Laser-shock compression experiments at 3rd and 4th generation light sources generally employ phase plates, which are inserted into the beamline to achieve a repeatable intensity distribution at the focal plane. Here, the laser intensity profile is characterized by a high-contrast, high-frequency laser speckle. Without sufficient smoothing, these laser non-uniformities can translate to a significant pressure distribution within the sample layer and can affect data interpretation in x-ray diffraction experiments. Here, we use a combination of one- and two-dimensional velocity interferometry to directly measure the extent to which spatial frequencies within the laser focal spot intensity pattern are smoothed out during propagation within the laser plasma and a polyimide ablator. We find that the use of thicker polyimide layers results in spatially smoother shock fronts, with the greatest degree of smoothing associated with the highest spatial frequencies. Focal spots with the smallest initial speckle separation produce the most rapid smoothing. Laser systems that employ smoothing by spectral dispersion techniques to rapidly modulate the focal plane intensity distribution are shown to be the most effective ones in producing a spatially smooth shock front. We show that a simple transport model combined with the known polyimide Hugoniot adequately describes the extent of shock smoothness as a function of polyimide thickness. Our results provide a description of spatial structure smoothing across a shock front, which can be used to design targets on x-ray free electron laser facilities.

## I. INTRODUCTION

The effect of laser imprinting, whereby spatial non-uniformities in laser intensity incident on a material lead to perturbations in the resulting ablatively driven shock front, has received much attention over the last few decades due to the negative impact this process can have on laser direct-drive (LDD), inertial confinement fusion (ICF) experiments. Such perturbations in the shock front can lead to the growth of Rayleigh–Taylor and Richtmyer–Meshkov instabilities in the ablator layer of an ICF capsule, which causes mixing of ablator and fuel layers and, ultimately, may lead to a reduced nuclear fusion yield.^{1–3} The development of laser beam smoothing techniques, such as the use of continuous phase plates (CPPs)^{4,5} and smoothing by spectral dispersion (SSD),^{1,6–8} has been effective in greatly reducing the magnitude of laser imprinting in LDD ICF experiments. However, there has been considerably less attention paid to the negative impact laser imprinting may have on planar shock compression experiments, which use *in situ* x-ray diffraction to investigate the structural behavior of matter at extreme pressures and temperatures.^{9–14}

The commissioning of high-power laser systems at fourth generation x-ray free electron lasers (XFELs) in the last decade has yielded important results in our understanding of materials under dynamic compression.^{11–13,15–20} In these experiments, where x-ray diffraction data are volume-integrated, the presence of spatial and temporal non-uniformities in the laser drive can potentially result in a significant distribution of pressure states in the sample, which can complicate the interpretation of diffraction data, especially in the vicinity of a phase boundary. Temporal uniformity of the sample pressure states (shock-steadiness) can be controlled by laser pulse shaping techniques. However, the effect of pressure non-uniformity due to laser imprint has never been quantified in such experiments to-date. This is of particular concern as the typical specifications of the laser systems used at XFEL facilities are not optimized for achieving spatially uniform focal spot intensities, as opposed to larger laser facilities dedicated to LDD ICF experiments (see Table I).

While the use of CPPs is common practice in laser experiments at XFELs, in order to generate a spatially averaged uniform laser intensity profile with shot-to-shot consistency, the use of SSD is typically not employed. Furthermore, unlike laser facilities dedicated to LDD ICF experiments, where multiple beams are overlapped to achieve a more uniform laser intensity drive (imprint reduced by a factor of $N$ where $N$ is the number of overlapped beams^{32,33}), many XFEL facilities employ single beam optical irradiation geometries. Finally, while the shock roughness imparted by laser imprint is known to be smoothed out with increasing ablator thickness,^{34,35} the specifications of the laser system at XFEL facilities (pulse energy $\u223c$50–100 J, pulse duration $\u223c$10–30 ns) and the desire to reach very high $(>100$ GPa) pressures lead to the use of small focal spots (to maximize laser intensity) combined with thin $(<25\mu m)$ ablator layers (to eliminate the effects of lateral pressure release waves^{36}). Recent experiments have reported using ablators as thin as 5 $\mu m$,^{9} where smoothing effects may not be sufficient to mitigate sample pressure non-uniformities associated with phase plate imprint.

