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Abstract
Laser-induced damage experiments on ${{\rm HfO}_2}$ and ${{\rm Nb}_2}{{\rm O}_5}$ thin films were performed with 500 fs pulse duration at 1030 nm wavelength. Threshold fluences as a function of beam size have been determined for effective beam diameters ranging from 40 to 220 µm, in a single shot regime. The results suggest no beam-size effect related to material properties in the investigated range, but size effects related to the metrology. The results indicate the importance of appropriate focusing conditions and beam measurement to qualify the optics for use in lasers with large beam sizes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Since the early years of laser research, laser damage has been an extensively studied phenomenon [1]. Studies focusing on determination of the laser-induced damage threshold (LIDT) are important for proper handling of optical components in laser systems [2] and for research on material processing [3], comprising ablation and laser machining [4]. It was soon recognized that laser-induced damage in optical components is usually initiated by defects such as pits, grooves, cracks, absorbing inclusions, scratches, pores, impurities, or material contamination [5–10]. Defects that act as laser damage precursors are inherently stochastically distributed and thus provide an explanation of non-deterministic behavior and damage threshold dependence on laser beam size [6,11]. Larger beams increase the probability that a defect is present within the irradiated area. The defect dominated damage behavior was observed in the long-pulse (nanosecond) regime, in which damage is a consequence of several physical processes involving absorption, heating, phase changes of materials, hydrodynamic processes, and plasma formation. Since nanosecond pulses are relatively long compared to the time scales of these processes, small defect precursors can trigger a cascade of events that can lead to micro-explosion and damage [12,13]. However, if the pulse duration is shorter than the relaxation time, i.e., energy transfer from electrons to an atomic network, which lasts several picoseconds for dielectrics, the processes of excitation and relaxation are decoupled in time [14]. In such an ultrashort regime, the damage is mainly driven by multiphotonic absorption in irradiated material because the other processes cannot be involved within a short pulse duration. Laser damage with sub-ps pulses has therefore a strong nonlinear dependence on intensity, and the damage threshold fluence is deterministic without significant statistical variations, as opposed to nanosecond pulses [15–19]. Evidence of a deterministic damage threshold suggests that damage initiation is given by fundamental intrinsic material properties (energy bandgap, refractive index) rather than by stochastically distributed defects. Thus, if the limiting factor of material damage resistance seems to be the intrinsic material properties, then the laser damage threshold is expected to be independent of laser beam size. This was confirmed in early studies on this topic with fused silica irradiated by 400 fs pulses within beam diameters ranging from 0.4 to 1.0 mm [20]. The fused silica LIDT independent of beam size was confirmed at 100 fs with the number of pulses ranging from 1 to 1000 [21].
However, this concept of damage onset initiated by ultrashort pulses might not be entirely correct. There are experimental studies employing pulses of duration between 30 fs and 1 ps showing damage/ablation thresholds dependent on beam size. These experiments were done on stainless steel [22,23], silicon [22,23], or even dielectric materials (fused silica [24], barium borosilicate glass [25], ion phosphate glass [26], dentin [27], sapphire monocrystal [28], and polystyrene [29]). In [22,24,25,26,29], the beam-size dependence on laser damage was described using defect-site models distinguishing two laser-induced damage regimes: an extrinsic defect-dominated regime for larger beam sizes and intrinsic regime for smaller ones. The defect-site models fitted well the experimental results, even though the nature of defect sites initiating damage remains unclear [25,26,30]. In [23], the effect of material treatment on the damage threshold was studied. Both the ${{\rm AlO}_x}$ slurry treatment on silicon and grit sandpaper treatment on stainless steel led to an increase in defect density, increasing the effect of beam size on LIDT. The effects of defects in the studies in Table 1 are potential explanations, but they are not demonstrated. The role of defects in dentin [27], stainless steel [22,23], polystyrene [29], or slurry treated silicon [23] could be affected by the fact that opaque materials do not show optical quality or sample homogeneity. In addition, multiple shot tests reflect the presence of cumulative effects including laser-induced defects that facilitate the process of electronic excitation, which differs from fundamental interaction of a sub-ps pulse with a monolayer of dielectric material that we aim to study in this work. In the study of the damage threshold on dentin [27], the effect of beam size was significantly affected by repetition rate. At 100 Hz, the ablation threshold was almost independent of beam size, whereas at higher repetition rates, 1 and 500 kHz, the beam size dependence on ablation thresholds was evident, and heat accumulation was proposed as an explanation. Recently, E-beam deposited and PIAD silica thin films together with fused silica were tested by a broad range of picosecond pulse durations (1–60 ps), and results were compared for three sizes of beam waists (30, 50, 100 µm in FWHM) [31]. The results obtained by 1 ps pulses are ambiguous with respect to the beam-size effect. The highest threshold fluences were measured with the smallest beams (30 µm) in all three optical materials, but the thresholds achieved with 50 µm beam waists were lower than the 100 µm ones for both coatings.
