Generation of necklace-shaped high harmonics in a two-color vortex field

Jinxing Xue; Candong Liu; Cangtao Zhou; Cangtao Zhou; Shuangchen Ruan; Shuangchen Ruan

doi:10.1364/OE.427595

1. Introduction

Structured light beams play an important role in several spectroscopic and imaging applications [1–3]. Structured light improves the ability to manipulate macro- and micro-objects, such as particles, molecules, atoms, and electrons. In recent years, there has been growing interest in the manipulation and control of the distinctive phase and intensity properties of structured beams, based on exploitation of the angular momentum of the light field. The most common structured beams are those that possess OAM, also referred to as vortex or twisted beams [4]. Vortex beams are usually generated from wave plates or diffractive elements [5–7]. The field of an OAM beam has an azimuthal angle that is dependent on the phase of exp$(il\theta )$, where $l$ is the azimuthal index or topological charge, and $\theta$ is the azimuthal coordinate. The linear relationship between the azimuthal phase and the azimuthal angle leads to a spirally rotated wavefront [8]. Additionally, this azimuthal phase has a singularity at the center of the beam, which results in a ring-shaped intensity profile. Owing to these unique properties, OAM beams have a broad range of applications, such as optical communication [9], micromanipulation, microscopy [10–12], and quantum information [13].

High-order harmonics are generated from a gas medium when it is illuminated by infrared or mid-infrared laser pulses [14]. This provides a fully coherent light source in the extreme ultraviolet (XUV) or soft X-ray range, which has been used extensively for investigating the ultrafast dynamics of electrons at the atomic scale. High harmonics of vortices can be generated when pure vortex pulses interact with atoms [15]. In addition, the current research focuses on the OAM of vortex high-order harmonics, and proposes a variety of methods to control its OAM [16–22]. However, the generated vortex harmonics inherit the ring-shaped intensity profile of the driving vortex pulse, which cannot be modulated in a controllable manner. Intensity distribution is another dimension for high harmonics (HH) manipulation, which is essential for completely describing the electric field of high harmonics.

In this theoretical study, a two-color vortex field composed of an 800 nm Laguerre-Gaussian (LG) and a 400 nm Gaussian fields were used to manipulate the intensity profile of high harmonics. The two vortex fields have the same linear polarization and propagate in the same direction. Driven by this field, the intensity distribution of high harmonics contains multiple lobes, which are referred to as necklace harmonics owing to the azimuthally and periodically modulated intensity profile along the azimuthal coordinate. The phase profile indicates that the generated harmonics exhibit multiple discrete OAMs. Our results provide a new method for manipulating the intensity profile of HH in a controllable manner, which is useful for exploring the ultrafast dynamics of electrons.

2. Theoretical model

Existing simulations based on the single-active-electron approximation [23] in the non-adiabatic Lewenstein model have considered the single-atom dipole response [24]. Macroscopic responses for both fundamental and harmonic beams have been numerically calculated by solving the three-dimensional propagation equation in cylindrical coordinates, which has resulted in macroscopic polarization. At the gas cell exit, harmonic spectra are obtained via Fourier transformation of the time-dependent polarization. A more detailed description of the theoretical model can be found in Ref. [25]. The two-color vortex-driving field with angular frequency $\omega _1$ and $\omega _2$ can be expressed as

(1)$$E(t,\theta)=Re\{E_{1}(t)e^{i(\omega_{1}t+l_{1}\theta)}+E_{2}(t)e^{i[\omega_{2}(t+t_{d})+l_{2}\theta]} \}$$

where $E_1(t)$ and $E_2(t)$ are the envelopes of $\omega _1$ and $\omega _2$ fields, respectively; $l_1$ and $l_2$ are the topological charges (TC) of the two fields; and $t_d$ is the time delay between $E_1(t)$ and $E_2(t)$. In this study, the central frequency of the first vortex-driving pulse was 2.35 rad/fs (i.e., 800 nm in wavelength), and the peak intensity was $I_1=2.6\times 10^{13}$ $W/cm^2$ with a full-width-at-half maximum (FWHM) pulse duration $\tau$ = 26 fs and a focal spot size of 21 $\mu m$ FWHM. The high harmonic orders were defined with respect to the $1\omega$ driving frequency. The $1\omega$ vortex field can be described using the LG mode as $LG_{l,p}^{\omega }$, where $l$ is the azimuthal index (i.e., TC), and $p$ is the radial index. In this study, the second driving field was set as a pure Gaussian field, where the frequency is doubled ($2\omega$ field), the pulse duration remains the same with respect to the first driving field, and the peak intensity was set to $I_2=1.4\times 10^{14}$ $W/cm^2$. To ensure a sufficient overlapping area for the two fields, the FWHM of the focal spot size of the $2\omega$ field was set to 30 $\mu m$. A 1 $\mu m$-long gas cell filled with 3 $torr$ argon was used as the interaction medium for the subsequent simulations. Our model yielded indistinguishable results when using longer gas cells of up to 100 $\mu m$ and higher gas pressures of up to 100 $torr$. Hence, thin and low-pressure gas was used to avoid the propagation effect.

