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Abstract
High peak and average power lasers with high wall-plug efficiency, like the Big Aperture Thulium (BAT) laser, have garnered tremendous attention in laser technology. To meet the requirements of the BAT laser, we have developed low-dispersion reflection multilayer dielectric (MLD) gratings suitable for compression of high-energy pulses for operations at 2 micron wavelength. We carried out 10000-on-1 damage tests to investigate the fluence damage thresholds of the designed MLD gratings and mirrors, which were found between 100-230 mJ/cm2. An ultrashort pulsed laser (FWHM = 53 fs, λ = 1.9 μm) operating at 500 Hz was used in the serpentine raster scans. The atomic force microscope images of the damage sites show blister formation of the underlying layers at lower fluences but ablation of the grating pillars at higher fluences. We simulated the dynamic electronic excitation in the MLD optics with a finite-difference in the time domain approach in 2D. The simulation results agree well with the LIDT measurements and the observed blister formation. This model is able to evaluate the absolute LIDT of MLD gratings.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
To meet the requirements for driving compact particle accelerators based on laser wakefield acceleration (LWFA), the repetition rate, and therefore the average power, of PW-class lasers must be increased by several orders of magnitude beyond the state-of-the-art. Accordingly, to make these accelerators practical, the driver lasers must operate with high overall wall plug efficiency [1]. To meet these needs, the high-average-power petawatt-class Big Aperture Thulium (BAT) laser concept has been proposed [2]. This laser concept is based on directly diode-pumped Tm:YLF operating near $\lambda =2\mu$m and is capable of supporting high energy pulses with sub-100 fs durations [3]. A very recent work by our co-authors in this work and their collaborators demonstrating amplification of multi-joule nanosecond pulses in the BAT architecture corroborates the importance of development of high power laser optics at these wavelengths [4]. However, development of efficient, high damage threshold pulse compression gratings (PCGs) for operation near 2 $\mu$m has lagged behind development for operation near 1 $\mu$m [5] and 800 nm [6–8]. The PCGs are typically planar reflection gratings which are essential in chirped pulse amplification (CPA) setups [9,10]. Since PCGs are exposed to the highest fluence in a CPA system, their laser-induced damage threshold (LIDT) becomes the main limitation of the output power [11]. To achieve a higher LIDT than that of the metal-coated PCGs, the multi-layer dielectric (MLD) PCG was proposed [12]. Since the working wavelengths of Nd:glass/Yb-doped materials and Ti:Sapphire lasers are near 1053 nm and 800 nm, respectively, MLD PCGs are well developed to operate at both wavelengths [6,12–15]. MLD gratings have been developed for compression of low energy fiber lasers at wavelengths near 2 $\mu$m [16,17], but, to date, there have been no reports of MLD PCGs designed for high energy 2 $\mu$m lasers.
The LIDT of MLD PCGs is associated with the inherent electric field modulation. The steady-state modeling of the field modulation in a PCG shows periodic maxima [14]. Stuart et al. found that the damage sites (@825 nm, 140 fs) localized at the back edges of grating ridges in a HfO$_2$ layer, which was consistent with the "hot" zones from their simulation results [18]. Similar damage morphologies were observed in SiO$_2$ [19] and HfO$_2$ [20] in grating pillars. By simulating field enhancement for various angles of incidence (AOI), Britten et al. fitted the experimental LIDTs (@1053 nm, 1 ps) with inverse law of the maximum field enhancement ($1/|E|^{2}$) slightly better than the area projection [14]. Neauport et al. found a linear dependence between LIDT (@1053 nm, 500 fs) and $1/|E|^{2}$ for different grating designs [21]. Therefore, criteria of designing MLD PCGs often aim to minimize the field enhancement and maximize the efficiency [14,22,23].
For 100 fs or shorter laser pulses, the LIDT of dielectrics is increasingly affected by the wide bandwidth of laser pulses and material properties. Unfortunately, these factors cannot be modeled by the steady-state approach. The high intensity up to $10^{13}-10^{15}$ W/cm$^{2}$ can lead to the photoionization in dielectrics, and the excited electrons may initiate the avalanche. The dielectrics are converted to a metallic state within the first a few tens of fs of an intense pulse. This transient modification of materials also dramatically affects pulse propagation [24]. As the laser pulse keeps heating the dense plasma, the hot electrons will eventually cause a material breakdown.
