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Generation of ultra-intense vortex laser from a binary phase square spiral zone plate

Lingyu Zhang, Hao Zhang, Hongtao Huang, Jingyi Wang, Hongyu Zhou, and Tongpu Yu

Open Access Open Access

Abstract

With the development of ultra-intense laser technology, the manipulation of relativistic laser pulses has become progressively challenging due to the limitations of damage thresholds for traditional optical devices. In recent years, the generation and manipulation of ultra-intense vortex laser pulses by plasma has attracted a great deal of attention. Here, we propose a new scheme to produce a relativistic vortex laser. This is achieved by using a relativistic Gaussian drive laser to irradiate a plasma binary phase square spiral zone plate (BPSSZP). Based on three-dimensional particle-in-cell (3D-PIC) simulations, we find that the drive laser has a phase difference of π after passing through the BPSSZP, ultimately generating the vortex laser with unique square symmetry. Quantitatively, by employing a drive laser pulse with intensity of 1.3 × 1018~W/cm2, a vortex laser with intensity up to 1.8 × 1019~W/cm2, and energy conversion efficiency of 18.61% can be obtained. The vortex lasers generated using the BPSSZP follow the modulo-4 transmutation rule when varying the topological charge of BPSSZP. Furthermore, the plasma-based BPSSZP has exhibited robustness and the ability to withstand multiple ultra-intense laser pulses. As the vortex laser generated via the BPSSZP has high intensity and large energy conversion efficiency, our scheme may hold potential applications in the community of laser-plasma, such as particles acceleration, intense high-order vortex harmonic generation, and vortex X/γ-ray sources.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

As ultra-intense laser technology has advanced, particularly with the advent of chirped pulse amplification (CPA) techniques [1], there has been a rapid escalation in both peak laser power and focus intensity [2,3]. At the same time, the exciting potential of intense laser-matter interactions has attracted a great deal of interest [4,5]. However, due to the limited damage thresholds of conventional optical devices [6], generating and manipulating intense laser pulses has become an increasingly daunting task as the laser power has increased. In contrast, using plasma as a medium to manipulate intense laser pulses offers distinct advantages over traditional solid-state materials, owing to the ability of plasma to sustain ultra-high intensities. Over the past few decades, plasma optics has become an important way for manipulating and further investigating ultra-intense laser pulses. The remarkable applications of plasma optics encompass backward Raman amplification [7–9], laser pulse amplification [10–13], plasma holography [11,14], and plasma gratings [15–17]. Recently, there has been growing interest in generating and manipulating the relativistic vortex lasers carrying orbital angular momentum (OAM) [18]. Such vortex lasers have a wide range of significant applications in the laser-plasma community [19–38], including vortex X$/ \gamma$-ray emission [19–23], intense high-order vortex harmonic generation [24–27], and particles acceleration [28–38].

Previous studies have demonstrated that the non-relativistic vortex laser can be generated using various diffraction optical elements, such as the spiral phase plate (SPP) [39], the spiral photon sieves (SPS) [40], and the spiral zone plate (SZP) [41–43]. Over the past few years, great attention has been given to the study of circular SZPs. Moreover, it has been discovered that the square spiral zone plate (SSZP) has significant potential in some specific applications [44–49]. Since the square symmetry [44] has ideal characteristics in adapting to the inherent geometry of rectangular pixel arrays, it usually performs better in the far field and has potential applications in optical image processing [45], depth estimation [46], automatic focusing [47], quantum computing [48] and precision alignment of accelerator [49]. In contrast to circular SZPs, SSZPs offer the advantage of simplified fabrication processes and eliminate the presence of curved shape stripes [50]. Especially, the SSZP has a two-arms-cross structure in the focal plane, and the two-arms-cross pattern is also helpful for parallel light aggregation in three-dimensional microstructure manufacturing [51]. Given these advantages of SSZPs, Gao et al. designed a new type of binary phase square spiral zone plate (BPSSZP) [44]. The diffraction field center of this new SSZP exhibits low vortex charges and follows the modulo-4 transmutation rule [52]. The BPSSZP diffraction field changes regularly with the topological charge at a period of 4. When $l = 1, 2, 3,$ the center of the diffraction field presents a dark core, and a bright spot appears in the center of the diffraction field when $l = 4$. This phenomenon is previously unobserved in circular SZPs. These relativistic vortex lasers generated via the BPSSZP may hold potential for applications in mitigating the laser-plasma instability (LPI). It has been shown that the relativistic vortex laser could suppress the growth of stimulated Raman scattering (SRS) to promote ignition via a low LPI laser system composed of a super light spring [53]. Moreover, the unique square symmetry would make the vortex laser formate a more spatial uniform cluster beam to raise its inhibitory effect.

