Slits have been widely used in laser–plasma interactions as plasma optical components for generating high-harmonic light and controlling laser-driven particle beams. Here, we propose and demonstrate that periodic thin slits can be regarded as a new breed of optical elements for efficient focusing and guiding of intense laser pulse. The fundamental physics of intense laser interaction with thin slits is studied, and it is revealed that relativistic effects can lead to enhanced laser focusing far beyond the pure diffractive focusing regime. In addition, the interaction of an intense laser pulse with periodic thin slits makes it feasible to achieve multifold enhancement in both laser intensity and energy transfer efficiency compared with conventional waveguides. These results provide a novel method for manipulating ultra-intense laser pulses and should be of interest for many laser-based applications.

Mourou and Strickland were jointly awarded the 2018 Nobel Physics Prize for the chirped pulse amplification (CPA) technique, which was proposed in the 1980s.1 Relativistic field intensities of more than 1018 W/cm2 can be produced by this technique, enabling the investigation of brand new physics and strong-field phenomena in laser–matter interactions.2 Nowadays, the most powerful laboratory laser systems can already reach intensities of the order of 1023 W/cm2.3 The utilization of high-intensity lasers interacting with matter drives the generation of many particles and types of radiation, such as high-energy electrons, ions, neutrons, and X/γ rays.1,2 Featuring short pulses, small focal spots, high brightness, and flexible tunability, these particle and radiation sources have attracted great interest in a wide range of research areas.

Particles and radiation generated by relativistic lasers are highly dependent on the spatial and temporal properties of the laser field, and therefore effective manipulation of lasers at relativistic intensities is crucial for the study of intense laser–matter interactions.4 The conventional approach to manipulating light is based on the use of solid-state optical media, but this is limited by the amount of power that can be applied to such media. By contrast, plasmas can tolerate much higher laser intensities than conventional optical elements, which makes it possible to manipulate light at extreme intensities and energy densities. A variety of plasma-based optical components, including amplifiers,5–8 gratings,9–13 mirrors,14–17 waveplates,18–20 and plasma waveguides,21,22 have been studied.

In recent years, the interaction of intense lasers with slits has attracted much attention. It has been demonstrated that slits can be used as plasma optical devices for applications to high-order harmonic generation23,24 and modulation of laser-driven electron25 or proton26 beams. The phenomenon of single-slit diffraction, first observed by Francesco Grimaldi in 1665,27 has undergone extensive analysis with regard to both the near-field and far-field diffraction patterns. Recent experiments have revealed the diffractive mechanism of focusing from a single slit28 and have demonstrated several aspects of this effect for different types of waves.29 In addition, diffractive guidance of waves by utilizing slits to periodically truncate their edges has also been achieved.30 However, up until now, investigations of the use of slits to achieve focusing and guidance have been limited mostly to nonrelativistic light-wave conditions, and the laser–slit interaction in relativistic regimes remains unclear.

In this paper, we investigate the fundamental physics of the interaction of an intense laser with thin slits, and we propose that the employment of periodic thin slits can simultaneously achieve focusing enhancement and long-distance guidance of the intense laser. It is revealed that the relativistically induced transparency at the edges of the slits and the interference between diffractive and transmitted beams contribute to the enhancement of focusing. Additionally, the high transmission rate during induced transparency leads to multiple focusings and superior energy transfer efficiency. It is shown that by appropriately adjusting the parameters, multifold enhancement in both laser intensity and energy transfer efficiency can be achieved compared with the conventional waveguides, where in the latter case the laser energy is strongly depleted.

The present scheme is mainly demonstrated using a two-dimensional particle-in-cell (PIC) simulation with the EPOCH code.31 The simulation box is from 0 to 40 µm in the x direction and from −20 to 20 µm in the y direction. The number of sampling cells in the simulation box is determined by the slit thickness and the skin depth lskin = c/ωp, where c is the light speed in vacuum and ωp is the plasma frequency. There are at least ten cells in the longitudinal laser propagation direction to resolve individual slits, in addition to resolving the skin depth. For example, for a slit thickness of d = 20 nm, when the simulation box is sampled by 20 000 × 2000 cells, the corresponding spatial resolution is 2 nm longitudinally and 20 nm transversally to resolve the slit thickness and skin depth, respectively. There are 30 macroparticles of electrons and 10 macroparticles of both H+ and C6+ per cell.

