3.1 Regression analysis of the relationship between the photothermal weak absorptions and the LIDTs based on machine learning

The photothermal weak absorption and the LIDT of NYCF crystals were measured using the proposed system to elucidate the mapping relationship between these parameters. Previous work^[ Reference Lu, Zhang, Jiang, Kou, Zhang, Xu, Zhao, Wu and Su^20] indicated that the relationship between photothermal weak absorption and LIDTs on CaF₂ crystal surfaces followed a Chapman function trend. In contrast, the dataset collected in this study is considerably richer, resulting in fitted data that exhibit substantial dispersion. In fact, when handling a large dataset, the limited fitting parameters of the traditional Chapman function become insufficient to capture all the corresponding relationships and nuances between photothermal weak absorption and LIDTs^[ Reference Williams and Rasmussen^21]. Consequently, this approach does not reliably support the non-destructive assessment of the surface damage threshold of CaF₂ crystals. Compared with relying on a specific functional form, machine learning algorithms offer a more robust nonlinear fitting tool for analyzing variable mapping relationships under complex trends^[ Reference Breiman^22, Reference Tipping^23].

From a machine learning perspective, the mapping between photothermal weak absorption values and LIDTs represents a regression task. Given that the inputs to the machine learning architecture are characterized by diverse data types, intricate nonlinear relationships and supplementary physical prior knowledge, a fully connected neural network (FCNN) – a classical algorithm commonly employed for regression tasks – was selected as the architecture of choice. Figure 4(a) presents a schematic diagram of the FCNN architecture used to establish the mapping between photothermal weak absorptions and LIDTs in an NYCF crystal (111) sample (NYCF-2). The network accepts the experimentally measured photothermal weak absorption values as input and, through iterative computation, outputs a predicted LIDT. A mean squared error (MSE) function, defined between these predicted LIDTs and the experimental LIDT measurements, serves as the loss function. Previous studies^[ Reference Williams and Rasmussen^21] have indicated an exponential relationship between photothermal weak absorption and LIDTs; therefore, the data were augmented via a logarithmic transformation^[ Reference Shorten and Khoshgoftaar^24] to reduce the complexity of network training. The hidden layer dimensions were configured as 128, 64, 32 and 16, respectively. It is worth noting that the number of computational layers plays a critical role in determining prediction outcomes. Although increasing the depth of these layers can reduce training loss and yield relatively accurate predictions on the dataset, overly deep architectures run the risk of overfitting. Consequently, indefinitely increasing the number of layers does not necessarily lead to enhanced prediction accuracy. A total of 200 training samples were employed, with an initial learning rate of 0.0005. The Adam optimizer was used to compute gradients and update network parameters. The network implementation was carried out using PyTorch and accelerated with a GeForce RTX 3060 GPU.

Figure 4

(a) Schematic of the FCNN for predicting photothermal weak absorption-LIDT mapping. (b) Predicted LIDT results from models trained on single-crystal CaF₂ (111) axes versus experimental data. (c) Removal effect of strong photothermal absorption points after laser irradiation.

Figure 4(b) displays the relationship between the LIDT values of NYCF crystal (111) predicted by the neural network model and the scatter plot data of damage thresholds obtained through empirical testing. The curve indicates that the mapping from photothermal weak absorption to the LIDTs, as predicted by the neural network, does not entirely correspond with the Chapman function. Specifically, in the high photothermal weak absorption region (above 100 parts per million (ppm)), the LIDTs do not continue to decrease indefinitely; rather, they tend to stabilize. Furthermore, there were some distortion points with relatively high LIDT test values. This observation is primarily attributed to the fact that, during photothermal weak absorption testing, the sample surface often accumulates impurities with strong absorption characteristics, such as dust. Although these impurities typically exhibit weak adhesion, they can nonetheless produce significant photothermal weak absorption effects. In contrast, during laser damage testing, such impurities are removed by laser irradiation and therefore do not play a determinative role in the laser-induced damage process of NYCF crystals. This phenomenon is frequently observed in LIDT testing, where impurities visible via CCD imaging prior to irradiation disappear afterward (as shown in Figure 4(c)), a removal effect also exploited in laser conditioning to cleanse the sample surface through the laser shock effect^[ Reference Runkel, Neeb, Staggs, Auerbach and Burnham^25, Reference Adams, Weiland, Stanley, Sell, Luthi, Vickers, Carr, Feit, Rubenchik, Spaeth and Hackel^26]. Although all the NYCF crystal surfaces underwent isopropyl alcohol cleaning before both the photothermal weak absorption and LIDT assessments, not all surface defects were completely removed, indicating that these defects possess a notable degree of adhesion. Following laser irradiation, subsequent CCD observations confirmed the removal of these defects, and no laser-induced damage was detected on the NYCF crystal surfaces. This mechanism demonstrates a viable method for improving the LIDT of optical component surfaces via laser conditioning, specifically, a laser cleaning process. In this process, the laser shock effect generated during laser irradiation effectively removes impurities with strong absorption characteristics, thereby explaining why the removal of such defects results in higher LIDT values at the laser damage tested locations. However, laser conditioning treatment cannot entirely eliminate all surface defects; rather, it primarily removes those with relatively low adhesion and simpler structural characteristics. Notably, the LIDT values maintained a stable dynamic range in regions with higher weak absorption, and these values are critical parameters in determining the sample’s ultimate resistance to laser irradiation.