In this paper, we extend for the first time the vortex laser with unique square symmetry to a relativistic regime by using such kind of BPSSZPs. By using three-dimensional particle-in-cell (3D-PIC) simulations, it is shown that when a Gaussian laser irradiates the BPSSZP, the BPSSZP not only converts the input drive laser into a vortex laser with square symmetry, but also concurrently focuses the vortex laser. The BPSSZP exhibits a distinctive diffraction pattern characterized by four bright spots of the focused vortex laser. This contrastive pattern diverges from the focused vortex laser pattern of traditional circular spiral zone plates. The schematic representation of the scheme is shown in Fig. 1. Especially, this vortex laser pulse follows the modulo-4 transmutation rule and is characterized by relativistic intensity, large angular momentum (AM), and high energy conversion efficiency. These features provide an efficient solution for producing relativistic vortex lasers with square symmetry in future experiments.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of a drive laser irradiating binary phase square spiral zone plate (BPSSZP) to produce a vortex laser with square symmetry. The projections in front of and behind the simulation box are the laser intensity of the Gaussian laser and the vortex laser, respectively. The electric field distributions of the Gaussian vortex and vortex laser are shown along the direction of laser propagation. (b) The profile of the left-handed (LH) BPSSZP with topological charge $l=-1$ and (c) the right-handed (RH) BPSSZP with topological charge $l=1$, where $d$ is the thickness of RH and LH BPSSZP, the black area is the plasma region, and the white area is the vacuum region.

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2. Methodology

In order to demonstrate the proposed scheme, we perform 3D-PIC simulations with the relativistic electromagnetic code EPOCH [54]. Here, the BPSSZP’s transmittance function $t_{\textrm {BPSSZP}}$ is expressed as a binarization of a spiral square lens $t_{\textrm {SSL}}$ as follows [44]:

$${t}_{\textrm{BPSSZP}}( {r,\theta}) = \textrm{Bin}[t_{\textrm{SSL}}({r,\theta})]= \textrm{Bin}\left\{ \textrm{exp}[- ik~\textrm{max}(y^{2},z^{2})/2f]\textrm{exp}(il\theta)\right\},$$
with the following binarization function:
$$\textrm{Bin}{(t)} = \left\{ \begin{matrix} {~~1~~~~~\textrm{if}~~~~\textrm{Im}(t) > 0} \\ {- 1~~~~~\textrm{if}~~~~\textrm{Im}(t) < 0} \\ \end{matrix} \right.~~.$$
Here, $k=2\pi /\lambda _0$ is the wave number of the incident laser, $\lambda _{0} = cT_{0} = 1~\mu m$ is the laser wavelength, $T_{0}$ is the laser cycle, $f$ is the focal length of BPSSZP with $\lambda _0$, $l$ being the topological charge of BPSSZP, $(r,\theta )$ is the polar coordinate, and $(y,z)$ is the Cartesian coordinate. The BPSSZP is divided into two parts: vacuum area and plasma area. The laser pulse can propagate through the vacuum area of the target when $\textrm {Im}(t)>0$. By contrast, it brings a phase difference of $\pi$ after the laser pulse passes through the target’s plasma area when $\textrm {Im}(t)<0$. In our simulations, a linearly polarized (LP) Gaussian laser pulse with focal spot size $\sigma _{0}=10~\lambda _{0}$ is incident along the $x$-axis from the left boundary at $t=0$. The laser electric field normalized amplitude is $a_{0} = \left (eE_{0}\right )/\left (m_{e}c\omega _{0}\right ) = 1$, corresponding to the relativistic threshold of $1.3\times 10^{18}~$W/cm$^2$. Here $e$ is the unit charge, $E_0$ is the electric field amplitude, $m_e$ is the electron mass, $c$ is the speed of light in vacuum, and $\omega _0$ is the laser frequency, respectively. The drive laser pulse has a Gaussian time profile, the incident laser pulse full duration is $6T_0$ (the full width at half maxima is $5T_{0}$), and the pulse energy of the incident laser is $0.026~\textrm {J}$. The grid size of the simulation box is $20~\mu m\times 36~\mu m\times 36~\mu m$ in the $x\times y\times z$ directions, which is sampled by $600\times 1080\times 1080$ cells with 6 macro-particles per cell. The focal length of the right-handed (RH) BPSSZP is set as $f = 10\lambda _0$. The target is schematically shown in Fig. 1, which is composed of fully ionized carbon ions, hydrogen ions, and electrons with a carbon to hydrogen density ratio of 1:4. The electron density of the plasma is $n_e = 0.9n_c$, the corresponding density of carbon ions ($C^{6+}$) and protons ($H^+$) are $0.09n_c$ and $0.36n_c$, respectively, which enables the incident drive laser pulse to pass through target easily. Here $n_{c} = \left ( m_{e}\omega _{0}^{2} \right )/(4\pi e^{2}) = 1.1 \times 10^{21}~{cm}^{- 3}$ is the critical density for $\lambda _{0}$ = $1~\mu m$ laser. The BPSSZP is located between $x = 3\lambda _0$ and $x = 4\lambda _0$. Through the utilization of Eq. (3), it can be calculated that the thickness of BPSSZP is $d = 1\lambda _0$ when $l = 1$. We adjust the phase by varying the thickness of the target as follows: $d(\phi )=d_{0}(2\pi - \phi )/2\pi$, where $\phi$ is the phase difference, $d_{0}=\left ( 1 + \frac {c}{v_{p} - c} \right )l\lambda _0$ [55]. The BPSSZP can convert the incident Gaussian laser into a vortex laser, and the incident laser will produce a phase difference of $\pi$ as it passes through the plasma region of BPSSZP. Based on the property of BPSSZP, the thickness $d$ of BPSSZP is set as $d(\phi )=d_{0}/2$. Here, the thickness $d$ of BPSSZP can be expressed as:
$$d = \frac{1}{2}\left( 1 + \frac{c}{v_{p} - c} \right)l\lambda_0.$$
The phase velocity of intense light propagating in the low-density plasma is $v_{p} = c/\sqrt {1 - n_{e}/(\gamma _{L}n_{c})}$ [56], where $\gamma _{L}$ is given by $\gamma _{L} = \sqrt {1 + a_{0}^{2}/\alpha }$ with $\alpha = 1$ for circularly polarized (CP) lasers case and $\alpha = 2$ for LP lasers case. The phase velocity of light $v_p$ depends on the plasma density and is greater than the vacuum speed of light $c$. Therefore, the laser pulse progressively assumes a helical phase front as it exits from the plasma slab. Presently, upon full ionization, the lightest aerogel is of density $0.028n_c$ [55], and the thickness of the aerogel can reach the nanometer scale [57]. With an electron density of $0.9n_c$ and a thickness of approximately $1~\mu m$ for BPSSZP, the aerogel can be chosen as the target material in experiments. The intricate architecture of the BPSSZP with a near-critical-density plasma can be effectively fabricated using the 3D printing [58] applied to aerogel in the current laboratories [59].

3. Simulation results

Figure 2(a) shows the resultant distribution of the 3D isosurface distribution of the electric field $E_y$ after the drive Gaussian laser passes through the plasma zone. There is no filtering for the laser electric field. It is indicated that the output laser pulse has a vortex feature. The vortex electric field $E_y$ generated by the circular SZP exhibits a circular helical profile and possesses a hollow intensity distribution [60]. In contrast, the vortex electric field $E_y$ generated by the BPSSZP exhibits a distinctive square characteristic due to BPSSZP’s binary phase square profile. For comparison, the Fresnel-Kirchhoff’s diffraction formula is used to predict analytically the diffraction electric field component of the laser. Here, the diffracted electric field can be expressed as