Numerous publications have been devoted for studying the electronic excitation in bulk dielectrics and surfaces, indicating that the LIDT is associated with material properties (e.g., bandgaps) and laser parameters (e.g., duration, wavelength) [18,25–27]. However, to the best of our knowledge, such modeling has not been performed for MLD gratings. Instead, LIDT of MLD gratings is usually evaluated numerically by steady-state field enhancement without considering the electronic excitation and its effect on propagation [18,21,28]. It is well accepted that LIDT of MLD gratings is dependent on $|E|^{2}$, and damages should initiate at grating ridges opposite to the incident beam since $|E|^{2}$ is highest there [18,21]. Nevertheless, this conclusion is based on two assumptions: (1) the difference in the femtosecond-laser damage resistance of different materials is negligible; (2) the gratings maintain a linear optical response under LIDT intensity. However, these assumptions are not realistic since it has been observed that the materials with wide bandgap tend to have high LIDT [29]. Moreover, subject to LIDT intensity, the optical response of the grating is transiently modified due to the plasma generation, so is the electric field.
Our previous works evaluated the LIDT of MLD mirrors and short pulses (<100 fs), where the electronic excitation, modification of optical response, and Maxwell’s equations are solved using the finite-difference-in-time-domain (FDTD) algorithm [30,31]. In this paper, we first present a 600 lines/mm MLD PCG design with high diffraction efficiency over 100 nm of bandwidth. Secondly, we perform the LIDT test with femtosecond laser at the operating wavelength for both gratings and mirrors with the same interference coating design (no groove on top layer). Thirdly, we perform a dynamic model based on the FDTD framework incorporating the Keldysh ionization theory and non-linear absorption and propagation in plasma. LIDTs yielded from the simulations are in good agreement with experimental thresholds for both gratings and mirrors.
2. Experimental setup and procedure
Figure 1 shows the setup used for the damage study. We use a TOPAS Prime Light Conversion optical parametric amplifier (OPA) to convert the output of the OSU GRAY Ti:Sapphire laser system at $\sim$780 nm to an idler output at 1.9 $\mu$m central wavelength. The testing MLD samples are 2-inch-diameter round and 1-cm thick. The resolution of the in situ imaging system to observe the damage onset is 1.1 $\mu$m. The laser damage tests were performed in in ambient air in a Class-100,000-cleanroom lab. Figure 2(a) presents the idler spectrum (FWHM $=110$ nm). Unwanted signal from the OPA along with sum frequency generated light between pump and idler are separated out with the help of a wavelength separator and silicon plate at Brewster’s angle for 1.9 $\mu$m. The pulse energies are adjusted using a waveplate-polarizer and monitored in situ using a calibrated PbS photodiode measuring the pick-off from a CaF$_2$ window. The mid-IR pulses then pass through a polarization changing periscope converting them to s-polarization. Finally, they are focused onto the sample surface with a 68-$\mu$m-diameter (FWHM) focal spot and 53-fs pulse width. Figure 2(b) presents the autocorrelation trace of the pulse.
Fig. 1. Experimental damage testing setup showing both in situ imaging of the sample surface as well as focal spot imaging. A typical focal spot captured using mid-IR camera showing 68-$\mu$m-spot diameter (FWHM).
Fig. 2. (a) Spectrum of mid-IR pulses; (b) Autocorrelation trace of input pulse pulse indicating a duration of 53 fs FWHM.
Figure 3 shows that the raster scans were preceded by 10000-on-1 multishot tests at various fluences to estimate the LIDT of the samples using the in situ imaging system. Each row consisted of a fiducial crater and one 10000-on-1 site. The fiducial craters were produced by exposing the sample to high fluence pulses over several seconds while each 10000-on-1 site was irradiated by 10000 pulses at a particular fluence. Test sites in each row underwent different fluence interactions to ascertain proper fluence intervals below and above LIDT. These 10000-on-1 damage tests provided us with a reliable range of fluences with which to perform the area raster scans below and above the LIDT. The raster scan technique was chosen because it is ideally suited for ’real world’ testing of femtosecond optics for high repetition applications, sampling an area much larger than that of a typical focal spot, while capturing the S-on-1 aspect of laser damage as well, and demonstrated in our previous works [32,33]. Several raster scans were conducted at pulse energies straddling the estimated LIDTs. Top and bottom edges of the raster scan areas are excluded from the analysis because they received excessive irradiance as the sample stage motors slowed down to switch its direction. Each 2 mm $\times$2 mm square was uniformly illuminated with 133956 shots, and each single site in the squares was illuminated by approximately 129 shots on average.
Fig. 3. (a) Sample surface map; (b) Completed raster map; (c) Raster map showing a completed column and an adjacent uncompleted column; (d) Zoom-in view of the area in the red box. The red arrows indicate the raster direction. Overlap between rows as well as columns were roughly 90$\%$ to achieve uniform irradiance over the entire square.
