Method and system for evaluating the quality of a photolithography mask

Abstract

The invention relates to a method for emulating an aerial image (40) of a photolithography mask obtained by a photolithography system, the method comprising: a) obtaining one or more parameters (q) describing the photolithography process in the photolithography system; b) acquiring an aerial image (24) of the photolithography mask using an inspection system; c) obtaining an underlying design (20) of the photolithography mask; d) generating a plausible design (38) of the acquired aerial image (24) using the underlying design (20) by solving an optimization problem; and e) emulating an aerial image (40) of the photolithography mask using the plausible design (38) and the one or more parameters (q) describing the photolithography process. The invention also relates to a method for evaluating the quality of a photolithography mask and a corresponding system.

Description

[0001] 22.06.2025 h- 1 -Method and system for evaluating the quality of a photolithography maskRelated ApplicationsThis application claims benefit of the German patent application No.102024118195.8filed on June 27th 2024, which is hereby incorporated by reference in its entirety.Field of the Invention The invention relates to methods and systems for quality control and quality assur-ance in photolithography masks, more specifically to a method for emulating an aerialimage of a photolithography mask, to a method for evaluating the quality of a photoli-thography mask and to a corresponding system. The methods and systems can beutilized for quantitative metrology, process monitoring, defect detection and defect review in photolithography masks. Background of the Invention Semiconductor manufacturing involves precise manipulation, e.g., etching, of materi- als such as silicon or oxide at very fine scales in the range of nm. Therefore, a quality management process comprising quality assurance and quality control is important for ensuring high quality standards of the manufactured wafers. Quality assurance refers to a set of activities for ensuring high-quality products by preventing any defects that may occur in the development process. Quality control refers to a system of in- specting the final quality of the product. Quality control is part of the quality assurance process. A wafer made of a thin slice of silicon serves as the substrate for microelectronic devices containing semiconductor structures built in and upon the wafer. The semi- conductor structures are constructed layer by layer using repeated processing steps that involve repeated chemical, mechanical, thermal and optical processes. Dimen- sions, shapes and placements of the semiconductor structures and patterns are sub- ject to several influences. One of the most crucial steps is the photolithography pro- cess. Photolithography is a process used to produce patterns on the substrate. The patterns to be printed on the surface of the substrate are generated by computer-aided-design (CAD). From the design, for each layer a photolithography mask is generated, whichcontains a magnified image of the computer-generated pattern to be etched into thesubstrate. The photolithography mask can be further adapted, e.g., by means of opti- cal proximity correction techniques. During the printing process an illuminated image projected from the photolithography mask is focused onto a photoresist thin film formed on the substrate. A semiconductor chip powering mobile phones or tablets comprises, for example, approximately between 80 and 120 patterned layers. Due to the growing integration density in the semiconductor industry, photolithography masks have to image increasingly smaller structures onto wafers. The aspect ratio and the number of layers of integrated circuits constantly increases and the structures are growing into 3rd(vertical) dimension. The current height of the memory stacks is exceeding a dozen of microns. In contrast, the feature size is becoming smaller. The minimum feature size or critical dimension is below 10nm, for example 7nm or 5nm, and is approaching feature sizes below 3nm in near future. While the complexity and dimensions of the semiconductor structures are growing into the 3rddimension, the lateral dimensions of integrated semiconductor structures are becoming smaller. Pro- ducing the small structure dimensions imaged onto the wafer requires photolitho- graphic masks or templates for nanoimprint photolithography with ever smaller struc- tures or pattern elements. The production process of photolithographic masks and templates for nanoimprint photolithography is, therefore, becoming increasingly morecomplex and, as a result, more time-consuming and ultimately also more expensive.With the advent of EUV photolithography scanners, the nature of masks changed from transmission-based to reflection-based patterning. On account of the tiny structure sizes of the pattern elements of photolithographic masks or templates, it is not possible to exclude errors during mask or template pro- duction. The resulting defects can, for example, arise from degeneration of photoli- thography masks or particle contamination. Of the various defects occurring during semiconductor structure manufacturing, photolithography related defects make up nearly half of the number of