$$E\left( {y,z} \right) = \frac{1}{i\lambda_{0}}{\iint u_{0}}\left( y',z' \right)t\left( y',z' \right)k(\theta)\frac{\exp(ik\rho)}{\rho}dy'dz',$$
where $\rho = \sqrt {(x - x')^2 + (y - y')^2 + (z - z')^2}$, $u_{0}({y',z'}) = C\exp ( - {r^{2}}/{\sigma _{0}^{2}})$ represents the complex amplitude of the incident Gaussian laser, $C$ is a constant, and $t(y',z')$ is the transmittance of BPSSZP. $k(\theta ) = \frac {{\cos \left ( {n,r} \right )} - \cos \left ( n,r_{0} \right )}{2}$ is the inclination factor, where $(n,r)$ is the angle at which the laser deviates from the original optical path after diffraction occurs in BPSSZP, $(n,r_{0})$ is the angle at which the laser reaches the BPSSZP from the laser source. Three positions between $x=12.7\lambda _0$ and $x=13.7\lambda _0$ near the focal plane are chosen for comparison between the simulation results and the theoretical calculations of the electric field $E_y$. As shown in Fig. 2(b)-(g), the electric field obtained from the simulations (top row) exhibits the characteristic of the vortex laser and fits the theoretical calculations (bottom row) very well. This indicates that the BPSSZP is capable of converting the incident Gaussian laser into a vortex laser. The projection of the laser intensity on the $(x,y)$ and $(x,z)$ plane in Fig. 2(a) shows that the intensity of the vortex laser can reach $1.8\times 10^{19}~\textrm {W/cm}^2$, which is magnified by 13 times compared to the incident drive laser.

 figure: Fig. 2.

Fig. 2. (a) 3D isosurface distribution of the electric field $E_y$ at $t = 17 T_0$. The projection of $E_y$ on the $(y,z)$ plane at $x = 13.5 \lambda _0$ is shown on the right. The projection of laser intensity on the $(x,y)$ plane at the bottom is taken at $z = 0\lambda _0$. The projection plane of $(x,z)$ at the back is taken at $y = 0\lambda _0$. Here, $I$ represents the intensity of the output vortex laser. (b)-(d) The transverse distribution of $E_y$ at different sections of $x = 12.7\lambda _0$ to $13.7\lambda _0$ at $t = 17 T_0$ (simulation results). (e)-(g) Same as (b)-(d), but from the Fresnel-Kirchhoff diffraction formula.

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Figure 3(a)-(d) illustrate the geometric profile of BPSSZP with different topological charges $l$, where the white area represents a phase delay of zero and the black area represents a phase delay of $\pi$. To further investigate the focusing behavior of BPSSZP, we calculated the resultant vortex laser pulse intensity along the $x$-axis in the simulation with different topological charges at the position of $x = 13.5\lambda _0$ at $t= 18T_0$, as shown on the $(y,z)$ plane in Fig. 3(a)-(d). It is shown that the vortex laser intensity generated via the BPSSZP with different topological charges $l$ can reach $10^{19}~$W/cm$^2$, which is about an order of magnitude higher than the incident Gaussian laser pulse. As $l=1,2$ or $3$, the center of the laser intensity focus is characterized by a square dark core with crossed bright arms and four bright spots. When $l=4$, the axial intensity is no longer zero. Instead, a bright spot is generated at the center, as shown in Fig. 3(d). Unlike the vortex laser generated via a circular SZP, this SSZP can generate the vortex laser with unique square symmetry, and its diffraction characteristics change periodically with the topological charge $l$, which is an integer in a cycle of 4. This special feature satisfies the modulo-4 transmutation rule of the SSZP [52].

 figure: Fig. 3.

Fig. 3. BPSSZP with different topological charges and the diffraction pattern of laser intensity near the focal plane of $x = 13.5 \lambda _0$ at $t = 18T_0$. (a) $l=1$ case, (b) $l=2$ case, (c) $l=3$ case, (d) $l=4$ case.