The y-polarized laser is focused to a spot of radius w0 = 7 µm located at x = 20 µm, with a Gaussian temporal envelope of FWHM duration τ0 = 33.3 fs. It has a wavelength λ = 1 µm and a normalized laser amplitude a0 = eE0/meL = 5, where e, E0, me, and ωL are the elementary charge, laser amplitude, electron mass, and laser frequency, respectively. A moving window that follows the propagation of the laser pulse after t = 40T0, with T0 = λ/c the laser cycle, is used to improve computational efficiency.

The target comprises a periodic array of slits with a total length of L = 160 µm, greater than the Rayleigh length ZR=πw02/λ=153.9 µm. The interval distance between adjacent slits is ds, and each slit has the same width w, thickness d, and electron density ne = 50nc (the effects of electron density will be discussed later), where nc=ϵ0meωL2/e2 is the critical density. The initial position of the target is located at x = 20 µm, which is the same as the focusing position of the laser pulse.

Here we start by studying the interaction process of an intense laser with a single slit. To clarify the relativistic effects in intense laser–slit interactions, the results of nonrelativistic laser interaction with a single slit are also shown for comparison. Figures 1(a) and 1(b) show that there is little difference between the two cases of low-intensity and high-intensity lasers at t = 24T0, when the spatial structure of the slit is unchanged, as presented in Fig. 1(c) for the electron density distribution along the y axis. The reflected laser field is self-consistently included. Figures 1(d)1(f) show the results at t = 44T0, when the laser pulse has interacted with the slit for 24 cycles. One can clearly see that under the interaction of the relativistically intense laser, the slit undergoes a significant expansion and the laser field can penetrate through the slit edges. This is due to the relativistic motion of plasma electrons, which reduces the effective target electron density and thus the corresponding plasma frequency,
ωpγ2=nee2γϵ0me,
(1)
where γ is the Lorentz factor of the electrons. This corresponds to a relativistically corrected critical density n = γnc. As can be seen in Fig. 1(f), the electron density is decreased notably, and relativistically induced transparency occurs at the edges of the slit. Because the corrected plasma frequency in Eq. (1) is below the laser frequency ωL,32 the effective width indicated by ne = n (the red dotted line) is wider than the initial one (the black dotted line).
FIG. 1.

The top row shows PIC simulation results for a single slit with w = 2.6 µm and d = 70 nm at t = 24T0 (79.92 fs): (a) and (b) electron density and Ey field distributions in the (x, y) plane for a0 = 0.5 and a0 = 5, respectively; (c) electron density for an intense laser field (a0 = 5) along the y axis averaged over x = 2–2.07 µm where the single slit was initially located on the x axis. The bottom row (d)–(f) shows the corresponding results at t = 44T0 (146.52 fs). The dashed black lines indicate the initial positions of the single-slit edges.

FIG. 1.

The top row shows PIC simulation results for a single slit with w = 2.6 µm and d = 70 nm at t = 24T0 (79.92 fs): (a) and (b) electron density and Ey field distributions in the (x, y) plane for a0 = 0.5 and a0 = 5, respectively; (c) electron density for an intense laser field (a0 = 5) along the y axis averaged over x = 2–2.07 µm where the single slit was initially located on the x axis. The bottom row (d)–(f) shows the corresponding results at t = 44T0 (146.52 fs). The dashed black lines indicate the initial positions of the single-slit edges.

Close modal

It should be noted that the field enhancement for the high-intensity laser is larger than that in the case of the nonrelativistic laser. This can be understood from two aspects. On the one hand, for ne < n, the phase velocity of the laser propagating in the plasma, vph=c/1ωpγ2/ωL2, is larger than the speed of light in vacuum, and thus the phase difference between the edge part and central part of the laser will lead to a further focusing of the laser field. On the other hand, the relativistically induced transparency leads to an enhanced laser transmission rate (which will be discussed later), and then the laser intensity will be increased by the constructive interference of the diffracted and transmitted beams.33 As the slit evolves spatially and temporally, the focusing mechanism of the intense laser gradually changes from classical diffractive focusing to well-enhanced focusing owing to relativistic plasma effects. It is clear that the fundamental laser frequency component dominates the focusing process.