3. Methods
3.1 MLD grating development
To meet the requirements of the high average power lasers at wavelengths near 1.9 $\mu$m, we have developed MLD reflection gratings for the efficient compression of high energy. These gratings were designed and produced by the Diffractive Optics Group at Lawrence Livermore National Laboratory. Gratings with a groove density of 600 lines/mm were selected as a compromise between high dispersion and moderate Littrow angle. For a wavelength of $\lambda =1.9 \mu$m this groove density results in a Littrow angle of $34.75 ^{\circ }$. The MLD grating design criteria seeks to minimize the electric field existing in the solid grating material and the underlying multilayers, while at the same time maximizing the efficiency. Hybrid 18-layer pairs of alternating high-low indexes mirror stacks are deposited on the substrate. The topmost 5-layer pairs consist of HfO$_2$/SiO$_2$ for improved laser damage resistance and the remaining 13-layer pairs consist of Ta$_2$O$_5$/SiO$_2$ for high reflectivity and low absorption. On top of the mirror, a solid SiO$_2$ layer was deposited, which was etched into for the fabrication of the grating structure (this layer remains unetched for the mirror samples).
The key design criteria for these gratings is a design which is free of guided mode resonances which cause narrow and deep dips in the diffraction efficiency and are suspected to reduce laser damage performance. We designed the grating by using a computer code based on rigorous coupled-wave analysis (RCWA). The choice of the number of layer pairs of the MLD mirror stack was made as a compromise between having a sufficiently high reflectance over the desired wavelength range, high LIDTs, and assuring good stability of the thin-film stack. The grating is required to operate at the Littrow angle due to the sensitivity of the bandwidth with AOI. Figure 4(a) shows the modeled diffraction efficiency for $s$-polarized light as function of wavelength and grating duty cycle at Littrow conditions. The gratings consist of an ion beam sputtered hafnia-silica multilayer mirror stack and a single corrugated top silica layer. The fabrication procedure of the grating etched into the top layer of the MLD stack consists of two basic steps: the lithographic generation of a grating in a resist layer by interference laser lithography and the transfer of this grating mask into the dielectric top layer by reactive ion beam etching using equipment that supports $\sim$1-meter sized optics. For the work presented here, gratings were fabricated on 50.8 mm fused silica and silicon substrates. After grating structures are plasma etched into topmost layer of the multilayer stack, we utilize semiconductor-grade tetramethylammonium hydroxide solvent and sulfuric acid to remove the unused etch mask and as a final clean step. Figure 4(b) shows the measured reflection efficiency of MLD coated substrate prior to etching for TE polarized light at a near Littrow AOI. As can be seen from this figure, the MLD coating itself supports a broad bandwidth of $\sim$400 nm centered near $\lambda =1.9 \mu$m. Measurements of the diffraction efficiency of the etched MLD gratings using a narrow band $\lambda =1.88 \mu$m Tm:YLF laser yielded a diffraction efficiency of 96.7$\%$ at an AOI near Littrow.
Fig. 4. (a) Theoretical grating efficiency as function of wavelength and grating duty cycle at Littrow conditions. (b) Measured reflection efficiency as a function of wavelength of the MLD grating coating prior to etching at an AOI near Littrow.
3.2 Method for determining damage threshold
Significant instability in beam energy throughout the raster scans leads to difficulty in labeling whole raster scans as either damaged or not and necessitates a more precise assessment method. To achieve this, the in situ energy values determined by a photodiode are plotted in two dimensions resembling the physical images of a raster scan itself with excellent agreement. Figure 5 shows that sites of damage match up with high energy spikes in the gray-scale intensity plot, and sites of little to no damage matched up with energy dips.
Fig. 5. Column based damage assessment by comparing post-test optical microscopy images of a typical raster scan damage sites (top image) to gray-scale pulse energy readouts obtained in real time during that specific scan (bottom image). Peak to peak energy fluctuation is 2.3 $\mu$J in row lineout and 0.6 $\mu$J in column lineout (marked by red arrow)
This method allows us to pinpoint the LIDTs possibly down to few shots within a column. However, the reasonable stability within each column of the raster scans deems a column-by-column assessment sufficient. The assessed LIDTs are then converted into peak fluence damage thresholds using focal spot images collected at regular intervals throughout the experiment.
