A hollow, cylindrical, micron-scale structure is proposed to enhance and manipulate the laser plasma interaction. It is shown through 3-D particle-in-cell simulations that the incident laser pulse intensity is enhanced within the tube. A detailed study of the intensification optimizes the tube dimensions and provides a characterization of the in-tube intensity. By coupling the micro-tube plasma lens to a traditional flat interface, we show an increase in on-target intensity. We detail proton energy enhancement as a potential application of the micro-tube plasma lens target, where the tube structure focuses the light and provides additional electrons that enhance the accelerating sheath field.

The interaction of intense laser pulses with structured interfaces is of great interest due to its applications in the generation of high energy electrons, protons, and x-rays. Experimentally, structured target interactions with high intensity laser pulses have shown an enhancement in laser absorption.1–3 Recently, snow targets have shown the ability to enhance proton energies when compared with flat interfaces.4 Silicon micro-wire array towers have been used to demonstrate an enhancement in electron energy and high energy electron yield.5 Cone targets have long been studied for their effects on electron transport.6–8 

As technology for target fabrication has progressed,9 advanced structured interfaces have become an active area of theoretical and computational research. The use of cone targets and cone-tube targets has shown an enhancement in energy and directionality of proton beams.10–14 Front surface structures, such as towers and near-critical density plasmas, have shown an enhancement in proton energy as well.15–18 Micro-wire array targets were suggested to create a highly directed energetic electron beam with enhanced bremsstrahlung radiation.19,20 The use of hollow cylindrical targets has been demonstrated to produce bright x-rays and attosecond pulses.21,22 Additionally, rear-surface structures have been demonstrated to collimate secondary electron and ion beams.23–25 By making use of ultrahigh contrast lasers made possible by cross-polarized wave generation (XPW)26 or plasma mirrors,27,28 it is possible for structured interfaces to effectively manipulate the laser-plasma interaction in ways that still remain unexplored and underutilized.

In previous work, we described a novel mechanism whereby front surface micro-tube plasma (MTP) structures caused an intensification of the incident relativistic laser pulse.29 The MTP acts to redistribute the incident pulse intensity within the tube, resulting in a localized enhancement of the laser intensity. In that work, we also modeled the effect on the generation of secondary radiation for a 5 × 1022 W cm−2 pulse incident on an 8 μm long MTP target, observing an enhancement in peak proton energy compared to a flat target. In this article, we perform a series of 3D particle-in-cell (PIC) simulations to better understand and optimize the enhancement mechanisms with a focus on using MTP lenses to improve ion acceleration via enhanced target normal sheath acceleration (TNSA). We present a full characterization of a high intensity (∼5 × 1021 W cm−2), short pulse (∼32 fs) laser inside the MTP lens targets. We explore the effects of varying the MTP diameter, which allow us to determine the distance from the entrance of the tube to the intensified hot spot. Then, by placing a foil at the rear of the optimized MTP lens, we show that the on-target intensity can be greatly increased. Enhanced proton energy is studied as a potential application of the MTP lens, where we show an increase in maximum attainable proton energies up to a factor of 3.5, an enhancement factor comparable to previous work while using a laser pulse that is an order of magnitude lower in intensity.

Simulations were performed with the 3D PIC code VLPL.30 For ion acceleration simulations, a y-polarized laser pulse propagates along the x-direction in a simulation box of 75λ × 12λ × 12λ extent in x × y × z, where λ = 0.8 μm is the laser wavelength. During MTP optimization simulations, the x-dimension of the simulation box was varied to decrease simulation time, while the y- and z-dimensions remained constant. The cell size was held constant at 0.02λ × 0.1λ × 0.1λ in the x × y × z dimensions for all simulations. A time step of Δt = 0.008 T, where T = 2π/ω0 (ω0 is the laser frequency) was chosen to meet the resolution criterion for relativistic electron motion.31 A transversely super-Gaussian laser pulse with pulse profile ay=a0e(r/σ)4(t/τ)2 is focused onto a CH2 substrate foil with thickness 5λ located 29λ from the left boundary of the simulation box. The foil thickness was chosen such that for all geometries studied, the dominant ion acceleration mechanism is expected to be TNSA. The laser amplitude is initially a0=50(I5×1021Wcm2) where a0 = eEl/meω0c, pulse duration τ = 12 T (≃32 fs), and focal spot σ = 3λ, which were chosen to closely match the optimized parameters of the Scarlet laser housed at The Ohio State University.32 On the front surface of the foil, we positioned carbon micro-tubes of variable length (L) and inner diameter (ID) with a wall thickness of λ, as shown in Figure 1(a). The length of a given micro-tube ID was chosen based on optimization of the intensification factor as discussed below. The electron density of the tube was chosen to be ne = 180nc (nc = meϵ0ω2/e2 is the critical plasma density) to match the electron density of tubes that can be easily manufactured using 3-D printing techniques. The rear CH2 foil is initialized to ne = 150nc. The whole target is initially cold and fully ionized.