3. Results and discussion

Higher-order harmonic signals (up to the $53^{rd}$ order) were observed in the simulations. A plateau conversion efficiency of $10^{-5}$ and a divergence of 3 $mrad$ at the $12^{th}$ HH order were achieved. In this study, we observed HH signals of both even and odd orders owing to the breakdown of time symmetry of the total electric field. Moreover, at all HH orders, strong spatial modulation was attained. Figure 1 displays the intensity and phase distributions of the $12^{th}$ ($1^{st}$ row) and $19^{th}$ ($2^{nd}$ row) HH orders. A necklace-like spatial pattern is observed for both HH orders. The intensity profile depicts four lobes distributed equally along the azimuthal angle. In fact, the spatial profile of all the HH signals (from HH3 to HH53) exhibited analog features. In the case of the HH vortex field, the OAM value is normally determined by the phase distribution. The phase maps of HH12 and HH19 are shown in Figs. 1(b) and 1(e), respectively. The dashed circular line in Fig. 1(b) is plotted out in Fig. 1(c). In Fig. 1(c), the phase linearly changes twice from $-\pi$ to $\pi$ in a full azimuthal circle, which is a typical characteristic of vortex beams with OAM. This indicates that HH12 possess an OAM of 2. The phase distribution of the odd-order HH19 is presented in Fig. 1(e), which shows that the phase changes thrice from $-\pi$ to $\pi$, resulting in an OAM of 3. Figure 1(f) presents the plotting of the dashed circle line in Fig. 1(e). The space-integrated HH spectrum is shown in Fig. 1(g). The position of HH12 and HH19 are indicated by the dashed-black lines.

Fig. 1. Intensity and phase distribution for HH12 (first row) and HH19 (second row). Figs. 1(a) and 1(d) denote the intensity distributions for HH12 and HH19. Figs. 1(b) and 1(e) show the phase distributions of HH12 and HH19, respectively. Figs. 1(c) and 1(f) show the phase evolutions for HH12 and HH19 along the azimuthal angle. Fig. 1(g) is the HH spectrum.

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As is known, the spatial profile of the harmonics closely matches the distribution of the emitting dipoles in the medium [26]. Furthermore, the emitting dipoles are directly related to the spatial profile of the driving field. Therefore, high harmonics driven by a traditional Gaussian laser exhibit a Gaussian spatial distribution [27], which cannot be modulated controllably. However, for an LG laser, the spatial phase exp$(il\theta )$ imposes an additional phase on the electric field at all time instants. This additional spatial phase will extensively alter the spatial profile of high harmonics owing to the sensitivity of the generation process to the phase of the driving field. Hence, the LG field is naturally time–space-coupled because the electric field evolves differently at different positions. As high harmonics are efficiently generated near the peaks of the driving field, it can be stated that

(2)$$\left\{ \begin{aligned} & cos(\omega_{1}t+l_{1}\theta)=\pm1\\ & cos[\omega_{2}(t+t_{d})+l_{2}\theta]=\pm1 \end{aligned} \right.$$

The fields $\omega _1$ and $\omega _2$ have peaks or valleys that occur at the same time, as well azimuthal angles. Therefore, $\omega _{1}t+l_{1}\theta =k_{1}\pi$, and $\omega _{2}(t+t_{d})+l_{2}\theta =(k_{1}+2m)\pi$, where $k_1$ and $m$ are integers, and the value of $k_1$ is limited to -1, 0, and 1. From the time–space coupling perspective, the spatial phase $l_{1,2}\theta$ is confined to the time at which the field reaches its maximum value, and the time phase $\omega _{1,2}t$ restricts the azimuthal angle at which the electric field reaches its peak value. This mutual restriction leads to a specific time-space structure for the driving field, and the time-space structure is then transferred to the high harmonics. Furthermore, the time delay shifts the time and the azimuthal angle accordingly. After eliminating the temporal terms, the azimuthal angle of each lobe is expressed as