defects. Hence, in semiconductor process control, pho- tolithography mask inspection, review, and metrology play a crucial role to monitor systematic defects. Defects detected during quality assurance processes can be used for root cause analysis, for example, to modify or repair the photolithography mask. The defects can also serve as feedback to improve the process parameters of the manufacturing process, e.g., exposure time, focus variation, etc. Each defect in the photolithography mask can lead to unwanted behavior of the pro- duced wafer, or a wafer can be significantly damaged. Therefore, each defect must be found and repaired if possible and necessary. Reliable and fast defect detection methods are, therefore, important for photolithography masks. To analyze the quality of photolithography masks inspection systems can be used.Inspection systems acquire an aerial image of a photolithography mask and detectdefects in the aerial image. An aerial image indicates the intensity distribution at sub-strate level of a wafer, when illuminating the photolithography mask with light of spe-cific wavelengths. The aerial image is then analyzed using defect detection methods.However, the aerial image acquired with an inspection system usually differs from theaerial image generated in a photolithography system or optical mask qualification sys- tem due to various reasons, for example due to a different illumination, due to the imaging optics in the photolithography system, due to the wavelength of the illumina- tion in the inspection tool, if the inspection tool is not actinic, etc. Thus, to determine if a defect in a photolithography mask actually prints on a wafer in the photolithographysystem and requires repair, a separate system – a photolithography system or a sys-tem that can emulate the optics of the photolithography system such as an opticalmask qualification system – is required.Therefore, it is an objective of the invention to emulate an aerial image of a photoli-thography system or optical mask qualification system using only an inspection sys-tem. It is another objective of the invention to evaluate the quality of a photolithogra-phy mask that is to be used in a photolithography system or in an optical mask quali-fication system, in particular, if potential defects in the photolithography mask actuallyprint on a wafer during the photolithography process and need to be repaired.The objectives are achieved by the invention specified in the independent claims. Ad- vantageous embodiments and further developments of the invention are specified in the dependent claims. Summary of the inventionEmbodiments of the invention concern methods and systems for emulating an aerialimage of a photolithography mask obtained by a photolithography system or optical mask qualification system and for evaluating the quality of a photolithography mask using an emulated aerial image. A first embodiment involves a method for emulating an aerial image of a photolithog-raphy mask obtained by a photolithography system or optical mask qualification sys-tem. The method comprises: i.) obtaining one or more parameters describing the pho-tolithography process in the photolithography system or optical mask qualification sys-tem; ii.) acquiring an aerial image of the photolithography mask using an inspectionsystem; iii.) obtaining an underlying design of the photolithography mask; iv.) gener-ating a plausible design of the photolithography mask using the acquired aerial imageand the underlying design by solving an optimization problem that minimizes the de-viation of a simulated aerial image of the plausible design from the acquired aerialimage; and v.) emulating an aerial image of the photolithography mask using the gen-erated plausible design of the acquired aerial image and the one or more parametersdescribing the photolithography process in the photolithography system or opticalmask qualification system.To evaluate the quality of a photolithography mask, an inspection system can be used.The inspection system acquires an aerial image of the photolithography mask and analyzes the aerial image for potential defects. To further examine potential defects ifthey actually print on a wafer, a second system is commonly used. One option is aphotolithography system that is used to print a wafer from the photolithography maskand inspect the potential defect locations on the printed wafer. Another option is anoptical mask qualification system that is used to acquire an aerial image of the regionscontaining the potential defects on the photolithography mask, thereby emulating thephotolithography process, e.g., illumination parameters or imaging parameters usedduring the photolithography process. The invention describes a third option that doesnot require the use of a second system (such as a photolithography system or anoptical mask qualification system) for quality evaluation. Instead, the aerial image ofthe inspection system is used to emulate an aerial image of the photolithography maskunder photolithography process conditions using parameters of the photolithographyprocess, i.e., to emulate an aerial image of an optical mask qualification system. Usingthe emulated aerial image for quality evaluation, computation time and resources canbe saved as unnecessary repairs of defects are prevented. In addition, no second system