Furthermore, data acquired from experimental observations exhibit significant oscillatory distribution characteristics across various photothermal weak absorption intervals. Attempts to model all LIDT values using a single curve risk overfitting the prediction model. Accordingly, a more robust approach involves providing a confidence interval for the prediction curve based on the intrinsic characteristics of the data distribution, thereby offering statistically meaningful guidance for predicting the LIDT from photothermal weak absorption. Notably, the recorded photothermal weak absorption values are not uniformly distributed between 0 and 300 ppm but rather follow a random density distribution. In this context, the density of the experimentally acquired weak absorption data, ${\rho}_{\mathrm{data}}(x)$ , is defined using Gaussian kernel density estimation (KDE):

(1) $$\begin{align}{\rho}_{\mathrm{data}}(x)=\frac{1}{N}\sum \limits_N^{i\in N}\exp \left[-{\left(\frac{x_i-x}{\omega}\right)}^2\right],\end{align}$$

where $\omega$ denotes the bandwidth of the photothermal weak absorption interval used for calculation and $N$ represents the number of photothermal weak absorption data points within that interval. Here, ${x}_i$ indicates the photothermal weak absorption value at the current iteration step $i$ in the summation, while $x$ corresponds to the photothermal weak absorption value at the center of the current interval. The parameter $\omega$ was set to 3% of the interpolation between the maximum and minimum photothermal weak absorption values.

After computing ${\rho}_{\mathrm{data}}(x)$ for all photothermal weak absorption values, the envelope width of the network prediction at the current photothermal weak absorption value $x$ , denoted as ${W}_{\mathrm{envelop}}(x)$ , is defined as follows:

(2) $$\begin{align}{W}_{\mathrm{envelop}}(x)=\frac{1}{2}+\frac{1}{2}\left[1-{\tilde{\rho}}_{\mathrm{data}}(x)\right]\times \mathrm{std}\left(x,\omega \right),\end{align}$$

where $\tilde{\rho}_{\mathrm{data}}$ represents the normalized data density and $\omega$ represents the bandwidth of the interval for the current photothermal weak absorption value.

The region bounded by the dashed lines in Figure 4(b) indicates the fluctuation range of the LIDT predicted by the trained model for the NYCF crystal. This signifies that the objective of the current model is not to provide a one-to-one mapping from photothermal weak absorption to the LIDT test results, but rather to provide a confidence interval. Consequently, the model achieves sufficient prevention and suppression of overfitting^[ Reference Barber, Candes, Ramdas and Tibshirani^27, Reference Gal and Ghahramani^28]. Moreover, as Figure 4(b) demonstrates, the discrete points near 175 and 275 ppm, which fall outside the predicted confidence interval, strongly validate the efficacy and rationale of our confidence interval approach based on data density. These outliers result from the uneven distribution of data across intervals. Specifically, the outlier near 175 ppm is situated in a sparsely populated region, where insufficient neighboring data lead the model to assign a narrower confidence interval, causing the observation to fall outside this interval. This behavior illustrates that the proposed method does not force a fit for extreme fluctuations in regions with inadequate data support; rather, it authentically signals such risks through the confidence interval. In contrast, the anomaly near 275 ppm is found in a densely populated region, prompting the model to assign a wider confidence interval that effectively encompasses most data fluctuations, including the anomaly itself.