FIG. 1.

Target conditions for the simulation. Distributions of target electron density (gray), in-tube electron density (purple), proton density (green), and laser intensity (red) at t = 10 T (a), 30 T (b), and 50 T (c), respectively. The laser enters from the left of the simulation and is incident on a micro-tube plasma (MTP) lens coupled to a 5λ thick CH2 foil. The length (L) and the inner diameter (ID) of the MTP lens are varied to locate the intensity spike on the rear substrate and optimize the on-target intensity.

FIG. 1.

Target conditions for the simulation. Distributions of target electron density (gray), in-tube electron density (purple), proton density (green), and laser intensity (red) at t = 10 T (a), 30 T (b), and 50 T (c), respectively. The laser enters from the left of the simulation and is incident on a micro-tube plasma (MTP) lens coupled to a 5λ thick CH2 foil. The length (L) and the inner diameter (ID) of the MTP lens are varied to locate the intensity spike on the rear substrate and optimize the on-target intensity.

Close modal

A detailed discussion of light intensification within an MTP lens can be found elsewhere.29 In short, a laser pulse incident on a hollow cylindrical aperture will undergo diffraction. In the near-field Fresnel region, the result is a boost of the intensity within the cylinder resulting from a redistribution of the incident pulse. Previous work showed that, by varying the pulse intensity while maintaining the tube geometry, there is a relativistic intensity dependent intensification factor, where, as the incident pulse becomes more intense, the intensification factor increases. The intensity dependence results from additional focusing caused by tube electrons being dragged into the hollow region of the MTP. In order to optimize the intensification effect, we perform simulations on MTP targets with variable dimensions. Since diffraction is dependent on the geometrical properties of the hollow cylinder, we fix the incident pulse parameters while varying the tube dimensions to characterize the in-tube laser profile. Then, using the MTP structure as a focusing element, we place an optimized MTP on the front surface of a flat CH2 foil and investigate the MTP effects on proton acceleration.

By varying the ID of the MTP target, the location and level of intensification vary greatly. As such, we conducted an extended study on the optimal relation between the tube ID and length. The tube ID was studied without the rear CH2 in order to determine the field at the location of peak intensity, the results of which are summarized in Table I. The ID was varied from 2–7λ using a laser pulse as described above. The dimensionless intensification factor, defined as ηpeak=Ipeak,intube/Ipeak,input, is noted for each tube diameter studied. With a smaller tube, the diffraction effect coupled with the background plasma focusing effect causes the peak intensity to form closer to the entrance of the tube. As the ID is increased, the focusing position is shifted farther from the entrance of the tube. A peak intensity snapshot is shown in Figures 2(a)–2(c). As shown, the highest peak intensity from the geometries studied is found with the 2λ ID tube (ηpeak = 8.56), although simulation results suggest this intensification is short lived, as detailed later in the text. The 4λ ID tube also demonstrates an exemplary peak intensification (ηpeak = 8.36), with the peak intensity falling lower for the larger ID tubes. In order to fully characterize the in-tube pulse profile, we examine the lifetime of the focusing effect to determine the influence of increased background plasma that is found with smaller tube ID.

TABLE I.

Summary of simulation results. All simulations unless otherwise noted have an input pulse of a0 = 50. Case 0 details the input pulse while cases 1–6 show the effects of variable micro-tube plasma lens inner diameters (ID) on the input pulse. The length column represents the distance from the entrance of the MTP lens where the peak intensity occurs. The peak intensification is defined as ηpeak=Ipeak,intube/Ipeak,input. The average intensification, ηave=Iave,intube/Iave,input, is calculated by an averaging technique described in the text. The spot size is listed as the FWHM in y × z for the pulse averaged in space over a half wavelength at the time of peak intensity. For proton acceleration simulations, a 5λ foil is placed at the location of peak intensity in each case and the resulting proton maximum energy is given by Ep,max. Case 7 details a pulse with a0 = 86 that is used on a flat CH2 target for ion acceleration, the purpose of which is described later in the text.

