Using both experiment and 2D3V particle-in-cell (PIC) simulations, we describe the use of specular reflectivity measurements to study relativistic (2 > 1018 W/cm2μm2) laser-plasma interactions for both high and low-contrast 527 nm laser pulses on initially solid density aluminum targets. In the context of hot-electron generation, studies typically rely on diagnostics which, more-often-than-not, represent indirect processes driven by fast electrons transiting through solid density materials. Specular reflectivity measurements, however, can provide a direct measure of the interaction that is highly sensitive to how the EM fields and plasma profiles, critical input parameters for modeling of hot-electron generation, evolve near the interaction region. While the fields of interest occur near the relativistic critical electron density, experimental reflectivity measurements are obtained centimeters away from the interaction region, well after diffraction has fully manifested itself. Using a combination of PIC simulations with experimentally inspired conditions and an analytic, non-paraxial, pulse propagation algorithm, we calculate reflected pulse properties, both near and far from the interaction region, and compare with specular reflectivity measurements. The experiment results and PIC simulations demonstrate that specular reflectivity measurements are an extremely sensitive qualitative, and partially quantitative, indicator of initial laser/target conditions, ionization effects, and other details of intense laser-matter interactions. The techniques described can provide strong constraints on many systems of importance in ultra-intense laser interactions with matter.

Ultra-intense laser-plasma interactions are capable of producing relativistic electrons which have a variety of applications including the fast-ignition approach to inertial confinement fusion,1 creating ultra-short x-ray sources2 and isochorically heating matter to warm-dense states.3 Each application has specific requirements on energy spectrum, angular distribution, and conversion efficiency where control of various aspects of these relativistic electron distributions is essential to their effectiveness. As the laser-accelerated electron properties are extremely sensitive to the plasma environment in which they are born, laser pulse contrast is one of the more effective controls for shaping hot-electron distributions.

Ultra-intense short-pulse lasers are typically preceded by several millijoules or more of laser energy nanoseconds before the main pulse arrives at the target. This “pre-pulse” usually has sufficient intensity to create tens of microns of under-dense plasma in front of the target. Not only does this pre-plasma move the laser interaction interface from supra-critical solid density to the lower relativistic critical density but also subjects the main pulse to instabilities4–8 and quasi-static field generation,9,10 which can greatly modify relativistic electron generation and transport. High-contrast lasers minimize the laser energy before the main pulse and, therefore, limit pre-plasma formation. Several facilities, like LULI,7 Trident,11 HERCULES,12 and Titan already have this pulse cleaning capability typically obtained through nonlinear optical processes, such as harmonic generation and third order cross-polarized wave generation,13 or with plasma mirrors.14,15 The advent of laser systems with intense, but extremely clean, laser pulses has ushered in a new regime of experiments. The absence of pre-plasma, however, tends to reduce the overall laser coupling efficiency and, under these conditions, the interaction region more closely resembles the initial target interface, thus making surface conditions important.16–18 To study these phenomena, an experiment was performed to study relativistic electrons generation in the intense laser-plasma interactions (LPI) using both high and low-contrast laser pulses. Since these electrons are not directly observable in the region of interest, i.e., near the interaction region, studies typically rely on a conglomeration of indirect experimental measurements to constrain simulations used to study the source.

Historically, transport diagnostics have been almost exclusively used to study hot-electron generation, where the electrons are indirectly observed as they propagate through the target by measuring bremsstrahlung emission,19–21Kα emission from tracer fluor layers22,23 or wires6,24 generated when hot-electrons knock out K-shell electrons, and transition radiation at an interface25 while other measurements directly measure electrons that escape the target using magnetic spectrometers.26,27 Electron transport, however, can be quite sensitive to target geometry and refluxing,28 target heating,29 current30 and resistivity gradient31 driven magnetic fields inside the target, target charging effects,32 and beam filamentation instabilities.33,34 All these transport phenomena complicate the interpretation of the LPI generated source.

Although still an indirect measure of the hot-electrons of interest, diagnostics measuring the unabsorbed light (from either the specularly reflected pulse or harmonics generated near the interaction interface35) can provide a direct measurement of key properties of the laser-plasma interaction. The change in divergence between the incident and specularly reflected pulse, due to the shape of the relativistic critical surface, has been shown to be a strong indicator of pre-plasma scale length near critical density.36 Instantaneous spectral shifting and broadening due to motion of the critical surface,37 relativistic effects,38 and dynamically ionizing media39 have also been observed to be quite sensitive to pre-plasma environment, as well as temporal pulse front steepening due to group velocity dispersion.40 Spatial, spectral, and polarimetry measurements of harmonics generated near the critical surface have been found to be sensitive to pulse contrast41 and target surface roughness42,43 as well as indicative of magnetic fields in the under-dense pre-plasma environment.44,45 This underlying theme of sensitivity to the initial laser and plasma environments, therefore, suggests that specular reflectivity measurements can provide alternative constraints on the possible interaction profiles at the source of the hot-electron generation. Indeed, since these laser-fields near the interaction region are directly responsible for accelerating these electrons to relativistic energies, it is critical that they be accurately represented in simulations studying hot-electron generation at the source.

All the interesting physical phenomena that affect the specular pulse properties such as absorption, relativistic non-linearities, hole-boring, field ionization, etc., occur near the interaction region around the relativistic critical surface (dubbed “near-field”). Many of these phenomena are nonlinearly dependent on laser intensity and electron density profiles—so, since the pulse has a finite focal spot, we expect there to be some spatial dependence on these effects in the near-field. However, specular reflectivity measurements are typically obtained centimeters away well after diffraction has fully manifested itself (dubbed “far-field”), and it is a priori unknown the extent to which the far-field observations can be used to constrain the near-field interaction conditions; in order for these specular reflectivity measurements to be useful in benchmarking simulations, they must be indicative of the near-field physics.

In this work, we address how well far-field specular reflectivity measurements, as an alternative to electron transport diagnostics, can constrain simulations studying near-field physics and hot-electron generation. In Sec. II, we discuss the setup and specular reflectivity measurements from the high vs. low-contrast experiment that inspired this simulation study. Using somewhat simplified 2D3V particle-in-cell simulations, discussed in Sec. III, both high and low-contrast pulse interactions are modeled using laser and plasma conditions characteristic of the experiment. In Sec. IV, estimates of various pulse-altering phenomena are made to determine their relative influences on the interaction in the near-field. Propagating the light to the far-field within these same simulations, where the experimental measurements were made, would be prohibitively expensive since we would have to resolve the light propagation over millimeters or even centimeters of space before the diffraction pattern would converge. Instead, far-field properties of the specularly reflected pulse are obtained by analytically propagating the near-field simulation results to the far-field, millimeters away (Many techniques exist and several are discussed, along with their validity under these conditions, in the Appendix). Far-field properties of the pulse are addressed in Sec. V and implications of diffraction dominated pulse propagation on the interpretation of experimental specular reflectivity measurements are discussed in Sec. VI.