To analyze the underlying physical mechanisms of damage and to validate the accuracy of our artificial intelligence analysis model, we conducted a theoretical analysis of the relationship between photothermal weak absorption and the LIDT. Laser-induced damage in CaF₂ primarily arises from temperature elevation due to surface defects that absorb laser energy. This absorption leads to the formation of temperature gradients, which in turn create thermal stresses resulting in mechanical damage^[ Reference Li, Kang, Han, Zheng and Su^29]. Consequently, we integrated the photothermal weak absorption data obtained from the damage threshold test sites into our damage theory analyses and computational models. Initially, the material heating process during laser irradiation of the NYCF crystal was analyzed using the Fourier heat conduction differential equation with appropriate boundary and initial conditions. By employing Green’s function method, we derived the analytical solution to this differential equation as follows:

(3) $$\begin{align}&\Delta T\left(\overrightarrow{r},t\right)=\frac{A_i\left(1-{r}_f\right){I}_0{R}^2}{k}\nonumber\\& \quad \times\sqrt{\frac{\alpha }{\pi }}{\int}_{\tau =0}^t \frac{\mathrm{exp}\left[-\frac{x^2+{y}^2}{4\alpha \left(t-\tau \right)+{R}^2}\right]\mathrm{exp}\left[-\frac{z^2}{4\alpha \left(t-\tau \right)}\right]}{\sqrt{t-\tau}\times \left[4\alpha \left(t-\tau \right)+{R}^2\right]}f\left(\tau \right)\mathrm{d}\tau,\end{align}$$

where ${A}_i$ , ${r}_f$ , ${I}_0$ , $R$ and $\tau$ represent the absorption coefficient, reflectance, incident laser intensity, radius of incident laser beam and the pulsed duration, respectively; $f\left(\tau \right)$ represents the temporal function of incident laser and $\alpha =k/\rho c$ denotes the thermal diffusivity (where $k$ , $\rho$ and $c$ represent the thermal conductivity, density and specific heat, respectively). From this, we can derive the temperature gradient induced under fixed incident laser fluence conditions, as illustrated in Figure 5(a).

Figure 5

(a) Laser-induced temperature variation in NYCF crystals. (b) The associated variation in thermal stress within these crystals. (c) The relationship between photothermal weak absorption and the theoretical laser-induced damage threshold.

Subsequently, we substitute the temperature gradient $\Delta T$ into the thermal-elastic theory, enabling the calculation of thermal stresses induced by the thermal gradient:

(4) $$\begin{align}\sigma =\frac{\beta E}{1-\upsilon}\left(\frac{1}{R^2}{\int}_0^R\Delta T\times r\mathrm{d}r+\frac{1}{r^2}{\int}_0^r\Delta T\times r\mathrm{d}r-\Delta T\right),\end{align}$$

where $\beta$ , $E$ and $\upsilon$ are the thermal expansion coefficient, Young’s modulus and Poisson’s ratio, respectively. This enables us to obtain the specific numerical values for the thermal gradient induced by the incident laser intensity converted into thermal stress, as illustrated in Figure 5(b).

Because the tensile strength (34 MPa) of the CaF₂ crystal is significantly lower than its compressive strength (300 MPa), the hoop stress is the most critical factor likely to initiate mechanical failure. Accordingly, by using the hoop stress threshold (34 MPa) as a reference and accounting for the spatial variations in photothermal weak absorption, the corresponding LIDT can be theoretically derived via inverse calculation. It is important to note that the photothermal weak absorption testing method employed in this study is a relative measurement technique. Initially, the theoretical absorption coefficient ( $T={e}^{- Ad}$ ) for the NYCF crystal is calculated based on its previously recorded transmission spectrum. The photothermal weak absorption values measured at the transmission position of the NYCF sample serve as the standard reference. Under identical testing conditions – using a 2.5 W Nd:YAG quasi-continuous pump beam and a 0.6 W He-Ne probe beam at 632.8 nm – the photothermal weak absorption values obtained from different positions on other NYCF crystal samples are scaled proportionally against the standard. This process yields the corresponding absorption coefficients ( ${A}_i$ ) for each test position^[ Reference Welsch and Ristau^30], which are then incorporated into the calculation formula. In this study, 10 distinct positions from the same NYCF crystal sample were selected to convert the photothermal weak absorption values into theoretical laser damage thresholds ( $\mathrm{LIDT}_{\mathrm{theory}}$ ), as illustrated in Figure 5(c).