CaseIDLengthηpeakηaveσ (FWHM)Ep,max (MeV)
… … … … 3.6λ × 3.6λ 66 
2λ 1.6λ 8.56 3.20 0.7λ × 0.8λ 104 
3λ 2.2λ 7.40 3.44 0.9λ × 1.0λ 123 
4λ 4.0λ 8.36 5.00 0.9λ × 1.0λ 167 
5λ 5.1λ 5.68 3.44 1.3λ × 1.4λ 180 
6λ 8.0λ 4.22 3.10 1.4λ × 1.5λ 232 
7λ 10.5λ 2.41 2.15 1.7λ × 1.9λ 228 
7a … … … … 1.4λ × 1.5λ 73 
CaseIDLengthηpeakηaveσ (FWHM)Ep,max (MeV)
… … … … 3.6λ × 3.6λ 66 
2λ 1.6λ 8.56 3.20 0.7λ × 0.8λ 104 
3λ 2.2λ 7.40 3.44 0.9λ × 1.0λ 123 
4λ 4.0λ 8.36 5.00 0.9λ × 1.0λ 167 
5λ 5.1λ 5.68 3.44 1.3λ × 1.4λ 180 
6λ 8.0λ 4.22 3.10 1.4λ × 1.5λ 232 
7λ 10.5λ 2.41 2.15 1.7λ × 1.9λ 228 
7a … … … … 1.4λ × 1.5λ 73 
a

Case 7 uses an input pulse of a0 = 86.

FIG. 2.

Intensity distribution at z = 0 in the x-y plane at the time of peak intensity for inner diameters of 2λ (a), 4λ (b), and 6λ (c). (d)–(f) show the transverse intensity distribution for the corresponding tube dimensions given in (a)–(c), averaged in the laser propagation direction within the green box as described in the text. Transverse intensity lineouts for the average intensity are shown in (g)–(i) on an arbitrary scale.

FIG. 2.

Intensity distribution at z = 0 in the x-y plane at the time of peak intensity for inner diameters of 2λ (a), 4λ (b), and 6λ (c). (d)–(f) show the transverse intensity distribution for the corresponding tube dimensions given in (a)–(c), averaged in the laser propagation direction within the green box as described in the text. Transverse intensity lineouts for the average intensity are shown in (g)–(i) on an arbitrary scale.

Close modal

In order to characterize the focal spot lifetime, we locate the position and time of peak intensity within the micro-tube. We record the intensity in the entire simulation every half laser period. At the time of the peak intensification, we average the intensity longitudinally in space over x = ±λ/4 from the location of peak intensity, as shown in Figures 2(a)–2(c), where the green box indicates the location where the averaging takes place. This is repeated for ±12 laser periods from the peak time in order to capture the majority of the beam. After this averaging, we note the peak value of the averaged intensity, Iave,intube. We define the intensification of the beam averaged in this way as the dimensionless ηave=Iave,intube/Iave,input, where Iave,input is the peak value of the input pulse average in the same manner as Iave,intube. The results from this averaging technique are shown in Figures 2(d)–2(f), while lineouts of these intensity distributions are shown in Figs. 2(g)–2(i). We find that ηave is best for ID = 4λ with a value of 5.0, while the average intensification falls off on either side of this peak. The 4λ ID case causes a peak intensity that is similar to that of the 2λ ID case; however, the hot-spot duration for the 2λ ID case is short lived due to an overwhelming background plasma filling the tube, causing more of the laser pulse to be absorbed and/or reflected. We find that the 4λ ID case creates the best conditions for enhancing the in-tube intensity, resulting in an ηpeak greater than 8 and an ηave up to 5.

