformation of the complex amplitude of the ${q^{th}}$ harmonic in the near field [11,14].

In this work, based on the SFA theory, we simulate HHG from argon gas jet in the bicircular driving field. The electric-field vector of this field is $\textbf{E}(t) = {E_x}(t){\hat{\textbf{e}}_x} + {E_y}(t){\hat{\textbf{e}}_y}$, with

(4)$$\begin{aligned} &{E_x}(t) = {{[{E_r}\sin (r\omega t) + {E_s}\sin (s\omega t + {\varphi _0})]} / {\sqrt 2 }}\\ &{E_y}(t) = {{[ - {E_r}\cos (r\omega t) + {E_s}\cos (s\omega t + {\varphi _0})]} / {\sqrt 2 }} \end{aligned}$$

where the component-frequencies are $r\omega$ and $s\omega$ and ${\varphi _0}$ is the relative phase. Here, the fundamental frequency $\omega$ is 0.0365 a.u. (i.e., the wavelength ${\lambda _0} = 1250\; nm$), which is the reference for defining the high harmonic orders, and $T = 2\pi /\omega$ is the laser electric field period. The amplitudes of the component-fields ${E_r}$ and ${E_s}$ have the same peak intensity of ${I_j} = 4.0 \times {10^{14}} W/c{m^2}$ (${I_j} = E_j^2$, $j = r$ or $s$).

In order to consider the OAM of the driving field, the spatial structure of the vortex beam is represented as a LG beam propagating in the z-direction with wavelength ${\lambda _0}$, topological charge $l$ and radial nodes index $p$ (written in cylindrical coordinates)

(5)$$\begin{aligned} L{G_{l,p}}(\rho ,\varphi ,z) = &{E_j}\frac{{{W _0}}}{{W (z)}}{(\frac{{\sqrt 2 \rho }}{{W (z)}})^{|l |}}L_p^{|l |}[\frac{{2{\rho ^2}}}{{{W ^2}(z)}}]\\ &\times \exp ( - \frac{{{\rho ^2}}}{{{W ^2}(z)}})\exp (ik\frac{{{\rho ^2}}}{{2R (z)}} + i{\Phi _G}(z) + i\ell \varphi ). \end{aligned}$$

Here, ${W _0}$ is the beam waist, $W (z) = {W _0}\sqrt {1 + {{{z^2}} / {z_r^2}}}$ is the beam width, ${z_r} = {{{k_0}W _0^2} / 2}$ is the Rayleigh range with the wave number ${k_0} = {{2\pi } / {{\lambda _0}}}$, $R(z) = z(1 + {{{z^2}} / {z_r^2}})$ is the wave front radius, ${\Phi _G}(z) ={-} (|l |+ 2p + 1)\arctan ({z / {{z_r}}})$ is the Gouy phase, $L_p^{|l |}[x]$ are the associated Laguerre polynomials and the index $l$ denotes the OAM of the beam.

2.2 Quantum-orbit theory within SFA

In order to understand the SFA simulation results, quantum-orbit theory within SFA is employed. The corresponding transition amplitude for the ${q^{th}}$ order harmonic is given by [26,30]:

(6)$${\boldsymbol{T}_{\boldsymbol{q}}} \propto \mathop \sum \limits_s \frac{1}{[i({t_s} - {t_{0s}})]}^{3/2}\frac{\left\langle {{\psi_0}} \right|\boldsymbol{r}|{\boldsymbol{k} + \boldsymbol{A}({t_s})} \rangle \left\langle {\boldsymbol{k} + \boldsymbol{A}({t_{0s}})} \right|\boldsymbol{r} \cdot \boldsymbol{E}({t_{0s}})|{{\psi_0}} \rangle }{\sqrt {\det S^{\prime\prime}(q,{t_{0s}},{t_s})} }e^{ - iS(q,{t_{0s}},{t_s})}$$

