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Abstract
Relativistic vortex laser has drawn increasing attention in the laser-plasma community owing to its potential applications in various domains, e.g., generation of energetic charged particles with orbital angular momentum (OAM), high OAM X/γ-ray emission, high harmonics generation, and strong axial magnetic-field production. However, the generation of such relativistic vortex laser is still a challenge to the current laser technology. Using micro-structure targets named axial line-focused spiral zone plate (ALFSZP), we propose a novel scheme for ultra-intense vortex laser generation. In the scheme, a relativistic Gaussian laser pulse irradiates an ALFSZP, and diffracts as it passes through the ALFSZP. Due to the focusing and radial Hilbert transform capabilities of the ALFSZP, the seed laser is converted efficiently to a vortex one which is then well focused in a tunable focal volume. Three-dimensional particle-in-cell simulations indicate that using a seed laser pulse with intensity of 1.3 × 1020 W/cm2, the vortex laser intensity achieved is as high as 1.3 × 1021 W/cm2 with the averaged angular momentum per photon up to $0.73\hslash$, promising diverse applications in various fields aforementioned.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since Allen et al. [1] found the vortex beams carrying orbital angular momentum (OAM) along their propagation axis in 1992, the vortex laser has been widely applied in many domains spread over the microhertz to megaelectronvolt–gigaelectronvolt frequency ranges [2–10], such as microscopic particle control [3,4], microscopy imaging [5,6], and optical communication [7,8]. Unlike the normal Gaussian laser having nearly flat wave front, the vortex lasers have a helical wave front and hollow transverse intensity distribution. With these unusual properties, the vortex laser has been a unique tool to gain insight into the dynamics of particles. Over the past decade, vortex laser has also drawn wide attention in the research area of laser-plasma interaction [11–43]. Ones have extensively investigated theoretically and/or numerically the generation of energetic charged particles with OAM [14–16,20,24–28,32], high OAM X/$\gamma$-ray [11,29,43] emission and high harmonics generation [12,17,41] by using vortex laser-plasma interaction. Though the results are interesting and appealing, the required intensity of the vortex laser is around $10^{21}$ $\textrm{W}/\textrm{cm}^2$ in these studies. Unfortunately, such a relativistic vortex laser is still a challenging work for the current laser technology, principally due to the limit of the damage threshold of optical modulators [39]. Although a couple of feasible schemes have been reported recently, like light fan [13], Raman amplification [18], plasma holograms [22,33], plasma $q$-plate [23], spiral-shaped foil [31], spiral phase plasma [37], and azimuthal plasma phase slab [38], the highest intensity of vortex laser pulse achieved experimentally is still lower than $10^{20}$ $\textrm{W}/\textrm{cm}^{2}$ [21,39,40]. New schemes for the generation of ultra-intense vortex laser with intensity above $10^{21}$ $\textrm{W/cm}^2$ are in high demand.
Spiral zone plate (SZP) is a kind of diffraction optical elements to generate optical vortexes [44–46]. The SZP is modified from Fresnel zone plate, which can perform radial Hilbert filtering operation and the focusing operation in one step. The SZPs have widely applications in micro-manipulation [3,4,47], microscopy [5,48], and X-ray astronomy [49] due to their compact sizes, small weights, and design flexibility. However, the optical vortexes diverge rapidly after passing through the focal point owing to the SZP structure. In order to overcome this limitation, Zheng et al. [50] designed a novel diffraction optical element called axial line-focused spiral zone plates (ALFSZP), to create optical vortexes with a long focal length along the propagation axis in tradition optics. Compared to the SZP, the change of ALFSZP relies on the adjustment of the number of wave zones $N$ and the width variation between adjacent wave zones $\delta$d of the SZPs. Note that, the fabricating technology of this diffraction optical element has also been possible [50]. For example, with the laser etching technology, the fabrication process of these optical elements includes four steps consisting of (1) chromium layer and photoresist coatings, (2) exposure, (3) chromium etching, and (4) resist removal.
