Neutral particle acceleration by spatially modulated laser pulses

The velocity gain of neutral particles (atoms, molecules, etc) from laser acceleration is always small. A possible scheme to obtain a high speed neutral particle beam is multistage acceleration. However, according to previous theoretical and experimental studies, generally, lateral acceleration is larger than longitudinal acceleration. These transverse velocities destroy the expected quality of the longitudinally transmitted neutral particle beam. In order to realize multistage accelerations of neutral particle, it is necessary to restrain the beam divergence caused by lateral acceleration. How to optimize and utilize these laterally accelerated neutral particles is worthy of in-depth study. In this paper, we use a multi-mode combined laser pulse and a flattened Gaussian laser pulse to accelerate the neutral atoms. The transverse divergence of the beam is well controlled while the longitudinal acceleration is retained, which provides the possibility for improving the beam quality of neutral particles as well as the corresponding multistage acceleration.


Introduction
Neutral particle beams, including neutrons, atoms, molecules, and mesoscopic scale particles, play a crucial role in a diverse range of fields, such as fundamental physics, chemical reactions, materials synthesis, neutron source technology, and magnetic confinement fusion [1][2][3][4]. It is crucial to control the motion and achieve higher speeds of neutral particles. Unlike charged particles, neutral particles cannot be easily manipulated by electromagnetic fields, posing greater technical challenges. Nevertheless, numerous methods of neutral particle acceleration have been developed. Early approaches, such as optical tweezers and laser cooling, utilize relatively weak optical fields to manipulate neutral particles. Additionally, advancements in strong laser technology have paved the way for various techniques in laser-based manipulation of neutral particles [5]. For instance, two decades ago, Stapelfeld et al utilized the dipole force generated by a non-resonant optical field for the manipulation of molecules [6]. Recently, Maher-McWilliams et al accomplished precise manipulation of atom beams with strong laser light lattices [7]. Rajeev et al proposed an indirect collision-method to acquire directional atom beams. In their scheme, the high-speed positive ions obtained from laser acceleration, collide with slow-moving Rydberg atoms. Then neutralization occurs when electrons of the Rydberg atoms transfer to the ions [8]. Similarly, a neutron source technique for laser targeting has also been proposed [9,10].
Especially, Eichmann's team have conducted experiments which demonstrated that laser pulses in free space can directly accelerate neutral atoms with ultra-high acceleration. The principle behind this phenomenon involves exciting the atoms to Rydberg states through a strong laser field, causing the quasi-free electrons to experience an apparent ponderomotive force. The atoms are then accelerated through the Coulomb traction force between the electrons and their corresponding nucleus [11,12]. Unlike the electromagnetic force of an optical field, the ponderomotive force depends only on the intensity of the optical field and averages the phase [13]. We further investigated this scheme theoretically [14]. However, the interaction between light and atoms in this scheme is not a simple process [15]. Experimental results have shown that some atoms are ionized, while others remain highly excited and stable in the strong field [16]. In our study, we only consider highly excited neutral particles that are not ionized, while the ionized particles are separated using an external electric field. Additionally, although highly excited atoms in a strong field are subjected to the Kramers-Henneberger force, it is negligible in the frequency range we used, contributing less than 5% of the ponderomotive force and thus ignored [17]. Despite the significant acceleration achieved in this scheme, the final exit velocity is not sufficiently high, primarily due to various factors limiting greater acceleration [18]. Moreover, the transverse acceleration effect is much stronger than the longitudinal acceleration, causing the particle beam to have a transverse divergence that hinders the realization of multi-level acceleration. This transverse divergence is a result of the transverse gradient force (ponderomotive force) exerted by the optical field [19]. If the transverse intensity distribution of the optical field is flattened through the spatial modulation of optical beams [20], it leads to a significant reduction in the transverse gradient force of the optical field. This technique has been applied in charged particle laser acceleration, where a laser beam with a non-Gaussian profile in the transverse distribution is used to effectively suppress the transverse divergence of the particle beam [21].