Table 1. Review of Some Studies on Beam-Size Effect on Damage Threshold by Ultrashort Pulsesa
A comparison of the abovementioned published results (Table 1) is difficult since the experiments have not been done in the same conditions. The laser parameters differed in pulse duration, laser wavelength, number of pulses, repetition rate, or spot size range. The tests were done on various samples of different properties (material, surface state, fabrication, polishing, cleaning, contamination, prior history, etc.). The study of beam-size dependence in femtosecond range could be also affected by nonlinear effects, which were observed in bulk fused silica below the damage threshold in the case of smaller numerical apertures, i.e., larger beam sizes [33]. Apart from experimental conditions, the laser damage dependence on beam size is a function of the used threshold definition. The results published in a nanosecond study [34] suggest damage threshold independent of beam size if the damage threshold is defined as a fluence of 0% damage probability. However, if the damage threshold is considered as a fluence of 50% damage probability, the effect of beam size is evident. In general, the effect of beam size grows with the increasing probability of damage used for threshold definition. Additionally, the damage threshold results may vary by a few percent because of used calorimeters or beam profiling cameras that have influence on possible deviations of measured pulse energies or beam areas [35].
The motivation of this study is to test laser damage resistance of dielectric materials used in coatings of optical components that determine the limit of reliable operation of high-energy sub-picosecond solid-state lasers [2]. The effect of beam size on the damage threshold is extremely important for qualifying optical components for mass use in high-power lasers. The results in this work should indicate whether small beam sizes can be used for the testing of optical components that will be implemented in high-energy large-beam lasers. The limit to this approach is the role of macroscopic defects, such as nodules, that were evidenced in the sub-ps regime [36]. In that case, only Raster scan testing procedures are relevant for laser damage testing [37]. We will focus on the single shot testing method to study fundamental interaction of a sub-picosecond pulse with a monolayer of dielectric material (${{\rm HfO}_2}$, ${{\rm Nb}_2}{{\rm O}_5}$) and to exclude the complexity of cumulative effects and interference phenomenon in a multilayer stack.
2. DESCRIPTION OF EXPERIMENT AND METHODS
A. Experimental Setup
Laser damage tests were done by using a commercial diode pumped Yb:KYW laser (Amplitude Systemes S-pulse HP). The system emits radiation of a nearly Gaussian spatial profile in the near-infrared wavelength around 1030 nm. The emitted pulses have a pulse duration of $500 \pm 50$ fs, measured by a single shot autocorrelator (AVESTA ASF 70 fs–3 ps, Acore software). The experimental setup is described in Fig. 1.
Fig. 1. Schematic drawing of sub-ps near-infrared LIDT station. He–Ne laser is intended for station alignment. SHTR, shutter; HR1/HR2, high reflective flip-flop mirrors; HWP, half-wave plate; TFP, thin-film polarizer; BS, beam splitter; PY1/PY2, pyroelectric detectors; PR-ND, partially reflective at 1030 nm and neutral density filters; LENS, focusing lens; S, sample; BP, beam profiling camera; SSA, single shot autocorrelator. More details are given in Section 2.A.
We used an Yb:KYW laser at a 10 Hz repetition rate. The maximum pulse energy was 1 mJ. The single shot mode was achieved using a mechanical shutter (SHTR, Thorlabs SH05). The pulse energy was adjusted by a zeroth-order half-wave plate (HWP), mounted on a motorized rotation stage, and a thin-film polarizer (TFP). The beam that passed through the polarizer falls on the beam splitter (BS), which directs a small part of pulse energy (5%) to a pyroelectric energy meter (OPHIR PE9) recording the energy of each pulse. The energy meter is calibrated to the energy incident on the tested sample (S), which was measured using a second pyroelectric meter (PY2, OPHIR PE9F) placed behind a focusing lens (LENS).
Fig. 2. Effective beam diameter (${d_{{\rm eff}}}$) and ${3\times }$ standard deviation ($3\sigma$) of effective area as a function of lens position, with typical normalized beam profiles at different positions. For $z \lt 0$, the camera is close to the focusing lens (before the focal plane). Comparison of lenses with different focal lengths: (a) ${f_{30}} = 30\;{\rm cm} $, ${d_{{\rm eff,min}}} \approx 40\; {\unicode{x00B5}{\rm m}}$; (b) ${f_{15}} = 15\;{\rm cm} $, ${d_{{\rm eff,min}}} \approx 86\; {\unicode{x00B5}{\rm m}}$.
The beam used for LIDT testing is linearly polarized and focused by a plano–convex lens on the tested sample (S), which was placed at a 45° incidence angle. LIDT testing was performed in ambient air at room temperature. Positioning of the tested sample near the focal plane is done using a motorized 2D translation stage. The laser damage station is equipped with a He–Ne laser, which is used for beam alignment.
The focused laser radiation of pulse energy, reduced by six to seven orders of magnitude using a combination of partial reflection at 1030 nm and neutral density filters (PR-ND), was analyzed by a beam profiling camera (BP) connected to an imaging software. The camera was placed with the sample (S) (see Fig. 1), and its sensor was oriented perpendicularly to the beam direction. In this study, two focusing plano–convex lenses with focal lengths of 30 cm and 15 cm were used separately. Both lenses had 25 mm diameter and were AR coated. Examples of measured beam profiles at different lens positions for both lenses are shown in Fig. 2. For $z \lt 0$, the camera is close to the focusing lens (before the focal plane).
B. Tested Samples
The tested samples were monolayers of ${{\rm HfO}_2}$ and ${{\rm Nb}_2}{{\rm O}_5}$. The ${{\rm HfO}_2}$ sample of 150 nm thickness was deposited by electron-beam evaporation with ion assistance on BK7 substrate. The refractive index was 1.93 determined at 1053 nm with spectrophotometry [38]. The ${{\rm Nb}_2}{{\rm O}_5}$ monolayer was deposited on fused silica substrate with a magnetron sputtering process controlled by the HELIOS system [39]. The refractive index was determined by spectrophotometry to be 2.26 at 1030 nm wavelength. The thicknesses of tested ${{\rm Nb}_2}{{\rm O}_5}$ layers were 150 and 450 nm.