The slit width and thickness have a significant influence on the focusing effect, which can be observed through two parameters: the maximum electric field Ey max and the transmission rate. Ey max is the maximum value of Ey that can be obtained at the rear side of the target during propagation of the laser after its interaction with the slit. The transmission rate is defined as the ratio of the total laser energy transmitted through the slit aperture WT to the input total laser energy, where the transmitted laser energy WT is calculated by integrating the Poynting vector P of the electric field over all transverse planes and the interaction duration after the laser has propagated through the target: WT = ∬P · dSdt. We mainly use the Poynting vector in the x direction for calculation, since the values in other directions are negligible.

To investigate the impact of these parameters, a detailed parameter scan was conducted. Figure 2(a) shows the maximum Ey as a function of slit thickness for w = 2.6 µm. The results from the PIC simulations are compared with the classical optical diffraction simulation based on the Kirchhoff diffraction formula in two dimensions:
U(M)=14πSUGnGUndS,
(2)
where U is the spatial part of the solution of the homogeneous scalar wave equation, M is an arbitrary observation point on an arbitrary closed surface S, and G is the Green’s function, taken to be a fundamental solution of the Laplace operator.
FIG. 2.

(a) Maximum Ey (a) at the rear side of the target and (b) transmission rate for a0 = 0.5 and a0 = 5 plotted as functions of thickness for w = 2.6 µm. (c) and (d) Corresponding quantities plotted as functions of width for d = 20 nm and d = 70 nm. The dashed black lines in all the plots indicate the corresponding results of a classical optical diffraction simulation obtained from the measurements using the Kirchhoff diffraction formula.

FIG. 2.

(a) Maximum Ey (a) at the rear side of the target and (b) transmission rate for a0 = 0.5 and a0 = 5 plotted as functions of thickness for w = 2.6 µm. (c) and (d) Corresponding quantities plotted as functions of width for d = 20 nm and d = 70 nm. The dashed black lines in all the plots indicate the corresponding results of a classical optical diffraction simulation obtained from the measurements using the Kirchhoff diffraction formula.

Close modal

As shown in Figs. 2(a) and 2(b), the laser focusing depends critically on the slit thickness in the high-intensity case (a0 = 5), and is greatly enhanced compared with the low-intensity case (a0 = 0.5), where the slit thickness has little impact on the results. This conclusion agrees well with the classical optical diffraction simulation results denoted by the dashed black line, because Kirchhoff’s theory of diffraction from Eq. (2) is independent of the slit thickness. However, there exists an optimal slit thickness for the high-intensity case (a0 = 5), as shown in Fig. 2(a). If the target is too thin, the intense laser can easily penetrate through the edges of the slit. In such a case, the transmission rate is very high, as shown in Fig. 2(b), but the diffraction focusing process becomes ineffective. On the contrary, for targets much thicker than the skin depth, the transmitted intensity is too weak to contribute to the interference.33 Therefore, there exists an optimal thickness around d ∼ 70 nm, as shown in Fig. 2(a), which is due to the noticeable interference of transmitted and diffracted beams.

The slit width is another factor that influences the maximum Ey strength and transmission rate, as shown in Figs. 2(c) and 2(d), respectively. Unlike the diffraction focusing of the low-intensity laser that is always consistent with Kirchhoff’s theory of diffraction, we find that a remarkable focusing gain of the relativistic laser can be obtained with suitable slit parameters. In particular, Fig. 2(c) shows that in the relativistic regime, the optimal slit width for the maximum focusing electric field also depends critically on the slit thickness. For an intense laser interacting with thin slits, the relativistic effects enable the laser to penetrate through the slit edges owing to the radiation pressure or the skin effect, and so the effective slit width weff is larger than its initial one w. The radiation pressure gives the effective width as
a0expw124w02athπnencdλ,
(3)
which means that the laser pulse is able to push all electrons away from the foil by its radiation pressure. The skin effect gives the effective width as
a0expw224w02expdlskin,γ1,
(4)
where lskin,γ = c/ω and γ=1+a2/2 is the average Lorentz factor of electrons in the local laser field of amplitude a. This condition means that even when the laser field amplitude a is smaller than ath, the laser can still penetrate through the slit foil when the slit is much thinner than the skin depth.