(3)$$\theta=\frac{[\omega_{2}k_{1}-\omega_{1}(k_{1}+2m)]\pi+\omega_{1}\omega_{2}t_{d}}{\omega_{2}l_{1}-\omega_{1}l_{2}} =\theta_0+\Delta\theta$$

where $\Delta \theta$ = $\omega _{1}\omega _{2}t_{d}/({\omega _{2}l_{1}-\omega _{1}l_{2}})$ is the azimuthal angle shift induced by the time delay. It can be inferred from Eq. (3) that within a 2$\pi$ azimuthal angle, the number of $\theta$ values determines the number of lobes of each harmonic. This is proved by the results shown in Fig. 1, where $\omega _2$ = 2$\omega _1$, $l_1$ = 1, $l_2$ = 0, and the time delay $t_d$ = 0. Hence $\theta$ = -0.5$\pi$, 0, 0.5$\pi$, and $\pi$. The number of $\theta$ values is four and corresponds to the four lobes shown in Figs. 1(a) and 1(d).

To illustrate the time–space structure of the two-color LG electric fields more clearly, the total electric field expressed by Eq. (1) is plotted in Fig. 2(a), wherein the azimuthal angle is -0.5$\pi$ (blue solid line) and 0.5$\pi$ (red solid line). The envelope is ignored, and the time delay is $t_d$ = 0. Clearly, for $\theta$ = 0.5$\pi$ and -0.5$\pi$, the waveforms are considerably different. Thus, the harmonics generated at these azimuthal angles are expected to be radically different. Figure 2(b) shows the total electric field within azimuthal angles of 0 to 2$\pi$. The vertical axis represents time in units of the optical cycle of the fundamental field, which is limited to a range of -0.5 $T$ to 0.5 $T$. $T$ is the optical cycle of the fundamental field. Four peaks are observed, at -0.5$\pi$, 0, 0.5$\pi$ and $\pi$, and these are denoted by the white dashed lines. At these four azimuthal angles, high harmonics can be efficiently produced, while at other angles, the intensities of the harmonics are considerably lower owing to the weak electric field.

Fig. 2. Time–space structure of a two-color LG field: $LG_{1,0}^{\omega }$ + $LG_{0,0}^{2\omega }$. (a) Waveforms for $\theta$ = -0.5$\pi$ (blue solid line) and $\theta$ = 0.5$\pi$ (red solid line); (b) total electric field for all azimuthal angles within -$\pi$ to $\pi$.

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A thorough examination of Eq. (3) reveals that the time delay can induce an azimuthal shift in the harmonics. Figure 3 shows the spatial profile of HH12 for variable time delays. The other simulation parameters were same as those in Fig. 1. Clearly, the four lobes, labeled by A, B, C, and D, rotate with the azimuthal angles of $\Delta \theta$ = $\frac {1}{16}\pi$, $\frac {2}{16}\pi$, and $\frac {3}{16}\pi$. The results agree with the predictions obtained using Eq. (3). The red lines in Fig. 3 represent the rotation of the four lobes.

Fig. 3. Rotation of intensity pattern for different time delays: (a) $t_d$ = $\frac {1}{16}T$, (b) $t_d$ = $\frac {2}{16}T$, and (c) $t_d$ = $\frac {3}{16}T$.

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According to Eq. (3), the number of $\theta$ values is