where $s$ denotes the different quantum orbits, the action of each quantum orbit $S(q,{t_{0s}},{t_s}) = \int_{{t_s}}^\infty {( - {I_p} + q\omega )dt} + {{\int_{{t_{0s}}}^{{t_s}} {{{(\boldsymbol{k} + \boldsymbol{A}(t))}^2}dt} } / 2} + \int_{ - \infty }^{{t_{0s}}} {( - {I_p})dt} ,$ and ${\psi _0}$ is the initial wave function of the atom. The quantum orbits are obtained by solving the saddle-point equations: ${[\boldsymbol{k} + \boldsymbol{A}({t_0})]^2}/2 ={-} {I_p}$, $\int_{{t_0}}^{{t_{}}} {(\boldsymbol{k} + \boldsymbol{A}(t^{\prime}))dt^{\prime}} = 0$ and ${[\boldsymbol{k} + \boldsymbol{A}(t)]^2}/2 = q\omega - {I_p}$, where ${t_0}$ and $t$ are the ionization and recombination times of the electron, respectively, and $\boldsymbol{k}$ is the stationary momentum. Because the right-hand side of the first equation is negative, the solutions of ${t_0}$, $t$ and $\boldsymbol{k}$ are complex. These three saddle-point equations have intuitive physical meaning. The first and third equation express energy conservation at times ${t_0}$ and t, respectively, and the second equation ensures that the electron returns to its parent ion. Usually, the electron can return to the parent ion many times before the recombination. However, due to the propagation effect of the wavepacket, the contributions of the single-return orbit with the shortest travel time $t - {t_0}$ are usually dominant [31]. In this work, for simplicity, we will consider only the single-return quantum orbits in our simulations. In addition, for the bicircular fields used in this work, there are many segments within one cycle. For each segment, we will calculate the corresponding single-return quantum orbits and then put all of them into Eq. (6) to obtain the HHG spectrum.

3. Results and discussion

In Fig. 1, we present the far-field intensity and phase profiles of, e.g., the 13^th high-harmonic vortex generated by the ω-2ω and ω-3ω bicircular fields with the combination of component-OAM $\textrm{(1,1)}$ and $p = 0$(i.e., $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{2\omega ,{\bf \circlearrowleft}}$ and $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{3\omega ,{\bf \circlearrowleft}}$), respectively. Just as shown in Figs. 1(a) and 1(d), the profiles of generated XUV vortex for both the ω-2ω and ω-3ω fields exhibit a doughnut shape, indicating that a perfect vortex is obtained. Similar doughnut shape has been observed in [18,19]. On the other hand, we analyze the OAM of the 13^th harmonic by counting the phase shifts in multiples of 2π from the phase profiles in Figs. 1(c) and 1(f). It clearly shows that the OAMs of the vortex beams from the bicircular fields of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{2\omega ,{\bf \circlearrowleft}}$ and $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{3\omega ,{\bf \circlearrowleft}}$ are 9 and 7, respectively. These OAM values are in a good agreement with the prediction of the rule in Eq. (1). For the laser field with a superposition of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{2\omega ,{\bf \circlearrowleft}}$, the 13^th harmonic is from the contribution of $m = 5$ photons of frequency ω and $n = 4$ photons of frequency 2ω, and thus the OAM of the 13^th harmonic is equal to ${l_{H13}} = 5 \times 1 + 4 \times 1 = 9$. Similarly, for the laser field with a superposition of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{3\omega ,{\bf \circlearrowleft}},$ the 13^th harmonic is from the contribution of $m = 4$ photons of frequency ω and $n = 3$ photons of frequency 3ω, and thus, the OAM ${l_{H13}} = 4 \times 1 + 3 \times 1 = 7$. Accordingly, the SAM of the vortex beams is $m - n = 1$ for both of the two cases. Therefore, our result clearly shows that the circularly polarized XUV vortices are obtained from HHG driven by bicircular fields.