For a thin slit with d = 20 nm, much smaller than the relativistically corrected skin depth lskin,γ, the target is almost transparent to the intense laser, which presents a very high transmission rate of more than 80% for different slit widths, as shown in Fig. 2(d). In the case of a thin slit, the effective width of the slit weff is actually determined by the relativistically induced transparency width given by Eqs. (3) and (4), which respectively give w1 ≈ 9.5 µm and w2 ≈ 13.5 µm. Therefore, for thin slits of widths w < max(w1, w2), the effective width is the same, and the focusing should depend slightly on the width, as shown in Fig. 2(c). It shows that the slit width w does not significantly affect the focusing gain, especially for w < w1. Although there exists a small peak on the curve, the discrepancy in the focusing gain is less than 10% for different widths within w < w1. The small peak corresponds to a width of w = 6 µm. For slit widths smaller than this value, the laser converts more energy into plasmas and the lower transmission rate leads to a smaller focusing gain, while for slit widths larger than this value, the diffraction effect and the light interference become relatively weak.

For a thick slit with d = 70 nm, much larger than the skin depth, there are no solutions to Eqs. (3) and (4), which means that the target is opaque to the laser pulse. In this case, the gain factor of the laser electric field decreases as the slit width increases, although the transmission rate increases, as shown in Figs. 2(c) and 2(d). This is mainly because, for a narrower width, the slit experiences a stronger laser field at its edges and expands more efficiently, which further leads to a reduction in the local effective electron density near the edge and then to enhanced focusing. In this sense, once the slit width satisfies a0exp(w2/4w02)1, i.e., the laser field at the slit edges is relativistic, focusing gain beyond pure diffractive focusing should be achievable. This gives a theoretical width of 2w0lna0, which is about 17 µm for the present parameters and matches well with the simulation results, as shown in Fig. 2(c). In particular, when the slit width approaches this value, the results for different laser intensities and slit thicknesses all become identical. It is interesting to note that in the thick-slit case, the transmission rate of the intense light is almost the same as that in the low-intensity case, as shown in Fig. 2(d). This is because the energy of the transmitted intense laser is partially converted into plasma near the slit edge, which in turn acts to enhance the focusing.

In this subsection, we discuss the interaction of an intense laser with periodic slits. It is found that periodic slits can further enhance the focusing effect compared with the single-slit case. Figure 3 shows a comparison of the interaction of an intense pulse with different targets. In contrast to the single slit in Fig. 3(a) and the waveguide in Fig. 3(c), it can be seen that the laser pulse interacting with periodic slits in Fig. 3(b) achieves a greater focusing gain. This is more notable after 80 cycles of propagation at t = 139T0, as shown in Figs. 3(d)3(f).

FIG. 3.

The top row shows simulation results comparing the interactions of a relativistically intense laser with different targets of the same width w = 2.6 µm at t = 59T0: (a) a single slit of thickness d = 20 nm; (b) periodic slits of thickness d = 20 nm and adjacent spacing ds = 16 µm; (c) a waveguide. The bottom row (d)–(f) shows the corresponding results at t = 139T0.

FIG. 3.

The top row shows simulation results comparing the interactions of a relativistically intense laser with different targets of the same width w = 2.6 µm at t = 59T0: (a) a single slit of thickness d = 20 nm; (b) periodic slits of thickness d = 20 nm and adjacent spacing ds = 16 µm; (c) a waveguide. The bottom row (d)–(f) shows the corresponding results at t = 139T0.

Close modal

As shown in Fig. 3(d) for the single-slit case, after the focusing process, the laser undergoes diffraction in vacuum, and its intensity is reduced. It can be seen from Fig. 3(f) that in the case of the waveguide, the laser is strongly depleted and has a rather weak intensity. By contrast, Fig. 3(e) shows that the laser is well-guided in periodic thin slits and retains a much higher intensity than in the other two cases. In this sense, the periodic thin slits can be regarded as a new breed of plasma optical device that has many advantages in manipulating intense laser pulses. First of all, the periodic thin slits can periodically truncate the edges of the laser pulse and lead to multiple focusings. In addition, the high transmission rate in each thin slit and short interaction time with the slit plasma ensure that the laser energy is not significantly decreased, leading to very high transfer efficiency.

To demonstrate that a system of periodic slits can further enhance the focusing effect compared with a single slit, we discuss the influence of the number of slits, N, on the time evolution of the electric field amplitude. As can be seen in Fig. 4, for targets with fewer slits such as N < 3, the maximum Ey becomes larger as the number of slits increases. This indicates that the electric field arriving at the later slits is higher than that at the earlier slits, leading to enhanced focusing but also significant diffraction. However, when the number of slits N ≥ 3, the largest Ey remains almost unchanged, because the electric field declines too quickly for the next focusing to be enhanced as effectively as at the earlier slits. Note that the electric field can be focused periodically and simultaneously maintain a high amplitude if there are enough slits (say N = 11). Therefore, compared with a single slit, the system of periodic slits offers a significant advantage in terms of focusing enhancement and maintenance over long distances through multiple focusings.