(4)$$N= \left\{ \begin{aligned} & \eta l_1-l_2, \,\eta \, is\, odd\\ & 2(\eta l_1-l_2 ), \,\eta \, is \,even\end{aligned} \right.$$

where $\eta$ = $\omega _2/\omega _1$ $>$ 1 is the frequency ratio. This equation indicates that the number of lobes is controlled by two parameters, namely, the ratio of the driving frequency, $\eta$ = $\omega _2/\omega _1$, and the topological charges $l_1$, $l_2$. Hence, additional simulations were performed. Figures 4(a) and 4(b) show the spatial profiles of HH19 driven by $LG_{1,0}^{\omega }$ + $LG_{0,0}^{3\omega }$ and $LG_{4,0}^{\omega }$ + $LG_{0,0}^{3\omega }$, respectively. The other simulation parameters, such as laser intensities, duration of the driving pulses, and beam radius, were kept the same as those in Fig. 1. From Eq. (4), the number of lobes should be N = 3 and N = 12 for $LG_{1,0}^{\omega }$ + $LG_{0,0}^{3\omega }$ and $LG_{4,0}^{\omega }$ + $LG_{0,0}^{3\omega }$, respectively. The $3^{rd}$ and $12^{th}$ lobes in Figs. 4(a) and 4(b) agree with these predictions. This also indicates that the number of lobes can be controlled by varying the frequency ratio and topological charges. This controllability may have applications in high-dimensional communications, wherein the intensity profile of the carrier wave may have to be modulated.

Fig. 4. Spatial distribution of high harmonics HH19 driven by (a) $LG_{1,0}^{\omega }$ + $LG_{0,0}^{3\omega }$ and (b) $LG_{4,0}^{\omega }$ + $LG_{0,0}^{3\omega }$.

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The OAM of harmonic photons can be determined based on parity and energy conservation. The generation of high harmonics is a parametric process, wherein the parity is conserved [28]. This requires an odd number of photons to be absorbed from the driving field. Therefore, for two-color LG fields, parity and energy conservation are expressed as

(5)$$\left\{\begin{aligned} & n_{1}+n_{2}=2k+1\\ & n_{1}\omega_{1}+n_{2}\omega_{2}=q\omega_{1} \end{aligned}\right.$$

where $n_1$ and $n_2$ are the numbers of photons absorbed from the $\omega _1$ and $\omega _2$ fields, respectively; $k$ is an integer, and $q$ represents the order of the harmonics. The frequency ratio is $\omega _2/\omega _1$ = 2. From the linear equations listed in Eq. (5), the number of photons absorbed is $n_1 = 2(2k + 1) - q$ and $n_2 = q - (2k + 1)$. As $n_1$ $\ge$ 0 and $n_2$ $\ge$ 0, ${\frac {q-2}{4}}\leq k \leq {\frac {q-1}{2}}$. Hence, this inequality defines all the possible photon channels ($n_1$, $n_2$) that an electron can select a channel from, to emit the $q$ harmonic. Additionally, all the possible photon channels can be distinguished in a straightforward manner as different channels exhibit different OAMs. Owing to OAM conservation, the OAM of the $q$ harmonic is $l_q$ = $n_1l_1$ + $n_2l_2$, which can be rewritten as

(6)$$l_{q}=[2(2k+1)-q]l_{1}+[q-(2k+1)]l_{2}$$

Thus, the integer $k$ determines the OAM of harmonic photons, where $\frac {q/\eta -1}{2}\leq k\leq \frac {q-1}{2}$. This is comprehensive because the physical meaning of $2k+1$ is the total number of photons absorbed from two-color LG fields. It’s worth commenting on the selection rules of Eq. (6), which are more general than those in Ref. [29]. The spin conservation rises a stronger confinement for the photon channels due to the bi-circular LG beams. This confinement reduces the number of the photon channels. For instance, the $12^{th}$ harmonic could not be generated from the bi-circular LG beams in Ref. [29] while from the selection rules of Eq. (6), the $12^{th}$ harmonic can still be generated. For a specific harmonic, $k$ may possess multiple values, which suggests that the OAM of high harmonics may have multiple values as well. For instance, for $\eta$ = 2 and $q$ = 12, the range of $k$ is ${\frac {5}{2}}\leq k \leq {\frac {11}{2}}$. Because $k$ is an integer, $k$ = 3, 4, 5, and the possible photon channels are: ($n_1$, $n_2$) = (2, 5), (6, 3), (10, 1). The corresponding OAM is $l_{12}$ = 2, 6, 10. However, only $l_{12}$ = 2 is observed in Fig. 1(b). This is attributed to the peak intensity of the $2\omega$ field being larger than that of $1\omega$ field, which results in the photon channel ($n_1$, $n_2$) = (2, 5) dominating the generation process [30]. When the intensity of the $2\omega$ field is weaker than that of the $1\omega$ field, the three distinct OAM values of HH12 can be observed. Figure 5 shows the intensity and phase profile of HH12, wherein the intensity of the $2\omega$ field is reduced by two orders, i.e., $I_2$ = $1.4\times 10^{12}$ $W/cm^2$, and the other simulation parameters are maintained the same as those in Fig. 1. Based on the number of phase jumps from -$\pi$ to $\pi$ in Fig. 5(b), the OAM of HH12 is: $l_{12}$ = 2, 6, 10, as indicated by the black dashed circles labeled as $a_1$, $a_2$, and $a_3$. Another remarkable observation is the intensity pattern shown in Fig. 5(a), which is more complicated than that in Fig. 1(a). In Fig. 1(a), four lobes are located on one ring. In contrast, there are three rings, indicated by the dashed-white circles in Fig. 5(a). On each ring, there are four lobes. An azimuthal angle shift can also be observed between each pair of rings. Moreover, the four lobes located on $c_1$ are much weaker than those located on $c_2$ and $c_3$. The OAM of HH12, which is located on $c_1$, $c_2$ and $c_3$, can be calculated to be 2, 10, and 6, respectively; these values agree with the predictions obtained using Eq. (6). Upon propagating the complex electric field of HH12 to the far field, it can be observed that $c_1$ and $c_2$ follow a short trajectory, while $c_3$ follows a long trajectory. Figure 5(c) shows the phase evolutions for $a_1$, $a_2$, and $a_3$ along the azimuthal angle.

