2.1.

Spallation and Vaporization

The significant differences in surface roughness after a single ablation event can be ascribed to the particular ablation mechanisms. Since for ultrashort laser pulses, almost the entire laser energy is temporarily stored in the electron gas before it is coupled into the ion lattice,⁸ stress confinement of the target provokes that spallation and, for higher fluences, phase explosion are the predominant mechanisms.⁹ The material is ablated when the rarefaction wave which follows on the initial shock wave exceeds the specific spall strength. Therefore, the heat affected zone of the propellant is rather small and the remaining target material is comparatively cold and exhibits a smooth surface. If the pulse duration is longer than the electron–phonon coupling time ${\tau }_i$ of the material, however, laser heating still continues during the laser pulse, which yields a rather large heat affected zone.¹⁰ This implies various thermal effects, e.g., vaporization and the formation of a crater rim. Taking into account the pulse durations ${\tau }_{L,1}$ and ${\tau }_{L,2}$ of the two laser systems in use and the material-specific electron–phonon coupling times,¹¹ copper and gold have ${\tau }_{L,1}=10\mathrm{ps}<{\tau }_i<{\tau }_{L,2}=500\mathrm{ps}$. This marks the so-called picosecond regime for ${\tau }_{L,1}$ for Cu, Au, and the nanosecond regime for ${\tau }_{L,2}$, respectively. In contrast to this, for Al, ${\tau }_{L,1}$ is in the range of ${\tau }_i$ which might suggest some kind of transition.

2.2.

Correlation Between Surface Roughness and Thrust Noise

To further motivate the determination of the surface roughness, one might take a look at theoretical predictions correlating the surface roughness with the absorbed energy.¹² As soon as multiple scattering takes place, the laser absorption is raised significantly, which in turn decreases the ablation threshold ${\mathrm{\Phi }}_{\mathrm{thr}}$.^5,6 The (cold) absorption on highly reflecting materials as, e.g., aluminum can be approximated linearly by the effective number of reflections.¹² For a given surface correlation length and a Gaussian height distribution, this results in a linear correlation to surface roughness $S_a$, easily doubling the absorbed laser intensity.¹² With the above knowledge and a few assumptions, a linear approximation can be deduced to indicate the amount of thrust noise $\sigma [F]$ introduced. ($\sigma [A]$ depicts the standard deviation of $A$, $E[A]$ its mean or expected value.) First, we may assume that the average surface roughness $E[S_a]$ and the surface correlation length ${\tau }_s$ are of the same size. ${\tau }_s$ represents the characteristic width of the autocorrelation function of the surface profile.¹² Keeping ${\tau }_s$ constant for all, $S_a$ introduces a linear shift of the absorbed fluence ${\mathrm{\Phi }}_{\mathrm{abs}}(S_a/{\tau }_s)$ depending on the surface roughness $S_a$. This again translates the fluence-dependent coupling coefficient $c_m[{\mathrm{\Phi }}_{\mathrm{eff}}(S_a)]$. The coupling coefficient depicts the ratio of generated thrust to the average laser power $P_{\text{laser}}$ used. Second, using the Sinko model,¹³ a realistic intermediate slope of $c_m(\mathrm{\Phi })$ half way to its maximizing fluence ${\mathrm{\Phi }}_{\max }$ ($2{\mathrm{\Phi }}_{\mathrm{thr}}\approx ({\mathrm{\Phi }}_{\max }/2)$) can be deduced dividing $c_m({\mathrm{\Phi }}_{\max })$ by ${\mathrm{\Phi }}_{\max }$. Finally, adding the relation from Bergström et al.¹² for aluminum, $\sigma [{\mathrm{\Phi }}_{\mathrm{abs}}]\approx 3\mathrm{\Phi }(\sigma [S_a]/E[S_a])$, leads to the following approximation:

3.1.

Experiments

The scanning of multiple layers of material was performed using hatching patterns. A hatching pattern describes the procedure of turning the scan direction $\alpha$, in our case by 10 deg, after every layer has been scanned. With this method, 18 layers were removed and the generated surface was analyzed using a white light interferometer (Veeco NT9100). While the hatching angle was fixed to 10 deg, the laser fluence, and therefore, the crater radius $r_c$ was varied. Furthermore, the line and spot displacement were kept equal $\mathrm{\Delta }y=\mathrm{\Delta }x$ and varied to create different amounts of ablation crater overlap (Fig. 2).

Fig. 2

Ablation pattern scheme.⁴ Parameters for a hatching scan are spot displacement $\mathrm{\Delta }x$, line displacement $\mathrm{\Delta }y$, hatching angle $\alpha$, and spot radius $r_s$.