Fig. 1. Far-field intensity (columns 1 and 2) and phase (column 3) profiles for 13^th harmonic vortex driven by the ω-2ω and ω-3ω bicircular LG fields with equal component-OAM of $\textrm{(1,1)}$, respectively. Upper row $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{2\omega ,{\bf \circlearrowleft}},$ lower row $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{3\omega ,{\bf \circlearrowleft}}$. The fundamental frequency ω is 0.0365 a.u. and the component-intensity is equal ${I_1} = {I_2} = {I_3} = 4.0 \times {10^{14}} W/c{m^2}$. The intensity of harmonic vortex for the ω-2ω field is multiplied by 10 for visual convenience.

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Furthermore, a closer inspection reveals that the amplitude of the generated circularly polarized XUV vortex for the ω-3ω field is much higher than that for the ω-2ω field. To more clearly show the difference of the vortex amplitudes, we plot the corresponding amplitude distributions from another view angle (i.e., 3D histograms) in Figs. 1(b) and 1(e). As one can see, the peak intensity of the XUV vortex for the ω-3ω field is about one order of magnitude larger than that for the ω-2ω field, indicating that the amplitude of the circularly polarized XUV vortex can be significantly influenced by the bicircular LG fields.

To show whether the great increase of the vortex amplitude for the ω-3ω field depends on the harmonic order, we also present, e.g., the 19^th order high-harmonic vortex. Figures 2(a) and 2(d) show that the doughnut-shape intensity profile can be also observed for the ω-2ω field and the ω-3ω field. In addition, as shown in Figs. 2(c) and 2(f), the OAMs of the 19^th harmonic vortex are 13 and 9 for the two fields, respectively. This is also in a good agreement with the anticipation of the selection rules in Eq. (1). For the field with the superposition of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{2\omega ,{\bf \circlearrowleft}}$, $m = 7$ photons of frequency ω and $n = 6$ photons of frequency 2ω are absorbed to generate the 19^th harmonic, and thus, ${l_{H19}} = 7 \times 1 + 6 \times 1 = 13$. Similarly, for the superposition of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{1,0}^{3\omega ,{\bf \circlearrowleft}}$, $m = 4$ photons of frequency ω and $n = 5$ photons of frequency 3ω contributes to the 19^th harmonic, and thus ${l_{H19}} = 4 \times 1 + 5 \times 1 = 9$. Accordingly, the SAM of the vortex beams is $m - n ={-} 1$ for both of the two cases, which is opposite to those of the 13^th harmonic. Furthermore, the 3D plots in Figs. 2(b) and 2(e) show that the harmonic vortex amplitude for the ω-3ω field is also about one order of magnitude larger than that for the ω-2ω field. Therefore, our results show that the dramatic enhancement of the harmonic vortex amplitude for ω-3ω driving fields can be found for different harmonic orders with opposite circular polarization.

Fig. 2. Same as in Fig. 1 but for the 19^th harmonic vortex.

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Furthermore, we also study the dependence of the vortex amplitude on the component-OAM of the driving bicircular LG fields. In Fig. 3, we present the 13^th harmonic vortex for ω-2ω and ω-3ω bicircular fields with the combination of unequal component-OAM $\textrm{(1,2)}$ (e.g., $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{2,0}^{2\omega ,{\bf \circlearrowleft}}$ and $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{2,0}^{3\omega ,{\bf \circlearrowleft}}$). As depicted in Figs. 3(a) and 3(d), the intensity profiles for both fields show a similar doughnut shape. In addition, the phase plots, as displayed in Figs. 3(c) and 3(f), show that the OAMs of the vortex are 13 and 10 for the ω-2ω and ω-3ω field, respectively, which are also consistent with the prediction of Eq. (1). For the field with a superposition of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{2,0}^{2\omega ,{\bf \circlearrowleft}},$ the number of photons absorbed to generate the 13^th harmonic is $m = 5$ and $n = 4$ for the photons of frequency ω and frequency 2ω, respectively, while $m = 4$ photons of frequency ω and $n = 3$ photons of frequency 3ω contribute to the 13^th harmonic in the field of $LG_{1,0}^{\omega ,{\bf \circlearrowright }} \oplus LG_{2,0}^{3\omega ,{\bf \circlearrowleft}}$. Furthermore, Figs. 3(d) and 3(e) show that the intensity profiles for the 13^th harmonic generated by the ω-3ω bicircular field also exhibits a pronounced enhancement of more than one order than that generated by the ω-2ω bicircular field. Therefore, our simulations clearly show that the enhancement of high-harmonic vortices generated by a ω-3ω bicircular LG field is universal, irrespective of the harmonic order and the component-OAM of the bicircular driving fields.