In a typical XFEL diffraction experiment, a high-power $\u223c$5–20 ns duration laser pulse, focused onto a 300 $\mu m$ diameter CPP-shaped focal spot, launches an ablatively driven shock wave into the target package consisting of a $\u223c$5–75 $\mu m$ thick low-Z ablator (e.g., polyimide), a sample layer (several to tens of $\mu m$ thick) and an LiF window (to permit velocimetry measurements to constrain sample pressure). The XFEL pulse is timed to probe the sample at peak compression, which results in a scattered volume-integrated x-ray diffraction pattern. Sample pressure is constrained by measuring the sample/LiF interface velocity $(ui)$ using a line-imaging VISAR (velocity interferometer system for any reflector^{37}) and impedance-matching techniques, with a prior knowledge of the sample and LiF Hugoniots.^{38} VISAR provides spatially resolved $ui$(time) data over several hundred micrometer lines at the target plane (integrated over a $\u223c10$ $\mu m$ linewidth) and with a spatial resolution of $\u223c20$ $\mu m$ (see Fig. S1 in the supplementary material).

Polyimide $(C22H10N2O5)$ is widely used as an ablator material on XFELs due to its low x-ray scattering power, amorphous material structure, and high ablation velocity, the latter ensuring that a strong shock wave is launched into the sample.^{11,15,16,19,39} Polyimide is also widely available in uniform thickness films of 13, 25, and 50 $\mu m$—standard thicknesses in laser-driven experiments. As ablation pressure in polyimide scales with laser intensity as $PAbl\u221dILaser0.67$,^{40} local non-uniformities in the laser intensity across the focused spot can cause localized spikes in pressure, which can propagate through the ablator and into the sample layer, causing a distribution of pressure states (Fig. 2).

Here, we use high spatial-resolution two-dimensional (2D) velocity interferometry to characterize the magnitude of shock roughness imparted on a sample by phase plate speckle imprinting and determine the magnitude of smoothing as the polyimide ablator thickness is varied between 13 and 75 $\mu m$. The effect of SSD in reducing shock roughness was also investigated. Our results show root-mean-squared (r.m.s.) pressure instabilities up to 7% for 13 $\mu m$ ablators at peak pressures of 60 GPa and a threefold reduction in shock roughness as the polyimide thickness was increased to 75 $\mu m$. SSD was found to greatly reduce the effect of laser imprint in thin (36 $\mu m$ ablators) but not in 75 $\mu m$ thick ablators where hydrodynamic smoothing is the dominant smoothing process. Our results confirm that the use of thin plastic ablators are problematic in laser-driven compression experiments at facilities that do not use SSD. Thicker ablators $(\u223c50$ $\mu m)$ should be employed to achieve a comparable reduction in shock roughness through hydrodynamic smoothing. These findings are intended to inform target designs for future experiments at XFEL facilities and to ensure optimal pressure uniformity in the sample under study.

### A. Techniques for laser intensity spatial smoothing

As laser focal spots are inherently non-planar, most laser-driven compression experiments employ CPPs, inserted into the beam path, to ensure a *spatially averaged* uniform intensity, which is stable in time, ensuring high reproducibility of drive performance.^{1,41,42} CPPs are transparent plates characterized by multiple adjacent regions, within the expanded beam area, which add a random phase delay to the transmitted light. This random distribution of phase delays effectively divides the beam into mutually spatially incoherent beamlets.^{43} The laser focal spot is a superposition of these beamlets and is characterized by a reproducible intensity envelope modulated by a fine speckle—due to inter-beamlet interference [see Figs. 1(b) and 1(c)]—where the minimum speckle size may be estimated as^{44}

Here, $\lambda $ is the laser wavelength, and $F#$ is defined as the focal length of the focusing lens divided by the beam diameter. For NIF (351 nm, $F#25)$, Janus (527 nm, $F#6.7)$, Omega-60 (351 nm, $F#6.7)$, and MEC (527 nm, $F#3.5)$, this equates to minimum speckle sizes of 19.8 $\mu m$, 8.0 $\mu m$, 5.3 $\mu m$, and 4.2 $\mu m$, respectively. These values agree well with the measured minimum speckle separation (see Fig. S9 in the supplementary material) and is expected to be independent of the overall focal spot diameter.

For experiments that employ CPPs, laser energy is deposited non-uniformly due to the high-frequency temporally static speckle pattern, resulting in ablation-surface modulations. Laser imprint occurs most strongly at early times until continued laser ablation produces a sufficiently large plasma in which the laser-absorption region (up to the critical density) becomes separated from the ablation surface. Thereafter, laser speckles are smoothed—to some extent—by lateral thermal conduction (the “cloudy day” effect of Ref. 45) in the region between these surfaces^{1} (see Fig. 2). During shock transit within the polyimide layer, vortices are established behind the corrugated shock front, which result in lateral material transport and, with time (propagation distance within the polyimide), serve to anneal the roughness of the shock front.^{46,47} It is expected that the extent of this smoothing effect is dependent on the viscosity of the compressed material.