To numerically solve and analyze the dynamics of neutral particle acceleration using flattened beams, it is necessary to obtain the optical intensity distribution across the beam's cross-section and its propagation evolution. These beams are commonly known as flattened Gaussian beams (FGBs) or super-Gaussian beams (SGBs) in the literature, and they have various descriptions [22], such as co-axis multi-Gaussian beams [23], off-axis multi-Gaussian beams [24], and SGBs described by the super-Gaussian function [25], and etc. Although the various descriptions mentioned above can provide high-quality flattened light intensity distributions at specific cross-sections, intensity expression during propagation is too complicated to deduce an analytic expression for the optical intensity gradients and the final exit velocity of accelerated particles. To simplify the analysis, we initially focus on neutral particle acceleration in a near-flat-top optical field created by superimposing a few modes of Hermite-Gaussian (HG) beams. Our preliminary calculation results confirm the expectation that transverse acceleration can be suppressed in beams with flattened transverse distributions. However, it is challenging to achieve a well-flattened distribution beam using this multi-mode superposition method. Additionally, the flat area in the center of this optical field is relatively small. Therefore, we use numerical methods to investigate the laser acceleration of neutral particles using FGBs and compare the results with those obtained using fundamental-mode Gaussian beams. Through this comparison, we observe that the suppression effect of transverse acceleration in FGBs is even better than that achieved using the above-mentioned multi-mode superposition Gaussian beams. Furthermore, the longitudinal acceleration in FGBs exhibits enhanced performance. Finally, we explore the acceleration using a long laser pulse, which provides the possibility of exciting atoms outside the laser field and subsequently accelerating them within the laser field, thereby allowing for second-stage acceleration. Figure 1 illustrates the interaction between a neutral atom and a laser pulse, with the laser pulse propagating along the z-axis. When atoms come into contact with the optical field, they are randomly excited to Rydberg states and subsequently accelerated by the ponderomotive force acting on the excited electrons. In our analysis, we define t = 0 as the moment when the center of the laser pulse reaches the origin. We use (x 0 , y 0 , z 0 ) to represent the initial position of an atom when it is excited to a Rydberg state, and set t 0 (maybe before t = 0 or after t = 0) as the excitation moment, which is also the time when the Rydberg atom acceleration starts. At this moment, the position of the laser pulse center is (0, 0, ct 0 ), then η 0 = ct 0 − z 0 can represent the relative position between the Rydberg atom and the laser pulse.

Calculation model and results
When the light intensity is relatively low (<< 10 18 W cm −2 ), the ponderomotive force on a Rydberg atom with single electron excitation can be approximated by [11,26]: M and m e represent the mass of atom and electron respectively.R(t) is the second-order partial derivatives of atomic center-of-mass positions with respect to time. E 0 (⃗ r, t) is the intensity of slow-varying electric field. ω represents the angular frequency of the electric-magnetic wave. After interaction with the laser, the velocity of the Rydberg atom is

Multimode combined beam acceleration
First, we use the superposition of HG modes to attain the laser pulse with nearly flat distribution in transverse intensity. HG laser beams are solutions of the Helmholtz equation under the slowly-varying-envelope approximation [27]. The laser beams we adopt are the combined beam of HG (0,0), (1,0) and (0,1) modes propagating along z axis. (0,0) mode is polarized along y axis, the other two modes are polarized along x axis. For HG(1,0) and (0,1) modes, they have the same field intensity with phase difference of π/2. Namely, E y = E 0,0 , E x = E 1,0 + E 1,0 exp(iπ /2). For the intensity of the laser pulse, we multiply the intensity expression of the stationary beam with a Gaussian profile factor f(η) = exp −(η/ cτ ) 2 , and τ FWHM = 2 √ ln 2 · τ is the full width at half maximum of the intensity of laser pulse. Neglecting the longitudinal component of the electric field, the intensity distribution of multi-mode combined Gaussian laser pulse is [28]: Among them, a 00 is the reference strength of the fundamental mode, ζ represents the relative coefficient of (1,0) and (0,1) modes' field amplitude to the fundamental mode, w 0 is the beam width at focus, Z R = kw 2 0 /2 is the Rayleigh length, k = 2π /λ is the wave number, and c is the velocity of light in vacuum. Figure 2 shows the transverse ponderomotive potential distribution within the focal plane of the multi-mode combined Gaussian beams with different value of ζ when the intensity is relatively low (stationary field τ FWHM → ∞). We find that with the propagation of the beam, the shape of the potential changes little within a distance of one Rayleigh length along z-axis.