Fig. 3. Same as in Fig. 1 but for the bicircular LG fields with unequal component-OAM $\textrm{(1,2)}$.

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In order to reveal the underlying physics of the enhancement of the circularly polarized XUV vortex amplitude from the ω-3ω bicircular field, we analyze the HHG spectra of a single atom. Figure 4 presents the HHG spectra of an atom in the ω-3ω bicircular field and ω-2ω bicircular field, respectively. In these two spectra, the harmonic appears at the different harmonic orders (see the inset in Fig. 4). Such difference can be derived from the selection rule in Eq. (1) [18]: only odd harmonics are emitted for ω-3ω, while it becomes $3\nu \pm 1,\nu = 0,1,2,\ldots $ for ω-2ω, which is well consistent with our simulation. In addition, the most eye catching feature is that the harmonic amplitude for the ω-3ω field is In addition, the most eye catching feature is that the harmonic amplitude for the ω-3ω field is about one order of magnitude larger than that for the ω-2ω field, in agreement with the amplitude enhancement of the circularly polarized XUV vortices shown in Figs. 1–3. Therefore, our result indicates that the amplitude enhancement of the circularly polarized XUV vortex for the ω-3ω driving fields origins from the harmonic enhancement of each atom in the bicircular LG fields.

Fig. 4. SFA-simulated HHG spectra of a single atom in the ω-3ω bicircular field and ω-2ω bicircular field, respectively. The fundamental frequency ω is 0.0365 a.u. and the component-intensity is equal ${I_1} = {I_2} = {I_3} = 4.0 \times {10^{14}} W/c{m^2}$. The inset shows the region with harmonic order from 13 to 20.

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In the following, we will further understand the underlying physics of the harmonic enhancement of a single atom in the ω-3ω driving field, by simulating the HHG spectra with quantum-orbit theory within SFA [26]. Figure 5(a) exhibits the simulated HHG spectra for the ω-3ω bicircular field and ω-2ω bicircular field, respectively. It clearly shows that there are only odd harmonics for the ω-3ω field, while for ω-2ω, the harmonic order becomes $3\nu \pm 1,\nu = 0,1,2,\ldots $. Moreover, the harmonic amplitude from ω-3ω driving field is one order of magnitude larger than that from ω-2ω field. Thus, our simulation with quantum-orbit theory is qualitatively well consistent with the SFA calculation shown in Fig. 4. Similar harmonic enhancement for the ω-3ω field has been also studied with the quantum-orbit theory [31,32] and was qualitatively ascribed to two factors. One is that the electric field vector of the ω-3ω field has four linear segments within one cycle, while the ω-2ω field has three segments. Since more such segments are available for the ω-3ω than for the ω-2ω field, the corresponding harmonic emission rate is higher. Another factor is that, according to semiclassical analysis, the time-dependent dipole is proportional to ${\tau ^{ - 3/2}}$, where $\tau = t - {t_0}$ is the travel time of electron between the ionization and the recombination. For the ω-3ω field, the travel time of electron is shorter than that for ω-2ω field and accordingly, due to the wavepacket spreading, the emission amplitude is larger for the ω-3ω field than for the ω-2ω field. In this work, we will further show that the strong enhancement of the harmonic amplitude for the ω-3ω field is also ascribed to the higher tunneling ionization rate.