In this paper, we present a plasma-based scheme to generate ultra-intense vortex laser pulse by employing an ALFSZP. In the scheme, a Gaussian laser pulse irradiates the ALFSZP. Since the ALFSZP is modified from SZP, it can convert the incident seed laser pulse to a vortex one and focus the vortex pulse simultaneously. After passing through the ALFSZP, the seed laser is converted to an ultra-intense vortex laser pulse in a tunable focal volume. Three-dimensional particle-in-cell (3D-PIC) simulations show that a vortex laser with the focused intensity above $10^{21} \textrm{W/cm}^2$ can be achieved. The averaged angular momentum (AM) of the vortex laser photon is up to $0.73\hslash$. Such an ultra-intense vortex laser may trigger many potential applications in various domains, enabling future experimental tests of vortex-laser-plasma interaction in the ultra relativistic regime and opening a practical avenue for investigating high AM physics in the future.
2. Results and discussions
2.1 Overview of the scheme
Since the ALFSZP is modified from SZP, we introduce SZP at first. The SZP is designed by combining the focusing capability of Fresnel zone plate and radial Hilbert transform. Therefore, SZP can generate and focus the vortex laser in its focal point. The radial Hilbert phase function can be expressed as $H_p(r,\phi )=\textrm{exp} (il\phi )$, where $l$ represents the topological charge, and ($r$, $\phi$) are the polar coordinates. The Fresnel zone plate phase function can be written as $F_p(r,\phi )=\textrm{exp} (-i\pi r^2/\lambda_0 f)$, where $\lambda_0$ is the wavelength of incident seed laser, and $f$ is the focal length of the Fresnel zone plate corresponding to the wavelength of $\lambda_0$. The phase function of the SZP is obtained by multiplying the radial Hilbert phase function with the Fresnel zone plate phase function, which can be expressed as [46]
The phase function of the SZP in Eq. (1) can be binarized as a transmittance function:For the ALFSZP, it has a small variation $\delta d$ of the widths of adjacent wave zones. This variation of the widths introduces a phase correction $\xi$ in the transmittance in Eq. (2). Using the relationship between the radius $r$ and azimuth $\phi$, the phase correction $\xi$ can be expressed as $\xi =2m\pi - \pi [2m\lambda_0 f_{min}+m(m-1)\lambda_0 \delta f /N]/(\lambda_0 f_m)$. Thus, the transmittance of the ALFSZP can be written as [50]
Figure 1 illustrates schematically our scheme and the key features of the produced vortex laser beams. In order to demonstrate the feasibility of the proposed scheme, we performed full 3D-PIC simulations with the open-source code EPOCH [51]. The whole process of the ultra-intense vortex laser pulse generation can be divided into two stages. In the first stage, a seeded linearly polarized (LP) Gaussian laser pulse with the dimensionless laser electric field amplitude $\textbf {a}=a_0 \textrm{exp} \big (-r^2/\sigma _0^2\big ) \textrm{cos} \varphi \textbf {e}_y$ is incident from the left side of the simulation box, where $a_0=(eE_0)/(m_e c \omega _0 )=10$, $\sigma _0=10\lambda _0$ is the laser focus spot size, $\lambda _0=cT_0=1$ $\mu m$ is the laser wavelength, $T_0$ is the laser cycle, $\varphi$ is the laser phase term, $\omega _0$ and $E_0$ are the laser frequency and the electric field amplitude. $e$, $m_e$, and $c$ are the unit charge, the electron mass, and the speed of light in vacuum, respectively. The seed pulse has a Gaussian time profile and duration is $6T_0$. The grid size of the simulation box is $30\lambda _0(x) \times 36\lambda _0(y) \times 36\lambda _0(z)$, sampled by $1200\times 1440\times 1440$ cells with 9 macro-particles per cell. In the simulations, the parameters of ALFSZP is settled as $f_{min}=12 \lambda _0$, $f_{max}=22 \lambda _0$, and $N=10$. Therefore the focal volume starts from $x=12 \lambda _0$ and ends at $x=22 \lambda _0$. The ALFSZP is located at $x=3\lambda _0$ initially with thickness of $d_0=0.1\lambda _0$. It is right-handed (RH) and consists of fully ionized carbon and hydrogen ions with a 1:4 density ratio of carbon to hydrogen. The corresponding density of electrons, carbon ions ($C^{6+}$), and protons ($H^+$) are $100n_c$, $10n_c$ and $40n_c$, respectively, where $n_c=(m_e \varepsilon _0 \omega _0^2)/e^2$ is the critical density and $\varepsilon _0$ is the vacuum dielectric constant. With the development of precision machining technology, it has shown that the precision of metal-assisted chemical etching [52] and electron-beam lithography [53] can reach 51 nm and 40 nm, respectively. The minimum width of ALFSZP micro-structured target employed in our scheme is around $\sim$0.5 $\mu m$. Therefore, such a micro-structured ALFSZP target can be fabricated by metal-assisted chemical etching and electron-beam lithography in the current laboratories.