Fig. 5. Spatial distribution of intensity (a) and phase (b) for HH12. The intensity of the $2\omega$ field is reduced to $I_2$ = $1.4\times 10^{12}$ $W/cm^2$. The dashed-white circles in (a) indicate the location of the lobes. In (b), the dashed-black circles are labeled by the arrows $a_1$, $a_2$, and $a_3$, indicating the three distinct OAM values of $l_{12}$ = 2, 6, and 10. Fig. 5(c) shows the phase evolutions for $a_1$, $a_2$, and $a_3$ along the azimuthal angle.

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This study focused on manipulating the intensity profile of harmonics in the XUV range. However, the manipulation is limited to the ring-shaped intensity profile, resulting in a periodic structure along the azimuthal axis. For the generation of isolated vortex attosecond pulse, the three distinct OAM of HH12 (Fig. 5(b)) indicates that the OAM of attosecond pulses cannot be well-defined.

4. Conclusions

In this study, a method was proposed to generate high harmonics with a necklace-shaped intensity profile from a two-color vortex field. Simulations based on the nonadiabatic Lewenstein model were performed. By analyzing the time–space structure of the driving field, it was determined that the number of lobes for each harmonic is controlled by the peaks of the driving field along the azimuthal angle. An analytical equation (Eq. (3)) was derived to determine the azimuthal angle at which the lobes emerge. Furthermore, the time delay between the two driving fields linearly induced an azimuthal angle rotation of the lobes. Moreover, the harmonics with necklace-like intensity profiles exhibited OAM. This was confirmed by the phase distribution of the harmonics and their spirally rotated wavefront. Based on the fundamental laws of parity and energy conservation, the OAM value of each harmonic was identified. The most interesting observation is that the harmonics possessed multiple distinct OAM values.

It can be expected that harmonic necklace beams may find applications in several fields such as optical communications, micromanipulation, microscopy and quantum information. The controllability of the harmonic intensity distribution provides another free dimension to manipulate macro- and micro-objects such as particles, molecules, atoms, and electrons. This could enable expansion of the messaging capacity in high-dimensional communication. Furthermore, driven by a field with a stronger intensity or longer wavelength, the photon energy of necklace harmonics can reach the soft X-ray range and may generate isolated vortex attosecond pulses. Our results also open a new way for advancing the extreme ultraviolet lithography as the intensity profile of extreme ultraviolet source is crucial in lithography.

Funding

Featured Innovation Project of Educational Commission of Guangdong Province of China (2018KTSCX352); National Key Research and Development Program of China (2016YFA0401100); National Natural Science Foundation of China (11575031, 11875092, 12005149).

Acknowledgments

The authors of this paper acknowledge helpful discussions with Prof. Chengpu Liu from Shanghai Institute of Optics and Fine Mechanics, as well as Doc. Honggeng Wang from Shenzhen 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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Generation of necklace-shaped high harmonics in a two-color vortex field

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

4. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (6)

Optics Express