4. Experimental results
Table 1 shows the results obtained from the measurements described above. In both simulation and experiment, the laser fluence is defined as beam normal. The samples A-C were manufactured with the intention of being identical, so they are very similar to each other. In the grating design, the etched depth is nominally 1250 nm and the duty cycle of the grooves is nominally 32%. Sample A is closest to the nominal values and the numerical simulation is also based on the nominal values. The unetched samples (D and E) have LIDTs generally higher than the grating samples (A-C), possibly due to field enhancement by the grating structure. There is no clear correlation between substrate choice and the LIDT of gratings. There are two primary sources of error in fluence calculation from LIDT measurements, i) focal spot area, and ii) energy/pulse fluctuation. Our focal spot area error ($< 1\%$) determined by the focal spot camera resolution is negligible compared to the error caused by peak-to-peak pulse energy fluctuation over a single column scan ($\sim 19\%$). Thus our measured LIDT fluence error mainly reflect the effect of peak-to-peak pulse energy fluctuation.
Table 1. Measurements of peak fluence damage threshold for the gratings (A-C) and mirrors (D-E).
The progression of damage as a function of fluence seems to obey a “blister model" [34] where the first high-index (low bandgap) layer that the pulse interacts with would absorb most of the pulse energy and expand, pushing the layers above it upward. In our case, the surface layer of the samples is SiO$_2$ followed by HfO$_2$, the HfO$_2$ layer may expand and push the surface SiO$_2$ layer upward. Figure 6 is the atomic force microscope (AFM) scans of the irradiated samples. They confirm the occurrence of blistering, where damaged areas of the sample rose in a height by more than 0.5$\mu$m. As the fluence increases, the top layers appear to crack and break, possibly due to large tensile stresses of the blistering process that surpasses mechanical limits of the top layers. At higher fluences, the material is ablated from the surface forming craters. The depth profiles in Fig. 6(f) clearly show the bulge around the crater, indicating the initial blister formation of the coating, followed by the ablation at pillars occurring at a higher fluence. Note that the AFM tip cannot go all the way down to the narrow troughs of the periodic grooves this high-aspect-ratio grating sample, causing some uncertainty in depth profiling in this case. However, it is not significant enough to make our observation of blister formation followed by crater formation questionable.
Fig. 6. Progression of damage morphology of sample B with increasing fluence captured via atomic force microscopy scans: (a) Blister formation on the grating after a raster scan showing line blistering along the path of the laser scan; (b) Cracking and heavy damage of the grating on a higher-fluence area of a raster scan; (c) Blister formation at a s-on-1 site at fluence of 130 mJ/cm$^{2}$; (d) Cracking formation in the grating structure at a higher fluence of 153 mJ/cm$^{2}$; (e) Crater formation at high fluence of 373 mJ/cm$^{2}$; (f) Profile of crater along ridges (red) and perpendicular to ridges (black). (g) Blister formation at an S-on-1 damage site in the mirror with Si substrate at fluence = 357 mJ/cm$^{2}$; (i) with SiO$_2$ substrate at fluence = 197 mJ/cm$^{2}$; (h) and (j) are the profiles of the lineouts in (g) and (h), respectively.
Figures 6(g)-(i) show the blister formation observed in samples D (Si substrate) and E (SiO$_2$ substrate), respectively. The coatings of the two mirror samples were manufactured with the intention of being identical. The surface RMS roughness is between 5 and 6 nm for both samples. Interestingly, none of the S-on-1 sites on the mirror samples show any crater formation near the LIDT fluences or at the maximum fluence tested in this work, which implies that the top silica layer (which was made extra-thick for grating etching purposes) is too strong/undamaged to break, as it had not been sufficiently ionized due to its large bandgap. This is corroborated in our simulations. The significant blistering implies that the high index layers (or a single high index layer) have been damaged via high-density carrier generation, causing ultrafast melting and ablation [35], detaching from the layers above while simultaneously pushing it (them) upwards, resulting in blister formation. Our previous work [34] has demonstrated that blister formation is a typical precursor to ablation and crater formation in S-on-1 femtosecond laser damage tests of MLD systems. Another work of ours has demonstrated with cross-sectional scanning electron microscopy (SEM) that damage of a high index layer causes blistering in an MLD chirped mirror damaged by 5 fs laser pulses [36].
Although the different substrates do not seem to affect the LIDT of gratings significantly, they do seem to affect the LIDT of the mirrors. Since we cannot distinguish the mirror samples with different substrates using the surface roughness parameter (or morphology) of the damaged area, other factors most likely contributed to the LIDT difference between them. One possible cause is the difference in inherent stress in the coatings, which is usually not apparent from a surface scan. While the coatings may be identical (presumably coated during the same run), inherent stress accumulation depending on the choice of substrate is different. Sometimes substrates are heated and cooled during (and/or before/after the actual coating run) the coating process to control the coating quality, which also may play a role when the substrates are different. The inherent stress related to damage/delamination can manifest itself when the sample undergoes large environmental changes, such as humidity, temperature [37] or, in this case, the laser irradiation treatment. In our case, any environmental changes besides laser irradiation were the same for all samples. Careful measurement of inherent stress in addition to cutting open the blisters will be necessary to determine the actual cause of this difference. As it is beyond the scope of this paper, we plan to study this interesting topic in a future work.