With a well characterized interaction between the incident pulse and variable MTP lenses, a series of simulations coupling a CH2 foil to the location of the peak intensity have been performed. The proton energy spectrum for four representative cases is shown in Figure 3 at t = 90 T, where the incident pulse is focused on the front of the CH2 foil at t = 30 T. For reasons shown below, although the optimized 4λ gives the best on-target intensity, this does not necessarily equate to the best ion energy spectrum with the rear foil parameters studied in this work. Instead, the highest energy protons found in our study arise from the 6λ ID MTP. This result indicates that the enhancement of TNSA does not solely arise from the intensified on-target laser field, but is more likely a combination of several effects and is likely not optimized for ion acceleration with the current MTP lens configurations under investigation. As is known, TNSA relies upon hot electrons, which, for traditionally flat interfaces, scales as the square root of the laser intensity. Structured interfaces have shown an enhancement in conversion efficiency to electrons; thus, it is natural to assume that introducing the tube also enhances the generation of energetic electrons.

FIG. 3.

Proton energy distribution for a flat interface with a0 = 50 (case 0, black), a flat interface with a0 = 86 (case 7, green), a 4λ ID MTP lens target (case 3, red), and a 6λ ID MTP lens target (case 5, blue) at t = 90 T. The most energetic ions are found using the optimized parameters for a 6λ ID MTP target.

FIG. 3.

Proton energy distribution for a flat interface with a0 = 50 (case 0, black), a flat interface with a0 = 86 (case 7, green), a 4λ ID MTP lens target (case 3, red), and a 6λ ID MTP lens target (case 5, blue) at t = 90 T. The most energetic ions are found using the optimized parameters for a 6λ ID MTP target.

Close modal

We identify two main components that are responsible for the enhanced ion energy: a higher on-target intensity and an enhanced sheath field. In an effort to elucidate the effects from the contributing factors, an additional simulation with an altered incident pulse onto a flat CH2 was performed. In order to determine the effects of the increased on-target intensity independently of the plasma effects of the tube, we used the beam conditions listed in Table I, case 7. The incident pulse in this case has a0 = 86 and a FWHM spot size of 1.4λ × 1.5λ, resulting in a peak intensity increase by a factor of 2.97 and an intensity increase averaged with the same method described above by a factor of 3.07. We have chosen these laser parameters to match the ηave and spot size of the in-tube pulse from the 6λ ID MTP case (Table I, case 5) because this case resulted in the highest proton energies. We note that we have chosen to match the average intensity increase to ηave of case 5 due to the well behaved nature of ηave with respect to the tube ID.

1. Increased on-target intensity

In the TNSA model, the maximum ion energy scales as Eion,max α I1∕2.33 If the enhancement in maximum ion energy were due only to the intensification within the tube, the highest proton energy would arise from the 4λ ID tube. This is not the case, as the 6λ ID tube produces a maximum proton energy of 232 MeV, whereas the 4λ ID MTP target achieves a peak proton energy of 167 MeV. In an effort to understand the effect of intensification on the proton energy distribution, we perform a simulation using a laser pulse with a0 = 86 interacting with a flat CH2. As shown in Table I, the pulse matches reasonably well with the characterized pulse for our initial investigations with the 6λ ID MTP lens. The proton energy spectrum from this simulation is shown in green in Figure 3. The maximum proton energy from the a0 = 86 simulation reaches 73 MeV, which is slightly higher than that of the a0 = 50 simulation (black, 66 MeV), but not nearly as high as the maximum energy achieved with optimized MTP targets. Clearly, the dominant effect of proton enhancement cannot be attributed solely to the intensification in this target regime and additional plasma effects induced by including the MTP lens must play a critical role in the proton energy enhancement.

2. Enhanced sheath field

The substrate thickness dictates that the majority of high energy protons originate from the rear of the CH2 foil. As such, the dominant factor in the proton acceleration is the sheath field. In order to understand the sheath field effect, we look at the electric fields that arise at the rear of the foil in the 6λ ID MTP target and compare this to the flat CH2 foil. Figure 4 shows the longitudinal electric fields for the two cases at t = 30 T and t = 50 T. The 6λ ID MTP target has a sheath field that not only has a higher peak value (>100 MV/μm) as evident by Figures 4(a) and 4(b), but also has an extended longitudinal range as seen in Figs. 4(c) and 4(d) of the same figure. Additionally, we compare the maximum sheath field value for the two cases every two laser periods through the interaction of the laser pulse (Figure 5). The sheath field in the 6λ ID case develops at an earlier time than the flat CH2 and reaches a peak value that is nearly double that of the flat foil. It is worth noting that while the increased on-target intensity is not the primary contributing factor to enhanced TNSA, the strong deformation of the substrate interface in the MTP case in Figure 4(d) indicates an increased hole-boring velocity compared to the flat interface in Fig. 4(c). This is a direct consequence of the enhanced on-target intensity with the MTP lens target.