C. Beam-Size Measurement
Particular attention in this study was given to the determination of beam size with its statistical deviation dependent on lens position. The beam size is expressed using the effective diameter (${d_{{\rm eff}}}$), defined using the square root of the effective area (${A_{{\rm eff}}}$) divided by $\pi$: [40]
For a beam with a transverse profile of optical intensity described using a Gaussian function, the diameter at $1/{e^2}$ peak intensity is ${d_{1/{e^2}}} = \sqrt 2 \cdot {d_{{\rm eff}}}$. The effective area is obtained by the ratio of pulse energy $E$ and maximum energy density (${F_{{\rm max}}}$) of the laser pulse in the target plane, i.e., [40]For both used lenses, the beam profiles were measured at discrete lens positions with a maximum lens position step of 0.5 mm. The beam profile after a lens of 30 cm focal length [see Fig. 2(a)] was measured by a WinCam UCD23 camera (DataRay Inc.) with a CCD sensor of 6.45 µm pixel length. In the case of a lens with 15 cm focal length, the beam profiles were analyzed by two different cameras: the WinCam UCD23 and BP87 (Femto Easy), whose parameters are listed in Table 2. The results recorded by these cameras were analogous, but those of BP87 were preferred because the used CMOS sensor provided higher lateral resolution due to its 3.45 µm pixel length, which was important for the smallest beam around the focal plane. Another advantage of the BP87 beam profiler was its lower noise in comparison to the WinCam. The statistical results [see Fig. 2(b)] were derived from 100 frames per one lens position.
Table 2. Parameters of Beam Profilers
Measurement of the effective area dependent on lens position allowed us to determine accurately the focal plane corresponding to the lens position with the minimum of the effective area. The obtained data points of the effective beam area were then linearly interpolated between each measured point to determine subsequently the effective beam areas of specific lens positions corresponding to the LIDT tests.
The values of three standard deviations ($3\sigma$) from a mean were calculated using the formula
D. Laser Stability
The accuracy of the damage threshold depends on laser stability. Instabilities in the temporal or spatial beam profile can affect the damage threshold and lead to erroneous results [41]. The stability parameters of the used near-infrared LIDT station were measured, and their $3\sigma$ deviations are summarized in Table 3. Due to the low variations in pulse energy, pulse duration, and beam size, the station enables to perform laser damage tests with high accuracy and limits the errors in measurement.
Table 3. Variations of Laser Stability Parameters, Expressed in Pulse-to-Pulse $3\sigma$ Deviationsa
E. Damage Test Procedure
The LIDT test consisted of a procedure adapted to the study of the beam-size effect on the damage threshold. The different beam sizes were achieved by focal lens positioning on the motorized stage. At a given lens position, the sample was irradiated at different spots with unique pulse energies that were changed with ${\sim}1\%$ energy increment. The damage threshold was then determined as an average between the lowest fluence with a damaged spot and the highest fluence with zero probability of damage. The LIDT results on both ${{\rm Nb}_2}{{\rm O}_5}$ and ${{\rm HfO}_2}$ were deterministic.
F. Damage Detection
The laser damage was detected in situ by optical microscopy with a ${20\times }$ magnification of the objective mounted on a BXFM Olympus microscope. The technique allows real time estimation of the irradiated sample surface state. After the LIDT testing, an ex situ damage inspection was performed using a Zeiss Axiotech differential interference contrast microscope with an objective of ${20\times}$ magnification. The ex situ observation technique was preferred for the determination of damage threshold results presented in this work.
G. Fluence Evaluation
The ${F_{{\rm ext}}}$ external energy density (or fluence) at the laser induced damage threshold is obtained by dividing $E$ pulse energy by ${A_{{\rm eff}}}$ effective area of the beam [see Eq. (2)], as defined in the international standards: [40]
The relative uncertainty of the damage threshold fluence is thus obtained from the uncertainties of pulse energy and effective area as
In this work, we express the inaccuracies using $3\sigma$ values.
H. Intrinsic LIDT Fluence
Since the optical layers are the scene of interferential effects, distribution of the electric field inside a layer irradiated by a laser is not homogeneous. Electric field distribution is critical for understanding the sub-ps LIDT results because the excitation of dielectrics is governed by electronic processes [42]. To compare LIDT results with different conditions having an influence on electric field distribution, e.g., angle of incidence, polarization, layer thickness, or refractive index, it is necessary to rescale the LIDT results with the electric field intensity (EFI) maximum ($\textit{EFI}_{{{\rm max}}}$) within the given layer. Therefore, the fluence values reported in this study correspond to ${F_{{\rm int}}}$ intrinsic fluence determined using ${F_{{\rm ext}}}$ external fluence and the $\textit{EFI}_{{{\rm max}}}$:
Fig. 3. Distribution of electric field intensity (EFI) inside ${{\rm Nb}_2}{{\rm O}_5}$ layer of 450 nm thickness (refractive index 2.26 at 1030 nm). Fused silica substrate (FS, refractive index 1.45 at 1030 nm). Polarization P, angle of incidence 45°. The EFI is normalized to the incident electric field amplitude in air.