CPP speckle patterns are static in time, and therefore, there is a cumulative effect on an imprint structure over the duration of the drive pulse. On LDD ICF facilities, additional SSD techniques are employed to smooth the speckle on a timescale that is shorter than the ICF target’s hydrodynamics, i.e., the timescale for a significant separation between the laser-absorption region and the ablation surface.^{1,43,45,48,49} With the SSD approach,^{1,6–8} a THz phase-modulated bandwidth is imposed upon the laser beam by an electro-optic modulator, which is then dispersed by a grating, resulting in color variations across the beam. The changing phase distribution across the beam, combined with the use of CPPs, produces a $\u223c$picosecond time-varying speckle pattern, which breaks up the cumulative effects associated with the sole use of CPPs. An extension of SSD to two dimensions was described by Skupsky and Craxton.^{50} Two-dimensional SSD employs two modulators with their THz bandwidths dispersed in orthogonal directions. Additional smoothing is available on the Omega-60 facility by employing polarization smoothing, produced by distributed polarization rotators (DPRs), which provides an *instantaneous* factor of $2$ reduction in the laser nonuniformity.^{51} For all the Omega-60 shots considered in this study, DPRs were selected. Single beam equivalent-target-plane intensity distributions, integrated over time, are shown in Fig. 1 from the Omega-60 laser facility^{28} for (c) a CPP, (d) a CPP with 1D SSD, and (e) a CPP with 2D SSD. The smoothest beam is produced with a CPP with 2D SSD. In this geometry, it is estimated that the integration time required to achieve 2% intensity nonuniformity is $\u223c$70 ps.^{52} XFEL facilities currently do not employ SSD techniques (see Table I) and, therefore, are more sensitive to the negative effects of direct phase plate imprinting.

## II. EXPERIMENTS

Experiments were performed at the Jupiter Laser Facility using the Janus laser, situated at the Lawrence Livermore National Laboratory, Livermore, CA, and at the Omega-60 laser facility located at the Laboratory for Laser Energetics in the University of Rochester, Rochester, NY. A schematic of the target design is shown in Fig. 2. At Janus, polyimide ablators of 13, 25, 50, and 75 $\mu m$ were used. While a number of different types of polyimide are commercially available, the most widely used are Kapton-HN and Kapton-CB/B (where Kapton is the commercial name from the manufacturer). As Kapton-CB/B is more opaque to visible light, we choose this for our ablator material. Each ablator was coated with 0.2 $\mu m$ of Al on the drive side to ensure that the ablator is opaque to the incoming laser light and was bonded to a 500 $\mu m$ thick LiF [100] window, which had a 0.2 $\mu m$ coating of Al on the bonded side to ensure that the polyimide/window interface had a reflecting surface suitable for making velocimetry measurements.^{37,53} On Janus, a frequency doubled, Nd:glass laser at 527-nm produced a $\u223c$7-ns square laser pulse with a pulse energy of 50–200 J in a 1-mm diameter laser spot [Fig. 1(a)], which drove an ablatively driven shock wave into the polyimide ablator (e.g., see Fig. S4).