FIG. 4.

Time evolution of electric field amplitude for targets with different numbers N of slits with w = 2.6 µm, d = 20 nm, and ds = 16 µm. In all other cases of periodic slits, the number of slits varies for different values of ds, because we have fixed the propagation distance with a total length of L = 160 µm.

FIG. 4.

Time evolution of electric field amplitude for targets with different numbers N of slits with w = 2.6 µm, d = 20 nm, and ds = 16 µm. In all other cases of periodic slits, the number of slits varies for different values of ds, because we have fixed the propagation distance with a total length of L = 160 µm.

Close modal

Next, a detailed parameter scan on slit thickness d, slit spacing ds, and normalized laser amplitude a0 is conducted to further explore their effects on focusing enhancement using periodic slits. Figure 5(a) shows the maximum Ey as a function of slit thickness, where it can be seen that the results for the periodic slits are always greater than that for the waveguide indicated by the black horizontal line. In addition, periodic slits of lower thicknesses such as d = 20–50 nm allow multiple focusings and lead to a much greater focusing gain than a single slit, as shown in Fig. 5(a). This is mainly caused by the dependence of the focusing length on slit thickness. For thinner slits with d = 20–50 nm, the high transmittance of the intense laser leads to the effective slit width weff becoming larger than the slit width w. In this case, the focusing length lf, which is proportional to weff2 according to diffraction theory,34 is also increased, as shown in Fig. 5(b). Once the slit spacing distance is comparable to the focusing length, the periodic slits will lead to multiple focusings by periodically diffracting the laser pulse, whereas for periodic slits of greater thicknesses, the maximum focusing field strength is almost the same as that of a single slit. In these cases, the focusing lengths are almost the same as that obtained from classical diffraction theory, as shown in Fig. 5(b), and they are much smaller than the slit spacing. Then, the laser reaches its maximum long before it arrives at the second slit.

FIG. 5.

The top row shows the effects of thickness on (a) the maximum focusing Ey and (b) its corresponding focal positions, defined as the distances on the x axis away from the initial position of the slits, x = 20 µm. The bottom row shows the effects of (c) adjacent distance and (d) normalized laser amplitude on the maximum focusing Ey. The maximum focusing fields of the periodic slits in (a) and (c) are compared with the corresponding results for a single slit and a waveguide of the same width. The focal positions of the periodic slits in (b) are compared with those for a single slit and the result of optical diffraction focusing. To ensure the validity of the results of the interaction between the ultraintense laser and the slits, the electron density in (d) is set to ne = 100nc, whereas ne = 50nc is always set in all the other simulations.

FIG. 5.

The top row shows the effects of thickness on (a) the maximum focusing Ey and (b) its corresponding focal positions, defined as the distances on the x axis away from the initial position of the slits, x = 20 µm. The bottom row shows the effects of (c) adjacent distance and (d) normalized laser amplitude on the maximum focusing Ey. The maximum focusing fields of the periodic slits in (a) and (c) are compared with the corresponding results for a single slit and a waveguide of the same width. The focal positions of the periodic slits in (b) are compared with those for a single slit and the result of optical diffraction focusing. To ensure the validity of the results of the interaction between the ultraintense laser and the slits, the electron density in (d) is set to ne = 100nc, whereas ne = 50nc is always set in all the other simulations.

Close modal

The influence of the slit spacing ds on the maximum focusing laser field for two different widths is presented in Fig. 5(c). It can be seen that the lasers focused by the periodic slits are always stronger than those focused by a single slit or a waveguide for a wide range of parameters. For a0 = 5 and width w = 2.6 µm, the maximum focusing gain Ey/E0 is about 2.2, which corresponds to about a fivefold enhancement in laser intensity. The magnitude of the laser electric field also influences the focusing process significantly. Figure 5(d) shows the maximum Ey as a function of the normalized laser amplitude a0 for two different thicknesses. For a thin slit with d = 20 nm, the optimal laser amplitude is around a0 = 10. In this case, larger a0 leads to enhanced transmission but weakened diffraction, while lower a0 leads to reduction of the transmitted laser. Therefore, in both cases, the focusing effect caused by the interference of diffracted and transmitted beams is somewhat limited. However, if the target is thick enough, such as d = 100 nm, the maximum focusing Ey increases as a0 increases, owing to the enhanced relativistically induced transparency and plasma effects.