Fig. 4. LIDT results of ${{\rm Nb}_2}{{\rm O}_5}$ sample tested with lens of ${f_{15}} = 15\;{\rm cm} $ focal length. (a) Fitting of LIDT pulse energy dataset to the ${A_{{\rm eff}}}$ effective area curve using a linear relationship between them; (b) influence of shift in lens position by 0.1 mm to the intrinsic LIDT fluence with respect to effective beam radius. ${z_0}$ means one specific lens position; $l$ is distance from lens to the surface of tested sample.
I. Match LIDT Pulse Energies to Effective Areas
Since the effective area was measured at discrete lens positions before or after the LIDT tests, it was needed to match the effective area data to the LIDT energies. To do that, we first determined the $z$ lens position coordinate, for which the effective beam area was smallest ($z = 0$). Knowing the effective areas before ($z \lt 0$) and after ($z \gt 0$) the lens focal length, we described the evolution of effective areas in both directions from the waist. Then we tried to shift the data of LIDT pulse energies to correspond well to the evolution of effective beam areas as shown in Fig. 4. The results clearly show high sensitivity of determined fluences on the lens position shift and should be considered as a significant source of fluence inaccuracy in this work.
3. EXPERIMENTAL RESULTS AND DISCUSSION
A. Results with 30 cm Focal Length
The LIDT testing by a lens with 30 cm focal length was repeated three times for the ${{\rm HfO}_2}$ sample. The results of tests 1 and 2 were evaluated using the preferred ex situ DIC microscopy, while those of test 3 correspond to in situ damage detection. The LIDT results in Fig. 5 underline the critical effect of damage detection on LIDT determination. In our case, it adds an offset that seems consistent. Figure 5(a) illustrates the damage threshold pulse energies together with effective area values dependent on lens position. Both effective area and damage threshold energies indicate similar dependence on increasing distance from the focal plane. The behavior can be evidenced by independence of intrinsic LIDT fluence on the beam size as shown in Fig. 5(b). The small deviations (${\lt}10\%$) for larger effective beam radii (${\gt}65\,{\unicode{x00B5}{\rm m}}$) could be connected with beam divergence influencing the angle of incidence and thus EFI maxima. Also the real beam size at the spot on the tested sample can be different since the sample was inclined at 45°, and beam position slightly shifts, dependent on lens position. However, the observed deviations of intrinsic LIDT fluences are still in compliance with the shown error bars summarizing $3\sigma$ deviation of the effective beam area, $3\sigma$ deviation of pulse energy, and uncertainty given by ${\sim}1\%$ energy increment in the damage test procedure. For this measurement, we thus do not see a significant beam-size effect on intrinsic LIDT fluence.
Fig. 5. Summary of LIDT results with ${{\rm HfO}_2}$ sample tested by lens of ${f_{30}} = 30\;{\rm cm} $ focal length. (a) LIDT energy and effective area as a function of lens position. The two $y$ axes are linked by linear scaling law. (b) Intrinsic LIDT fluence dependent on effective beam radius. Tests 1 and 2 were evaluated using the preferred ex situ microscopy. Test 3 corresponds to the in situ damage detection. Length $l$ means distance from lens to the tested sample surface.
B. Results with 15 cm Focal Length
The beam-size effect on LIDT fluence was studied also with lens of 15 cm focal length. The results for ${{\rm Nb}_2}{{\rm O}_5}$ and ${{\rm HfO}_2}$ samples are shown in Figs. 6(a) and 6(b), respectively. In contrast to the previous results with the lens of 30 cm focal length (Fig. 5), the interpretation of LIDT fluences dependent on beam size is difficult in the case of a lens with 15 cm focal length since we observe differences of threshold values up to 20% around focal plane. Also, for large beam sizes, we see differences in LIDT values between tests performed with lens-sample distances smaller and larger than the focal length. In the following sections, we shall analyze the possible cause of LIDT deviation when changing the spot size.
Fig. 6. Summary of intrinsic LIDT results obtained using the lens of ${f_{15}} = 15\;{\rm cm} $ focal length. (a) Intrinsic LIDT of two ${{\rm Nb}_2}{{\rm O}_5}$ coatings of different thicknesses ($\nabla$ 150 nm, $\Delta$ 450 nm) obtained in two different LIDT test campaigns; (b) intrinsic LIDT fluence of ${{\rm HfO}_2}$ sample determined in two different LIDT test campaigns. The $\Delta$ and $\diamondsuit$ datasets were obtained with a very accurately aligned beam whose maximal shift of peak caused by lens positioning was around 30 µm. In the case of $\nabla$ and tests, it was around 400 µm.
C. Analysis of Potential Self-Focusing Effects in Air
Since sub-picosecond systems have high peak powers of pulses, they can create conditions for nonlinear effects that can modify the beam profile. The important phenomenon that can introduce errors in the damage testing is self-focusing. The evaluation of self-focusing for Gaussian beams is possible by estimating self-focusing power [44]:
where ${n_0}$ signs for a linear refractive index, and ${n_2}$ is the nonlinear refractive index of air, defined by $n = {n_0} + {n_2}I$, where $I$ means intensity.For femtosecond pulses (${\le} 200\;{\rm fs} $) at 800 nm wavelength, the nonlinear refractive index ${n_2}$ of air can be found in several publications [45–48], in which its value ranges between ${10^{- 23}}$ and $6 \cdot {10^{- 23}} \;{{\rm m}^2}/{\rm W}$, dependent on wavelength, pulse duration, or refractive index measurement method [49]. In 2014, Mitrofanov et al. [50] determined the nonlinear refractive index of air to be ${n_2} \sim 5 \cdot {10^{- 23}}\;{{\rm m}^2}/{\rm W}$ at 1030 nm wavelength, 200 fs pulse durations, which are parameters close to the irradiation conditions of our LIDT setup (540 fs pulse duration, 1030 nm wavelength). Substituting the value in Eq. (7), the self-focusing power for our setup is ${P_{\rm{SF}}} \sim 3.2 \;{\rm GW}$, which is two times larger than the highest used peak power of 1.6 GW, corresponding to pulse energy of 0.85 mJ. Thus, the beam propagation should not be exposed to the self-focusing phenomenon in air. In addition, this question is relevant only in the case of LIDT testing at sample-lens distances larger than focal length with pulse energies close to our maximum, i.e., for the largest beam sizes.