At Omega-60, polyimide thicknesses of 36, 50, and 75 $\mu m$ were used, which were coated and bonded to 500 $\mu m$ thick LiF [100] windows. Two laser beams with 3.7 ns duration were stacked in time to produce a composite laser drive of $\u223c7$ ns (see Fig. S6 in the supplementary material). At any given time, the target was irradiated by a single beam. These experiments employed CPPs [see Fig. 1(c)] and for some shots 1D and 2D SSD smoothing [Figs. 1(d) and 1(e)]. The Janus and Omega-60 beams [Fig. 1(a)] were incident onto the target package at angles of $12.5\xb0$ and $23\xb0$, respectively. At both facilities, the polyimide-LiF interface velocity $(ui$(t)) was recorded on each shot by a line-imaging VISAR, which sampled a 1 mm line along the target plane with a spatial resolution of $\u223c30$ $\mu m$^{37} (estimated to be slightly less resolving than the VISAR system at MEC; see Fig. S1 in the supplementary material). The measured $ui$(t) is related to pressure in the polyimide using standard impedance-matching techniques^{38} and the known Hugoniots for LiF^{54} and polyimide.^{19} For the experiments on Janus and Omega-60, we shock compressed the polyimide to $\u223c60$ and $\u223c90$ GPa, respectively (see Figs. S4 and S6 in the supplementary material). These pressures are above the reported melt pressure for polyimide reported in recent x-ray diffraction and laser-shock compression experiments.^{19} In addition to the line-imaging 1D VISAR system, a high-resolution 2D imaging velocimetry system^{34,35,53,55,56} was used. The 2D VISAR uses a 3 ps 400 nm input laser pulse from a frequency-tripled Ti:sapphire laser system to provide a two-dimensional snapshot of the velocity distribution at the polyimide–LiF interface with a 10 m/s velocity resolution. The field of view is a $800\xd7800\mu m2$ square with a spatial resolution of 2 $\mu m$.^{53,55,56} The measured $ui$(t) distribution is related to the Kapton pressure and the velocity distribution by the same impedance-matching methods used for 1D VISAR data. The timing of the 2D VISAR snapshot relative to shock breakout was known accurately with the use of a timing fiducial, which was visible on the streak image of the VISAR. The experimental geometry on Janus is described in detail in Ref. 53. The experimental setup on Omega-60 is shown in Fig. S3 of the supplementary material.

For all experiments, the 2D VISAR probes the roughness of the polyimide/LiF interface after shock arrival. Therefore, the measurement represents the spatial smoothness behind the shock front. Furthermore, once the shock arrives at the polyimide/LiF interface, the polyimide experiences a re-shock wave propagating back toward the ablation surface (see Fig. S4 in the supplementary material), which is also typical for standard XFEL diffraction targets. Therefore, the smoothness of the shock states reported here is a combination of smoothing processes within (i) the laser produced plasma, from propagation within (ii) the initial shocked polyimide volume, and (iii) the volume of double-shocked polyimide. The shock/reshock pressures were $\u223c60/90$ and $\u223c90/135$ GPa for the Janus and Omega-60 experiments, respectively.

## III. RESULTS

### A. Janus laser (527 nm, no SSD)

A total of 18 experiments were performed on the Janus laser. The obtained 2D $ui$ maps of the shocked sample show clear differences in roughness as a function of thickness [Fig. 3(a)]. From the $ui$ maps, we can extract the r.m.s. velocity perturbation $(Vrms)$ as a function of spatial scale. Here, $Vrms$ represents the particle velocity in the polyimide. Determination of velocity roughness follows the 2D VISAR analysis description in Refs. 34, 35, 53, 55, and 56 (see also the supplementary material for details). Typical $Vrms$ for 13, 25, 50, and 75 $\mu m$ polyimide ablators, as a function of spatial mode, is shown in Fig. 3(a). The $Vrms$ calculated from the initial Janus laser beam profile imprint is also plotted in Fig. 3(a) and is equivalent to the case where no ablator is used (see Fig. S7 in the supplementary material for calculation details). Two clear trends are immediately clear from the data. First, for each thickness of polyimide, the dominant contributions to the overall shock roughness come from small spatial scale modes. This is expected as the initial phase plate imprint is dominated by high frequency modes, with a minimum speckle separation of 10.5(1.8) $\mu m$ (Fig. S9 in the supplementary material). Second, the magnitude of $Vrms$ as a function of spatial scale is observed to decrease with increasing polyimide ablator thickness.

By summing the $Vrms$ contributions at each spatial mode, we determine the cumulative $Vrms$ for each ablator thickness [Fig. 3(b)]. By referring to the known polyimide Hugoniot relations,^{19} the cumulative rms pressure perturbation $Prms$ is also calculated [Fig. 3(c)]. As shown, in the case for the calculated initial laser imprint (no ablator, Fig. S7 in the supplementary material), the shock roughness is considerable with $Prms$ totaling 25% of the peak pressure determined from line VISAR measurements. Perturbations are reduced dramatically when a 13 $\mu m$ ablator is used but are still significant (4%–7%). Increasing the ablator thickness further decreases the $Prms$ to 3%–4% at 25 $\mu m$, 1.5%–2.5% at 50 $\mu m$, and $\u223c1.5$% at 75 $\mu m$. A transport model that takes into account the initial phase plate structure gives a reasonable agreement with the measurement $Vrms$ values as a function of polyimide thickness (see Sec. IV for details).