The experiment was performed on Titan at the Jupiter Laser Facility (JLF) at Lawrence Livermore National Laboratory (LLNL) in Livermore, CA. Titan is a two beam platform with a nanosecond scale “long-pulse” and a picosecond Petawatt scale “short-pulse,” centered around 1.053 μm, which can be used independently or co-linearly with controllable relative delay. The second harmonic of both pulses is also available, generated using a 2 mm thick KDP crystal. Since second harmonic generation is a nonlinear process (i.e., E(2ω)χ(2)|E(ω)|2), lower intensity portions of the pulse are not converted as efficiently as the higher intensity portions, resulting in a higher contrast pulse at the second harmonic (wavelength λL = 0.527 μm, angular frequency ωL = 2πc/λL = 3.57 rad/fs, period TL = λL/c = 1.76 fs). To have controllable “pre-pulse” for a low-contrast pulse comparison, the long-pulse was injected coaxially with the short pulse (converted with the same crystal as the short-pulse).

Shown in Fig. 1 is an illustration of the experimental setup showing the layout of the Titan target chamber and the paths of the incident (green) and reflected (blue) laser pulses. The incident pulse characteristics were obtained in situ for each and every shot from leakage (purple) through the last turning mirror before the parabola; the laser diagnostics included temporally resolved intensity and phase characterization using polarization-gated, frequency resolved optical gating (PG FROG),46 pulse energy, integrated spectrum, a pre-pulse monitor, and a full-aperture equivalent-plane focal spot monitor.

FIG. 1.

Illustration of the experimental setup on Titan at the Jupiter Laser Facility. The incident pulse (green) was characterized using leakage (purple) through the last turning mirror before the parabola and the specular pulse path is shown in blue.

FIG. 1.

Illustration of the experimental setup on Titan at the Jupiter Laser Facility. The incident pulse (green) was characterized using leakage (purple) through the last turning mirror before the parabola and the specular pulse path is shown in blue.

Close modal

The resulting second harmonic, high-contrast, short-pulse had approximately 35 J of energy with less than 10 μJ (Ref. 47) of intrinsic pre-pulse delivered over 3 ns before the main pulse; this is a significant improvement in contrast over the typical 15 mJ of intrinsic pre-pulse in the fundamental. The long-pulse delivered approximately 3 mJ of “pre-pulse” at the second harmonic in the 3 ns before the short-pulse arrived. The incident laser pulses were measured to have ≈500fs full-width at half-maximum (fwhm) duration with spectral bandwidth of around 1.7 nm fwhm. The incident laser was then focused down onto the target using an off-axis parabola (13.5°, f/2.5, f = 60 cm); the spot sizes of the short and long-pulses on target were approximately Gaussian with 8 μm and 14 μm fwhm, respectively. The peak intensity of the short(long)-pulse was 4.6 × 1019 W/cm2(6.4 × 1011 W/cm2); peak intensity contrast ratios for the high and low-contrast pulses were less than a few times 10−11 and 10−8, respectively. The interaction was p-polarized, incident on the target at 13.5° with respect to target normal.

In previous work,16,48 we have shown how target surface morphology can strongly influence the hot-electron generation and specular reflectivity measurements in high-contrast short-pulse interactions. Indeed, for targets with initial surface perturbations (i.e., roughness), significant enhancements in absorption have been reported which have typically been attributed to enhanced Brunel absorption,30,34 resonant excitation of surface plasma waves,49–52 or local field enhancement via Mie resonance,53,54 depending on the specifics of each study. Great diligence has therefore been taken to characterize the aluminum target's surface morphology on both large (∼1μm) and a sub-wavelength (∼100 nm) scales using atomic force microscopy.55 

After interacting, the light energy that is not absorbed by the target is either scattered or specularly reflected, illustrated by the blue path in Fig. 1. Using a variety of diagnostics, we studied the spatially resolved reflectivity, spatially integrated spectra, and temporally resolved instantaneous intensity and wavelength shifts of the specularly reflected pulse. For the purposes of this study, and due to large fluctuations in initial target conditions and incident laser parameters, only two shots have been chosen to be shown and modeled. In particular, the following data are characteristic of the the target features for these shots (similar spectral distributions with ∼200 nm rms surface perturbations); these results are qualitatively similar across nominally identical shots despite quantitative variations.

1. Spatially resolved reflectivity and integrated spectra

The energy and spatial profile of the specularly reflected pulse was obtained by imaging the light that is diffusively reflected off a calibrated Lambertian scattering plate made of optical grade Spectralon®, located along the specular axis, with an Andor DV434 CCD (as indicated in Fig. 1). The 25.4 cm × 25.4 cm screen was located 35.4 cm away from the interaction region (f/1.4 minimum solid angle); shown in Fig. 2(a) are the high (left) and low-contrast (right) shots of interest. The hard edges in the upper-right corners are shadows cast from other diagnostic apparatus inside the chamber, and the three holes near the center were used to send light to other diagnostics. The experimental reflectivity (defined as the ratio of the energy incident on the scattering plate to that of the original laser pulse) for the high(low)-contrast interaction was 41.6 ± 8.3% (14.6 ± 2.5%) where the error bars include uncertainty from the interpolation over the holes and spectral response of the detector. Aside from the nearly factor of 3 difference in reflectivity, the spatial profile of the high-contrast shot is quite structured compared to the much smoother low-contrast shot. The white circle indicates the approximate size of the incident f/2.5 top-hat laser profile at this position; both the high and low-contrast specular pulses appear to have increased contributions outside the incident beam cone.

FIG. 2.

(a) Experimental Spectralon® data of the high (left) and low-contrast (right) shots; the incident f/2.5 top-hot laser profile is indicated by the white circle. (b) Spatially and temporally integrated spectra of incident (green), high (blue) and low-contrast (red) specular pulses.

FIG. 2.

(a) Experimental Spectralon® data of the high (left) and low-contrast (right) shots; the incident f/2.5 top-hot laser profile is indicated by the white circle. (b) Spatially and temporally integrated spectra of incident (green), high (blue) and low-contrast (red) specular pulses.

Close modal

Shown in Fig. 2(b) are the time-integrated spectra, integrated over the entire spatial profile, of the incident (green) and specular pulses (blue = high-contrast, red = low-contrast). In either scenario, the specular pulse is blueshifted by ≈ 1 nm (i.e., an intensity weighted average of 526 nm) with ≳2.5 times broader spectra than the incident pulse (∼4.2 nm fwhm). We note the increased contribution of “red” wavelengths, relative to the carrier, in the low-contrast pulse as compared to the high-contrast pulse.