Suppose the atom is initially stationary. Our previous study has shown that, for an ultra-short laser pulse, the atom will be almost motionless during the acceleration, as R(t) ≡ R 0 [12]. Then we substitute equation (3) into equation (1), using equation (2) we can get the exit transverse velocity and longitudinal velocity of the Rydberg atom with the initial position (x 0 , y 0 , z 0 ):  (7) and (8), we can get two fully analytical solutions.
another two fully analytical solutions of the equations can also be obtained. According to equation (7), the transverse exit velocityv ⊥ ∝ τ , so compressing the laser pulse is favorable to transverse acceleration suppression. We consider the product term ℑ = 1 + 2 2 outside the integral sign in equation (7). If ℑ = 0, it comes to the zero v ⊥ of the Rydberg atom when the following condition is satisfied: When ζ > √ 2, r 0 has non-zero solution. It can be expected that particles nearly satisfying the condition (9) will have a quite low v ⊥ . We substitute equation (9) into the integral term of equation (8): Since t is a variable during the interaction, z 0 and η 0 are also randomly change for different particles. It is difficult to match the condition that for each particle the entire integral is zero. That means particles satisfying the condition (9) will most likely end up with On the other hand, as the integral has different values for different parameters, it shows the bad single-value characteristic of v // . Based on the figure 2, we can find the intensity of light (corresponding to ponderomotive potential) is nearly flat in transverse distribution when ζ = 1.5. In this case, when the Rydberg atom is excited at the central region, its v ⊥ is theoretically small. Figures 3 and 4 show the v ⊥ (V r ) and v // (V z ) distribution of excited 4 He atom when ζ = 0 (fundamental mode beam) and ζ = 1.5 (multi-mode combined beam) respectively. In this situation, we only consider the 4 He atom excited on the cross section, which is over the center-axis and near the focal spot. The other parameters of laser pulse are: w 0 = 16 µm, λ = 0.814 µm, τ FWHM = 240 fs, I 0 = 2.8 × 10 15 W cm −2 corresponding to a 00 .
From figures 3 and 4, compared to the fundamental Gaussian beam (ζ = 0), multi-mode combined Gaussian beam (ζ = 1.5) can provide better longitudinal acceleration and lower transverse acceleration even when 4 He atom is excited in a larger transverse distribution area (r 0 < w 0 /4). However, for fundamental Gaussian beam, it leads to a non-negligible transverse acceleration when the atom is excited in the off-axis area. We also find multi-mode combined Gaussian beam gives no significant improvement for the acceleration in longitudinal distribution area. However, figures 3 and 4 are both under the situation η 0 = 0, which means atoms are always excited in the center of the laser pulse. In reality, atoms are randomly excited in the optical field, so the excitation positions relative to the center of laser pulse will vary. Figure 5 presents the transverse and longitudinal acceleration results of 4 He atom excited with different η 0 of fundamental mode and multi-mode combined Gaussian beam.
Comparing the results of figure 5(b) to figure 5(a), under each different η 0 condition, for 4 He atoms excited in the central area (marked with the arrow), the final transverse exit velocities in multi-mode combined laser pulse ( figure 5(b)) are much smaller than those in fundamental mode laser pulse ( figure 5(a)). In the meantime, longitudinal exit velocities are almost the same for the both cases (refer to figures 5(c) and (d)), indicating that the longitudinal acceleration effect is maintained in the multi-mode combined laser pulses.
From the above discussion, it is evident that the acceleration effect is influenced by various parameters, making it challenging to estimate the overall impact. However, we can assume that the helium atoms are uniformly distributed around the focus of the laser beam. Since the electrons in a helium atom are tightly bound before excitation, the dragging mechanism mentioned earlier does not apply. To simplify the process, we performed a random uniform sampling in four dimensions and recorded the space-time coordinates (x 0 , y 0 , z 0 , t 0 ) for each excitation event. Figure 6 illustrates the statistical distribution of transverse and  longitudinal exit velocities for excited atoms after interacting with a fundamental mode Gaussian beam (ζ = 0) and a multi-mode combined Gaussian beam (ζ = 1.5). In figure 6, a negative value of V r indicates that the direction of the transverse exit velocity is opposite to the radial position vector⃗ r of the excited atom. From the distribution of figure 6(a), generally, atoms accelerated by the multi-mode combined Gaussian beam rarely have transverse exit velocities faster than 30 m s −1 . However, for atoms accelerated by the fundamental mode Gaussian beam, a considerable number of atoms exhibit transverse exit velocities up to . Other laser parameters are the same as figure 3. 40 m s −1 . Furthermore, in figure 6(b), the distribution of longitudinal exit velocities in the multi-mode Gaussian beam resembles that of the fundamental mode Gaussian beam. However, the acceleration effect of the multi-mode Gaussian beam is enhanced in the high-speed range (-14 to −10 m s −1 ).