Fig. 5. (a) HHG spectra of a single atom simulated with quantum-orbit theory within SFA, for the ω-3ω driving field and ω-2ω driving field, respectively. (b) HHG spectra calculated with the quantum orbits from only one segment of the laser field and moreover, the term related to travel time ${(t - {t_0})^{ - 3/2}}$ in the transition dipole is neglected. (c) The electric fields at the tunneling time of short quantum orbits as the function of the harmonic order. (d) Same as (b), but let the ionization transition amplitude equal to 1.

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In Fig. 5(b), we present the HHG spectra calculated with the quantum orbits from only one segment of each laser fields and moreover, the term related to travel time ${(t - {t_0})^{ - 3/2}}$ in the transition dipole is neglected in our simulations. In this case, the harmonic spectra are no longer discrete and the harmonic order becomes a continuous parameter [27]. As expected, in comparison with the result in Fig. 5(a), the amplitude difference between the spectra from ω-3ω driving field and ω-2ω driving field decreases greatly. However, it is found that the harmonics from ω-3ω driving field are still much stronger than that from ω-2ω field. In order to further understand the difference, we analyze the tunneling ionization rate of quantum orbits. Our analysis shows that, for each segment, there are two quantum orbits, called long and short orbits [26,30]. Due to the propagation effect of the wavepacket, the contribution of the short orbit is dominant, while the contribution of long orbit can be neglected, which is well consistent with our simulation. In Fig. 5(c), we present the electric fields at the tunneling time of the short orbits as the function of the harmonic order. It clearly shows that the electric field for the ω-3ω field is larger than that for the ω-2ω field. According to the ADK theory [33], the larger the electric field, the higher the tunneling ionization rate, and hence, the tunneling ionization rate is higher for ω-3ω field than for the ω-2ω field. In Fig. 5(d), we show the HHG spectra of a single atom simulated with quantum-orbit theory, in which we further neglect the difference of the tunneling ionization rate with the assumption that the ionization transition amplitude term $\left\langle {\textbf{k} + \textbf{A}({t_{0s}})} \right|\textbf{r} \cdot \textbf{E}({t_{0s}})|{{\psi_\textrm{0}}} \rangle \exp ( - i\int_{ - \infty }^{{t_{0s}}} {( - {I_p})dt} )$ in Eq. (6) is equal to 1. It is found that the harmonic amplitude from ω-3ω field becomes lower than that from ω-2ω field. Therefore, our work clearly shows that, in comparison with the results of the ω-2ω field, the enhancement of the harmonic amplitude for the ω-3ω field is also ascribed to the higher tunneling ionization yield.

4. Conclusions

We have theoretically studied the generation of the circularly polarized XUV vortices from HHG driven by bicircular LG fields with a frequency ratio of, e.g., 1:2 ($r = 1$, $s = 2$) and 1:3 ($r = 1$, $s = 3$), respectively. Our SFA simulation shows that the amplitude of the circularly polarized XUV vortices from the ω-3ω bicircular LG field is about one order of magnitude larger than that from the ω-2ω field, irrespective of the harmonic order and the component-OAM of the bicircular driving fields. Our analysis shows that the great increase of the vortex amplitude for the ω-3ω field origins from the harmonic enhancement of single atom. Furthermore, in terms of quantum-orbit theory, the underlying physics of the harmonic enhancement of single atom is ascribed to that, in comparison with the ω-2ω field, the ω-3ω field has more segments within one cycle and the corresponding electron orbit has the shorter travel time and the higher tunneling ionization rate. Our work provides a simple and robust method to greatly enhance circularly polarized XUV vortices from bicircular LG fields.

Funding

National Natural Science Foundation of China (11504116, 11874392, 11922413).

Acknowledgments

The authors of this paper acknowledge helpful discussions with Doc. Cheng Gong from Central China Normal University.

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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Enhancing circularly polarized XUV vortices from bicircular Laguerre-Gaussian fields

Abstract

1. Introduction

2. Theoretical methods

2.1 SFA theory for the high-harmonic vortex

2.2 Quantum-orbit theory within SFA

3. Results and discussion

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (6)

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