### B. Omega-60 laser (351 nm, 0/1/2D SSD)

Eight experiments were performed at the Omega-60 laser, which shock compressed polyimide ablators to $\u223c90$ GPa. In experiments where no SSD was used, the dominant contribution to the overall $Vrms$ is determined to originate from spatial modes of $<10\mu m$, as in our Janus experiments [Fig. 4(a)]. The cumulative $Prms$ is observed to decrease from 3% of the peak pressure with a 36 $\mu m$ ablator to 0.3% with a 75 $\mu m$ ablator [Fig. 4(e)]. When SSD was used with a 36 $\mu m$ ablator, a significant decrease in $Vrms$ across all spatial modes was observed relative to the no SSD case. The cumulative $Prms$ was decreased to 0.3% in the SSD case for a 36 $\mu m$ ablator compared with 3% in the no SSD case. 1D and 2D SSD were observed to be comparable in their effect on shock roughness (Fig. S8 in the supplementary material). However, the effect of SSD was much less dramatic with thicker ablators. Indeed, with 75 $\mu m$ of polyimide, the $Prms$ in experiments where SSD was not used (0.3%) was measured to be the same as when 2D SSD was used [Fig. 4(c)], indicating that at such thicknesses of polyimide, hydrodynamic smoothing of laser imprint is effective and cannot be improved upon by additional smoothing from SSD.

For the experiments on Janus and Omega-60, the measurement floor (gray dashed) was determined through analysis of static (uncompressed) samples. The fact that the driven sample $Vrms$ always lies above this floor may be related to microstructural texture in the sample, which evolves during shock compression.

In Fig. 5, we compare the calculated $Vrms$ for the non-SSD drives for Janus (solid curves) and Omega-60 (dashed curves). For comparable ablator thicknesses, we see improved smoothing at Omega-60. Besides the difference in laser wavelengths between Janus (527 nm) and Omega-60 (351 nm), the phase plates used at the two facilities produce different characteristic minimum speckle separation [Janus: 10.5(1.8) $\mu m$; Omega-60: 5.9(2.0) $\mu m$]. We note that experiments on Omega-60 also employ DPR polarization smoothing, which gives an instantaneous reduction in the speckle contrast of $2$.^{51} In addition, as the Omega-60 drive is comprised of two separate beams offset in time, each with a unique speckle pattern, there is a breakup in the cumulative imprint, which occurs for the single beam Janus drive. Even with these differences, the trends in Fig. 5 suggest that smaller speckle separations result in more rapid smoothing. This is expected given the lateral mass transport processes that drive the smoothing (Fig. 2).

## IV. DISCUSSION

A key finding of this study is that in laser-driven compression experiments, laser imprinting due to a phase plate speckle can cause significant sample pressure perturbations when thin polyimide ablators are used without SSD smoothing. This is an important finding as the majority of laser facilities coupled to 3rd and 4th generation light sources do not currently employ SSD smoothing (see Table I), and thin $(\u226425\mu m)$ plastic ablators have frequently been used in target designs in experiments.^{9,13,57–59}

While 2D VISAR systems are not currently used on XFEL facilities, we note that for sufficient shock roughness, there is an observable degradation in 1D VISAR fringe quality. For example, the raw 1D VISAR images for all the Omega-60 shots are shown in Fig. S6 in the supplementary material. Here, fringe movement is directly proportional to $ui(x,t)$. For the 36 and 50 $\mu m$ thick polyimide shots, there is a larger distribution of fringe movement at any given time (and, therefore, $ui)$ for the non-SSD drive when compared to similar SSD smoothing shots—a clear consequence of the shock roughness quantified in Fig. 4(a). It is important to note, however, that 1D VISAR fringe quality alone is not sufficient to quantify shock roughness. Our study shows that the dominant contributions to shock roughness come from modes of spatial scale $<20\mu m$. Such roughness is not resolvable by line-imaging VISAR systems (a typical spatial resolution is $>20\mu m$; see, for example, Fig. S1 in the supplementary material). In addition, the 1D resolved path of a line-imaging VISAR has an associated integration width over which the effects of small scale shock roughness are further smoothed out.

### A. Rippled shock front transport model

We can model the effect of the phase plate intensity distribution on the shock front structure by combining a measured phase plate intensity distribution with rippled shock theory. Analytic expressions for the evolution of a transverse sinusoidal perturbation on a shock front within an inviscid fluid have been reported by Miller and Ahrens^{46} and independently by Bates.^{60} The mathematical derivations in these works follow different approaches with seemingly different final expressions, but numerical evaluation reveals that they produce identical results.