In this section, we discuss the guiding efficiency of an intense laser through periodic slits. Figure 3 shows that periodic slits can stably guide the intense laser over long distances, in addition to enhancing its intensity. In previous subsections, we have already shown that the transmittance of the laser through a slit can be greatly enhanced by using thinner slits. Here we fix the slit thickness as d = 20 nm and analyze the impact of the adjacent spacing ds on the guiding efficiency, which is defined as the ratio of the output laser energy WE to the input laser energy within the slit width w0: WE=w0/2w0/2PdSdt, where P is the Poynting vector of the laser field transmitted through the periodic slits. It should be noted that this definition of the guiding efficiency is different from that of the transmission rate from a single slit, as given in Sec. III A.

Figure 6(a) shows that the periodic slits always provide a greater guiding efficiency than a waveguide of the same width. For a shorter ds, the laser interacts with more slits in the same duration, thus losing more energy to the plasma, as indicated in Figs. 6(d)6(f). Conversely, if the adjacent spacing ds is too long, the diffraction effect in vacuum before propagation to the next slit dominates and leads to a reduction in the transmission efficiency. Therefore, there is an optimal value of ds ∼ 30 µm, where the guiding efficiency of periodic slits for w = 2.6 µm and w = 8 µm are respectively about 72.9 and 1.7 times higher than that of the waveguide, as shown in Fig. 6(a). In particular, for the narrow slits of w = 2.6 µm, the guiding efficiency remains at a rather high value owing to the superior transmission rate through each thin slit, while only a small portion of the laser can propagate through a waveguide of the same width.

FIG. 6.

(a) Guiding efficiency of periodic slits, with constant thickness d = 20 nm, plotted as a function of adjacent spacing ds for various widths w = 2.6 µm and w = 8 µm, compared with the results for waveguides with the same width. (b) Time envelope of electric field amplitude for targets of 8 µm width: periodic slits with three different adjacent spacings ds = 16, and 30, and 40 µm, and a channel. (c) Corresponding results for width w = 2.6 µm. (d), (e), and (f) Variations of field energy, particle energy, and electron energy, respectively, with time for the same targets as in (b).

FIG. 6.

(a) Guiding efficiency of periodic slits, with constant thickness d = 20 nm, plotted as a function of adjacent spacing ds for various widths w = 2.6 µm and w = 8 µm, compared with the results for waveguides with the same width. (b) Time envelope of electric field amplitude for targets of 8 µm width: periodic slits with three different adjacent spacings ds = 16, and 30, and 40 µm, and a channel. (c) Corresponding results for width w = 2.6 µm. (d), (e), and (f) Variations of field energy, particle energy, and electron energy, respectively, with time for the same targets as in (b).

Close modal

To clearly see the difference between periodic slits and a waveguide, we track the time evolution of field, particle, and electron energies in the simulation, as presented in Figs. 6(d)6(f). Here, “particles” refers to both electrons and ions. When the laser enters the waveguide before t = 60T0, the electric field energy decreases rapidly because part of the intense laser is very strongly reflected. On full entry of the laser pulse into the waveguide, the laser electric field continuously pulls electrons out of the inner wall of the waveguide and then accelerates them. This continuous electron ejection and acceleration can result in severe dissipation as they carry energy away from the laser field. In addition, the ions can also carry part of the energy by charge separation in the electric field. In the case of the waveguide, Fig. 6(e) shows that more than 50% of the laser energy is converted into plasma or particles. By contrast, the field energy in the case of periodic thin slits varies slowly over time with periodic oscillations, as shown in Fig. 6(d). Correspondingly, fewer electrons are ejected and accelerated in periodic slits, and less than 5% of the laser energy is converted into electron energy, as shown in Fig. 6(f). This is because the intense laser periodically interacts with the thin slits and has a very high transmission rate through each slit. After its transmission through each slit, the laser loses only a small portion of its energy, and then propagates in vacuum before reaching the next slit. As a result, the guiding efficiency in periodic thin slits is remarkably higher than that in a conventional waveguide.