needs to be available for quality evaluation. Thus, the method can be used even if only a mask inspection system is available at a site. Furthermore, effort and costs can be saved in this way. An aerial image indicates the radiation intensity distribution of a photolithography sys- tem in a wafer plane for a given photolithography mask. The aerial image is, thus, used to simulate the structures on the surface of a wafer when printing the wafer using the photolithography mask in the photolithography system. A wafer plane refers to a plane within the resist on top of the wafer in the photolithography system. An aerialimage can also be generated by applying an inspection system or an optical maskqualification system to a photolithography mask. An aerial image can be simulated using a design of a photolithography mask and an aerial image simulation method. An aerial image can refer to the aerial image of a complete photolithography mask, or it can refer to the aerial image of a section of the photolithography mask. The photolithography mask may have an aspect ratio of between 1:1 and 1:4, prefer- ably between 1:1 and 1:2, most preferably of 1:1 or 1:2. The photolithography mask may have a nearly rectangular shape. The photolithography mask may be preferably5 to 7 inch long and wide, most preferably 6 inch long and wide. Alternatively, thephotolithography mask may be 5 to 7 inch long and 10 to 14 inch wide, preferably 6 inch long and 12 inch wide. An optical system refers to a system that uses light to inspect a photolithography mask or wafer or to generate an integrated circuit pattern on a wafer. Optical systems com- prise, for example, inspection systems, optical mask qualification systems, photoli-thography systems and metrology systems.A photolithography system refers to a system that is used to fabricate integrated cir-cuits. To this end, during the photolithography process a photolithography mask isirradiated with light to transfer the integrated circuit pattern via the photolithographymask to a substrate by means of a light-sensitive chemical process.An inspection system refers to an optical system used to detect defects in a photoli-thography mask by acquiring an aerial image of the photolithography maskAn optical mask qualification system refers to a system that is used to acquire anaerial image of a portion of a photomask, in particular of potential defects detectedusing an inspection system. The optical mask qualification system emulates settingsof a photolithography system, e.g. illumination and imaging parameters, to examine the effect of a potential defect on a printed wafer, to verify that photolithography masks are defect-free or whether a repair attempt has been successful.A design of a photolithography mask refers to a representation of the photolithographymask or a section thereof. The design can, for example, comprise a computer reada-ble file, such as a CAD file or a GDS file, or a technical drawing, a set of polygonsrepresenting the structures of the photolithography mask or a section thereof. A de-sign of a photolithography mask can comprise material information, e.g., complex re-fractive indices of materials contained in the photolithography mask, electric permit-tivities, magnetic permeabilities, or derived representations. A design of a photolithog-raphy mask can comprise parameters describing dimensions of structures in the pho- tolithography mask, e.g., the thicknesses of the layers in the multilayer of an EUV mask or the thickness of absorber layers, or the dimension of the absorber structures.A design of a photolithography mask can comprise parameters describing the locationof structures in the photolithography mask, e.g., the location of absorber structures orlayers in the multilayer. A design of a photolithography mask can comprise parame-ters describing the shape of structures in the photolithography mask, e.g., the shape of the absorber structures such as side wall angles or corner rounding, etc. A designof a photolithography mask can comprise an image, e.g., a 2D image or a 3D image(e.g., a volume of voxels or a number of 2D slices of a volume), that represents prop- erties of the photolithography mask. The image can contain one, two or more chan- nels. The image can comprise image elements, e.g., pixels or voxels. The properties of the photolithography mask can comprise material properties, e.g., refractive indi- ces, electric permittivities, magnetic permeabilities, or derived representations. A de-sign of a photolithography mask can comprise descriptions of the structures within thephotolithography mask, e.g., in the form of curves, contours, polygons, Splines, NURBS, Bézier curves, etc. An “underlying design” of a photolithography mask refers to a design that was used to fabricate the photolithography mask, e.g., a model such as a CAD model. An un- derlying design can be provided for a photolithography mask, or it can be derived from an image of the photolithography mask, e.g., using image processing or machine learning methods.A “plausible design” of an aerial image refers to a design of a photolithography maskthat can be used by an aerial image simulation method to simulate the aerial image. A design is a plausible design of an aerial image, if the aerial image is a plausibleresult of an aerial image simulation method applied to the