D. Self-Focusing Effects in the Lens
A potential cause of the evolution of LIDT with a spot size could be self-focusing in the lens material. If our results are affected by this, the influence is largest for the highest pulse energies that correspond to the farthest lens positions from the focus. For sample-lens distances closer than focal length ($l \lt {f_{15}}$), the effect can cause more intense focusing and lower damage threshold energy. Since the beam profile measurement was performed with pulse energies reduced by six to seven orders of magnitude compared to the LIDT tests, the determined effective areas may not correspond to the real ones affected by self-focusing. The effect could thus be interpreted as a decrease in intrinsic LIDT fluence for the lowest sample-lens distances ($z \lt 0$) as shown in Fig. 7.
Fig. 7. Intrinsic LIDT fluence results for (a) ${{\rm Nb}_2}{{\rm O}_5}$ and (b) ${{\rm HfO}_2}$. The results are the same as in Fig. 6, but here they are plotted as a function of $z$ lens position.
More intense focusing can thus lead to a shift in the focal plane to shorter sample-lens distances but also to faster defocusing, dependent on $z$ position. Therefore, the LIDT thresholds could be higher for the largest sample-lens distances ($z \gt 0$), which correlates with the results in Fig. 7.
For the effective beam diameters in the range from 75 µm to 175 µm (see Fig. 6) the more pronounced discrepancy between the damage thresholds before and after the focal length in the case of the ${{\rm HfO}_2}$ sample than in the ${{\rm Nb}_2}{{\rm O}_5}$ tests correlates with the higher pulse energies in the ${{\rm HfO}_2}$ tests. This also suggests a potential self-focusing effect since the damage threshold energies of ${{\rm HfO}_2}$ were approximately three times larger for the same beam diameter.
E. Effect of Beam Divergence
In the LIDT determination procedure, it is assumed that a plane wave propagates in the sample to calculate the electric field distribution. Because of the Gaussian nature of the laser beam, this is not the case when LIDT tests are performed out of the focal plane. We have therefore tried to estimate the consequences on EFI calculation.
Assuming that our beam is close to the Gaussian beam profile [see Fig. 2(b)] and characterized with a certain ${{\rm M}^2}$ factor, we can calculate the $\theta$ divergence half-angle using the relation
where ${w_0}$ stands for $1/{{\rm e}^2}$ beam radius at the beam waist. The meaning of a half-divergence angle is that the angle from the beam axis should cover 86.5% of the pulse energy in the case of an ideal Gaussian beam in the far field from the focal plane. It could thus be assumed that the half-divergence angle would also define the range of incidence angles and positions where the material damage is initiated. For our beam, the half-divergence angle of the lens with a focal length of 15 cm is around 0.7°. Performing the calculation of EFI maxima for different incidence angles around 45°, we estimate the maximum difference in EFI maxima to be 1.4%. Thus, we cannot explain the observed differences in LIDT fluences using the beam divergence determined by the Gaussian beam approximation. However, the difference in EFI maxima could be added into the error bars when the sample plane is far from focus.F. Alignment
During the LIDT test, we changed the positions of the focusing lens along the beam axis. As the lens was moving step by step from one extreme position to another, there was a gradual movement of the beam peak in the plane perpendicular to the beam propagation. This movement occurred in both horizontal and vertical coordinates and was recorded by a beam profiling camera. Since the LIDT tests were done at a 45° incidence angle (see Fig. 8) the change of peak position in the horizontal plane can be projected also along the beam axis and thus influence the lens-sample distance.
Assuming approximately the same distance changes along the beam axis as in horizontal coordinates ($\Delta x \approx \Delta z$), we can estimate the error in LIDT fluence by expressing the ratio between effective areas located at lens positions of $\Delta z$ difference. We estimate the effective area (${A_{{\rm eff}}}$) difference as
The move of peak position indicates that there could also be some influence on the incidence angle (and EFI maximum) when the lens is moved. We can estimate the incidence angle change as
In the case of $\Delta x = 400\,\,{\unicode{x00B5}{\rm m}}$, $\Delta \theta \sim {0.15^ \circ}$ corresponds to a difference in EFI maxima ${\le} 0.3 \%$. Thus, the influence of beam positioning on EFI maxima does not play an important role. However, the effective area differences due to beam displacement calculated using Eq. (9) could be used for explanation of the differences between the LIDT test campaigns presented in Fig. 6.
G. Camera Errors
1. Noise Error
The BP87 camera shows a very low noise level around 0.1% of signal maximum. Furthermore, several beam profile measurements with different background area selections confirmed the same LIDT results. Thus, we do not consider the noise error as important.