5. Simulation model
To better guide the grating design in the future, we propose solving Maxwell’s equations with the FDTD algorithm, in which the photoionization, collisional ionization, and plasma heating are considered in each time step. Figure 7 presents a schematic of the modeling of dynamic electronic excitation, where the designed MLD grating and mirror are irradiated by fs pulses ($\lambda$=1.9 $\mu$m, FWHM=53 fs) at an AOI of $35^{\circ }$. In the propagation of an electromagnetic field in TE mode, the magnetic field $\vec {B} = B_x(x,y)\vec {x}+B_y(x,y)\vec {y}$ and the electric field $\vec {E} = E_z(x,y)\vec {z}$ can be expressed by Maxwell’s equations
It is assumed that the electrons instantly generated through photoionization have zero mean momentum. The current density can be expressed as
Fig. 7. Schematic of simulation model. The top layer is not etched for MLD mirrors. The MLD stack has 18 alternating HfO$_2$/SiO$_2$ and Ta$_2$O$_5$/SiO$_2$ layer pairs and an SiO2 layer on top where the grating structure is etched into.
The excited electron density can be expressed as
The parameters used in this simulation are all based on measurements and fundamental physics considerations and listed in Table 2. No fitting parameters have been used to match the experimental results. Due to the lack of measurements of the avalanche rate constants for hafnia and tantala, we applied the constant of silica. The distributions of electric field and electron density are obtained by solving Eqs. 1–6 using the 2D FDTD algorithm with a uniaxial perfectly matched layer (PML) boundary. The radius of the Gaussian focal spot is 5.6 $\mu$m. The FWHM of intensity of the sine$^{2}$ temporal intensity profile is 53 fs. The LIDT fluence is evaluated by the critical plasma density criterion [18,29]. The choice of a smaller spot size in the simulation than in the experiment is a compromise in achieving high precision and acceptable computational cost at the same time. One may expand the simulation space on high performance computing clusters in the future for simulations with laser foci sizes better matched with experimental parameters. Since our simulation captures dynamics on the timescale of the fs laser pulse and does not include the effect of defects, we believe the results would not significantly change in a simulation with a more realistic spot size. Animation/movies of the simulation results and their corresponding description can be found in the supplementary materials.
6. Simulation results
Figure 8 shows that the mirror reflects the short pulse efficiently, and as the pulse advances, the excited electron density rises gradually. In order to study the relationship between the field enhancement and the electronic excitation, a lineout of the spatial distribution of $|E_{\textrm {max}}|$ is presented in Fig. 8(h), which is the maximum of electric field strength captured at each grid asynchronously during the entire simulation. $|E_{\textrm {max}}|$ is normalized to the initial envelope amplitude of the incident pulse. The distribution of $|E_{\textrm {max}}|$ in the affected area is horizontally periodic due to the interference of incident light and diffracted light. Fig. 8(h) shows that the peaks of $|E_{\textrm {max}}|$ are between 1-1.5 in the top seven layers, and the maximum occurs at the top silica layer. According to the $1/|E|^{2}$ damage threshold criteria, the top silica layer is most prone to damage. However, the maximum $n_e$ that exceeds the critical value ($n_{\textrm {e,crit}}$=$10^{20.5}$ cm$^{-3}$ @ 1.9 $\mu$m) occurs in the fourth layer(HfO$_2$), followed by the sixth layer (HfO$_2$). Based on the critical electron density criterion, this result indicates that the short pulse initiates damage at layers beneath the surface. Although the distribution of $n_{\textrm {e}}$ is very similar to that of $|E_{\textrm {max}}|$, this correlation follows different scaling for different materials. It is noticed that $n_{\textrm {e}}$ drops sharply from a HfO$_2$ layer to an adjacent SiO$_2$ layer even when the electrical field strength remains almost constant. The striking difference can be explained by different bandgaps of HfO$_2$ (5.7 eV) and SiO$_2$ (9 eV). Since low-bandgap materials require fewer photons to free a bound electron, they tend to have lower photoionization rates and lower LIDTs [29]. Figure 8(b) shows the transient electric field distribution with horizontal and vertical interference fringes, but the electron density only appears as vertical stripes. This is because the horizontal interference fringes shift downward as the pulse proceeds in the dynamic propagation. Therefore, the principal maxima do not localize during the propagation but "weep" uniformly through the "hot zone" where the electron density is highest in the direction parallel to the mirror surface, which is clearly demonstrated in the supplemental video of the mirror.