Fig. 1. (a) Schematic of ultra-intense vortex laser beam generation from a seed laser-illuminated axial line-focused spiral zone plate (ALFSZP). A Gaussian laser pulse is incident from the left and irradiates the ALFSZP. This finally results in the generation of an ultra-intense vortex laser pulse. Here, the projections are the electric fields of incident seed laser and vortex laser, respectively. (b) The transverse profile of an ALFSZP with the topological charge $l=1$.
2.2 Simulation results
When the incident seed laser pulse irradiates the ALFSZP, the laser can pass through only the transparent region of the ALFSZP. Since the width of wave zones of ALFSZP varies with the radius and azimuth, the output laser pulse will obtain the azimuth phase difference, so that the seed pulse can be converted to a vortex laser pulse in the focal volume. Figure 2(a) shows the 3D isosurface distribution of the electric field $E_y$ at $t=19T_0$. One sees that the output laser pulse has an obvious vortex electric field $E_y$ with a hollow intensity distribution. In order to evaluate the performance of ALFSZP, the diffraction electric field is calculated by the Fresnel-Kirchoff’s diffraction formula. Here, the diffraction field can be expressed as
Fig. 2. (a) 3D isosurface distribution of the electric field $E_y$ at $t=19T_0$. The ($y$, $z$) projection plane of electric field $E_y$ on the right side is taken at $x=14.5\lambda _0$. The ($x$, $y$) projection plane of laser intensity at the bottom is taken at $z=0\lambda _0$, and the ($x$, $z$) projection plane at the backside is taken at $y=0\lambda _0$. Here $I$ and $I_0$ represent the intensity of output vortex laser and incident seed laser, respectively. (b)-(d) The distribution of transverse electric field $E_y$ at different cross-sections ranging from $x=13\lambda _0$ to $15\lambda _0$ at $t=18T_0$ (simulations results). (e)-(g) The same to (b)-(d) but from the Fresnel-Kirchoff’s diffraction formula in Eq. (4).
Note that the output laser passing through ALFSZP is usually not a pure Laguerre-Gaussian (LG) mode. To evaluate the weight of different LG modes in the output vortex laser, we select the cross section of electric field $E_y$ of the output laser at the position of $x=14\lambda _0$ at $t=19T_0$, and calculate the corresponding intensity ratio of different LG modes. Here, the weight of LG mode is defined as [54]
where $E_y(r,\phi,x)$ represents the cross section of the electric field $E_y$ of the output laser, and $E_{lp}(r,\phi,x)$ is the cross section of the electric field of LG$_{lp}$ mode laser. Thus the $E_{lp}(r,\phi,x)$ can be defined in the cylindrical coordinate system $(r,\phi,x)$ byFig. 3. (a) Laguerre-Gaussian (LG) mode spectrum at $x=14\lambda _0$ at $t=19T_0$. The azimuthal mode number $l$ is on the $x$-axis, the radial mode number $p$ is on the $y$-axis, and the weight of the LG$_{lp}$ modes is displayed on the vertical $z$-axis. (b) Transverse section of the vortex laser’s intensity at $x=14\lambda _0$ at $t=19T_0$. (c) Distribution of the laser intensity along $x$-axis at $t=12T_0$, $16T_0$, $19T_0$, and $22T_0$. Here the gray area marks the distribution of the focal volume along the $x$-axis. (d) Chart of input seed laser intensity and output vortex laser intensity (W/cm$^2$) via light fan [13], Raman scattering [18], plasma $q$-plate [23], spiral-shaped foil [31], plasma volume holograms [33], spiral phase plasma [37], azimuthal plasma phase slab [38], and our ALFSZP scheme, respectively. The three dotted lines represent only the output vortex laser intensity achieved in three experiments [21,39,40] without the input seed laser intensity reported.