Figures 6(b) and 6(c) show the time evolution of the Ey amplitude for w = 8 µm and w = 2.6 µm, respectively. It should be noted that in both cases, the laser propagating within periodic slits is considerably more intense than that in free space, denoted by the black dashed line. In Fig. 6(b), there is little difference in the field strength between the results of the periodic slits and the waveguide, but the periodic slits always have a much higher transfer efficiency than the waveguide. Figure 6(c) reveals that the Ey amplitude of the waveguide declines sharply and ultimately falls far below that in other cases. On comparing these two situations, it can be seen that the field for the narrower width w = 2.6 µm in Fig. 6(c) has a notably higher focusing gain than in the wider case with w = 8 µm in Fig. 6(b). Hence, periodic slits allow the intense laser to simultaneously maintain a notably greater laser field and a higher energy transfer efficiency, particularly in the case of narrower widths.

In addition to the main parameters of the slit and the normalized amplitude of the laser, there are several other factors, such as the electron density ne and the duration of the laser pulse τ0, that can also affect the interaction between the intense laser and the slits. In this section, we discuss these effects as well as three-dimensional effects. As we mentioned in Sec. II, in our simulations, we need to resolve both the slit thickness and the skin depth of the solid-density target. To save computational resources, we choose a relatively low density of 50nc for the slit target. Here, we examine the effect of electron density ne, which is experimentally related to the material of the target. In this case, we consider that the parameter that actually represents the effect of electron density on slit focusing is the surface density ned, since it is this that determines the relativistic transparency through Eq. (3). As can be seen in Figs. 7(a) and 7(b), by comparing the results for slits with electron densities of ne = 100nc and ne = 50nc, it is found that both the maximum Ey and the transmission rate are almost identical for the same surface density ned, and their dependences on the target thickness are also similar.

FIG. 7.

(a) Maximum Ey and (b) transmission rate for ne = 50nc and ne = 100nc, plotted as functions of thickness d with w = 2.6 µm.

FIG. 7.

(a) Maximum Ey and (b) transmission rate for ne = 50nc and ne = 100nc, plotted as functions of thickness d with w = 2.6 µm.

Close modal

To understand the influence of laser pulse duration, we compare here the results for a longer pulse with FWHM duration τ0 = 66.6 fs and those for one with τ0 = 33.3 fs. As can be seen from Fig. 8(a), for a very thin slit, there is little difference in the maximum Ey between the two laser pulses. This is because in this case, the maximum focusing result can be achieved with a shorter duration of interaction. On the other hand, for a thicker slit d > 70 nm, as the thickness increases, the difference between the results for a longer pulse and those for a shorter one becomes larger. This is because for τ0 = 66.6 fs, thicker slits are subjected to a longer duration of radiation pressure, resulting in an effective increase in transparency compared with the cases with shorter pulse duration, as shown in Fig. 8(b). For thick slits interacting with a longer laser pulse, the interference between diffracted and transmitted beams is enhanced with a higher transmission rate, as well as the focusing electric field. Figure 8(c) shows that there still exists an optimal thickness for the laser focusing in the case of a longer laser pulse, and the corresponding optimal value is greater than that with a shorter laser pulse, but the overall trend is similar to that for a shorter pulse.

FIG. 8.

(a) Maximum Ey for τ0 = 66.6 fs and τ0 = 33.3 fs plotted as a function of thickness d = 20–120 nm with w = 2.6 µm. (b) Corresponding transmission rate under the same conditions as in (a). (c) Maximum Ey for τ0 = 66.6 fs, plotted as a function of thickness d = 20–200 nm with w = 2.6 µm.

FIG. 8.

(a) Maximum Ey for τ0 = 66.6 fs and τ0 = 33.3 fs plotted as a function of thickness d = 20–120 nm with w = 2.6 µm. (b) Corresponding transmission rate under the same conditions as in (a). (c) Maximum Ey for τ0 = 66.6 fs, plotted as a function of thickness d = 20–200 nm with w = 2.6 µm.