Fig. 8. Simulation results for MLD mirror: (a)-(d) Normalized electric field during propagation. The white lines are the vacuum-mirror and film-substrate interfaces. (e)-(h) Evolving $(n_{\textrm {e}})$ in the red box. The green, orange, and red lines indicate SiO$_2$, HfO$_2$, and Ta$_2$O$_5$ layers, respectively. (i) Lineouts of $n_{\textrm {e}}$ and $|E_{\textrm {max}}|$ along the red line including the maximum. The vertical blue lines are interfaces. The peak fluence of incident pulse is 0.36 J/cm$^{2}$. The peak of the pulse arrives the target surface at t = 64 fs. For the time resolved dynamics capturing the relationship between maximum electric field strengths and maximum electron densities in each layer, see Visualization 1.
Figure 9 shows that the incident pulse is diffracted mainly at $m=-1$ order (Littrow mount). Compared to the results for the MLD mirror, the local field enhancement in the grating is higher, and the magnitudes of the second and third peaks are much greater than the other peaks (see Fig. 9(j)). The distribution of $|E_{\textrm {max}}|$ in the affected area is both vertically and horizontally periodic as the diffracted light causes more complex interference pattern. Figure 9(j) shows that the electron density follows the periodic distribution of $|E_{\textrm {max}}|$. Similar to the results of the mirror, the maximum electron density that exceeds the critical value also occurs at the fourth layer (HfO$_2$), followed by the sixth layer (HfO$_2$)(see sites $C$ and $D$ in Fig. 9(i)). Also, the grating pillars in the top silica layer has a relatively low electron density in spite of the highest $|E_{\textrm {max}}|$ (see site $A$ in Figs. 9(i) and (j)). This result indicates that the HfO$_2$ layers below the grating pillars are most susceptible to laser induced damage. The stark contrast between electron densities at sites $B$ and $C$ also highlights the effect of material properties on the electronic excitation induced by femtosecond pulses.
Fig. 9. Simulation results for MLD grating: (a)-(d) Normalized electric fields during propagation. The white lines are the vacuum-grating and film-substrate interfaces. (e)-(h) Evolving $n_{\textrm {e}}$ in the red box. The green, orange, and red lines indicate SiO$_2$, HfO$_2$, and Ta$_2$O$_5$ layers, respectively. (i)-(j) Electron density and normalized asynchronous maximal electric field strength in the white box, respectively. The peak fluence of incident pulse is 0.13 J/cm$^{2}$. The peak of the pulse arrives the target surface at t = 64 fs. For the time resolved dynamics capturing the relationship between maximum electric field strengths and maximum electron densities in each layer, see Visualization 2.
The temporal electric field profiles in Figs. 10 (a) and (b) show that the pulse propagation is significantly affected on the trailing edge by the solid-density plasma. Note that the electron density is about four times the critical density at the incident fluence of 0.26 J/cm$^{2}$ in Fig. 9 (a) and (b). By contrast, the maximum electron density at 0.01 J/cm$^{2}$ is negligible compared to the critical density and has a minute impact on pulse propagation. As shown in Fig. 10 (d), the pulse amplitude becomes lower in the trailing edge after the transient modification.
Fig. 10. Transient E field normalized to the peak field strength of the incident pulse at fluences below and above the LIDT fluence. (a) and (b) are the temporal E fields at the same positions of sites A and C in Fig. 9, respectively. (c) a cartoon of the path of the pulse incident on the grating (point 3 to 2 to 1) and diffracted from it (1 to 2 to 3) along the Littrow angle from t = 0 fs to t = 160 fs; (d) shows the E field spatial profiles of the diffracted pulse along the path 1-2-3 in (c), frozen in time at t = 160 fs, with the difference in the two field profiles for two respective fluences plotted in yellow on a 10x exaggerated scale for clarity. Note the effect of ionization in the grating that shows up at the trailing edge of the pulse leaving the grating surface at point 2.
7. Discussion
Field enhancement modeling cannot absolutely estimate the LIDT value. It is often used to compare the relative merits among similar grating designs. Based on a known LIDT measurement of a sample, LIDT of a similar design can be estimated from the field enhancement modeling [21]. By contrast, this 2D FDTD modeling predicts the absolute LIDT of the same order of magnitude as the measurements without any post-damage information of similar MLD grating samples as reference. The estimated LIDT fluences of the MLD gratings and mirrors suggested by the modeling are 0.13 J/cm$^{2}$ and 0.32 J/cm$^{2}$, respectively (see Fig. 11).