It could be observed that higher photothermal weak absorption values correspond to lower theoretical fluences ( $\mathrm{LIDT}_{\mathrm{theory}}$ ) required to induce laser damage. Furthermore, $\mathrm{LIDT}_{\mathrm{theory}}$ approximates the Chapman function with respect to photothermal weak absorption ( $\mathrm{LIDT}_{\mathrm{theory}}=a{\left(1-{e}^{-b\times x}\right)}^c$ , where $\mathrm{LIDT}_{\mathrm{theory}}$ is the theoretical laser damage threshold, $x$ is the photothermal weak absorption, $a$ , $b$ and $c$ represent the fitting parameters), a trend that aligns closely with experimental findings. It was worth noting that the theoretical calculations yield values of the LIDTs slightly lower than those obtained by experiments. This discrepancy primarily arises because the theoretical model pertains to single-pulse laser damage (i.e., a one-on-one test method where each test position is irradiated by a single shot), whereas the experimental results employed the R-on-1 test method, which incorporates laser conditioning effects^[ Reference Runkel, Neeb, Staggs, Auerbach and Burnham^25, Reference Adams, Weiland, Stanley, Sell, Luthi, Vickers, Carr, Feit, Rubenchik, Spaeth and Hackel^26]; meanwhile, the theoretical calculations for single-pulse laser damage thresholds assume that the laser damage occurs at the center of the incident laser beam with a peak fluence. In practical LIDT testing, defect damage occurs randomly within the laser beam, with some defects manifesting at the beam periphery where the fluence is lower. Inducing such peripheral defects requires a higher incident laser fluence, thereby leading to an overestimated laser damage threshold in practical testing. In Ref. [Reference Lu, Zhang, Jiang, Kou, Zhang, Xu, Zhao, Wu and Su20], a similar experimental phenomenon was observed, indicating that the photothermal weak absorption characteristics of NYCF play a pivotal role in determining its LIDT. This paper further elucidates the physical processes and mechanisms by which photothermal weak absorption influences LIDT test results.

However, the experimental results presented in this study indicate that the relationship between the photothermal weak absorption of the NYCF crystals and the LIDT does not fully adhere to the predictions of the Chapman function, and the research paradigm adopted in this study differs fundamentally from that of Ref. [Reference Laurence, Bude, Ly, Shen and Feit19]. By employing our well-established in situ photothermal weak absorption and LIDT testing system, we extracted over 200 one-to-one corresponding data points from a single test specimen. This enabled an in-depth investigation of the statistical distribution patterns of the LIDT within a fixed material system. We utilized an FCNN as a robust, data-driven tool to perform machine learning regression analysis on the customized dataset. Unlike the predefined functional forms used in Ref. [Reference Laurence, Bude, Ly, Shen and Feit19], our neural network autonomously learned the mapping relationships from the data through complex, nonlinear transformations. Moreover, the network’s input layer incorporated three critical features to comprehensively characterize the physical state: the measured photothermal weak absorption values ( $x$ ), their logarithmic transformation ( $\log \left(x\right)$ ) and the crystal orientation (encoded as a categorical variable). The network architecture consists of input, hidden and output layers:

(5) $$\begin{align}{\boldsymbol{x}}_{\mathrm{input}}={\left[x,\log (x),\mathrm{OneHot}\left(\mathrm{Orientation}\right)\right]}^{\mathrm{T}}.\end{align}$$

The input vector ${\boldsymbol{x}}_{\mathrm{input}}\in {\mathbb{R}}^{{3}}$ comprises four hidden layers:

