Deep learning for laser beam imprinting

: Methods of ablation imprints in solid targets are widely used to characterize focused 15 X-ray laser beams due to a remarkable dynamic range and resolving power. A detailed description 16 of intense beam profiles is especially important in high-energy-density physics aiming at nonlinear 17 phenomena. Complex interaction experiments require an enormous number of imprints to be 18 created under all desired conditions making the analysis demanding and requiring a huge amount 19 of human work. Here, for the first time, we present ablation imprinting methods assisted 20 by deep learning approaches. Employing a multi-layer convolutional neural network (U-Net) 21 trained on thousands of manually annotated ablation imprints in poly(methyl methacrylate), we 22 characterize a


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With the advent of intense X-ray free-electron lasers (XFELs), new unique experimental 29 opportunities emerged. Most of the experiments benefit from the extraordinary beam parameters 30 and versatility of these sources. Ultra-short pulses and high transverse coherence allow 31 investigating ultra-fast processes with excellent spatio-temporal resolution utilizing, for example, 32 methods of coherent diffractive imaging [1,2] or two-color pump-probe diffraction techniques [3].
( ( , ), ) = ( , ). Integrating the fluence profile over the entire transverse plane, we get 112 the pulse energy: being independent on the -position due to the energy conservation law. From this identity the 114 definition of the effective beam area eff ( ) follows as the "volume" below the normalized 2D 115 profile ( , ) or as the area below the normalized -scan curve ( , ) at a given -position: The boundary condition, the solution of which determines the locus of the threshold contour and 117 its area at a given -position, can be respectively expressed as: th = 0 ( ) ( , ) = 0 ( ) ( , ). a long working distance microscope intended for in-situ inspection and precise target alignment.

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The center of the ultrahigh vacuum interaction chamber was located approx.  Fig. 1. Schematic of the U-Net architecture with an input image size = 1024, = 4 descent levels, = 2 encoding layers per level, and = 16 convolution kernels in the first level. Light blue and gray boxes represent three-dimensional matrices (multi-channel feature maps) with dimensions indicated on the left and above the box. The left and right part of the image depicts the encoder (contracting path) and decoder (expansive path), respectively. Convolutions with learnable 3x3 kernels combined with the ReLU operation are indicated by blue arrows. Red downward arrows depict the max-pooling operation halving the image resolution. Green upward arrows represent 2x2 up-convolutions (upsampling) doubling the image resolution. In each descent level, skip connections (grey arrows) transfer information from encoder to decoder to better localize fine features.
There are now many variants of the U-Net architecture. As input we use square images of size time. It also requires more training data to avoid overfitting.

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Once the shape of the network is fixed, the network is parameterized by a weight vector .

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Given a set T ={ , } of pairs of training images ( ) and manually annotated (ground-truth) 196 binary masks ( ), the network is trained by minimizing the total loss function , 197 quantifying the sum of differences between network predictions = ( ) (probability maps) cross-entropy loss function [41]: The summation is done over the set Ω of all elements (pixels indexed by ) of the mask image represents a parametrization of the trained network. that fit into a single tile, no aggregation is necessary.

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The contribution of GTA 0,2,4,5 to the training dataset was 50%, 37%, 8%, 5% of tiled images, and Dice score: shapes. Unity-valued Dice score thus means an identical shape of both masks implying zero 274 RAD but the reverse statement is not necessarily true.

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Averaging over the R dataset, we obtain the mean relative area deviation (MRAD) as: Evaluating the mean squared RAD ⟨ 2 ⟩ = 1 2 , we calculate the standard 277 deviation of the mean: In a similar way, the mean Dice score ⟨ ⟩ and its 278 standard deviation is evaluated.     Table 1. Results of ground-truth analysts and three best performing U-Net models and comparison to reference GTA 0 . eff and th denote the resultant effective area and extrapolated threshold pulse energy and ⟨ ⟩ and ⟨ ⟩ represent the mean Dice score and relative area deviation related to the reference analyst, respectively.
A graphical comparison of the UNET{7, 8} model prediction maps with reference (GTA 0 ) 325 masks is presented in Fig. 5b for 5 different pulse energies ranging from full power down to nearly     To prove this, an additional thorough testing of the network performance trained using datasets of 419 varied compositions is necessary. Moreover, in order to independently evaluate the performance 420 of both the U-Net models and human annotators, a precise standardized measurement of imprint 421 contours, e.g. using the atomic force microscopy, is required. This is, however, an extensive 422 topic for another systematic study which would deserve a separate article.

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We applied the U-Net convolutional neural segmentation network to microscopy images of 425 ablative imprints in PMMA in order to extract ablation threshold contours and their areas. We 426 performed a comprehensive benchmark test by comparing its results with manual annotations 427 carried out by four independent human analysts. We may conclude that the performance of (of length 2 ) and sorted in descending order. We interpret the resulting sequence as the 451 probability that the object area contains at least pixels. Equivalently, also represents the 452 probability that an iso-fluence contour of the area = Δ is a subset of the threshold contour.

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Here Δ is an area occupied by a single image pixel. We can therefore write as a discrete Here the upper integration limit max is the area of the processed image and Ω denotes the set of 466 image pixels. It follows from Eq. (10) that the mean ⟨ ⟩ is equal to the area below the cumulative 467 function ( ) or to the "volume" below the predicted probability map . The variance 2 (central second order moment) is defined by an integral: which can be evaluated numerically, e.g. by the trapezoidal rule. For this purpose, can be 470 calculated, for example, by taking first order finite differences of . Alternatively, we can apply 471 per partes integration again. In Fig. 8 476 To extrapolate the threshold pulse energy, a generalized and more robust approach of the line fit 477 (Deming regression) is used. Contrary to the standard method, the generalized procedure fits the 478 data with a line in an implicit form + + = 0 by minimizing the squared orthogonal distance 479 of fitted data points to that line. Given a set of coordinate pairs { , } =1 , an averaged sum of 480 squared differences 2 can be defined as:

Appendix B. Threshold extrapolation
Here the fitting parameters , are components of a unit vector normal to the fitted line satisfying a condition 2 + 2 = 1 and is a negated normal distance between the line and origin. The minimum of the sum of squared normal differences occurs for parameters: Here ⟨ ⟩, ⟨ ⟩ are mean values of the variables, 2 = ⟨ 2 ⟩ − ⟨ ⟩ 2 and 2 = ⟨ 2 ⟩ − ⟨ ⟩ 2 are the  An uncertainty of the fit is estimated using an iterative statistical approach of random choice.

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From the set of first = 50 data points (the value to be justified below) a subset of ⌊ /2⌋ = 25 489 points is randomly selected and fitted. In total, 100000 iterations were performed to obtain a good extrapolation. In order to find the optimum , we repeat the above mentioned approach for varied 497 subset sizes gradually increased in range ∈ {4 . . . }. In each step the minimum mean squared 498 normal difference, an indicator of the fit goodness, is evaluated as: 499 2 min = 2 ⟨ 2 ⟩ + 2 ⟨ 2 ⟩ + 2 + 2 ⟨ ⟩ + 2 ⟨ ⟩ + 2 ⟨ ⟩, where the angle brackets denote averaging over all subset points and the parameters , , are 500 the fitting parameters evaluated using Eqs. 13-15. In total 5000 iterations were carried out for 501 each subset size while performing the fits on ⌊ /2⌋ points. The minimum subset size is = 4 since the linear fit requires at least 2 points. Fig. 9 shows the calculated minimum -squared Here we use the fact that the zeroth point of the -scan sequence is fixed and therefore its 516 uncertainty is zero.