FIG. 4.

Electron density (grayscale) and the longitudinal electric field Ex (color scale) in the x-y plane at z = 0 for a flat foil ((a) and (c)) and a 6λ ID MTP lens target ((b) and (d), case 5) at the time the peak of the laser pulse reaches the front of the CH2 foil ((a) and (b)) and 20 T later in the simulation ((c) and (d)). Note the log scale of electron density in (a) and (b) to show the periodic electron bunches located in the tube, while the linear scale of electron density in (c) and (d) highlight the increased front-surface target deformation when using the MTP lens target. The proton distribution on a log scale 20 T after the peak reaches the front of the CH2 foil is shown in (e) and (f).

FIG. 4.

Electron density (grayscale) and the longitudinal electric field Ex (color scale) in the x-y plane at z = 0 for a flat foil ((a) and (c)) and a 6λ ID MTP lens target ((b) and (d), case 5) at the time the peak of the laser pulse reaches the front of the CH2 foil ((a) and (b)) and 20 T later in the simulation ((c) and (d)). Note the log scale of electron density in (a) and (b) to show the periodic electron bunches located in the tube, while the linear scale of electron density in (c) and (d) highlight the increased front-surface target deformation when using the MTP lens target. The proton distribution on a log scale 20 T after the peak reaches the front of the CH2 foil is shown in (e) and (f).

Close modal
FIG. 5.

Maximum sheath field (solid line) and proton energy (dashed line) evolution for the flat CH2 foil (case 0, blue) and the optimized 6λ MTP lens target(case 5, red) where 0 T corresponds to the peak of the incident pulse reaching the front surface of the CH2 foil.

FIG. 5.

Maximum sheath field (solid line) and proton energy (dashed line) evolution for the flat CH2 foil (case 0, blue) and the optimized 6λ MTP lens target(case 5, red) where 0 T corresponds to the peak of the incident pulse reaching the front surface of the CH2 foil.

Close modal

The sheath field enhancement is attributed to high density electron bunches that are pulled out of the tube by the transverse laser electric field and accelerated forward via direct laser acceleration (DLA) of the laser pulse. When the laser pulse is sufficiently intense, these electrons can stay in the acceleration phase for a longer time. As the laser reaches the critical surface of the rear foil, the electrons decouple from the laser, and move through the rear foil, as seen in Figure 1(b). These electrons are clearly visible in Figure 4(b) and act to periodically enhance the sheath field as shown in the colorscale of Figure 4(b). The high density, highly localized electrons result in an electric sheath field that is far greater than one would expect from bulk hot electrons typically associated with TNSA of ions as described by the ponderomotive scaling.34 Additionally, the sheath field peak value at later times (Figures 4(c) and 4(d)) is similar in the two cases; however, the longitudinal extent of the sheath field is greatly enhanced in the case of the MTP lens target. These results are consistent with the findings of Refs. 12 and 13. Further, we have developed optimized structures to ensure the incident pulse interacts with both the MTP wall and rear foil substrate. The simulation results demonstrate that the confinement of localized electron bunches that result from the MTP lens structured interface gives rise to enhanced sheath fields and ultimately, much higher proton energies.

In conclusion, we investigated the interaction of a highly relativistic laser pulse with micro-tube plasma lenses coupled to traditional flat interfaces. By varying the dimensions of the MTP lens, we characterize the in-tube laser pulse to optimize the on-target intensity. Finally, we establish a potential application of the MTP lens target by demonstrating an enhancement in proton energy when comparing with traditional flat interfaces. The enhanced ion acceleration results from laser pulse intensification as well as localized electron bunches that are guided by the MTP walls. Other intensity favorable mechanisms such as radiation pressure acceleration (RPA),35,36 break-out afterburner acceleration (BOA),37 and X-ray generation38 would likely benefit more so than TNSA from the MTP target, and these studies are currently underway. Our results suggest a way to make use of current laser and 3D printing technology to increase the on-target intensity and efficiently accelerate ions to significantly higher energies.

This material is based upon work supported by the Air Force Office of Scientific Research under Award No. FA9550-14-1-0085. The authors would like to thank Alexander Pukhov for the use of the code VLPL and Douglass Schumacher for his helpful discussions.

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