Figure 3(d) presents the recent progress on the vortex laser generation via various schemes. According to the way of vortex phase introduced, these schemes could be classified into four types: (1) Using the azimuthal optical path difference to introduce the vortex phase, such as light fan [13], spiral-shaped foil [31], spiral phase plasma [37], and azimuthal plasma phase slab [38]. Compared with the seed laser, the intensity of the output vortex lasers almost unchanges in these schemes. (2) Combining and/or amplifying the seed lasers by Raman scattering [18]. The intensity of the output vortex laser in this regime is one order of magnitude higher than that of the seed laser. (3) Using the anisotropic parameter space of light propagation path, like plasma $q$-plate [23]. Due to the interaction between the seed laser and plasmas, the output vortex laser intensity is one order of magnitude lower than that of the seed laser. (4) Using the diffraction to reproduce or introduce the vortex phase, like plasma volume holograms [33] and ALFSZP. Owing to the focusing capability of the hologram and ALFSZP, the intensity of the output vortex laser can increase by 100 or 10 times compared to the seed laser, significantly higher than the light fan [13]. This indicates that our scheme is very competing in terms of enhancing the intensity of the vortex laser. To the best of our knowledge, the highest intensity of vortex laser pulse achieved experimentally is around 10$^{20}$ W/cm$^2$ with a total power of tens of TW [40]. Our scheme thus provides a possible way to achieve ultra-intense vortex laser with an intensity above $10^{21}$ $\textrm{W/cm}^2$ in a new regime. Actually, the enhanced vortex laser can be also achieved by using multilevel diffractive optical elements (DOEs), since the DOEs play an important role in increasing the diffraction efficiency as compared to the binary DOEs. Recently, various DOEs have been proposed, for example, to generate square optical vortices [55]. This may be also interesting for the laser-plasma community.
In order to demonstrate that the ALFSZP is capable of increasing the laser photon’s AM, we calculate the averaged AM of laser photons in the focal volume. For a laser pulse, its total electromagnetic AM and total electromagnetic energy can be expressed as ${\textbf{L}}_{laser}=\varepsilon _0 \int \textbf {r} \times (\textbf {E} \times \textbf {B})dV={\textbf{L}}_x+{\textbf{L}}_y+{\textbf{L}}_z$ and $E_{laser}=\frac {1} {2} \int \big (\varepsilon _0 \textbf {E}^2+ \frac {1} {\mu _0} \textbf {B}^2\big )\textrm{d}V$, respectively. Here, $\mu _0$ is the vacuum permeability. Therefore, each photon’s averaged AM can be written as
Fig. 4. (a) Evolution of averaged angular momentum (AM) of laser photons (black line) and energy conversion efficiency to the vortex laser pulse (red line). (b) Evolution of laser total AM in the right-handed (RH) ALFSZP (black line) and left-handed (LH) ALFSZP (red line). Here the gray area marks the stage when the vortex laser is in the focal volume.
Additionally, we also considered a left-handed (LH) ALFSZP ($l=-1$) in our simulations, but keep all other parameters unchanged. Figure 4(b) shows the evolution of laser total AM in the RH ALFSZP and LH ALFSZP, respectively. As expected, one sees a basically symmetric evolution of the $L_x$. While the laser is in the focal volume, the total AM of the vortex pulse increases and reaches $9.37 \times 10^{-17} \textrm{kg} \cdot \textrm{m}^2/\textrm{s}$. When the laser pulse leaves the focal volume, the total AM of the vortex laser decreases. This is also in excellent with the predictions as analyzed above.
3. Discussions
We first investigate the generation of high order mode vortex laser pulse by using such a micro-structured target. In order to verify the robustness of the ALFSZP, we change the target parameters $l$ to 2 and 3, respectively. Figure 5(a) and 5(b) show the transverse structure of ALFSZPs with $l=2$ and $l=3$, respectively. We here keep all other parameters unchanged. As we can see, the two targets have different transverse profiles which depends on the topological charge $l$. Figure 5(c)–5(f) show the distribution of 3D isosurface and cross-sections of the electric field $E_y$ with $l=2$ and $l=3$, respectively. With different parameter $l$, the dominant mode of the vortex laser changes. When $l=2$, the dominating mode of the vortex laser is LG$_{20}$. When we change $l$ to $3$, the dominating mode of the vortex laser is LG$_{30}$. Therefore, we can manipulate the laser mode easily by changing the target parameter $l$.