2. Temporally resolved instantaneous wavelength shifts

Specularly reflected light that passed through the hole in the Spectralon® closest to the specular axis (see Fig. 2(a)) was collected and collimated using a Galilean telescope (lenses L1 and L2 in Fig. 1) before being sent to another PG FROG to obtain the temporal profile of the specular pulses (albeit a spatial sub-sample, slightly off true specular). Shown in Fig. 3(a) are the experimental FROG traces for the incident, high-contrast and low-contrast specular pulses (left, middle and right columns, respectively), shown on a log10 color scale to emphasize low intensity portions of the signal. Using an iterative phase-retrieval algorithm, the pulses were reconstructed (with rms FROG errors of a few percent) to produce the instantaneous intensity (black)/spectral shifts (red) in Fig. 3(b) and spectrum in Fig. 3(c). The shaded areas (gray and pink for intensity and spectral shift, respectively) indicate uncertainty due to both experimental data and pulse retrieval algorithms; the wavelength shift error is given as a quadratic sum of the uncertainty in the absolute spectral reference (≲0.5 nm) and the standard deviation (σSTD) about the mean value of 100 unique reconstructions. Note that since each reconstruction is obtained by minimizing the rms FROG error by randomly switching between several iterative phase-retrieval algorithms (including the vanilla, power, and principle-component generalized projections algorithms56) and is initialized with a randomized seed guess, each reconstruction is unique and degenerate given the data constraint.

FIG. 3.

(a) Experimental PG FROG traces, (b) retrieved temporal, and (c) spectral pulse profiles; the incident laser pulse, high and low-contrast specular pulses are shown on the left, middle, and right columns, respectively. The shaded areas in (b) and (c) indicate uncertainty in intensity (gray) and spectral shifts (pink). The red curves in (c) are independent spectral measurements, averaged over the entire spatial profile.

FIG. 3.

(a) Experimental PG FROG traces, (b) retrieved temporal, and (c) spectral pulse profiles; the incident laser pulse, high and low-contrast specular pulses are shown on the left, middle, and right columns, respectively. The shaded areas in (b) and (c) indicate uncertainty in intensity (gray) and spectral shifts (pink). The red curves in (c) are independent spectral measurements, averaged over the entire spatial profile.

Close modal

Since the incident pulses (left column) had very little instantaneous wavelength shifts (|Δλ/λL|0.25%,Δλ=λinstλL), any significant shift in the specular pulses was predominantly due to the interaction with the target. Very little spectral shifting of the high-contrast specular pulse (middle column) was observed, suggesting that there was likely very little pre-plasma, while the low-contrast shot (right column) showed an early redshift of about +3.2 ± 1%, which gradually decreased to 0 by the peak of the pulse. By the end of the interaction, both pulses had slightly blueshifted to approximately −0.8 ± 0.2%. If the shifting was solely due to Doppler shifts from the motion of the critical surface, like previous studies,37 these trends would physically correspond to an initially receding critical surface on the rising-edge of the pulse followed by an expanding surface on the trailing edge.

Shown in Fig. 3(c) are also the independently and simultaneously measured, spatially integrated spectra (red) from Fig. 2(b). While these independent measurements of the spectra fit within the error bars of the retrieved FROG pulses for the incident and high-contrast specular pulses, the low-contrast measurement differed significantly on the red side of the spectrum. Since the FROG measurement only sampled an ∼f/14 window and the independent spectrum measurement was integrated over the entire ∼f/1.4 scattering plate, one possible explanation for this discrepancy could be spatial non-uniformities in the far-field instantaneous wavelength shifts.

2D3V Cartesian fully-kinetic particle-in-cell (PIC) simulations were performed, using the commercially available code Lsp,57 to address how well specular reflectivity measurements can constrain simulations used to study relativistic LPI and hot-electron generation. To this end, we modeled both the high and low-contrast interactions discussed in Sec. II. Since these interactions are known to be quite sensitive to initial laser and plasma conditions, great diligence has been taken to reasonably represent the high and low-contrast interactions, but—due to present simulation and computational limitations—several simplifying assumptions have been asserted to make the calculations feasible. While producing a quantitative match with all the experimental data is beyond the scope of this work, we believe that our simulations are qualitatively capturing the essence of these interactions and sufficient for our goal of studying trends in near and far-field specular pulse properties.

For the low-contrast, injected pre-pulse shots, radiation-hydrodynamic calculations were performed using Multi2D58 to estimate the pre-plasma environment conditions. The calculation was performed with laser and target conditions modeled after the experiment using cylindrically symmetric RZ geometry (normal laser incidence). Shown in Fig. 4(a) is the total electron density on a log10 color scale (as ne/nc, normalized to the critical electron density nc=meωL2/4πe2=4.0×1021cm3), (b) the average ion charge state Z*, and (c) the electron temperature Te at 3 ns, just before the short pulse arrived at the target. Absorption at these intensities is largely dominated by collisional processes,59,60 and nearly 100% laser absorption was observed in these simulations due to inverse-bremsstrahlung.

FIG. 4.

Modeled pre-plasma environment using a 2D RZ radiation hydrodynamics Lagrangian code Multi2D at 3 ns: (a) electron density (normalized to the critical density, ne/nc), (b) average charge state Z* and (c) electron temperature Te in eV. The three black curves in all subplots are the nc, nc/10, and nc/100 contours.

FIG. 4.

Modeled pre-plasma environment using a 2D RZ radiation hydrodynamics Lagrangian code Multi2D at 3 ns: (a) electron density (normalized to the critical density, ne/nc), (b) average charge state Z* and (c) electron temperature Te in eV. The three black curves in all subplots are the nc, nc/10, and nc/100 contours.

Close modal

Originally, the solid density aluminum interface (ne/nc ≈ 45) was perfectly flat (initially the x = 0 μm plane). Now, 3 ns later and just before the main pulse arrived, the pre-pulse has ablated away enough material to create a 5μm dimple in the originally flat solid density interface and created under-dense plasma that extends out nearly 100 μm with spatially varying charge state and temperature distributions. The pre-plasma electron density profile can be described by various exponential density profiles depending on where you look. Some of these local characteristics are summarized in Table I at various electron densities (relative to the critical density) using a line-out along the symmetry axis (z = 0 μm). The location x of these densities are given along with the local exponential scale length L where ne(x)exp(x/L); also shown are the local average charge states and electron temperatures. Although the under-dense plasma tens of microns from solid density is nearly 10× ionized, these simulations would suggest that it is only 5–6× ionized near the critical density.

TABLE I.