(6) $$\begin{align}\begin{array}{@{}lll@{}}{\boldsymbol{h}}^{(1)}=\mathrm{Relu}\left({\boldsymbol{W}}^{(1)}{\boldsymbol{x}}_{\mathrm{input}}+{\boldsymbol{b}}^{(1)}\right),& {\boldsymbol{W}}^{(1)}\in{\mathbb{R}}^{128\times 3},& {\boldsymbol{b}}^{(1)}\in{\mathbb{R}}^{128},\\ {\boldsymbol{h}}^{(2)}=\mathrm{Relu}\left({\boldsymbol{W}}^{(2)}{\boldsymbol{h}}^{(1)}+{\boldsymbol{b}}^{(2)}\right),& {\boldsymbol{W}}^{(2)}\in{\mathbb{R}}^{64\times 128},& {\boldsymbol{b}}^{(2)}\in{\mathbb{R}}^{64},\\ {\boldsymbol{h}}^{(3)}=\mathrm{Relu}\left({\boldsymbol{W}}^{(3)}{\boldsymbol{h}}^{\left({2}\right)}+{\boldsymbol{b}}^{(3)}\right),& {\boldsymbol{W}}^{(3)}\in{\mathbb{R}}^{32\times 64},& {\boldsymbol{b}}^{(3)}\in{\mathbb{R}}^{32},\\ {\boldsymbol{h}}^{(4)}=\mathrm{Relu}\left({\boldsymbol{W}}^{(4)}{\boldsymbol{h}}^{(3)}+{\boldsymbol{b}}^{(4)}\right),& {\boldsymbol{W}}^{(4)}\in{\mathbb{R}}^{16\times 32},& {\boldsymbol{b}}^{(4)}\in{\mathbb{R}}^{16}.\end{array}\end{align}$$

Here, $\boldsymbol{h}$ denotes the intermediate layer output and $\mathrm{Relu}(.)$ represents the nonlinear activation function. In addition, $\boldsymbol{W}$ and $\boldsymbol{b}$ correspond to the weighting factors and bias, respectively, for the network under training. Finally, the output layer is defined as follows:

(7) $$\begin{align}\mathrm{LIDT}=\mathrm{Relu}\left({\boldsymbol{W}}^{(5)}{\boldsymbol{h}}^{(4)}+{\boldsymbol{b}}^{(5)}\right),{\boldsymbol{W}}^{(5)}\in{\mathbb{R}}^{1\times 16},\ {\boldsymbol{b}}^{(5)}\in{\mathbb{R}}^1.\end{align}$$

The network is trained by minimizing the MSE between the predicted LIDT values and the corresponding measured values. Through model training and analysis of an experiment-specific customized dataset, we found that the resulting relationship is far more complex than the simple Chapman function described in Ref. [Reference Laurence, Bude, Ly, Shen and Feit19]. Instead, it manifests as a complex nonlinear mapping function, ${\mathbf{\mathcal{F}}}_{\mathrm{NN}}$ :

(8) $$\begin{align}\mathrm{LIDT}={\mathbf{\mathcal{F}}}_{\mathrm{NN}}\left\{x,\log (x),\mathrm{Orientation};\left[{\boldsymbol{W}}^{\left({i}\right)},{\boldsymbol{b}}^{\left({i}\right)}\right]\right\}.\end{align}$$

It is evident that the relationships learned by this complex nonlinear model surpass the Chapman function in both form and statistical substance. Rather than yielding a deterministic value, the model produces a conditional distribution that accounts for the inherent discreteness of the data. Consequently, the neural network analysis enables a more efficient and accurate non-destructive assessment of a crystal’s resistance to laser damage, while also providing deeper insights into crystal defect information through residual prediction analysis. This approach offers direct empirical support and theoretical guidance for developing suppression defect methods and enhancing crystal fabrication quality.

3.2 The effect of test beam size on non-destructive evaluation results

The previous discussion employed machine learning algorithms to systematically analyze the relationship between the photothermal weak absorption and the LIDTs. However, during practical LIDT testing, the incident beam size would significantly influence the experimental outcomes^[ Reference Foltyn^31]. Thus, to achieve a relatively accurate non-destructive prediction of the laser damage resistance of NYCF crystal surfaces, it is essential to analyze the characteristics and underlying mechanisms by which the beam size affects the evaluation results.

Photothermal weak absorption was used to characterize the absorption properties of individual defects within materials. Consequently, a smaller detection spot more accurately reflects the absorption values of these unit defects, yielding more precise test results. Therefore, the spot size for photothermal weak absorption testing in this study is set at 80 μm, ensuring that the test data accurately records the absorption behavior of defects within an 80 μm detection spot. In LIDT testing, an excessively small test spot size results in low sampling rates that lead to deviations in the test results, preventing an accurate reflection of the sample’s LIDT. Conversely, an overly large test spot size hinders the establishment of a dimensional correlation with weak photothermal absorption. To objectively elucidate the relationship between weak photothermal absorption and the LIDT while considering the testing characteristics inherent to both, this study employs LIDT test spot sizes of 210, 300 and 380 μm. This methodology effectively balances the requirements of weak photothermal absorption and LIDT testing.