Fig. 5. Transverse structure of an ALFSZP with $l=2$ (a) and $l=3$ (b), respectively. 3D isosurface distribution of electric field $E_y$ in the $l=2$ case (c) and $l=3$ case (d) at $t=18T_0$, respectively. Distribution of the transverse electric field $E_y$ at $x=12.5\lambda _0$ in the $l=2$ case (e) and $l=3$ case (f) at $t=18T_0$, respectively.
The scheme proposed is also valid for the circularly polarized (CP) laser case. We here considered a LH CP seed laser in our simulations with all other parameters unchanged. As we can see in Fig. 6(a) and 6(b), both the distributions of the $E_y$ and $E_z$ show a typical vortex laser feature and the phase displacement between these two components is $\pi /2$. To illustrate the feature of such a LH CP vortex laser in our scheme, we convert the transverse electric fields into $E_r$ and $E_\phi$ in the cylindrical coordinate system which can be expressed to $E_r=E_y\textrm{cos}\phi +E_z \textrm{sin}\phi $ and $E_\phi =-E_y\textrm{sin}\phi + E_z\textrm{cos}\phi $, respectively. As shown in Fig. 6(c)–6(d), both the distribution of the $E_r$ and $E_\phi$ are basically circularly-symmetric in the transverse plane. The $E_r$ and $E_\phi$ of the LH CP LG$_{10}$ laser can be expressed as
Fig. 6. Distribution of electric fields $E_y$ (a) and $E_z$ (b) on the ($y$, $z$) plane at $x=13.5 \lambda _0$ in the circularly polarized output vortex laser case at $t=20T_0$. Distribution of electric fields $E_r$ (c) and $E_{\phi }$ (d) in a cylindrical coordinate system.
We investigated the influences of the laser and target parameters on the final vortex laser generation. First we discuss the effect of the laser intensity on the vortex beam generation, where we keep all other parameters unchanged but vary the normalized seed laser amplitude from $a_0=1$ to $20$. Figure 7(a) shows the scaling of the laser total AM ($L_{x}$, black circles), the energy conversion efficiency to the vortex laser pulse ($\eta$, red circles), and the ratio of output vortex laser intensity to the seed laser intensity ($I/I_{0}$, blue circles) with the laser electric field amplitude $a_0$. It indicates that the laser total AM rises with the increase of the seed laser intensity. As we know, the laser total AM depends on the photon number and the averaged AM per photon, which can be expressed as $L_x \propto N(\delta + l)\hslash$. When the laser wavelength is constant, the relation between the photon number and $a_0$ is $N=E_{laser}/ \hslash \omega _0 \propto a_0^2$. The laser total AM can therefore be approximated by $L_x \propto a_0^2$, which is in excellent agreement with the results as shown in Fig. 7(a). However, the energy conversion efficiency to the vortex laser pulse, and the ratio of vortex laser intensity to the seed laser intensity are insensitive to the laser electric field amplitude $a_0$. This indicates that the ALFSZP is fit for a wide laser intensity range with the condition $a_0 < \pi n_e(n_c) d_0(\lambda _0)$, where $n_e$ is the electron density of the ALFSZP. Taking $a_0=20$ for example, the laser total AM, energy conversion efficiency to the vortex laser pulse, and the ratio of vortex laser intensity to the seed laser intensity can reach $3.57 \times 10^{-16} \textrm{kg} \cdot \textrm{m}^2/\textrm{s}$, 8.8$\%$ and 9.4, respectively, indicating an efficient vortex laser generation.
Fig. 7. Scaling of the laser total AM ($L_{x}$, black circles), the energy conversion efficiency to the vortex laser ($\eta$, red circles), and the ratio of vortex laser intensity to driven laser intensity ($I/I_0$, blue circles) with the laser electric field amplitude $a_0$ (a) and the thickness of ALFSZP $d_0$ (b). Here, the solid black curve in (a) is the fitting result.