Summary of the simulated pre-plasma environment from the Multi2D calculation, along the symmetry axis (z=0μm).

ne/ncx (μm)L (μm)Z*Te (eV)
+4.7 1.3 5.5 30 
0.1 −6.5 10.6 10.2 62 
0.01 −36.8 18 9.1 36 
ne/ncx (μm)L (μm)Z*Te (eV)
+4.7 1.3 5.5 30 
0.1 −6.5 10.6 10.2 62 
0.01 −36.8 18 9.1 36 

The setups for the high and low-contrast LPI simulations are illustrated in Fig. 5. For the (a) high-contrast simulations, we used multi-modal, realistic target perturbations where the interface began sharp and free of under-dense pre-plasma. The target was modeled as a solid density (ne = 1.81 × 1023 cm−3), initially Al3+ charge state aluminum slab, 75 μm deep by 150 μm wide with absorbing boundary conditions. Similarly, for the (b) low-contrast interactions, simulations were initialized with electron density, temperature, and average charge state distributions from the aforementioned Multi2D simulation of the pre-pulse; since only species with integer charge states can exist in Lsp, the continuous Z* distribution for each cell was approximated using a linear combination of the bounding integer charge states, i.e., Z* and Z*. The pre-plasma was modeled out to nc/100, nearly 37 μm away from the original solid density interface and laser focal plane.

FIG. 5.

Illustration of the (a) high and (b) low-contrast LPI simulations; electron density is shown on the log10 gray color scale and the normally incident laser is shown in green. Specular reflectivity measurements were made at the left boundary of the simulation box.

FIG. 5.

Illustration of the (a) high and (b) low-contrast LPI simulations; electron density is shown on the log10 gray color scale and the normally incident laser is shown in green. Specular reflectivity measurements were made at the left boundary of the simulation box.

Close modal

To reduce computational cost, the incident laser pulse was modeled with a 1 ps duration sin2 intensity profile with a 500 fs fwhm. We note here that we are not accurately representing the rise or fall of the incident pulse where there is clearly non-negligible energy according to the incident and specular PG FROG measurements (see Fig. 3(b)). For simplicity, we model the interaction at normal incidence (the laser is polarized in the simulation plane) rather than imposing the relatively small 13.5° incidence angle of the experiment. We expect that this deviation from the experimental conditions will play little to no role in these interactions, however, as initial surface roughness and rippling of the relativistic critical surface (from laser filamentation) introduce larger variations in the local incidence angle at different points across the profile than 13.5°. The laser was focused in each scenario at x = z = 0 μm with a focal spot characteristic of the experiment; a superposition of 3 Gaussians fit to the symmetrized experimental focal spot was used, resulting in an intensity fwhm of ≈8 μm and peak intensity of approximately 4.6 × 1019 W/cm2. The peak quiver energy εp of electrons in the oscillating electric field is given by εp = (γo – 1)mec2 = 1.1 MeV, where γo=1+ao2=3.2 is the peak relativistic Lorentz factor and ao = eEo/meωLc = 3.1 is the peak normalized vector potential. Both the incident and specularly reflected pulse field quantities were recorded at the laser inlet/outlet boundary of the simulation, approximately 25 μm from any initial plasma.

The vacuum gap on the front of the target served dual purposes by simultaneously reducing the influence of plasma-driven fields on the laser-field measurements at the boundary and accommodating hot-electron refluxing near the front of the target. We used absorbing boundary conditions; while this can result in artificial electrostatic fields at the boundaries of the simulation, we note that these fields are several orders of magnitude weaker than the laser-driven fields. Additionally, these electrostatic fields only develop at the rear side of the target (since the hot-electrons generated in the intense LPI are primarily forward going) and only after ∼1 ps into the interaction, so it is highly unlikely that they play any significant role in the interaction region.

The simulations were performed using a direct implicit algorithm incorporating an energy-conserving particle push57 with a temporal step of 27.5 as (≈TL/64) and regions of the grid where light was present had a spatial resolution of 31.2 nm (≈λL/16), growing linearly to ∼λL/2 inside solid density. Through a separate study, specular pulse properties and hot-electron distributions in time, space, energy, and angle were found to be sufficiently converged at these resolutions. Electron macro-particle densities ranged from 144 to 196/cell, and the ions ranged from 25 to 49/cell resulting in ∼500–750 M macro-particles in the simulation; all species are fully kinetic, initialized with T = 5 eV, and collisionless. Since electrons in regions where the laser can directly heat can reach ∼keV energies within the first tens of femtoseconds of the interactions, the collisionless assumption is commonly used for studying electrons born in the ultra-intense laser region to reduce computational cost.16,59,61 Indeed, we have verified that running with collisions (via the Jones algorithm62) produces nominally identical specular pulse and hot-electron properties as the collisionless case.

For low-Z targets, a dynamically ionizing medium may have minimal influence on the interaction, but in mid to high-Z materials, field and collisional ionization/recombination may cause the local electron density to fluctuate by an order of magnitude. Clearly, the high-contrast interactions should be initialized with the room temperature free-electron density and effective charge state (≈Al3+), but the extent to which these low-contrast interactions with aluminum are sensitive to ionization dynamics is a priori unknown. To partially account for dynamic ionization effects, sequential tunneling field ionization is included (using Ammosov, Delone, and Krainov ionization rates for ions in an alternating electromagnetic field63), but no collisional ionization or recombination effects are modeled. For this work, we also discuss a fixed ionization state, low-contrast interaction for comparison which serves as an extreme case where collisional recombination strongly dominates over field ionization.

Previous work37 has shown that instantaneous wavelength shifts observed in the specularly reflected light (in low-contrast interactions with fully ionized pre-plasmas) are predominantly due to Doppler shifting from the motion of the relativistic critical surface, given by Eq. (1) where βc = vc/c is velocity of the critical surface (relative to the speed of light c); in this geometry, βc > 0 (βc < 0) corresponds to a receding (expanding) critical surface and a red (blue) spectral shift. However, depending on the circumstances of the interaction, other mechanisms can play a significant role in the shifting. Indeed, any spatial or temporal variations in the plasma index of refraction η=1ne/γnc can also introduce spectral shifts, and these variations can arise either through ne(x,t) or γ(x,t). The amount of phase ϕ accumulated by a pulse after propagating a distance L through a spatially and temporally varying index of refraction is given by Eq. (2),38,64 where the corresponding instantaneous frequency is simply ωinst=ωLϕ/t

ΔλλL=λinstλLλL=2βc1βc,
(1)
ϕ=ωLcxoxo+Lη(x,t)dx.
(2)

Aside from bulk electron motion, modifications to the local electron density can come about by either ionization (ṅe>0η̇<0ϕ̇<0ωinst>ωL, blueshift) or recombination (ṅe<0η̇>0ϕ̇>0ωinst<ωL, redshift). Considering only over-the-barrier field ionization,65 the peak charge state that the aluminum can reach in this field is Al11+, which occurs at intensities ≳ 1.6 × 1018 W/cm2. Given the temporal profile of the pulse (and lack of recombination), the field ionization should only produce a blueshift in the first several tens of femtoseconds on the rising-edge of the pulse for our aluminum targets. While field ionization will saturate for this pulse with aluminum, other materials, like gold for example, may continue to be influenced by ionization/recombination effects throughout the interaction (as has been observed in experiment66). Effects that rely on the relativistic modification of the electron plasma frequency ωp,e2=4πnee2/γ(x,t)me like self-phase modulation, self-focusing, pulse-front steepening, filamentation, etc., will also be sensitive to the spatial and temporal profiles of the pulse.