In the LIDT test platform, the distance between the lens and the test sample was adjusted by modifying the position of the focused lens, thereby controlling the beam size of the laser irradiation on the NYCF crystal input surface. The incident beam diameters at 1/e were set to 0.3 and 0.38 mm, respectively. By correlating the calibration relationship between the photothermal weak absorption of the marked point and the LIDT test position, the relationship between the photothermal weak absorption and the LIDT under different damage beam size conditions could be established. Furthermore, the trained FCNN was employed to predict the LIDT outcomes, as illustrated in Figure 6(a). It could be observed that under identical photothermal weak absorption conditions, the overall LIDT test results decrease as the experimental beam size increases. Specifically, for the same photothermal weak absorption value, a larger beam size in LIDT testing is associated with a correspondingly lower LIDT outcome. Moreover, an increase in the beam size during the damage threshold test results in a broader dynamic range between the weak absorption and the LIDT test results.

Figure 6

(a) Relationship between photothermal weak absorption and the LIDT for different test beam sizes. (b) The beam size effect during the damage testing process.

Mainly because the LIDT test spot is larger than the photothermal weak absorption test spot, defects not exposed by the photothermal weak absorption measurement may be included within the LIDT test area (see Figure 6(b)). If these defects exhibit high-absorption characteristics and a low laser damage threshold, and no corresponding high-absorption defects are present in the photothermal weak absorption test spot, then both a lower photothermal weak absorption value and a lower LIDT result will be observed. Meanwhile, during the R-on-1 LIDT testing process, laser conditioning treatment would effectively eliminate defects characterized by high-absorption and poor adhesion properties. These defects subsequently exhibited pronounced photothermal weak absorption while concurrently maintaining a high LIDT. These experimental phenomena constitute a wide variation in the test results, occurring outside the primary relationship between photothermal weak absorption and LIDT, as illustrated in Figure 6(a). Moreover, because the photothermal weak absorption test spot size is fixed, an increase in the LIDT test spot size corresponds to a greater probability of this phenomenon occurring.

Meanwhile, the various kinds of defects present on crystal surfaces exhibit variable damage threshold characteristics and are distributed nonuniformly in both lateral and longitudinal directions. Consequently, during photothermal weak absorption and LIDT testing, multiple defects with markedly different properties are encompassed within the test spot, thereby introducing randomness into the results. Furthermore, as the spot size in LIDT testing increases, it encompasses a greater variety and number of defects, leading to enhanced variability in the outcomes. This trend is consistent with the experimental findings (Figure 6(a)).

For the above reasons, studies investigating the relationship between photothermal weak absorption and LIDT test spot sizes may encounter significant discrepancies due to inconsistencies in the dimensions of photothermal weak absorption and LIDT test spots, compounded by the diverse and randomly distributed crystal surface defects. Moreover, these random effects become more pronounced as the LIDT test spot size increases.

For cases where the measured data fall below the predicted confidence interval of the LIDT, this discrepancy is primarily attributable to the effect of the spot size in both test methods. Specifically, the photothermal weak absorption test employs a smaller spot size compared to that used in the LIDT test. In instances where no strong absorption defects exist within the photothermal weak absorption spot, but critical defects with lower damage thresholds are present outside it yet within the LIDT test spot, damage to these defects can result in data values that fall below the neural network’s predicted LIDT. This effect becomes more pronounced as the LIDT test spot increases in size, thereby amplifying the disparity with the photothermal weak absorption spot. These data points are considered outliers. The neural network learning process autonomously classifies such anomalies based on the number of occurrences, recognizing that these outliers do not accurately reflect the intrinsic relationship between the material’s photothermal weak absorption characteristics and its LIDT. Consequently, they are excluded from the confidence interval.