FGB acceleration
As mentioned-above, there are various descriptions for FGBs. However, some of them are not the eigensolutions of paraxial wave equation in free space [25] and some descriptions are quite different from the actual beam propagation properties [24]. Similar to [29], we enact the vector potential ⃗ A(A x , A y = 0, A z = 0) of a FGB instead of the electric field as [23]. Using the Lorentz gauge, we can get the scalar potential Φ and the corresponding electromagnetic components of the FGB. Such a FGB is not completely linearly polarized, i.e. E x ≫ E y ̸ = 0, its longitudinal electric field E z ̸ = 0. Fortunately, here we simply need the intensity of a FGB pulse, which can be expressed as: where L n (ℜ) is the nth standard Laguerre polynomial, a 0 is the vector potential amplitude in the beams center, c is the velocity of light in vacuum, k = 2π /λ is the laser wave number, τ FWHM is the full width at half maximum of the intensity of laser pulse, w 0 is waist size of the Laguerre Gaussian beam,  figure 7. The transverse distribution at the beam waist is flat as shown in the right panel. However, the upper panel shows a self-focused high-intensity distribution area at both the front and the rear of the beam waist, where the intensity is much greater than that at the beam waist. From beam waist to the self-focused area, the flat transverse distribution of the optical field gradually evolves into a Gaussian-like transverse distribution. The bottom panels show the transverse distribution of FGBs with different N at the beam waist (z = 0) and near the focal-spot (z = 2Z RN ) of N = 8 respectively. As can be seen in the bottom-left panel, the transverse flat area at the beam waist becomes wider and wider with increasing N. When N ⩾ 8, the flattened areas of FGBs reaches almost 2w 0 in width. Here we fix the w N (0), which leads to a larger w 0 for a larger N, therefore, the actual flat area is larger than the apparent area that the panel shows. The bottom-right panel shows the situation beyond beam waist. The flattened area becomes smaller and even disappears when move away from beam waist along longitudinal direction. For instance, at z = 2Z RN , FGBs with N ⩽ 8 have no flattened distribution in transverse direction. Qualitative analysis shows that only the transverse acceleration of neutral particles excited in these transversely flattened areas will be well suppressed, while the transverse acceleration near the edge and focal spot will still be strong. Since the expressions for the intensity of FGBs contain multiple complex terms to sum up, the analytical expressions for the ponderomotive potentials and corresponding exit velocities cannot be obtained. In the following, we calculate the acceleration forces of the neutral particles via gradient of the ponderomotive potential, similarly to figure 5. Then we use equation (2) to obtain the velocities of the exit neutral particles by numerical integration.
Since our previous results show that the transverse acceleration has a greater performance when 4 He atoms are excited before the laser pulse [14,26], we choose η 0 = −2cτ for the transverse acceleration in figures 8(a) and (b). As shown in figure 8(a), at the beam waist (z = 0), the area where the transverse exit velocity of an excited atom is zero increases with the value of N. The transverse acceleration increases significantly in the edge of the flattened area but the increasing trend slows down with increasing N value. In the area that tends to focal spot (such as z = Z RN ), for a FGB with smaller N, the transverse flattened region quickly evolves to disappear. Then, the transverse exit velocity increases, as shown in figure 8(b). Nevertheless, a FGB with a large N (like N = 24) can still keep part of atoms having zero transverse exit velocity. Especially at z = Z RN , the field intensity of the fundamental mode Gaussian beam (N = 0) attenuates because it is away from its focal spot center (z = 0). However, for the FGB with N = 1, it is exactly at its self-focusing focal spot area (z ≈ Z RN ). For the FGB with N = 8, it is near its self-focusing focal spot area (z = (2 ∼ 5)Z RN ). Therefore, the variation of the corresponding field gradient leads to the results that the maximum transverse accelerations (off-axis) by the FGBs (N = 1,8) are even larger than that by the fundamental mode Gaussian beam (N = 0). These results are consistent with previous predictions by qualitative analysis.
Since our previous results show that the longitudinal acceleration has a greater performance when the 4 He atom is excited in the laser pulse center [14,26], we choose η 0 = 0 for the longitudinal acceleration in figures 8(c) and (d). Although for a wider range (|z 0 | > 2Z RN ) of the on-axis (r 0 = 0) and off-axis (r 0 = 0.5w N (0)), high-order FGBs (N > 0) induce a higher longitudinal exit velocities than the fundamental mode Gaussian beam. These atoms also experience a larger transverse acceleration, so they are not what we want. However, in the range (|z 0 | < 2Z RN ) around the beam waist transverse acceleration from high-order FGBs (N > 0) is small while longitudinal acceleration is large (shown with the dash-line circles).
. Other laser parameters are the same as figure 8.