Compared to Table 1, the predicted LIDT fluence of the grating is within the range of the measured LIDT fluences, but it is near the upper limit of the range, especially compared to the LIDT fluence of sample A. In the case of the mirrors, the LIDT fluence obtained from the simulations is higher than the measurement by about 40%. Many factors may cause the deviation of the simulation results from measurements, such as manufacturing procedures, experiment approaches, potential contamination [44], etc. Based on the critical density criterion, our simulation does not capture any reduction in LIDT fluence due to potential defects/traps generated by multiple pulses [45], which means that it may over-predict the LIDT. Moreover, the 2-$\mu$m wavelength is uncharted territory for fs testing of the coatings, and this wavelength region allow unique situations where critical density is lowered by a factor of 4-6 compared to typical NIR wavelengths of 1.0-0.8 $\mu$m, whereas electron ponderomotive energies are higher by the same factor for the MIR 2-$\mu$m wavelength [46], which may affect collisional ionization rates.
Meanwhile, the difference between simulation results and the measurements is greater for the mirror than for the grating. It is possible that that the S-on-1 procedure has a more significant effect on the LIDT of the mirror than the grating. Although the predicted LIDT fluence of the grating is three times lower than that of the mirror, the experimental LIDT of the grating is found to be even lower than that predicted grating LIDT. This is because the grating can cause a high E-field enhancement where the maximum E-field strength at the grating LIDT fluence can be as high as that in the mirror at the mirror LIDT fluence. However, as shown in Figs. 8 and 9, such enhanced area (volume in 3D) in the grating is much smaller than those in the mirrors. Moreover, the field enhancement distribution has a very high gradient near the hot spot in the grating, which shows that most areas are subject to much lower electric field strength than the hot spots. By contrast, the gradient of field enhancement distribution in the mirror is much smaller, indicating that the mirror is more uniformly irradiated by electric field strength closer to the maximum. During an S-on-1 testing event (S$\gg$1) at a fluence slightly below the 1-on-1 LIDT fluence, defects such as self-trapped excitons are more likely to be generated in the mirror during interaction with successive pulses, and develop a clearly observable damage site caused by continued interaction with many subsequent pulses. Similarly, the S-on-1 LIDT fluence of the mirror is also more susceptible to pre-existing defects. Moreover, it is consistent with the experimental result where the LIDT of the mirrors is more affected by the substrates than that of the gratings because the larger field-enhanced contiguous area in the mirrors can intensify the effect of laser irradiance on the inherent stress related to the choice of substrate.
It is notable that the blister and crack formations were observed in the MLD grating samples in this experiment, while the onset of laser induced damage in SiO$_2$/HfO$_2$-based MLD gratings was usually observed at pillars opposite to the incident light in the near-IR wavelength regime [18–20]. This damage in the pillars often occurred in hafnia [18,20], because hafnia has a lower LIDT than fused silica [29]. When the field enhancement is much higher in a silica pillars than the hafnia layers beneath it, the laser irradiation may also initiate damage on the pillars [19]. However, in our case, the field enhancement in the pillars is moderately higher than the hafnia layers beneath. Therefore, it is possible that the onset of damage occurs beneath the pillars at the LIDT fluence. Then, the damaged hafnia layer expands and pushes the surface upward, resulting the crack and break of surface when the tensile stress exceeds the mechanical limit (see Fig. 6). Similar blister formation was also observed in SiO$_2$/HfO$_2$-based reflective MLD thin film systems by fs laser pulses [47]. The simulation results also support this observation, as the maximal electron density occurs at the hafnia layer below the silica pillars (see Fig. 9(i)). Since the simulation results are consistent with the LIDT measurements and observed damage path, we conclude that it captures the major mechanisms of ultrafast-laser-induced damage. In future work, we will utilize the Focused Ion Beam (FIB) to cut the blistering spots open, analyze the structure of materials there (i.e. density changes, void formation) and measure the cross-section by Scanning Electron Microscope (SEM) or Transmission Electron Microscope (TEM). Such ultra-high-precision measurement will reveal more features of blistering and help to examine the numerical model more accurately.