Shown in Fig. 6 are spatially and temporally resolved estimates of the near-field spectral shifts (given as a percentage of the incident pulse carrier) due to (a) Doppler shifting from motion of the relativistic critical surface and (b) relativistic phase modulation from propagating through the under-dense plasma and (c) their combined effect for the simulated high-contrast, fixed and dynamically ionizing low-contrast interactions (left, middle and right columns, respectively). The Doppler shift was calculated by tracking the motion of the relativistic critical surface and Eq. (1). The phase modification due to ionization, redistribution of under-dense electron density, and relativistic effects was estimated using the temporally evolving under-dense plasma density from the simulation, Eq. (2), and the incident laser intensity profile (assumed to be unperturbed); the integral was only performed up to the reflection off the relativistic critical surface since the intensity of the reflected pulse is significantly less than the incident pulse. We note here that this is likely an overestimate of the self-phase modulation due to non-linear absorption of the laser energy as it propagates through the under-dense plasma. The combined influences of phase modulation from propagation through the plasma before being Doppler shifting from reflecting off the moving relativistic critical surface in (c) are given by Eq. (3). For clarity, the 1% and 50% peak intensity contours of the incident laser are indicated by the dashed-black curves and any intensity below 0.1% of the peak is truncated (shaded gray areas)

ΔλtotalλL=ΔλDopplerλL+ΔλSPMλL+ΔλDopplerλLΔλSPMλL.
(3)
FIG. 6.

Near-field estimates of spectral shifting of the specularly reflected pulses due to (a) Doppler shifts, (b) relativistic phase modulation, and (c) their combined influence in the high-contrast, low-contrast fixed field ionization interactions (left, middle, and right columns respectively). The spatial striations are due to laser filamentation and electron density modification.

FIG. 6.

Near-field estimates of spectral shifting of the specularly reflected pulses due to (a) Doppler shifts, (b) relativistic phase modulation, and (c) their combined influence in the high-contrast, low-contrast fixed field ionization interactions (left, middle, and right columns respectively). The spatial striations are due to laser filamentation and electron density modification.

Close modal

For the high-contrast interaction (left column), very little shifting from either Doppler motion or phase modulation are expected and only late blueshifts (∼1%) are observed at late times from the expansion of the critical surface. For the fixed ionization, low-contrast interaction (middle column), the estimation of the Doppler shift from the motion of the relativistic critical surface suggests an early redshift (∼+1.5%) that decreases in time to near zero (intensity weighted values). Spatial striations in the distribution on the rising-edge of the pulse, due to the rippling of the critical surface from laser filamentation, coalesce down to only a few by the trailing edge of the pulse. The self-phase modulation estimate suggests shifting of ∼±2%, comparable to that of the Doppler estimates.

For the low-contrast, dynamically ionizing simulation (right column), field ionization raised the local electron density near the relativistic critical interface and drove an early Doppler blueshift (up to ∼−5%, although short-lived) just from the motion of the critical surface away from solid density (which was absent in the fixed ionization case). Only lasting ≈50 fs, this is consistent with the rise time for the laser to reach ∼1018 W/cm2, an intensity sufficient to strip the aluminum ions to the highest charge state achievable with this laser through field ionization. The expansion of critical surface is then quickly overcome by the increasing laser intensity, turning into an even stronger redshift (∼+2.5%) than what was observed for the fixed ionization case. Once again, this redshift gradually disappeared by the peak of the pulse, turning into a slight blueshift on the trailing edge. Although still present, the spatial striations in the profile are less persistent and well defined than the fixed charge state simulation, consistent with ionization defocusing counteracting the laser filamentation and self-focusing which ultimately results in a less structured relativistic critical surface. The instantaneous wavelength shifts due to phase modulation were estimated to be less than approximately +2% on the rising-edge and −1% on the trailing. If the electron density profile was constant throughout the simulation, the relativistic phase modulation effect should have created equal strength red and blueshifts. This suggests that the under-dense electron density profile is dynamically evolving throughout the interaction (more so than the fixed ionization case). Significantly larger shifts are expected in either scenario of the low-contrast simulations as compared to the high-contrast interaction, consistent with what was observed in experiment.

In order to study the correlations between the near-field physics of interest and the diffraction driven far-field experiments measurements, we must be able to study the simulated specularly reflected pulse after diffraction has fully manifested itself millimeters or even centimeters away from the interaction region. Although being computationally intractable or prohibitively expensive in the PIC simulations, producing far-field pulse phenomenon is analytically approachable. Using the broad-band, non-paraxial algorithm outlined in the Appendix, we now discuss properties of the simulated near-field pulses after they have been propagated 10 mm away from the interaction region into the far-field; at this distance, the diffraction patterns have sufficiently converged. Quantities of particular interest, as inspired by the experiment, are the temporally integrated spatial intensity distribution, temporally and spatially integrated spectrum, and spatial uniformity of the temporally resolved instantaneous wavelength shifts.

Shown in Fig. 7(a) are the time-integrated far-field spatial intensity distributions of the high-contrast (blue) and low-contrast interactions (magenta = fixed ionization, red = field ionization). Also indicated by the shaded gray area is the solid angle of the Spectralon® from the experiment. Consistent with the experimental trends, we observe “spikier” structures in the high-contrast case and overall broader spots with finer scale features for the low-contrast interactions. In fact, the spiky nature of the high-contrast profile is characteristic of the initial surface perturbations (see the Appendix, Subsection 1), suggesting that the target surface roughness, to some extent, survived throughout the interaction and can be inferred from specular reflectivity measurements; this can be invaluable information for those studying high-harmonic generation.43 The differences in the spatial distributions between the two low-contrast interactions are likely due to differences in the shape of the relativistic critical surface (recall the evidence of significant filamentation observed in the near-field for the fixed ionization case that was less defined for the field ionization simulation), as has been previously observed.36 

FIG. 7.

(a) Simulated integrated intensity profiles, 10 mm in the far field of the high (blue) and low-contrast interactions (magenta = fixed ionization, red = field ionization); the experimental measurement solid angle indicated by the shaded gray area. (b) Spatially integrated spectra with the same color scheme as (a).

FIG. 7.

(a) Simulated integrated intensity profiles, 10 mm in the far field of the high (blue) and low-contrast interactions (magenta = fixed ionization, red = field ionization); the experimental measurement solid angle indicated by the shaded gray area. (b) Spatially integrated spectra with the same color scheme as (a).