The above calculations do not yet reveal the overall effectiveness of the acceleration by the FGBs. To better evaluate the overall effectiveness of the FGB acceleration, we conducted additional calculations using a random sampling of different initial conditions, focusing on FGB with N = 8. For comparison, we also calculated transverse and longitudinal acceleration using a fundamental mode Gaussian beam with N = 0. For the range of parameters, can refer to the calculation results in figure 8. As can be seen in figure 9(a), the overall transverse acceleration by the FGB is well suppressed as there are very few instances with exit speed greater than 6 m s −1 . From figure 9(b), it is easy to find the longitudinal acceleration of the FGB is well maintained. Different from the fundamental mode Gaussian beam acceleration, the distribution of longitudinal exit velocity of the FGB acceleration shows a distinct peak in the high-speed range (near 14 m s −1 ). This suggests that an actual FGB acceleration can produce a longitudinally monoenergetic neutral particle beam within this speed range. In order to further investigate this point or optimize the results, we conducted particle tracking and found that most exit particles with low transverse velocities were concentrated near the waist of the beam. Consequently, it can be inferred that a portion of the particle bunch can be considered as a collimated beam.

Longitudinal acceleration with different duration of laser pulse
According to equation (8), the longitudinal exit velocity can be simplified as v // ∝ (Aτ − B). A and B are nonnegative and irrelevant parameters to τ . For a short pulse, if Aτ ≪ B, the longitudinal exit velocity is negative (along −z axis) when η 0 = ct 0 − z 0 = 0, which means compressing the laser pulse is also profitable to improve the longitudinal acceleration. For a long pulse, if Aτ ≫ B, whether the longitudinal exit velocity is negative (along −z axis) or it is positive (along +z axis) depends on the initial longitudinal position of atomic excitation z 0 when η 0 = 0. For the convenience of discussion, we simply consider the case where atoms on the optical axis (r 0 = 0) are accelerated by fundamental mode beams (ζ = 0) when η 0 = 0, then Figures 10(a) and (b) show the results calculated with formula (20) for 4 He atoms, we can find that the results are consistent with the above analysis. For the first stage acceleration, particles are initially excited inside the laser pulse. In the case of short pulses, the particles typically acquire a negative longitudinal exit velocity. However, for long pulses, the longitudinal exit velocity can be both negative (excited behind the focal spot) and positive (excited before the focal spot). It is important to mention that in the case of multistage acceleration, after the first stage, the excited atoms with a specific longitudinal velocity must re-enter the pulse center of another laser pulse generated by the external field, namely, η 0 = ct 0 − z 0 = −∞. For the convenience of discussion, we also consider the case where atoms on the optical axis (r 0 = 0) are accelerated by fundamental mode beams (ζ = 0) when η 0 = −∞, then Here, v // ∝ τ . That is to say a longer laser pulse is advantageous for increasing the longitudinal exit speeds of excited atoms. Figures 10(c) and (d) show the calculated results for 4 He atoms with formula (21). It is observed that, the longitudinal exit velocity can be both negative (located behind the focal spot) and positive (located before the focal spot). At this point, both forward and backward propagating pulses can accelerate the speed of neutral particles emitted in both directions. It should be pointed out that using a long pulse of the multi-mode combined Gaussian laser beam can effectively achieve the desired second stage acceleration. However, with a long pulse of the FGB, the initial point for obtaining good longitudinal acceleration is close to the self-focused high intensity region of the FGB. Although the emitted neutral particles obtain good longitudinal acceleration, they also diverge laterally. A possible way to avoid the defect of the acceleration with FGB is to use the short pulse with pulse envelope shaping, that is, temporal modulation makes the front and rear edges of the pulse asymmetric.

Summary and discussions
In conclusion, we propose a solution to address the issue of transverse divergence in laser acceleration of neutral particles by utilizing a laser pulse with a flat transverse distribution. This beam can be achieved by spatial modulation. Our proposed scheme involves using a multi-mode combined Gaussian beam with three HG modes and a FGB to accelerate He atoms. After a brief analysis of the optical field distribution, we further calculate the transverse and longitudinal exit velocities of excited He atoms under different initial conditions and compare them with the acceleration results of the fundamental mode Gaussian beam.
In order to evaluate the overall effectiveness of acceleration, we make a uniform sampling for the initial conditions in a certain range and calculate the exit velocities. The results shows that the overall transverse acceleration by the FGB is better suppressed, and the transverse velocities distribute in a small range around zero. In particular, the longitudinal acceleration of FGB is not only well maintained, but also more concentrated in the high-speed range, which helps to obtain a longitudinal monoenergetic neutral particle beam.
If we use a long laser pulse for acceleration, we may well achieve second stage acceleration. Nevertheless, even without multistage acceleration, the improvement of beam quality by the FGB accelerations is valuable. We hope these results will provide some guidance for future related experiments.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).