Since the relative importance of impact ionization is a topic of ongoing discussion [24], we compare the simulation results with and without the impact ionization for the 53 fs pulse at the LIDT fluence. While most studies employ simplified impact ionization models solely depending on the electric field strength [48–50], it is important to know that the obtained $\nu _{\textrm {col}}$ is not entirely accurate. These models can result in nonphysically exaggerated $\nu _{\textrm {col}}$ and fail in situations near full ionization. These models are usually used when electron heating and energy distribution are not modeled, but these factors are crucial since only the portion of sufficiently energetic electrons can initiate the impact ionization. To address this problem, in this study, $\nu _{\textrm {col}}$ is not solely dependent on the electric field, but a function of electron density and temperature, assuming that the electron energy follows the Maxwellian distribution. Figure 12 illustrates that at the "hot" zones where the field enhancement is strong, the maximum electron density with the impact ionization is 5.4 times higher than that without. This is because, at a fluence near LIDT, sufficient electrons have been generated by photoionization in the rising edge of the laser pulse, making the electron heating and avalanche possible in the trailing edge. The excited electrons in the "hot" zones are significantly heated and energetic enough to initiate the avalanche. The estimated LIDT fluence without the impact ionization is 0.46 J/cm$^{2}$, which is much higher than the measurement.
Fig. 12. Excited electron densities in top layers of MLD mirror with collisional ionization and without. The peak fluence of incident pulse is 0.36 J/cm$^{2}$.
Since this numerical model is presented for the first time, the goal of the demonstration in this study is to present the physical validity, reliability, and general applicability of the model. Therefore, the parameters in this study are strictly based on fundamental physical considerations or from measurements reported from previous studies (please see Table 2). We note, however, that the parameters of amorphous materials in MLD mirrors and gratings can vary within a range due to different designs, manufacturing procedures, defect concentration, and aging. Ideally, a thorough measurement of the material properties (e.g., band structure, collision ionization constant, defect concentration, effective electron masses) in the MLD gratings and mirrors in each coating run should be carried out, and used as input to the model for most consistent results. However, this can be expensive and not practical. Given the very good agreement with the LIDT measurement achieved by the model using current set of parameters without using any fitting or optimization of any parameter(s), we predict that one can achieve even better agreements between the simulation and experiment of this type by fine-tuning the parameters in Table 2 based on measurements from the coating runs. Various fundamental physical aspects of the model can be further improved, for example, by capturing the non-thermal nature of the laser excited electron kinetics, which may affect the collisional ionization model/rate, energy transport by ballistic electrons, etc. Afterwards, one can utilize the improved numerical model inversely to estimate the deviation of the physical parameters of the materials in the coating from their values reported in previous studies.
The defects play an important role in laser damage, which has been experimentally observed in MLD gratings at a wavelength of 1053 nm in picosecond regime [51]. We expect that defects can also impact the LIDT of MLD gratings in the fs regime, but the effect remains unclear before further study. Moreover, there is a variety of defects that can cause different effects on the laser damage. For example, the self-trapped excitons can result in a low effective bandgap. A F-center can cause a high initial electron density locally. Geometric defects or contaminations like nodular, particles on surface, and pits in the MLD grating will directly affect the pulse propagation, which may lead to high local field enhancement or low diffraction efficiency. Since most defects have irregular geometry and concentration, our future work will focus on systematically studying their effects on LIDT of MLD gratings using this numerical model. Once the properties of a defect is defined, the bandgap, initial electron density, electron mass, refractive index, and geometry of the targets can all be modified in the initial setting to imitate the defect.
8. Conclusion
In this work, we successfully designed and manufactured a 600 l/mm low-dispersion MLD PCG operating at 1.9 $\mu$m with a high diffraction efficiency over a wide bandwidth of 100 nm. The 10000-on-1 laser damage tests were carried out with 53-fs laser pulses. The LIDT fluences were measured to be 100-129 mJ/cm$^{2}$ and 156-230 mJ/cm$^{2}$ for the gratings and mirrors, respectively. The AFM inspection of damage sites reveals the damage progression in the grating samples: blistering of the coating is observed at fluences close to the LIDT fluence, then the grating pillars are gradually ablated as the fluence increases. We have modeled the dynamics of electric field and electronic excitation in two dimensions. The LIDT fluences obtained from the simulation agree well with the measurements. The simulation results indicate that the grating pillars are subjected to the maximum field enhancement, but the highest electron density is found in a hafnia layer beneath the grating pillars, which supports the observed blister formation. This work will enable efficient, high average power, high energy short pulse lasers based on promising materials like Tm:YLF and others at mid-IR wavelengths. To further improve the LIDT of PCGs, future work is underway to guide the grating design with this theoretical model.
Funding
Azimuth Corporation (FA6850- 443 16-D-540-0012); Lawrence Livermore National Laboratory (17-ERD-033, 21-ERD-016); U.S. Department of Energy (DE-AC52-07NA27344); U.S. Department of Energy (DE-SC0019900).
Acknowledgments
The authors acknowledge valuable discussions with Dr. Vitaly Gruzdev and Dr. Joseph Smith, and thank Emma DeAngelis for help with the mirror AFM scans. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
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