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We also investigated the evolution of the laser AM and energy conversion efficiency to the vortex laser. In the scheme, we have calculated the BPSSZP for the single focal point, our calculation of the laser energy conversion efficiency and the laser total AM is based on the vortex laser in proximity to the focal point $(x = 12.5\lambda _{0}~\textrm {to}~14.5\lambda _{0})$. Here, the angular momentum $(\mathbf {J}_{laser})$ and energy $(E_{laser})$ of the laser pulse are calculated as: $\mathbf {J}_{laser} = \varepsilon _{0}{\int {\mathbf {r} \times \left ( {\mathbf {E} \times \mathbf {B}} \right )\textrm {d}V = \mathbf {J}_{x} + \mathbf {J}_{y} + \mathbf {J}_{z}}}$ and $E_{laser} = \frac {1}{2}{\int {\left. ( \varepsilon \right._{0}\mathbf {E}^{2} + (1/\mu _{0})\mathbf {B}^{2})\textrm {d}V}}$, respectively, with $\varepsilon _{0}$ the permittivity of vacuum, and $\mu _{0}$ the permeability of vacuum. The laser energy conversion efficiency to the vortex laser by using BPSSZP has also been calculated, as shown in Fig. 4(a). It is found that the maximum energy conversion efficiency is observed when the laser pulse is in proximity to the focus plane, which can reach 18.61%. The energy conversion efficiency of the vortex laser generated by the conventional circular SZP can reach 10% [60]. As expected, the conversion efficiency via the BPSSZP is significantly higher than that of the conventional circular SZP. As the laser pulse leaves the focus plane, the focused vortex laser begins to diverge, resulting in a gradual decline in energy conversion efficiency. Since the dominant part of AM in the laser is along the $x$-axis, only $J_x$ needs to be considered when we calculate the vortex laser AM. For comparison, we also considered the left-handed (LH) BPSSZP in our simulations while keeping other parameters unchanged. Figure 4(b) shows the total AM evolution of the laser in the RH BPSSZP and LH BPSSZP, respectively. It can be seen that the evolution of $J_x$ is essentially symmetrical. As expected, there is no AM for the incident Gaussian laser as shown in Fig. 4(b). Upon via the BPSSZP, a gradual acquisition of AM is observed in the incident Gaussian laser. Subsequently, the incident Gaussian laser is converted into a vortex laser and gradually focused. The AM procured during this transition from the incident laser to the vortex laser is actually OAM. It is shown that the total AM of the vortex laser pulse increases to $9.26 \times 10^{-19}~\textrm {kg}\cdot \textrm {m}^2/\textrm {s}$ near the focal plane. As the laser pulse leaves the focal plane, the AM of the vortex laser pulse decreases gradually.

 figure: Fig. 4.

Fig. 4. (a) Evolution of the laser energy conversion efficiency from the Gaussian laser pulse to the vortex laser pulse. (b) Evolution of the total laser angular momentum (AM) in right-handed (RH) BPSSZP (black line) case and left-handed (LH) BPSSZP (red line) case, respectively. The gray area here represents the stage when the vortex laser is passing through the focal plane.

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4. Discussions

Unlike the LP laser, the CP laser carries spin angular momentum (SAM), with $\pm$ $\hbar$ per photon for RH and LH CP laser, respectively. Thus, the feasibility of employing the CP incident laser in our proposed scheme is also discussed. According to Eq. (3), it can be seen that the thickness of BPSSZP will change with the variation of $\alpha$ and $a_0$, and $d$ is equal to $1.26~\mu m$ in the CP laser case. We conducted two additional simulations, wherein the RH CP laser passed through the RH BPSSZP and LH BPSSZP, respectively.

As seen from Fig. 5(a), the energy conversion efficiency of the vortex laser generated after the RH CP laser has passed through the RH and LH BPSSZP was calculated, respectively. It was observed that the evolution of laser energy conversion efficiency in both cases is consistent, and the energy conversion efficiency of the vortex laser is as high as 17.6%. It can be seen that the RH CP laser can get a high energy conversion efficiency through RH BPSSZP and LH BPSSZP, respectively. Figure 5(b) shows the total AM evolution of the RH CP laser passing through RH BPSSZP and LH BPSSZP, respectively. Contrary to the LP laser, the total AM evolution of the RH CP laser in the RH BPSSZP and LH BPSSZP does not exhibit a symmetric pattern. This is due to the fact that the RH CP laser carries the SAM, and the direction of the OAM obtained is different when the RH CP laser passes through the RH BPSSZP and LH BPSSZP, respectively. When the RH CP laser passes through the RH BPSSZP, the AM increases. However, when the RH CP laser passes through the LH BPSSZP, the AM diminishes. Consequently, the superimposed OAM contributes to distinct distributions of AM when combined with the initial SAM. As a result, the total AM of the generated vortex laser is much higher than that of the vortex laser generated by the LP laser, with a peak value of $43.3 \times 10^{-19}~\textrm {kg}\cdot \textrm {m}^2/\textrm {s}$. This is over four times larger than in the LP case. Figure 5(c) and (d) show the distribution of the electric field $E_y$ after the RH CP laser passes through RH BPSSZP and LH BPSSZP, respectively. Compared with Fig. 2(c), it can be seen that the electric field $E_y$ generated by the RH CP laser through BPSSZP is similar to that generated by the LP laser through BPSSZP. This indicates that our scheme is also valid for the RH CP laser case.

 figure: Fig. 5.