We then considered the effect of the thickness of ALFSZP on the vortex laser generation. Figure 7(b) shows the simulation results, where $d_0$ is varied in the range of $0.05\lambda _0$ to $0.5\lambda _0$, while all other parameters are unchanged. It shows that the laser total AM and energy conversion efficiency to the vortex laser are insensitive to the thickness of ALFSZP (within the range of lengths considered). Under the condition of $a_0 < \pi n_e(n_c) d_0(\lambda _0)$, the seed laser can not pass through the plasma. As long as the transparent region of the ALFSZP and the seed laser spot size are constant, the energy conversion efficiency from the seed laser to the vortex laser has no obvious changes. However, the ratio of vortex laser intensity to the seed laser intensity slightly decrease with the increase of the thickness of ALFSZP. Since such a micro-structure target can now be fabricated by vertical directionality controlled metal-assisted chemical etching [52] or electron beam lithography [53]. Our scheme thus provides a practical and efficient way to generate ultra-intense vortex laser by designing the laser and target parameters, which could be tested in the current 100 TW laser system.
Finally, we take into account of the damage of the ALFSZP target in experiments, which is one of challenges in experiments of ultra-intense laser interacting with micro-structured targets. The destruction of micro-structure targets may require the replacement in each experiments, which is undesirable and fatal to high-frequency lasers. Here, we consider the performance of the ALFSZP target under multi-laser irradiation. Figure 8 shows the distribution of electric field $E_y$ at the ($x$, $y$) plane by triple-laser pulses with the same delay of $\tau =10T_0$. As we can see, the output laser still has a hollow intensity distribution and keeps almost intact after the third laser pulse passes through the target. This is owing to the ALFSZP modulation capability determined by transverse profile while the laser damage is mainly in the longitudinal direction. Therefore, a thicker target can undergo the irradiation of multi-laser pulses in experiment. We also considered the preplasmas in additional simulation, where all other parameters are unchanged except the plasma distribution. In the simulation, the scale length $L$ of the preplasmas ($L=(d \textrm{ln}n/dx)^{-1}$) is set to be $L=0.1 \lambda _0$ ranging from $2.9$ to $3.0$ $\mu m$ in the $x$ direction. The simulation results show that the electric field $E_y$ distribution is in substantial agreement with the previous simulation results. This indicates that the influence of the preplasmas is wispy. By using the state-of-the-art plasma mirror technology, the influence of preplasmas can be further mitigated.
Fig. 8. Distribution of electric field $E_y$ on the ($x$, $y$) plane and ($y$, $z$) plane (at $x=14.5\lambda _0$) for the first pulse (a) ($t= 20 T_0$), second pulse (b) ($t= 36 T_0$), and third pulse (c) ($t=52 T_0$).
4. Conclusion
In conclusion, we have proposed and numerically demonstrated a laser-plasma scheme to produce ultra-intense vortex laser by high-power Gaussian laser pulse illuminating an ALFSZP target. 3D-PIC simulations show that the output vortex laser pulse is characterized by ultra-high intensity (∼ 1021 W/cm2), large AM (∼ 9.37 × 10−17kg · m2/s), and small focus spot size (∼ 1.5μm). The energy conversion efficiency to be vortex laser pulse can reach 10%. With tunable focal volume and topological charge l of ALFSZP, we can manipulate the focal area and the dominant mode of the vortex laser. Our scheme provides a promising and practical avenue to generate ultra-intense vortex laser pulse for multi applications in an ultra-relativistic regime, such as particles acceleration, high OAM X/γ-ray emission and harmonics generation.
Funding
National Key Research and Development Program of China (2018YFA0404802); National Natural Science Foundation of China (11875319, 12004433, 12135009); Science and Technology Innovation Program of Hunan Province (2020RC4020); The Open Fund of the State Key Laboratory of High Field Laser Physics (SIOM); Natural Science Foundation of Hunan Province (2020JJ5649, 2021JJ40657); Research Project of NUDT (ZK19-12, ZK20-36); Hunan Provincial Innovation Foundation for Postgraduate (CX20200002, CX20200038, CX20210062).
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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