2. Pixel-Size Error
The effective area determination is limited by the spatial resolution of the beam profiler given by the pixel size [51]. In the tests with a lens of 15 cm focal length, the pixel size ${l_{{\rm pixel}}} = 3.45\,\,{\unicode{x00B5}{\rm m}}$. Taking into account an absolute error of effective beam diameter determination, i.e., $\delta {d_{{\rm eff}}} = \pm {l_{{\rm pixel}}}$, we assess the relative error: [51]
The problem of spatial resolution is a serious issue when beam profilers are used to measure relatively small spots. For our lens with 15 cm focal length, the minimum effective beam diameter ${d_{{\rm eff}}} \approx 40\,{\unicode{x00B5}{\rm m}}$ corresponds to the relative error ${\epsilon _{{\rm pixel}}} \approx 17 \%$. This error could be one of the main reasons that the LIDT results in Figs. 6(a) and 6(b) show such high dispersion for smaller beam sizes.
3. Maximum Pixel Intensity Error
We estimate an error of maximum intensity value measured on one pixel of our camera by considering the difference between two extreme cases of maximum pixel positioning, i.e., the case of peak at the center of the pixel [Fig. 9(a)] and the case when maximum intensity is at the corner of the pixel [Fig. 9(b)]. Assuming a Gaussian intensity profile, we can determine $h$, the mean value of intensity within the ${S_{{\rm pixel}}}$ pixel area, by
Fig. 9. Schematic drawing of Gaussian beam intensity peak: (a) at the center of pixel and (b) at the corner of pixel.
4. Camera Contrast Error
The error resulting from the discrete irradiance levels can be assessed as [51]
where ${\tilde V_{\textit{xy}}}$ represents the average value of ${V_{\textit{xy}}}$ voltages inside the software aperture, expressed in digitized gray levels that are proportional to the light energy collected by each pixel of the array. We found ${\epsilon _{{\rm contrast}}} \approx 0.03 \%$.5. Camera Linearity Error
The nonlinearity error is expressed as [51]
where ${2^{(DR)}}$ means the dynamic range of the camera represented by the number of gray levels. We found ${\epsilon _{{\rm lin}}} \approx 0.01 \%$. Thus, the camera linearity and contrast errors are negligible.H. Other Errors
Other errors in measurement might include the accuracy of motorized stage movements or the accuracy of microscope observation. The latter we estimate to be around low percentage units.
4. CONCLUSION
In our particular case, the LIDT results, obtained by a lens with 30 cm focal length in the range of effective beam diameters between 80 and 160 µm, show that the sub-picosecond damage threshold of dielectric coatings is independent of beam size as shown in Fig. 5(b). To evaluate the tests of such optical components as accurately as possible, we provide in Table 4 a synthesis of identified contributors to errors. As the major error contributor in the best case scenario, we detected $3\sigma$ variations of beam size (6%). The influence of contributors related to beam alignment, i.e., beam positioning at a 45° incidence angle, pulse energies and effective areas matching, beam divergence, or effect of the incidence angle on EFI maximum, could be minimized by damage testing at normal incidence or by beam profile measurement at the same incidence angle as the tests are performed.
Table 4. Synthesis of Error Margins for Identified Contributors in the Best Case Scenario of LIDT Tests with Lens of 30 cm Focal Lengtha
In contrast to the results with a lens of 30 cm focal length, it is difficult to determine the relation between beam size and LIDT fluence in the case of a lens with a smaller focal length of 15 cm. The different fluences before and after the beam waist (Fig. 7) could suggest potential beam deformation related to self-focusing in the lens. Another error could rise from beam profile measurements, especially for smaller beam sizes. This could be related to pixel size error of 17% or maximum pixel intensity error of 1.5 %. Last but not least, it is necessary to emphasize the significant effect of the chosen shift in lens positions to the intrinsic LIDT fluence in respect to the effective beam diameters; see Fig. 4.
From a practical point of view, this study recommends in our case the lens of 30 cm focal length to be used for LIDT testing of optical components intended, e.g., for use in larger beam laser systems. The lens of 15 cm focal length, in contrast, should not be used for damage testing since the uncertainties in LIDT fluence, regardless of their nature, are too large.
From a more general perspective, this work underlines the difficulty of LIDT measurements with very focused laser beams. Despite our best efforts, the deviations of LIDT are quite large, and we believe similar issues should have been encountered in previous studies related to this topic (see Table 1) where spot size dependences were observed for highly focused beams.
Funding
European Union’s Horizon 2020 research and innovation programme, under Marie Skłodowska-Curie grant agreement no. (813159).
Acknowledgment
We thank L. Lamaignère for fruitful discussions and ideas about the work.
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.
REFERENCES
1. M. Soileau, “Laser-induced damage phenomena in optics: a historical overview,” in Laser-Induced Damage in Optical Materials, D. Ristau, ed. (CRC Press, 2014), pp. 3–8.
2. W. Koechner, “Damage of optical elements,” in Solid-State Laser Engineering, Springer series in optical sciences (Springer, 2006), Vol. 1, pp. 680–701.