Close modal

The total reflectivity (defined to be the unabsorbed light fraction, which includes scattered light contributions) in the high-contrast simulation was 62%. The low-contrast interactions had 20.8% and 14.3% reflectivity for the fixed and field ionization cases, respectively. Since 2D and 3D pulses propagate differently (see the Appendix, Subsection 2), it is non-trivial to report a comparable specular reflectivity value with experiment. If we were, however, to define the specular reflectivity to be only the light within the experimental solid angle (shaded gray area), the 2D specular reflectivity would be 51% for the high-contrast interaction and 7.7% (6.6%) for the fixed (field) ionization low-contrast interactions, respectively.

Shown in Fig. 7(b) are the spatially integrated spectra; we find significantly broader spectra for the low-contrast interactions than the high-contrast case (same color scheme as (a)), consistent with the larger spectral shifts expected from the near-field physics. While the spectra for the high-contrast interaction broadened and shifted in a way that was qualitatively and quantitatively consistent with the experimental data, both of the low-contrast interaction simulations differed significantly from the experiment. While the experimental data suggested an overall blueshift of 1 nm, centered at 526 nm with only ∼2.5× broadening for both the high and low-contrast interactions, the low-contrast simulations produce specular pulses with ≳10 × broader spectra that are strongly redshifted to ≈535 nm, an overall 1.5% redshift; this suggests that the LPI in these low-contrast simulations are not characteristic of the low-contrast experiment. We suspect that this discrepancy with the experimental shifting trend is likely due to not accurately representing the rising-edge of the experimental laser pulse.

Shown in Fig. 8(a) are the spatially and temporally resolved far-field instantaneous wavelength shifts of the specularly reflected pulses for the high-contrast, low-contrast fixed, and field ionization simulations (left, middle and right columns, respectively). Consistent with the near-field estimates, the high-contrast interaction produced a specular pulse with very little shifting, only turning blue near the tail of the pulse; the spatial-temporal distribution is also quite uniform in space like that of the near-field estimate in Fig. 6. We observe significantly stronger spectral shifting in the low-contrast interactions but with less spatial uniformity. Unlike the near-field estimates, the largest shifts actually appear away from the specular axis on the rising-edge of the pulse, becoming more spatially uniform after the peak of the pulse.

FIG. 8.

(a) Far-field instantaneous spectral shifts for the specularly reflected high-contrast (left) and low-contrast fixed (middle) and field (right) ionization cases. (b) Spatially averaged intensity (black) and spectral shifting (red); the shaded areas indicate a ±σSTD in the spatial non-uniformity of each quantity (gray and pink, respectively). Also shown are the spatially integrated near-field estimates of phase modulation (green), Doppler shifts (cyan) and their combined effect (blue).

FIG. 8.

(a) Far-field instantaneous spectral shifts for the specularly reflected high-contrast (left) and low-contrast fixed (middle) and field (right) ionization cases. (b) Spatially averaged intensity (black) and spectral shifting (red); the shaded areas indicate a ±σSTD in the spatial non-uniformity of each quantity (gray and pink, respectively). Also shown are the spatially integrated near-field estimates of phase modulation (green), Doppler shifts (cyan) and their combined effect (blue).

Close modal

Since the shape of the relativistic critical surface largely determines the far-field diffraction pattern,36,48 spatial non-uniformity in spectral shifting suggests that the interaction region is dynamically evolving throughout the interaction and that many of the strongest spectral shifts occur in non-paraxial k-modes. Indeed, the critical surface around the peak laser intensity, because of self-focusing and filamentation instabilities, is quite rippled with wavelength scale features and dynamically evolving which drives the spatial-temporal non-uniformities in the far-field. For the low-contrast dynamically ionizing simulation, we find more uniform intensity and spectral shifts in space which is consistent with the near-field picture of a less dynamic, smoother shape of the reflecting surface than in the fixed ionization case.

Due to the diffraction driven redistribution of energy in the far-field, a direct comparison of the spatial-temporal shifting between the near-field physics and the far-field distribution is non-trivial; instead, we compare intensity weighted near-field quantities to the spatially resolved far-field shifts. Shown in Fig. 8(b) are the intensity-weighted instantaneous intensity (black) and spectral shifting (red) curves for each of the distributions in (a); the ±σSTD variation in intensity and spectral shifting distributions (shaded gray and pink areas, respectively) along the spatial profile of the beam to indicate the extent of spatial uniformity. The intensity-weighted, near-field estimates of Doppler shifting (cyan), phase modulation (green) and their combined effect (blue) from Fig. 6 are also plotted. In each case, the intensity-weighted far-field spectral shifts are reasonably consistent with the near-field estimates from Doppler shifting (cyan) with the largest discrepancies occurring on the rising and falling edges of the pulse. At these early and late times, the addition of self-phase modulation pushes the shifting in the right direction but tends to overestimate its influence, likely due to the simplifying assumptions made in the calculation. We note here that the estimated Doppler shifting patterns, at all times, lies within the ±2σSTD spatial variation while the phase modulation alone does not, suggesting that the Doppler shifting from the motion of the relativistic critical surface is still likely the dominant mechanism.

Unlike the experiment, very little temporal broadening of the pulse occurs in either low-contrast simulation, but both appear to be slightly temporally steepened. If the electron density profile was constant in time and there was no laser absorption or modification due to the interaction, profile steepening of the peak due to group velocity dispersion (GVD)40 is estimated to be only a few femtoseconds, negligible compared to the ∼100 fs observed. Self-phase modulation can, however, broaden the spectrum, resulting in greater GVD and creating a stronger effect, but that influence is also negligible here. The temporal profile modification of the specular pulse is, therefore, likely due to a combination of non-linear absorption, self-focusing, and diffraction effects.

The experiment results and PIC simulations demonstrate that specular reflectivity measurements are an extremely sensitive indicator of initial laser/target conditions, ionization effects, and other details of intense laser-matter interactions. The techniques described can provide strong constraints on many systems of importance such as the generation of hot-electrons from ultra-intense lasers. For coupling experiments looking to study the hot-electron source generated in the intense LPI, these measurements (unlike most other transport diagnostics) are ideal constraints since they can provide a direct measure of the relativistic interaction. Since the near-field EM distribution is responsible for accelerating electrons to relativistic energies, it is critical that these fields are reasonably represented in the simulations. Indeed, simultaneous measurements of the spectrum, instantaneous wavelength shifts, divergence, and total energy of the specularly reflected pulse provide a very tight set of constraints on the LPI simulations in a way that transport diagnostics relying on secondary processes cannot.