The laser incidence angle $\theta$ (Fig. 1) was set to 0 deg. This allowed to use a conventional scanning setup and access the entire target surface. Laser absorption and coupling coefficient show only a minor dependency up to an incidence angle of 60 deg.¹² As a laser source, a solid-state 10-ps pulse duration system (Duetto, Time Bandwidth/Lumentum) was used, while a rather small microchip laser (PNP-B08010, Teem Photonics) was able to deliver a pulse duration of 0.5 ns. To omit laser power variations, a Faraday isolator was introduced after the microchip laserhead. Both laser systems emit at a wavelength of 1064 nm and are capable of delivering tens of microjoules pulse energy. Laser power was detected by a separated light beam and a pulse energy measuring head (PE9-F, Ophir). To calibrate the effective pulse energy available at the target, simultaneous power measurements with a second laser power head (12A-P, Ophir) placed in the target chamber were performed twice a day. With the microchip laser, a telescope was used to adjust the beam diameter. The $1/e^2$ laser spot diameter $d_s$ for the 10-ps system was 47 and $37\mu \mathrm{m}$ for the 0.5-ns system. Before entering the scan head, a quarter-wave plate was introduced and adjusted to generate circularly polarized light. Circular polarization will prevent the results from being biased by asymmetries due to beam polarization. The beam then was redirected into the scan head (intelliSCAN 14, Scanlab) and focused by a telecentric $f$-theta objective ($f=167\mathrm{mm}$) through an antireflective (AR)-coated vacuum window. In the case of the 0.5-ns pulse duration, an AR-coated window was added within the vacuum chamber to prevent coating of the chamber window by the ablation plume. The focus was placed on the targets’ surface and scan experiments were performed at a vacuum below $2.5·{10}^{-2}\mathrm{mbar}$.

As target materials, aluminum, copper, and gold were chosen. The surface was polished to optical mirror quality by KUGLER.

3.2.

Simulations

Hatching patterns were simulated by linear superposition of idealized single ablation crater profiles.⁴ The concept includes the introduction of an idealized ablation crater using the radially dependent fluence ${\mathrm{\Phi }}_L(r)$ of a Gaussian laser spot to calculate a radially dependent depth $d_a$

The function can be fitted to the experimental or simulation data via parameters a and ${\mathrm{\Phi }}_{\mathrm{thr}}$, where ${\mathrm{\Phi }}_{\text{peak}}$ denotes the peak fluence of a Gaussian-shaped laser spot and ${\mathrm{\Phi }}_{\mathrm{thr}}$ the material threshold fluence. This function always leads to a parabolic profile and is defined for a laser fluence above the threshold (${\mathrm{\Phi }}_L\ge {\mathrm{\Phi }}_{\mathrm{thr}}$). The crater radius $r_c$ then defines the distance of the center to the rim of the crater

Fig. 3

Estimated ablation crater from simulation data. Fluence profiles with corresponding crater profiles (2-D simulation results) for laser ablation of gold with 6 and $12\mathrm{J}/{\mathrm{cm}}^2$, resp. The crater diameters depend on the ablation threshold fluence and therefore vary with the laser pulse fluence, whereas the radius of the laser spot $r_s=25\mu \mathrm{m}$ is kept constant. Laser parameters: $\tau =10\mathrm{ps}$, $\lambda =1064\mathrm{nm}$, circular polarization.

3.3.

Evaluation

Figure 1 shows the surface of a processed cavity. To determine the surface roughness, a region of interest was chosen and the average deviation from the mean value of all chosen pixel values was interpreted as the surface roughness $S_a$.

4.1.

Fourier Interpretation

If the ablation depth can be added linearly to predict the performance of a chosen hatching pattern, a Fourier interpretation and prediction for the results can be performed. Figure 4 shows the Fourier transformation of the ablation depth of a single crater (similar to Fig. 3). It can be seen that the amplitude of the signal decays for higher spacial frequencies $k_x$ and $k_y$. The Fourier representation of a perfectly flat surface would be given by one single signal at the zero frequency ($k_x=k_y=0$). A simple hatching pattern as, e.g., a regular grid, can be interpreted as a folding of the crater with the grid. In Fourier space, this is equivalent to the multiplication of the Fourier spectrum of the crater profile (contour plot in Fig. 4) with the Fourier transformation of the hatching grid. Since the Fourier transformation of a regular grid with grid constant, $\mathrm{\Delta }x$ results in a regular grid, with inversely proportional grid constants $2\pi /\mathrm{\Delta }x$, a grid with a higher density or larger crater overlap $O_c=1-(\mathrm{\Delta }x/d_c)$ (smaller $\mathrm{\Delta }x$) will push the grid points away from the zero frequency (colored rectangles in Fig. 4). Due to the decay at higher frequencies and the multiplication in Fourier space, less and less amplitude next to the zero frequency will be accumulated and, due to the concentric rings of the FT crater profile, will produce modulations contributing to the surface roughness $S_a$ as shown in Fig. 5.