From a neural network algorithm perspective, the data sparsity is evident in Figure 6(a), specifically the concentration of data points below the predicted envelope, which originates from the data-driven operational logic of the neural network model. During training, the model learns the sampling density across various photothermal weak absorption intervals, thereby forming a priori judgments regarding prediction reliability. In regions with dense data sampling, the model has sufficient statistical information to generate highly confident predictions, resulting in stable and robust outputs (i.e., the principal section of the envelope). Conversely, in sparsely sampled regions or with isolated data points that markedly deviate from the main cluster, the model identifies their atypical characteristics. Drawing on patterns acquired from extensive data, the model does not compel the prediction envelope to accommodate these outliers, which may arise from experimental noise, unidentified extreme defects or transient measurement fluctuations. Consequently, data points appearing below the envelope do not signify model failure but rather illustrate the robustness of the proposed model in avoiding overfitting to potential noise. This envelope more accurately reflects the statistically reliable LIDT performance trend of the material, once random disturbances are excluded. Simultaneously, the outliers provide valuable diagnostic information, indicating potential anomalous damage mechanisms under specific conditions, and offering crucial insights for subsequent research aimed at enhancing the laser damage resistance of crystals.

3.3 The influence of crystal orientations on experimental results

In crystalline materials, the crystal orientation significantly influences damage characteristics^[ Reference Yoshida, Jitsuno, Fujita, Nakatsuka, Yoshimura, Sasaki and Yoshida^32]. Accordingly, this study investigates the relationship between photothermal weak absorption and the LIDT of NYCF crystals with different orientations. CaF₂ crystals possess a fluorite structure characterized by {111} cleavage plane and {100} <110> slip systems^[ Reference Muntoz, Rodriguez and Castaing^33]. Moreover, the anisotropy of the crystal would influence its mechanical and other physical properties^[ Reference Muntoz, Rodriguez and Castaing^33– Reference Chen, Hu, Xiao, Qi, Yu and Qian^35]. These features were essential not only for employing NYCF crystals as the host lattices for the laser-active ions, but also for determining the resistance of each crystal to laser-induced damage^[ Reference Shao, Liang, Lin, Wang, Deng, Meng and Fang^10, Reference Li, Kang, Han, Zheng and Su^29].

To optimize the testing system for non-destructive evaluation of NYCF crystal surface damage thresholds based on machine learning algorithms, the experiments on the photothermal weak absorption and LIDTs of NYCF crystal with {100}, {110} and {111} planes were conducted. Utilizing an FCNN, the relationship between the photothermal weak absorption and LIDT in the NYCF crystal with different orientations under the identical beam condition (D = 0.21 mm) was analyzed, as illustrated in Figure 7. It was observed that the relationship between photothermal weak absorption and the LIDT on the surfaces of NYCF crystals with three distinct crystal orientations remains largely consistent. Within the 0–75 ppm range, the LIDT decreased sharply as photothermal weak absorption increased. However, beyond 75 ppm, further increases in weak absorption did not cause the LIDT to decline indefinitely; instead, it approached a stable plateau. These findings indicate that the photothermal weak absorption properties of NYCF surfaces across different crystal orientations also play a pivotal role in determining their resistance to laser-induced damage. However, the variation in LIDT with respect to weak absorption values differs among crystal orientations. Specifically, as the photothermal weak absorption value increases, the reduction in LIDT is most pronounced in the {111} plane, followed by the {100} plane and then the {110} plane. Furthermore, when the photothermal weak absorption value exceeds 75 ppm, the damage thresholds for all three orientations tend to stabilize, with the {111} plane exhibiting the lowest LIDT and the {110} plane the highest. Furthermore, the application of neural network analysis in this study has led to the identification of novel experimental phenomena and underlying mechanisms. For instance, there was significant variation in the dispersion of data points among the crystal orientations as the LIDT shifted with the photothermal weak absorption ({111} plane > {100} plane > {110} plane). These findings suggest that even when some defects exhibit identical photothermal weak absorption values, their corresponding laser damage thresholds differ. Notably, the {111} plane demonstrated the greatest data dispersion, indicating that defect precursors on the {111} plane not only have lower damage thresholds compared to those on the other two crystal orientations but also encompass a wider variety of defect types with varying thresholds. Consequently, more stringent defect control is required during the growth and processing of {111} planes. The underlying physical mechanisms are analyzed below.

Figure 7

The relationship between the photothermal weak absorption and LIDT of NYCF crystals with different crystal orientations.

The damage morphology characteristics of NYCF crystals with three distinct crystal orientations were systematically analyzed. Figure 7 illustrates the damage morphology of NYCF crystals across various orientations under laser irradiation at positions with similar photothermal weak absorption (approximately 86.5 ppm) and identical laser fluence (53.6 J/cm²). Notably, all crystal surfaces exhibited features of crater damage and peeling damage^[ Reference Lu, Zhang, Jiang, Kou, Zhang, Xu, Zhao, Wu and Su^20].