The rippled shock solution provides the evolution of the amplitude of a sinusoidal perturbation on a shock front as a function of time or wavenumber. For inviscid materials, the solution is self-similar for all wavenumbers and can, therefore, be represented by a function of a single parameter: the ripple function $R(\tau )$. The self-similar coordinate $\tau $ is a dimensionless time or wavenumber, $\tau =kvt$. Here, $k=2\pi /\lambda r$ is the wavenumber of the ripple at transverse wavelength $\lambda r$, $v=us\u2212up$ is the downstream velocity of the fluid in the reference frame of the shock with shock velocity $us$ and particle velocity $up$ (measured in the lab frame), and $t$ is the time. The derivative $R\u2032=dR/d\tau $ gives the normalized velocity associated with the oscillation.

The parameters required to evaluate $R(\tau )$ are the shock compression, $\eta =us/(us\u2212up)$; the downstream sound speed, $cs$; and the D’yakov parameter, $dP/dV|Hugoniot$, which is a measure of the shock compressibility. Accurate evaluations of these parameters can be obtained from an equation of state model that has been matched to shock data. For our modeling, we use the LEOS 5040 table for polyimide. We then use Eq. (79) from Miller and Ahrens to evaluate $R(\tau )$.^{46} We note the work of Ishizaki and Nishihara^{61,62} who presented a similar analytical rippled shock model for studying hydrodynamic perturbation growth. However, we choose not to use this model as they assume a simplistic ideal gas equation of state to model the material response of the polyimide ablator.

To compute the propagation and evolution of the shock front ripple structure associated with the phase plate, we use the following procedure. From the intensity distribution of the phase plate and well-known scaling laws for ablation pressure as a function of intensity, we compute the spatially dependent pressure distribution at the ablation surface (see Fig. S7 in the supplementary material). The spatial variation of the pressure distribution launches a spatially varying shock with the variations correlated to the phase plate intensity according to the pressure scaling law. This calculation neglects the fact that expansion of the ablation plasma will eventually smooth out the applied pressure, but it does capture the strong variations that are imprinted onto the shock when it is launched. The spatially varying shock pressure field is then converted to a spatially varying shock velocity field using the Hugoniot, $us(P)$, given by the equation of state model.

A Fourier transform of the shock velocity field decomposes it into a collection of complex-valued plane-wave modes, $Uk$, in the Fourier domain, with a strong central mode at $k=0$. Each Fourier mode, $Uk$, can be evolved independently to a propagation depth $d$ in the target using the scaled ripple velocity function: $Uk(d)=Uk|d=0R\u2032(|k|vd/\eta +\tau 0)/|R\u2032(\tau 0)|$; the parameter $\tau 0$ will be discussed below. Here, we have used the fact that the propagation time is $t=dus$. An inverse Fourier transform of $Uk(d)$ produces the shock front velocity distribution at depth $d$. Note that since the initial velocity map is real-valued $Uk=U\u2212k\u2217$, the ripple evolution operator does not change this conjugate relationship and the inverse transformed result is also real-valued. The complex values of the individual mode amplitudes encode the spatial phase relationships of the modes. The mode at $k=0$ does not evolve because $\tau =0$.

The offset $\tau 0$ is needed because the rippled shock solutions assume an initial amplitude variation across a shock with a uniform shock front velocity. In our situation, the shock is launched at a planar surface (front surface of polyimide) with zero amplitude but a finite velocity variation. To approximate this situation using the existing $R(\tau )$ solution, we assume that our shock begins its evolution part way through the first oscillation of $R(\tau )$ at the dimensionless coordinate $\tau 0>0$. There are two plausible choices for this initialization: the first is at the first minimum in the $R\u2032(\tau )$, where the velocity oscillation $(R\u2032(\tau )$) has its maximum magnitude (green arrow in Fig. 6); however, this choice occurs when $R(\tau )$ still has a finite amplitude. The second possibility is at the first zero-crossing of $R(\tau )$ where the amplitude is zero and which is located slightly past the first minimum in $R\u2032(\tau )$ (black arrow in Fig. 6). In either case, we normalize the scaled velocity function by dividing by $|R\u2032(\tau 0)|$. Comparisons of calculations with either choice, referred in Figs. 3(b) and 4(b) as Model 1 and Model 2, show little difference at a large propagation depth. A comparison for the scaled velocity functions for shock perturbations initialized using Models 1 and 2 is shown in Fig. 7.