3. D. Bäuerle, “Introduction,” in Laser Processing and Chemistry (Springer, 2011), pp. 3–12.
4. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008). [CrossRef]
5. N. Bloembergen, “Role of cracks, pores, and absorbing inclusions on laser induced damage threshold at surfaces of transparent dielectrics,” Appl. Opt. 12, 661–664 (1973). [CrossRef]
6. L. G. DeShazer, B. E. Newnam, and K. M. Leung, “Role of coating defects in laser-induced damage to dielectric thin films,” Appl. Phys. Lett. 23, 607–609 (1973). [CrossRef]
7. N. L. Boling and G. Dubé, “Laser-induced inclusion damage at surfaces of transparent dielectrics,” Appl. Phys. Lett. 23, 658–660 (1973). [CrossRef]
8. R. H. Picard, D. Milam, and R. A. Bradbury, “Statistical analysis of defect-caused laser damage in thin films,” Appl. Opt. 16, 1563–1571 (1977). [CrossRef]
9. J. Reif, S. Petzoldt, A. P. Elg, and E. Matthias, “The role of defects in laser surface damage thresholds of fluoride crystals,” Appl. Phys. A 49, 199–204 (1989). [CrossRef]
10. F. E. Hovis, B. A. Shepherd, C. T. Radcliffe, A. L. Bailey, and W. T. Boswell, “Optical damage at the part per million level: the role of trace contamination in laser-induced optical damage,” Proc. SPIE 2114, 145–153 (1994). [CrossRef]
11. J.-Y. Natoli, L. Gallais, H. Akhouayri, and C. Amra, “Laser-induced damage of materials in bulk, thin-film, and liquid forms,” Appl. Opt. 41, 3156–3166 (2002). [CrossRef]
12. G. Duchateau, M. D. Feit, and S. G. Demos, “Strong nonlinear growth of energy coupling during laser irradiation of transparent dielectrics and its significance for laser induced damage,” J. Appl. Phys. 111, 093106 (2012). [CrossRef]
13. Y. Xu, D. H. Dunlap, L. A. Emmert, and W. Rudolph, “Laser-driven detonation wave in hafnium oxide film: defect controlled laser damage and ablation,” J. Appl. Phys. 128, 123101 (2020). [CrossRef]
14. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996). [CrossRef]
15. D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64, 3071–3073 (1994). [CrossRef]
16. A. Joglekar, H. Liu, G. Spooner, E. Meyhöfer, G. Mourou, and A. Hunt, “A study of the deterministic character of optical damage by femtosecond laser pulses and applications to nanomachining,” Appl. Phys. B 77, 25–30 (2003). [CrossRef]
17. B. Mangote, L. Gallais, M. Zerrad, F. Lemarchand, L. H. Gao, M. Commandré, and M. Lequime, “A high accuracy femto-/picosecond laser damage test facility dedicated to the study of optical thin films,” Rev. Sci. Instrum. 83, 013109 (2012). [CrossRef]
18. M. Garcia-Lechuga, G. Gebrayel El Reaidy, H. Ning, P. Delaporte, and D. Grojo, “Assessing the limits of determinism and precision in ultrafast laser ablation,” Appl. Phys. Lett. 117, 171604 (2020). [CrossRef]
19. B. Zhou, A. Kar, M. J. Soileau, and X. Yu, “Invariance of the r2-ln(F) relationship and attainable precision in ultrafast laser ablation experiments,” Opt. Express 29, 5635 (2021). [CrossRef]
20. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Optical ablation by high-power short-pulse lasers,” J. Opt. Soc. Am. B 13, 459–468 (1996). [CrossRef]
21. A. Rosenfeld, M. Lorenz, R. Stoian, and D. Ashkenasi, “Ultrashort-laser-pulse damage threshold of transparent materials and the role of incubation,” Appl. Phys. A 69, S373–S376 (1999). [CrossRef]
22. O. Armbruster, A. Naghilou, M. Kitzler, and W. Kautek, “Spot size and pulse number dependence of femtosecond laser ablation thresholds of silicon and stainless steel,” Appl. Surf. Sci. 396, 1736–1740 (2017). [CrossRef]
23. A. Naghilou, O. Armbruster, and W. Kautek, “Femto- and nanosecond pulse laser ablation dependence on irradiation area: the role of defects in metals and semiconductors,” Appl. Surf. Sci. 418, 487–490 (2017). [CrossRef]
24. N. Sanner, B. Bussiere, O. Utéza, A. Leray, T. Itina, M. Sentis, J. Y. Natoli, and M. Commandré, “Influence of the beam-focus size on femtosecond laser-induced damage threshold in fused silica,” Proc. SPIE 6881, 68810W (2008). [CrossRef]
25. S. Martin, A. Hertwig, M. Lenzner, J. Krüger, and W. Kautek, “Spot-size dependence of the ablation threshold in dielectrics for femtosecond laser pulses,” Appl. Phys. A 77, 883–884 (2003). [CrossRef]
26. A. Hertwig, S. Martin, J. Krüger, and W. Kautek, “Interaction area dependence of the ablation threshold of ion-doped glass,” Thin Solid Films 453-454, 527–530 (2004). [CrossRef]
27. B.-M. Kim, M. D. Feit, A. M. Rubenchik, E. J. Joslin, J. Eichler, P. C. Stoller, and L. B. Da Silva, “Effects of high repetition rate and beam size on hard tissue damage due to subpicosecond laser pulses,” Appl. Phys. Lett. 76, 4001–4003 (2000). [CrossRef]
28. O. Uteza, B. Bussiere, F. Canova, J.-P. Chambaret, P. Delaporte, T. Itina, and D. Zeitoun, “Laser-induced damage threshold of sapphire in nanosecond, picosecond and femtosecond regimes,” Appl. Surf. Sci. 254, 799–803 (2007). [CrossRef]
29. A. Naghilou, O. Armbruster, M. Kitzler, and W. Kautek, “Merging spot size and pulse number dependence of femtosecond laser ablation thresholds: modeling and demonstration with high impact polystyrene,” J. Phys. Chem. C 119, 22992–22998 (2015). [CrossRef]
30. O. Armbruster, A. Naghilou, and W. Kautek, “The role of defects in pulsed laser matter interaction,” in Advances in the Application of Lasers in Materials Science, , P. M. Ossi, ed., Springer series in materials science; excellent LID introduction (Springer, 2018), Vol. 274, pp. 39–61.