Fig. 5. (a) Evolution of the energy conversion efficiency from the right-handed (RH) circularly polarized (CP) laser pulse to the vortex laser pulse in right-handed (RH) BPSSZP (black line) and left-handed (LH) BPSSZP (red line), respectively. (b) Evolution of the total angular momentum (AM) of the laser in right-handed (RH) BPSSZP (black line) case and left-handed (LH) BPSSZP (red line) case. The gray area here represents the stage when the vortex laser is passing through the focal plane. (c) The transverse distribution $E_y$ after the RH CP laser passes through RH BPSSZP ($x = 13.2\lambda _{0}$, $t = 17T_0$). (d) The transverse distribution $E_y$ after the RH CP laser passes through LH BPSSZP ($x = 13.2\lambda _{0}$, $t = 17T_0$).

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The effect of laser parameters on the vortex laser generation is also examined. Here we first focus on the intensity of the generated vortex laser. The simulation parameters are unchanged except for the laser electric field normalized amplitude $a_0$, which varies from 0.1 to 5. According to Eq. (3), BPSSZP’s thickness also varies with $a_0$. The correspondence between the thickness of BPSSZP and the variation of $a_0$ is shown in Table 1.

Tables Icon

Table 1. The thickness of BPSSZP and the corresponding laser electric field normalized amplitude.

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The influence of $a_0$ on the total angular momentum $J_x$ (black circles), the energy conversion efficiency $\eta$ (red circles) of the vortex laser, and the ratio of the vortex laser intensity to the drive laser intensity $I / I_0$ (blue circles) are plotted in Fig. 6(a). The finding demonstrates a direct correlation between the laser intensity increment and the increase in total AM, and a fitting analysis indicates a proportional relation between $J_x$ and $a_{0}^{2}$. Theoretically, the total AM of the laser depends on the number of photons and the average AM of each photon, whereby each photon carries both OAM of $l\hbar$ and SAM of $\delta \hbar$. Here, $\delta$ and $l$ represent the spin and orbital angular momentum of a photon, and $\hbar$ is reduced Planck constant. The laser intensity $I$ is determined by the multiplication of the photon number and the energy of a single photon, exhibiting a positive correlation with $a_0$. It can be expressed as $I = Nh\nu \propto a_{0}^{2}$. Consequently, it can be inferred that there exists a positive association between the total AM of the vortex laser and $a_0$ by $N = I/h\nu \propto a_{0}^{2}$, validating the congruence with the simulations as presented in Fig. 6(a). It is also found that for non-relativistic laser, the energy conversion efficiency to the vortex laser improves significantly with the increase of $a_0$. However, the increase of $\eta$ becomes less pronounced as $a_0$ > 1, and the ratio of the vortex laser intensity to the incident Gaussian laser intensity $I / I_0$ decreases slightly. It is indicated that the thickness of BPSSZP affects the diffraction of the drive laser pulse significantly. With a thicker BPSSZP, the quality of the achieved vortex laser is lower. When $a_0 = 5$, the total AM of the vortex laser can reach $12.3 \times 10^{-18}~\textrm {kg}\cdot \textrm {m}^2/\textrm {s}$, with an energy conversion efficiency to the vortex laser reaching a high value of 21.01%. The corresponding ratio of the intensity of the vortex laser to the incident Gaussian laser is as high as 9.5. There has been a lot of previous work using plasma as a medium to achieve relativistic laser amplification. For example, the utilization of plasma lenses for laser relativistic self-focusing (RSF) amplification (25 times) [12] or the utilization of overdense plasma for achieving a $10^{22}~\textrm {W/cm}^2$ ultra-intense laser amplification (7 times) [13]. In our scheme, the laser intensity can be amplified up to 13 times for $a_{0}=1$. Figure 6(b) illustrates the evolution of the laser photon average AM $J_{\textrm {photon}}$ from the Gaussian laser pulse to the vortex laser pulse for different $a_0$. It can be seen that the evolution of the laser photon average AM follows a similar pattern as $a_0$ varies from 0.1 to 5. The maximum value of the laser photon average AM is reached at $19T_0$, with corresponding peaks at $0.63\hbar$, $0.71\hbar$, $0.75\hbar$, $0.65\hbar$, and $0.58\hbar$, respectively. However, there is a slight decrease in the peak of the laser photon average AM when $a_0$ > 1. It corresponds to the above cause that the thickness of BPSSZP affects the diffraction of the drive laser pulse.

 figure: Fig. 6.