According to the rippled shock theory, each mode at $|k|>0$ evolves independently. The assumption of an independent mode response follows from the linearization conditions used to derive the ripple response from the Euler equations; this assumption is justified when each mode amplitude is much smaller than $\lambda r$. It is easy to see that the velocity oscillation associated with a rippled mode of initial amplitude $A$ and wavenumber $k$ is $Uk=AkvR\u2032$, and from this, the mode amplitude to wavelength ratio is $A/\lambda r=Uk/2\pi v$. From our decompositions of the phase plate intensities at Janus and OMEGA for the 60 and 90 GPa shock cases, respectively, we find that the mode amplitude spectrum for modes at $|k|>0$ has $|Uk|<0.1$ km/s and, therefore, that $A/\lambda r<0.005$ over the entire spectrum. This is consistent with the small amplitude assumption underlying the rippled shock models.

Calculations of $Vrms$ as a function of polyimide thickness, for the rippled shock models, are compared to experimental data in Fig. 8. The input to the model is the focal spot intensity map produced at each facility. The equivalent plane image from Omega-60 was taken on a separate beam and did not include some of the final transport optics, and therefore, we consider that input to the model is idealized. As discussed above, additional smoothing within the laser plasma is not accounted for. This is expected to be a stronger effect on the Janus platform as the longer laser wavelength (532 nm) has an associated lower critical absorption density within the laser plasma. Therefore, the ablation surface is further removed from the original polyimide surface as compared to Omega-60 (351 nm) (see Fig. 2). On Omega-60, two 3.7 ns laser timesets were used (see Fig. S6 in the supplementary material) and offset in time to produce a composite 7.4 ns pulse with each timeset containing a different phase plate speckle pattern. Therefore, experimentally, there is less of a time-integrated speckle imprint on Omega-60 compared to the single timeset Janus drive. However, the model assumes a single timeset for both Janus and Omega-60. The primary difference between the two facilities is that the speckle separation on Omega-60 is significantly smaller than for Janus (see the upper table in Fig. S6 in the supplementary material). As the F#’s are the same on both facilities, the difference in the speckle size is driven by the laser wavelength difference [see Eq. (1)]. The transport models predict that smoother shock drives are achieved with a smaller initial phase plate speckle. While the model provides reasonable predictions for the rate of shock smoothness with polyimide thickness, the absolute values of the experiments are not reproduced precisely. Nevertheless, this use of these models is important to ascertain likely roughness on shock facilities, which do not have 2D Visar systems.

### B. Consequence of shock roughness on x-ray diffraction experiments

In an x-ray diffraction experiment, pressure perturbations could affect the interpretation of the measured diffraction signal in several ways. A distribution in sample pressure states will cause broadening of the measured diffraction peaks. Peak broadening of diffraction peaks has been used to determine the average sample grain size and r.m.s. strain distribution^{13,39} of shocked samples using Warren–Averbach analysis. Such methods depend on an accurate knowledge of instrument broadening, with additional broadening attributed with material factors, such as grain size and strain. However, shock roughness from laser imprinting can also cause diffraction peak broadening as shown in Fig. 9. Using an equation of state for Sn,^{63} we have calculated the peak broadening of a Sn sample at 45 GPa with Gaussian pressure distributions equal to the $Prms$ values of 1.5%, 2.5%, 4%, 7%, and 25% corresponding to the shock roughness measured with 75 $\mu m$, 50 $\mu m$, 25 $\mu m$, 13 $\mu m$, and 0 $\mu m$ ablators from our Janus experiments. As shown in Fig. 9, the full width at half maximum of the (110) reflection of bcc Sn increases $\u223c2$-fold due to the shock roughness in a 13 $\mu m$ ablator relative to a 75 $\mu m$ ablator and $\u223c3$-fold in the case without polyimide ablator smoothing. While these results assume no additional smoothing, which will occur in the sample layer, they do indicate that peak broadening due to shock roughness must also be considered a contributor when performing such grain size analysis in diffraction experiments.

In addition, for x-ray diffraction experiments on samples shock compressed to pressure–temperature states close to a phase boundary, an unaccounted for distribution of pressure states could result in an incorrect identification of a mixed phase.

Our results also have implications for the size of an x-ray spot that should be used in laser-driven diffraction experiments. As experiments aim for high pressure, the size of the x-ray spot is often focused down to diameters of $\u223c10\mu m$ to eliminate the effects of lateral pressure release waves.^{36} However, our results show that the dominant contribution to the overall shock roughness comes from modes of spatial scale $\u226410\mu m$ [Figs. 3(a) and 4(a)] meaning that x-ray probes of a similar size may sample pressure states at the extremes of the distribution [see the blue shaded region in Fig. S9(d)]. Larger x-ray spots of $\u223c50\mu m$ (FWHM) would ensure that more of the pressure perturbations are sampled, and thus, the average pressure state probed would be closer to the average bulk pressure.