The interesting physical phenomena of interest that modify the specularly reflected pulse, such as absorption, self-phase modulation, Doppler shifts, etc., are all non-linearly dependent on spatial and temporal intensity and electron density profiles in the near-field; spatial distributions in the far-field, however, are not necessarily characteristic of the spatial distributions in the near-field due to diffraction. This is particularly true when the shape of the reflecting relativistic critical surface is dynamically evolving throughout the interaction which, for example, can result from laser beam filamentation and other instabilities. Therefore, experimental measurements should either be spatially integrated or resolved (and not subsampled as it was for our experiment), so as to not misinterpret the dominating mechanisms in near-field interaction.

While our simulations of the high-contrast interactions qualitatively (and quite nearly quantitatively) reproduced all the experimental trends (from the overall reflectivity and spiky intensity distribution to spectral broadening and shifting), we found that our low-contrast interactions produced far-field properties that were significantly broader in both space and spectral bandwidth with overall redshifted spectra. One possible explanation for these discrepancies could be from assumptions in the model, such as the simplified laser temporal and spatial intensity profiles and the exclusion of collisional ionization/recombination or other uncertainties in the initial pre-plasma conditions. Another possibility is that the reduced dimensionality of these simulations (i.e., only modeling one transverse spatial dimension to the beam propagation direction) is inherently problematic. This can arise through physics, such as the differences in the filamentation instability in 2D and 3D, or modeling limitations. For these 2D simulations using intensity profiles characteristic of the laser spot size, the interaction simulation will always under-estimate the relative contribution of beam energy at lower intensities (with respect to higher intensities) compared to the fully 3D beam; this could be a significant source of error due to the non-linear nature of these interactions with laser intensity. As addressed in the Appendix, Subsection 2, propagation of non-paraxial modes in the specular pulse will also inevitably lead to increased beam divergence in 2D over an equivalent 3D beam, due to the reduced k-space. Although the goal of this study was to address the feasibility of studying near-field physics with far-field specularly reflectivity measurements, for which they can be well suited despite diffraction effects, future simulation studies hoping to quantitatively match experimental trends may indeed need to be performed in 3D.

We gratefully acknowledge many useful conversations with S. C. Wilks and A. J. Kemp, support from the Jupiter Laser Facility staff, and all the hard work from Experimental Campaign Team including K. U. Akli, L. D. Van Woerkom, C. Chen, D. Hey, D. P. Higginson, M. H. Key, B. Westover, A. Beaudry, J. Westwood, M. Z. Mo, H. Friesen, F. N. Beg, C. Jarrott, A. Sorokovikova, I. Bush, J. Pasley, and R. B. Stephens. This work was performed under DOE Contract DE-AC52-07NA27344 with support from the Lawrence Scholar Program, OFES-NNSA Joint Program in High-Energy-Density Laboratory Plasmas, an allocation of computing time from the LLNL Grand Challenge and funding from the Natural Sciences and Engineering Research Council of Canada.

Two techniques are addressed in this work: (1) paraxial propagation using the Fresnel kernel and (2) a non-paraxial approach simply using the wave-equation (Eq. (A1)). Both solutions start with the wave equation and the ansatz of a linearly polarized plane wave (parallel with the z-axis) electric field E with wavelength λL = 2π/kL, angular frequency ωL = ckL, spatial envelope u(x, y, z), and temporal envelope ϵ(t) propagating along the x-axis (Eq. (A2))

(2x2+2y2+2z21c22t2)E(x,y,z,t)=0,
(A1)
E(x,y,z,t)=u(x,y,z)ϵ(t)ei(ωLtkLx)ẑ.
(A2)

After a separation of space and time variables and a Fourier transform along the spatial dimensions transverse to beam propagation direction (i.e., along y and z), the evolution of the spatial envelope as a function of propagation distance x takes the form of Eq. (A3), where ũ(x,ky,kz) is the Fourier transform of the field along the transverse spatial dimensions

(2x22ikLx(ky2+kz2))ũ(x,ky,kz)=0,
(A3)
ũ(k,kL,L)=ũ(k,kL,xo)eik22kLL.
(A4)

If we assume a slowly varying transverse spatial envelope (i.e., k2=ky2+kz2kL2, where k=kyŷ+kzẑ), Eq. (A3) reduces down to the elegant form of the Fresnel propagator given by Eq. (A4).67 Simply put, the far-field spatial field distribution a distance L = xxo away from the measurement plane at xo can be obtained through an appropriate phase addition in k-space of the near-field answer. For small Fresnel numbers (F = a2/(L) ≪ 1 where a is the characteristic size of the aperture), this approach has been found to reliably reproduce the far-field Fraunhofer diffraction patterns produced by single and double slits, round and square apertures as well as the Gaussian beam solution.

Since short pulses have finite bandwidth, this propagator kernel may be applied to each frequency individually. Starting with the space/time domain of the field quantities at boundary of the simulation, a Fourier transform along both dimensions results in the wave-number/angular frequency (k-ω) domain of the pulse. The Fresnel propagator kernel can then be applied at each and every k-ω pair before being inverse-Fourier transformed to obtain the space/time domain of the pulse after propagation. Physically, this is equivalent to propagating each laser frequency-wave number pair individually, treating each wave-number/angular frequency component as a monochromatic plane-wave, then superimposing the solutions after propagation.

Shown in Fig. 9 are the power spectra of both the initial target surface perturbations (green) and of the specularly reflected pulse, as measured at the boundary of the simulation (black), for the high-contrast interaction. The spectrum of the rough target has clearly been mapped onto the reflected pulse and into spectral modes up to and beyond sub-wavelength features (kL ≥ 2π/λL = 11.9 rad/μm, indicated by the shaded gray area). Also indicated is the approximate valid range of the paraxial assumption (cross-hatched area), which only covers approximately an order of magnitude in spectral intensity. The specularly reflected pulse is inherently non-paraxial; depending on the relative contribution of these modes to the spectra, propagating these modes paraxially or simply ignoring them could potentially introduce significant error in the far-field.

FIG. 9.

Normalized spectral intensity distributions of the target surface (green, sampled over ±50 μm about the laser axis) and the specularly reflected pulse (black). The cross-hatched area indicates the approximate region of validity for paraxial propagation (k2kL2/10) and the shaded gray area the region of validity for this non-paraxial algorithm (k ≤ kL).

FIG. 9.

Normalized spectral intensity distributions of the target surface (green, sampled over ±50 μm about the laser axis) and the specularly reflected pulse (black). The cross-hatched area indicates the approximate region of validity for paraxial propagation (k2kL2/10) and the shaded gray area the region of validity for this non-paraxial algorithm (k ≤ kL).

Close modal

In order to appropriately handle these larger k-modes, we approached the propagation problem by first removing the paraxial assumption. The catch, however, in dropping the paraxial approximation is that the differential equation to solve becomes second order, requiring two boundary conditions to solve. Although not as elegant as the Fresnel propagator kernel, one solution of the non-paraxial wave equation (Eq. (A1)) takes the form of Eqs. (A5)(A8) for the case of having field measurements at two planes located at x1 and x2 where constant C is dependent only on the boundary conditions, ũ(k,x1) and ũ(k,x2) are the k-space fields at x1 and x2, respectively, and the x dependence comes in through the α(x) and β(x) coefficients. Alternative solutions exist for different boundary conditions, but the two plane approach was the most practical for these simulations. The procedure for propagating the pulse is identical to the paraxial case, again modifying the phase of the near-field solutions in 2D Fourier space and then returning to the space-time domain with inverse Fourier transforms.