Fig. 6. (a) The total angular momentum of the vortex laser ($J_x$, black circles), the laser energy conversion efficiency to the vortex laser ($\eta$, red circles), and the ratio of vortex laser intensity to the incident laser intensity ($I / I_0$, blue circles) with the laser electric field normalized amplitude $a_0$. The solid black line is the fitting result. (b) Evolution of the laser photon average AM from the Gaussian laser pulse to the vortex laser pulse as $a_0$ varies from 0.1 to 5. The gray area here represents the stage when the vortex laser passes through the focal plane.

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Since the high-intensity lasers can result in varying degrees of damage on targets after passing through the target, the damaged micro-structured BPSSZP target may have to be replaced in every experiment. This is impractical and fatal for high-frequency lasers. In order to demonstrate the robustness of the target, three continuous drive lasers are considered in simulations, which pass through the BPSSZP one by one to check the performance of the BPSSZP target under multi-laser irradiation. Figure 7 shows the electric field $E_y$ distribution on the $(x,y)$ plane after the three laser pulses passing through the BPSSZP with the same time delay $\tau = 10~T_{0}$. It reveals that the electric field $E_y$ of all three laser pulses passing through BPSSZP still retains a hollow intensity distribution. The electric field intensity distribution remains almost unaffected even after the third laser pulse passes through the target, and from the electron density of BPSSZP in the inset, we can see that the target retains its integrity. It is noted that after the third laser pulse passed through BPSSZP, the laser energy conversion efficiency to the electrons was about 3.6%. The BPSSZP can be irradiated by multiple laser pulses, enabling the reusability of the target in future experiments. Thus, it offers possibilities for the practical implementation of BPSSZP in experimental studies.

 figure: Fig. 7.

Fig. 7. The distribution of electric field $E_y$ on the $(x,y)$ plane for the first laser pulse (a) ($t = 18 T_0$), the second laser pulse (b) ($t = 28 T_0$), and the third laser pulse (c) ($t = 38 T_0$). The insets in (a), (b), and (c) show the electron density distribution of BPSSZP $(x = 3.5 \lambda _{0})$ and the distribution of electric field $E_y$ on the $(y,z)$ plane at $x = 13.5\lambda _0$.

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5. Conclusion

In summary, a laser-plasma based scenario of ultra-intense vortex laser generation is proposed by a high-power Gaussian laser pulse irradiating a BPSSZP target. 3D-PIC simulations show that the Gaussian drive laser pulse can be converted into a vortex laser pulse with a focusing intensity greater than $10^{19}~$W/cm$^2$ after passing through the BPSSZP. The AM of the vortex laser generated can reach $9.26 \times 10^{-19}~\textrm {kg}\cdot \textrm {m}^2/\textrm {s}$, and the laser energy conversion efficiency to the vortex laser is up to 18.61%. It has been a significant advancement over the conventional circular plasma SZP in laser energy conversion efficiency. Due to the special binary phase square structure of BPSSZP, the modulo-4 transmutation rule is observed in the generated vortex laser when changing the topological charge $l$ of BPSSZP. Additionally, the plasma BPSSZP exhibits robustness by maintaining its integrity even after the passage of multiple laser pulses. This scenario may provide a practical way for various applications, such as charged particles acceleration, high OAM X/$\gamma$-ray emission, high-order vortex harmonic generation, and suppression of LPI in the relativistic regime.

Funding

Science and Technology Innovation Program of Hunan Province (2020RC4020); National Natural Science Foundation of China (12375244, 12305265, 12135009).

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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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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Figures (7)
Fig. 1.
Fig. 1. (a) Schematic diagram of a drive laser irradiating binary phase square spiral zone plate (BPSSZP) to produce a vortex laser with square symmetry. The projections in front of and behind the simulation box are the laser intensity of the Gaussian laser and the vortex laser, respectively. The electric field distributions of the Gaussian vortex and vortex laser are shown along the direction of laser propagation. (b) The profile of the left-handed (LH) BPSSZP with topological charge $l=-1$ and (c) the right-handed (RH) BPSSZP with topological charge $l=1$, where $d$ is the thickness of RH and LH BPSSZP, the black area is the plasma region, and the white area is the vacuum region.