Our results demonstrate that at laser facilities where SSD is not available, 50–75 $\mu m$ polyimide ablators should be used in target designs to mitigate the effect of laser speckle imprinting. Our Omega results indicate that thinner $(<36\mu m)$ polyimide ablators can be viably used in experiments at facilities where SSD is available. Laser facilities should consider installing an SSD system to ensure that a smoother pressure impulse is imparted onto samples of interest.

The recent laser-shock compression and x-ray diffraction study from Katagiri *et al.*^{19} on polyimide suggests shock melting of polyimide at 32(3) GPa. Our data at 60+ GPa show the propagation of spatially rough shocks within this liquid phase. This suggests that the roughness vs propagation thickness we measure represents the ability of molten polyimide to support spatial perturbations.

## V. GUIDELINES FOR FUTURE EXPERIMENTS AT 3RD AND 4TH GENERATION LIGHT SOURCES

It is hoped that the findings presented here will help inform optimum target designs and laser configurations at future experiments on laser facilities, which are coupled to 3rd and 4th generation light sources. As such, we have summarized our key findings and guidelines below.

### A. Facilities

Laser facilities should consider installing SSD systems to their existing setup as it has been shown to have a dramatic effect on laser imprint smoothing.

Frequency doubling or tripling should be considered to reduce the effect of shock roughness as shown in Eq. (1).

Laser facilities should consider employing frequency doubling or tripling to reduce speckle size [Eq. (1)].

A small $F#$ focusing system should be used to minimize the speckle size (and speckle separation) in the focal spot [Eq. (1)]. This would promote more rapid smoothing.

The used of distributed polarization rotators (DPRs) can provide an instantaneous reduction in laser uniformity by a factor of $2$.

### B. Users

At laser facilities that do not employ SSD, consider using thicker ablators, such as 50 $\mu m$, to reduce the effect of laser imprint through hydrodynamic smoothing.

Thinner ablators can be used on laser facilities where SSD is available. However, the time period of the sample that is held at a steady shock pressure decreases commensurately with ablator thickness (due to the reduced reverberation roundtrip time in the ablator) [see Fig. S4(b) in the supplementary material]. For example, the time for a 100 GPa shock reverberation from a 10 $\mu m$ polyimide ablator to enter a sample layer is $\u223c1$ ns.

Consider the speckle separation of the drive beam when choosing an x-ray spot size. A larger x-ray spot size will achieve greater averaging of the probed sample’s pressure states.

Diffraction peak broadening due to shock roughness (which can be estimated using the rippled shock models 1 and 2 described in this work) must be considered when performing grain size analysis of x-ray diffraction data.

## VI. CONCLUSIONS

We have employed high-resolution 2D velocimetry measurements of shocked polyimide ablators to characterize the shock roughness due to laser speckle imprinting. Our results found that significant shock roughness occurs in ablators $<25\mu m$ in experiments where SSD is not used, but thick ablators (50–75 $\mu m)$ are sufficient to mitigate imprinting through hydrodynamic smoothing. We also demonstrate that SSD is effective in mitigating laser imprinting even in thinner ablators.

We hope that the work presented here will guide the laser compression community in future experiments at 3rd and 4th generation light sources and also demonstrate the importance of SSD to newly commissioned laser facilities.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional information on experimental details pertaining to the Janus and Omega laser facilities, hydrocode simulations of the experiments, and additional discussion on the laser setup at other laser facilities.

## ACKNOWLEDGMENTS

We thank the operations staff at the Jupiter Laser Facility at LLNL. We thank Carol Davis at LLNL for work in the target assembly. This research was supported by the Laboratory Directed Research and Development Program at LLNL (Project No. 15-ERD-014). This work was performed under the auspices of the U.S. Department of Energy by LLNL (Contract No. DE-AC52-07NA27344).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Martin G. Gorman:** Data curation (lead); Methodology (equal); Writing – original draft (lead); Writing – review & editing (lead). **Suzanne J. Ali:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Peter M. Celliers:** Conceptualization (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Jonathan L. Peebles:** Data curation (equal); Formal analysis (equal); Methodology (equal). **David J. Erskine:** Formal analysis (equal); Methodology (equal). **James M. McNaney:** Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). **Jon H. Eggert:** Conceptualization (equal); Methodology (equal); Visualization (equal); Writing – review & editing (equal). **Raymond F. Smith:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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

## REFERENCES

*et al.*, “

*et al.*,