31. T. A. Laurence, R. A. Negres, S. Ly, N. Shen, C. W. Carr, D. A. Alessi, A. Rigatti, and J. D. Bude, “Role of defects in laser-induced modifications of silica coatings and fused silica using picosecond pulses at 1053 nm: II. Scaling laws and the density of precursors,” Opt. Express 25, 15381–15401 (2017). [CrossRef]
32. D. Ashkenasi and A. Rosenfeld, “Material processing of dielectrics with femtosecond lasers,” Proc. SPIE 3618, 102–113 (1999). [CrossRef]
33. J. B. Ashcom, R. R. Gattass, C. B. Schaffer, and E. Mazur, “Numerical aperture dependence of damage and supercontinuum generation from femtosecond laser pulses in bulk fused silica,” J. Opt. Soc. Am. B 23, 2317–2322 (2006). [CrossRef]
34. S. R. Foltyn, “Spotsize effects in laser damage testing,” in Laser Induced Damage in Optical Materials (1982), pp. 368–379.
35. L. Lamaignère, M. Balas, R. Courchinoux, T. Donval, J. C. Poncetta, S. Reyné, B. Bertussi, and H. Bercegol, “Parametric study of laser-induced surface damage density measurements: toward reproducibility,” J. Appl. Phys. 107, 023105 (2010). [CrossRef]
36. L. Gallais, X. Cheng, and Z. Wang, “Influence of nodular defects on the laser damage resistance of optical coatings in the femtosecond regime,” Opt. Lett. 39, 1545–1548 (2014). [CrossRef]
37. M. Sozet, J. Néauport, E. Lavastre, N. Roquin, L. Gallais, and L. Lamaignère, “Laser damage density measurement of optical components in the sub-picosecond regime,” Opt. Lett. 40, 2091–2094 (2015). [CrossRef]
38. L. Lamaignère, A. Ollé, M. Chorel, N. Roquin, A. A. Kozlov, B. N. Hoffman, J. B. Oliver, S. G. Demos, L. Gallais, R. A. Negres, and A. Melninkaitis, “Round-robin measurements of the laser-induced damage threshold with sub-picosecond pulses on optical single layers,” Opt. Eng. 60, 031005 (2020). [CrossRef]
39. M. Scherer, “Magnetron sputter-deposition on atom layer scale,” Vakuum in Forschung und Praxis 21, 24–30 (2009). [CrossRef]
40. “Lasers and laser-related equipment–test methods for laser-induced damage threshold–part 1: definitions and general principles,” tech. rep. ISO 21254-1:2011 (International Organization for Standardization, 2011).
41. A. Ollé, J. Luce, N. Roquin, C. Rouyer, M. Sozet, L. Gallais, and L. Lamaignère, “Implications of laser beam metrology on laser damage temporal scaling law for dielectric materials in the picosecond regime,” Rev. Sci. Instrum. 90, 073001 (2019). [CrossRef]
42. L. Emmert and W. Rudolph, “Femtosecond laser-induced damage in dielectric materials,” in Laser-Induced Damage in Optical Materials, D. Ristau, ed. (CRC Press, 2014), pp. 127–152.
43. K. Ohta and H. Ishida, “Matrix formalism for calculation of electric field intensity of light in stratified multilayered films,” Appl. Opt. 29, 1952–1959 (1990). [CrossRef]
44. A. V. Smith and B. T. Do, “Bulk and surface laser damage of silica by picosecond and nanosecond pulses at 1064 nm,” Appl. Opt. 47, 4812–4832 (2008). [CrossRef]
45. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N_2, and O_2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–660 (1997). [CrossRef]
46. Y. E. Geints, A. M. Kabanov, A. A. Zemlyanov, E. E. Bykova, O. A. Bukin, and S. S. Golik, “Kerr-driven nonlinear refractive index of air at 800 and 400 nm measured through femtosecond laser pulse filamentation,” Appl. Phys. Lett. 99, 181114 (2011). [CrossRef]
47. A. Börzsönyi, Z. Heiner, A. Kovács, M. P. Kalashnikov, and K. Osvay, “Measurement of pressure dependent nonlinear refractive index of inert gases,” Opt. Express 18, 25847–25854 (2010). [CrossRef]
48. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components: erratum,” Opt. Express 18, 3011–3012 (2010). [CrossRef]
49. J. Schwarz, P. Rambo, M. Kimmel, and B. Atherton, “Measurement of nonlinear refractive index and ionization rates in air using a wavefront sensor,” Opt. Express 20, 8791–8803 (2012). [CrossRef]
50. A. V. Mitrofanov, A. A. Voronin, D. A. Sidorov-Biryukov, G. Andriukaitis, T. Flöry, A. Pugžlys, A. B. Fedotov, J. M. Mikhailova, V. Y. Panchenko, A. Baltuška, and A. M. Zheltikov, “Post-filament self-trapping of ultrashort laser pulses,” Opt. Lett. 39, 4659–4662 (2014). [CrossRef]
51. A. Stratan, A. Zorila, L. Rusen, and G. Nemes, “Measuring effective area of spots from pulsed laser beams,” Opt. Eng. 53, 122513 (2014). [CrossRef]