Close modal

To check the validity of the results and obtain more accurate information on the field amplification, we conducted an additional three-dimensional simulation for a laser with beam waist of w0 = 5 µm, a periodic thin slit of width 6 µm, thickness of 20 nm, and adjacent spacing of ds = 16 µm. As a comparison, the single-slit case was also simulated. The simulation box of 40 × 8 × 8 µm3 was resolved with 4000 × 800 × 800 cells. The temperature of all particles was set to 10 eV. As shown in Fig. 9, for a0 = 5, the maximum electric field obtained from periodic slits in Fig. 9(b) is increased by a factor of 2.63 (the corresponding intensity is increased by a factor of about 7), which is about 1.2 times larger than that from a single slit, as shown in Fig. 9(a). This verifies that the intense laser field can be more significantly enhanced through multiple focusings, which is consistent with the results of two-dimensional simulations. Relativistic effects are still the main mechanism responsible for enhancing the focusing process. Figure 9(b) shows that after the intense laser has completely interacted with the first slit, the electron density at the slit aperture decreases significantly. This leads to relativistic transparency at the edge of the aperture, causing curvature of the phase front and interference between diffracted and transmitted light. The laser intensity undergoes further enhancement after interacting with the second slit. In addition, as previously demonstrated in Fig. 5(d), it is possible to further enhance the laser intensity by an increasing factor as the laser amplitude increases if d is relatively large. Therefore, with more powerful lasers, focusing gains of an order of magnitude or more could be achieved through the use of appropriate target parameters. Moreover, in this case, the laser can be guided well in periodic slits over long distances of 160 µm, which is more than twice the Rayleigh length.

FIG. 9.

Distributions of Ey and ne obtained from 3D simulation at the moment of reaching the maximum value for (a) a single slit and (b) periodic slits with the same width and thickness. The laser propagates from right to left, aligned with the positive direction of the x axis.

FIG. 9.

Distributions of Ey and ne obtained from 3D simulation at the moment of reaching the maximum value for (a) a single slit and (b) periodic slits with the same width and thickness. The laser propagates from right to left, aligned with the positive direction of the x axis.

Close modal

Finally, we discuss some potential applications. The use of a system of periodic slits open a new approach to engineering light–matter interactions at ultrarelativistic intensities and is of practical interest for many laser-based applications. First of all, the periodic slits can provide multiple focusings to further improve laser intensity while also maintaining it at a high level over long propagation distances, much beyond the Rayleigh length. This is crucial for laser-driven particle acceleration, which depends critically on the laser intensity and interaction length. In addition, the extremely high guiding efficiency of the periodic slit system ensures that the laser pulse is effectively directed along the desired path by minimizing the loss of energy. This is advantageous for applications that require precise control and manipulation of laser beams. For example, it could be utilized in Thomson scattering or optical undulators35 to maintain the laser intensity over long distances, which is beneficial for the generation of bright X-ray sources by colliding an intense laser with a relativistic electron beam.

The fundamental physics of relativistically intense lasers interacting with thin slits has been studied via two- and three-dimensional PIC simulations. A well-enhanced laser focusing scheme beyond the classical diffractive focusing regime has been discovered. This enhancement is due mainly to relativistic plasma effects, which involve relativistically induced transparency at the edges of the slits and to interference between diffracted and transmitted beams. On this basis, it has been further demonstrated that periodic thin slits, as a new kind of plasma optical device, can simultaneously achieve focusing enhancement and long-distance guidance of an intense laser. In this case, the high transmission rate through each slit leads to multiple focusings of the laser, together with much higher energy transfer efficiency than that obtainable in conventional waveguides. This novel approach offers effective control over the spatiotemporal attributes of ultraintense laser pulses, which should be useful in many laser-based applications, such as high-order harmonic generation, laser-driven particle acceleration, Thomson/Compton scattering, and X-ray light sources.

This work is supported by the National Key R&D Program of China (Grant No. 2022YFA1603300) and the National Natural Science Foundation of China (Grant Nos. 12175154, 12205201, 12005149, and 11975214). T.W.H. acknowledges support from the Shenzhen Science and Technology Program (Grant No. RCYX20221008092851073). The EPOCH code is used under UK EPSRC Contract Nos. EP/G055165/1 and EP/G056803/1.

The authors have no conflicts to disclose.

L. Xu: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). T. W. Huang: Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (lead). K. Jiang: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). C. N. Wu: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Writing – review & editing (equal). H. Peng: Data curation (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Visualization (equal). P. Chen: Data curation (equal); Formal analysis (equal); Software (equal); Visualization (equal). R. Li: Formal analysis (equal); Methodology (equal); Validation (equal); Visualization (equal). H. B. Zhuo: Methodology (equal); Resources (equal); Software (equal); Writing – review & editing (equal). C. T. Zhou: Funding acquisition (equal); Project administration (equal); Resources (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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