Fig. 4

Fourier transformation of the depth response for a $24\text{-}\mu \mathrm{m}$ crater simulated with a fluence of $0.5\mathrm{J}/{\mathrm{cm}}^2$ and a pulse duration of $\tau =10\mathrm{ps}$ on aluminum. Top and right diagrams display cross sections through the central (zero) frequency in $k_x$ and $k_y$ direction. The rectangular markers in the color plot indicate the positions of the lowest grid frequencies of the Fourier transformed hatching grid pattern. The relative overlap of the corresponding hatching pattern is indicated in per cent on the right top diagonal. For visibility reasons, the rotated grids of the hatching pattern were omitted.

Fig. 5

Relative surface roughness created by folding the ablation crater with the hatching grid using FFT filtering. Furthermore, the maximum relative amplitude at the Fourier grid multiplication points (omitting the zero frequency $k_x=k_y=0$) plotted over the crater overlap. Simulated pulse duration is 10 ps, laser wavlength 1064 nm, fluence of $1\mathrm{J}/{\mathrm{cm}}^2$, and a spot radius of $5\mu \mathrm{m}$ on aluminum.

Compared to earlier findings,⁴ a Fourier domain filter approach shows a similar behavior ($S_a$ in Fig. 5). Furthermore, a strong correlation to the maximum amplitude in Fourier space (next to the zero frequency) after the grid folding process suggests that the inverse grid points closest to the zero frequency dominate the surface roughness (“normalized maximum” in Fig. 5). The Fourier transform of the crater profile (Fig. 4) shows radii with zero contribution. If the Fourier transformed grid points closest to the zero frequency roughly coincide with these areas, their missing contribution to the surface roughness creates a local minimum. This result is reproduced in the roughness plot of Fig. 5.

4.2.

Roughness Measurements

The surface roughness clearly shows modulations plotted against spot displacement (Fig. 6) as well as the laser fluence (Fig. 6 left aluminum). Since there is a modulation visible, optimal parameters for the scanning process can be defined. This behavior can be deduced from the simulations of the hatching patterns (Fig. 5). The simulation results also suggest that these optima correlate linearly with the effective crater diameter $d_c$ calculated by Eq. (6). Therefore, it should be possible to plot curves through the roughness plots, which dominantly follow a path of minimum roughness (compare dashed lines in Fig. 6). From Fig. 5, it can be deduced that the first resonance can be expected at 42% crater overlap ($\mathrm{\Delta }x=58\%$ crater diameter). Using the corresponding ablation threshold and spot diameter in Eq. (6) as determined by single ablation crater Liu plots,¹⁵ we introduced separate graphs (dashed lines) in Fig. 6.

Fig. 6

Surface roughness of copper, gold, and aluminum treated with a $18\times 10\mathrm{deg}$ hatching pattern and a laser pulse duration of 500 ps (left) and 10 ps (right). Dashed lines indicate expected optimum roughness, while the positions marked by an “x” indicate the parameter sets, which were experimentally investigated.

Spot displacements in both regimes are chosen slightly differently, since the spot diameter for both laser systems differed ($d_s(10\mathrm{ps})=47\mu \mathrm{m}$ and $d_s(500\mathrm{ps})=37\mu \mathrm{m}$). Therefore, the curves of the 500-ps experiments are drawn for smaller displacements. Surprisingly, the expected behavior can be seen at least partially for aluminum in both pulse regimes as well as for copper at the 500-ps pulse duration. The other measurements differ distinctly from the expected behavior.

The encountered differences can be explained by the incubation effect.⁵ This effect has a tendency to reduce the ablation threshold which, taking into account Eq. (6), should lead to an underestimation of the optimum spot displacement. This leads to curves that in the displayed plots (Fig. 6) always lie at lower spot displacements than the actually observed optimum.

Despite the surface roughness, other aspects of the laser-ablative process have to be taken into account, too. Optimum specific impulse $I_{\mathrm{sp}}$ and laser impulse coupling coefficient $c_m$ do not necessarily coincide with the optimum of the surface roughness. These values depend on the applied laser fluence ${\mathrm{\Phi }}_L$. For 500-ps pulses, we were able to observe a surface roughness which was almost independent from ${\mathrm{\Phi }}_L$. These optimum spot displacement values, concerning the surface roughness, might allow to benefit from a freely chosen laser fluence.

A surprising aspect in comparing the ultrashort-pulse ablation regime (here 10 ps) and the short-pulse (here 500 ps) ablation regime is the demonstration of similar roughness generated at the optimal spot displacement regions. Single ablation craters in the short-pulse regime, generated by vaporization and melting, typically lead to a chaotic raising of the crater rim. These effects lead to bigger surface modulations compared to the ultrashort-pulse regime dominated by spallation and phase explosion.⁴ The effect of applying multiple layers of densely placed ablation spots appears to compensate for this difference effectively.

The last aspect visible in all roughness plots is a low surface roughness for low laser fluence. This behavior can be explained by the minor amount of material ablated at the threshold fluence and below. With no ablation taking place, the surface is not expected to become rough. Therefore, these values are excluded from the discussion.