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Fig. 2.
Fig. 2. (a) 3D isosurface distribution of the electric field $E_y$ at $t = 17 T_0$. The projection of $E_y$ on the $(y,z)$ plane at $x = 13.5 \lambda _0$ is shown on the right. The projection of laser intensity on the $(x,y)$ plane at the bottom is taken at $z = 0\lambda _0$. The projection plane of $(x,z)$ at the back is taken at $y = 0\lambda _0$. Here, $I$ represents the intensity of the output vortex laser. (b)-(d) The transverse distribution of $E_y$ at different sections of $x = 12.7\lambda _0$ to $13.7\lambda _0$ at $t = 17 T_0$ (simulation results). (e)-(g) Same as (b)-(d), but from the Fresnel-Kirchhoff diffraction formula.

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Fig. 3.
Fig. 3. BPSSZP with different topological charges and the diffraction pattern of laser intensity near the focal plane of $x = 13.5 \lambda _0$ at $t = 18T_0$. (a) $l=1$ case, (b) $l=2$ case, (c) $l=3$ case, (d) $l=4$ case.

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Fig. 4.
Fig. 4. (a) Evolution of the laser energy conversion efficiency from the Gaussian laser pulse to the vortex laser pulse. (b) Evolution of the total laser angular momentum (AM) in right-handed (RH) BPSSZP (black line) case and left-handed (LH) BPSSZP (red line) case, respectively. The gray area here represents the stage when the vortex laser is passing through the focal plane.

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Fig. 5.
Fig. 5. (a) Evolution of the energy conversion efficiency from the right-handed (RH) circularly polarized (CP) laser pulse to the vortex laser pulse in right-handed (RH) BPSSZP (black line) and left-handed (LH) BPSSZP (red line), respectively. (b) Evolution of the total angular momentum (AM) of the laser in right-handed (RH) BPSSZP (black line) case and left-handed (LH) BPSSZP (red line) case. The gray area here represents the stage when the vortex laser is passing through the focal plane. (c) The transverse distribution $E_y$ after the RH CP laser passes through RH BPSSZP ($x = 13.2\lambda _{0}$, $t = 17T_0$). (d) The transverse distribution $E_y$ after the RH CP laser passes through LH BPSSZP ($x = 13.2\lambda _{0}$, $t = 17T_0$).

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Fig. 6.
Fig. 6. (a) The total angular momentum of the vortex laser ($J_x$, black circles), the laser energy conversion efficiency to the vortex laser ($\eta$, red circles), and the ratio of vortex laser intensity to the incident laser intensity ($I / I_0$, blue circles) with the laser electric field normalized amplitude $a_0$. The solid black line is the fitting result. (b) Evolution of the laser photon average AM from the Gaussian laser pulse to the vortex laser pulse as $a_0$ varies from 0.1 to 5. The gray area here represents the stage when the vortex laser passes through the focal plane.

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Fig. 7.
Fig. 7. The distribution of electric field $E_y$ on the $(x,y)$ plane for the first laser pulse (a) ($t = 18 T_0$), the second laser pulse (b) ($t = 28 T_0$), and the third laser pulse (c) ($t = 38 T_0$). The insets in (a), (b), and (c) show the electron density distribution of BPSSZP $(x = 3.5 \lambda _{0})$ and the distribution of electric field $E_y$ on the $(y,z)$ plane at $x = 13.5\lambda _0$.

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Tables (1)

Tables Icon

Table 1. The thickness of BPSSZP and the corresponding laser electric field normalized amplitude.

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Equations (4)

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t BPSSZP ( r , θ ) = Bin [ t SSL ( r , θ ) ] = Bin { exp [ − i k   max ( y 2 , z 2 ) / 2 f ] exp ( i l θ ) } ,
Bin ( t ) = {     1           if         Im ( t ) > 0 − 1           if         Im ( t ) < 0     .
d = 1 2 ( 1 + c v p − c ) l λ 0 .
E ( y , z ) = 1 i λ 0 ∬ u 0 ( y ′ , z ′ ) t ( y ′ , z ′ ) k ( θ ) exp ⁡ ( i k ρ ) ρ d y ′ d z ′ ,
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