The crater damage primarily arises when surface and subsurface defect precursors absorb laser energy during irradiation. These precursors rapidly increase in temperature, establishing a thermal gradient relative to the adjacent substrate and generating thermal stresses. When these stresses exceed the crystal’s bending strength and thermal shock resistance, the NYCF crystals manifest mechanical removal-type crater damage^[ Reference Lu, Zhang, Jiang, Kou, Zhang, Shan, Li, Xu, Zhao, Wu and Su^14]. In contrast, peeling damage predominantly results from mechanical cleavage stresses on the crystal’s surface, with each instance exhibiting distinct morphological characteristics. The peeling damage on the {110} plane is primarily distributed along the <110> and <112> directions. Given that the angle between these axes is 71°, the damage morphology manifests as parallelograms with a 71° angle, as shown in Figure 8(a). In contrast, the peeling damage on the {100} plane predominantly aligns along mutually perpendicular <110> directions, resulting in morphologies that are mutually perpendicular, as illustrated in Figure 8(b). On the {111} crystal plane, cleavage damage is oriented primarily along two <110> directions that form a 120° angle relative to each other along the slip direction of the (111) plane. Consequently, cleavage damage exhibits triangular flake structures with exfoliation angles of 60°, as depicted in Figure 8(c).

Figure 8

The laser-induced damage morphology at the approximate weak absorption position (approximately 86.5 ppm) on NYCF crystal surfaces with different crystal orientations by identical incident laser fluence (53.6 J/cm²): (a) {110} plane; (b) {100} plane; (c) {111} plane.

Although the damage morphology of NYCF crystals varies with crystal orientation, the primary laser damage mechanism in CaF₂ crystals is mechanical stress-induced damage. In addition, the tensile strength of CaF₂ crystals (34 MPa) is significantly lower than their compressive strength (300 MPa)^[ Reference Chen, Hu, Xiao, Qi, Yu and Qian^35], emphasizing the importance of tensile strength in determining the fracture resistance and laser damage threshold. For NYCF crystals, the {110} plane exhibits the highest flexural strength and thermal shock resistance^[ Reference Lu, Zhang, Jiang, Kou, Zhang, Xu, Zhao, Wu and Su^20], which enhances its laser damage resistance relative to the {100} and {111} planes. Conversely, the {111} plane shows the lowest thermal shock resistance and a tendency toward cleavage, resulting in the lowest LIDT. The laser damage behavior of the {100} plane is intermediate between that of the {111} and {110} planes. These observations are consistent with the experimental results reported here, which demonstrate that, under similar photothermal weak absorption conditions, the {110} plane achieves the highest LIDT while the {111} plane records the lowest.

Meanwhile, the underlying mechanism of this phenomenon could also be elucidated by examining the morphological characteristics of the laser-induced damage. Under identical laser fluence, laser irradiation on the similar photothermal weak absorption regions of the three crystal orientations in NYCF resulted in crater damage in the central surface damage zones across all crystal orientations. This damage arises from laser absorption by sub-substrate defect precursors, which leads to mechanical delamination driven by elevated temperatures and induced thermal stresses. Consequently, the extent of the crater damage is directly correlated with the crystal’s thermal shock resistance and bending strength. As shown in Figure 7, the crater damage size is largest on the {111} plane. Notably, the damage caused by the defect precursor on the substrate has resulted in the complete removal of the surrounding material; however, no evidence of thermal melting damage was observed. In contrast, for the {100} plane of the NYCF crystal, the crater damage dimensions are intermediate, with the substrate defect precursor just beginning to mechanically induce delamination of the adjacent substrate. Conversely, the crater damage on the {110} plane is the smallest, and the defect-induced damage did not inflict significant mechanical damage on the surrounding substrate. This observation aligns with the previously discussed relationship between thermal shock resistance and bending strength across different crystal orientations, which accounts for the varying photothermal absorption and the LIDTs observed. Therefore, when developing non-destructive evaluation thresholds for CaF₂ crystals of various orientations, it is essential to implement corresponding algorithms based on the crystal orientation of the test sample to achieve a more precise assessment of its laser damage resistance.