ũ(k,x)=C[α(x)ũ(k,x1)+β(x)ũ(k,x2)],
(A5)
C=eikL(x1+x2)2(1icot[(x1x2)kL2k2]),
(A6)
α(x)=eikL(x+x2)+i(x1x)kL2k2(1e2i(xx2)kL2k2),
(A7)
β(x)=eikL(x+x1)+i(xx2)kL2k2(1e2i(x+x1)kL2k2).
(A8)
1. Validity

Since these field propagators only work in vacuum, the validity of these algorithms is tested using the high-contrast interaction with a sharp interface, free of under-dense plasma, with initial target surface features characteristic of the experiment by propagating both forwards to the far-field and backwards toward the near-field at the original laser focal plane. By propagating the field measured at the boundary of the simulation backwards toward the original laser focal plane, we can directly observe how the surface perturbations are imprinted onto the spatial profile of the specularly reflected pulse.

Shown in Fig. 10(a) are the propagated pseudo-intensity distributions of the specularly reflected pulse (i.e., |Ez|2) as a function of space and time at the original laser focal plane using the two different approaches. These images only show small window that is centered in space and time around the peak laser intensity on a log10 color scale, so that the shape of the phase-fronts are apparent. The paraxial approach was used in the subplots on the left and middle and the non-paraxial approach on the right; for the paraxial propagations, the left is filtered to only contain paraxial modes (k2kL2/10), while the middle is unfiltered (i.e., non-paraxial modes are treated paraxially). Also shown (black area) is the original realistic surface roughness time of flight (i.e., 2δx/c, where δx is the deviation of the surface from the laser focal plane at x = 0 μm). We can now see how the surface perturbations are directly mapped onto the phase-fronts of the specularly reflected pulse. Clearly, the filtered and unfiltered paraxial approximations are incapable of capturing the small scale features of interest (again, not actually what it was intended to be used for), whereas the non-paraxial does in quite good detail. Propagation to and from the far-field of this non-paraxial technique was also tested with a paraxial pulse (super-position of 3 Gaussian pulses) and was found to reproduce both the PIC and Fresnel answers.

FIG. 10.

(a) Phase-front distortions in the specularly reflected pulse at the x = 0 μm focal plane, with simulated surface roughness shown in black, using the filtered and un-filtered paraxial (left and middle, respectively) and non-paraxial approach (right). (b) Far-field intensity distributions at L = 1 mm for the filtered paraxial (red), unfiltered paraxial (magenta), and non-paraxial (blue) propagations with the solid angle of the experimental measurement indicated by the shaded gray area.

FIG. 10.

(a) Phase-front distortions in the specularly reflected pulse at the x = 0 μm focal plane, with simulated surface roughness shown in black, using the filtered and un-filtered paraxial (left and middle, respectively) and non-paraxial approach (right). (b) Far-field intensity distributions at L = 1 mm for the filtered paraxial (red), unfiltered paraxial (magenta), and non-paraxial (blue) propagations with the solid angle of the experimental measurement indicated by the shaded gray area.

Close modal

Turning our attention now to the far-field, in this case L = 1 mm away, we can observe how these time-of-flight variations in the specular laser wave-fronts influence the far-field intensity distribution. Shown in Fig. 10(b) are those distributions using the filtered paraxial (red), unfiltered paraxial (magenta), and non-paraxial (blue) approaches. Unsurprisingly, the paraxial and non-paraxial approaches both work equally well near the specular axis but begin to differ beyond approximately ± 15° off specular. When unfiltered, the Fresnel approach treats all these non-paraxial modes as though they were paraxial, bringing them closer to the specular axis. If the experimental detector has a finite collection angle (as our experiment did, solid angle indicated by the shaded gray area), the choice of propagator algorithm can influence the simulated reflectivity. Note that these propagated intensity distributions can be re-created simply by adding the time-of-flight fluctuations (due to surface perturbations) to the flat phase-fronts of the incident laser in the near-field and then propagating to the far-field.

This improvement allowed for the propagation of light mode light modes up to kL, nearly an additional order of magnitude in spectral intensity over the Fresnel approach as seen from Fig. 9. It may be possible that this kind of approach could be extended to modes beyond the laser wave-number, but very little energy is present in these modes and our simulations cannot sufficiently resolve their propagation anyway. In practice, this approach has been found to work best when the two measurement planes in the simulation are separated by ∼λL.

2. Dimensionality

The reduced dimensionality of these simulations also brings into question how well we can represent the experimental specular data with a 2D3V geometry. To address this, we study how beams with one and two transverse spatial dimensions to the beam propagation direction behave in the far-field. Shown in Fig. 11(a) are simulated focal spots, with spatial features characteristic of the target surface roughness used in the experiment, calculated assuming two (“2D,” top) and one (“1D,” bottom) transverse dimensions to the beam propagation direction; the 1D spot was generated by streaking the horizontal lineout through of the center of the 2D spot. Shown in (b) are the far-field intensity distributions after being propagated a distance of L = 1 mm with the Fresnel propagator (Eq. (A4)), and (c) contains normalized horizontal lineouts through the center of each beam profile. Due to the reduced k-space, the propagated 1D beam (blue) will always overestimate the energy in the wings of the distribution (i.e., from the non-paraxial components) compared to the equivalent 2D beam (red). The simulations previously discussed, being dimensionally identical to the 1D case here, will therefore always overestimate the far-field beam divergence and underestimate specular reflectivity from non-paraxial modes. Since this dimensionality effect predominantly influences the non-paraxial modes, the increased beam divergence may be negligible when these modes contain only modest amounts of the total beam energy.

FIG. 11.

1D vs. 2D propagation using the Fresnel propagation algorithm. Shown in (a) is a near-field 2D (top) and equivalent 1D (bottom, generated using a horizontal lineout of the 2D distribution through the center) spatial intensity distributions. (b) Propagated intensity distributions at L = 1 mm and (c) normalized lineouts of the far-field intensity distributions for the 1D (blue) and 2D (red) distributions.

FIG. 11.

1D vs. 2D propagation using the Fresnel propagation algorithm. Shown in (a) is a near-field 2D (top) and equivalent 1D (bottom, generated using a horizontal lineout of the 2D distribution through the center) spatial intensity distributions. (b) Propagated intensity distributions at L = 1 mm and (c) normalized lineouts of the far-field intensity distributions for the 1D (blue) and 2D (red) distributions.

Close modal
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