plausible design. A plausibledesign of an aerial image can be an underlying design of a photolithography maskthat could be used to generate the aerial image, e.g., using an aerial image measure-ment system for a set of parameters. A design can refer to the design of a complete photolithography mask, or it can refer to the design of a section of the photolithography mask. In a preferred example, the underlying design and the plausible design are repre- sented in a vector format. A vector format of a design represents the structures of the design by continuous coordinates and connections between them. The advantage of using a vector format is a very accurate representation of the structures in the design due to the continuous coordinates in contrast to the limited accuracy due to the pixel size in a raster image. In addition, measurements of the structures can be determined with increased accuracy, e.g., distances between structures. In another example, the underlying design and the plausible design are represented by non-binary images. Non-binary images can be used to represent structures in the photolithography mask with sub-pixel accuracy. In this way, structure boundaries can, for example, be located between pixels. Thus, the accuracy of the method is im- proved. According to an aspect of the invention, the underlying design of the photolithography mask is generated from the acquired aerial image, e.g., using image processing or machine learning. Thus, the method for detecting defects can even be used if no un- derlying design is available for the photolithography mask. A simulated aerial image can be simulated by an aerial image simulation method. An aerial image simulation method simulates the generation of an aerial image of a pho- tolithography mask from a design of the photolithography mask. The aerial imagesimulation method can use physics-based models, e.g., physics-based models of thephotolithography mask and / or of the propagation of electromagnetic waves through the photolithography mask. The aerial image simulation method can also use non- physics-based models, e.g., machine learning models that are trained using training data. An optimization problem comprises an objective function that is to be maximized or minimized. The optimization problem can also comprise constraints. Solving the opti- mization problem means applying some kind of mathematical optimization method. The mathematical optimization method computes a point with an objective function value that is expected to be better than the objective function values for multiple other points. Solving the optimization problem can, for example, mean computing the global optimum or a local optimum of the objective function. The mathematical optimization method can comprise computing an analytical solution or applying an iterative method such as gradient descent, a Simplex method, a variational approach, a combinatorial optimization approach, etc. Iterative methods can use an initial solution and iteratively adapt the initial solution until a convergence criterion is met or for a predefined number of iterations. A mathematical optimization method can also comprise computing the output of a model, which was trained to optimize an objective function for a given input, e.g., applying a machine learning model to the input that was trained to minimize a loss function. The model is, thus, trained to directly predict a solution to the optimi- zation problem. Parameters describing the photolithography process comprise, for example, -Illumination parameters describing the illumination setting of the photolithog-raphy system, comprising the distribution and intensities of different illumina-tion angles, e.g., an annular illumination setting, a dipole illumination setting, a quasar illumination setting, etc., -imaging parameters such as the numerical aperture of the photolithographysystem and the magnification of the photolithography system, obscurations,aberrations, apodizations or distortions,- design parameters such as parameters describing the material of the photoli-thography mask, e.g., layer thicknesses, refractive indices of different layers, etc.According to an example of the invention, solving the optimization problem in step iv.)comprises applying a machine learning model to the acquired aerial image, wherein the machine learning model is trained to map an acquired aerial image to a plausibledesign of the acquired aerial image.Machine learning is a field of artificial intelligence. Machine learning methods gener- ally build a parametric machine learning model based on training data consisting of a large number of samples. During training, an objective function or loss function is op- timized. After training, the method is able to generalize the knowledge gained from the training data to new previously unencountered samples, thereby making predic- tions for new data. There are many machine learning methods, e.g., linear regression, k-means, support vector machines, decision trees, random forests, neural networks or deep learning approaches. Deep learning is a class of machine learning that uses artificial neural networks with numerous hidden layers between the input layer and the output layer. Due to this complex internal structure the networks are able to progressively extract higher-level features from the raw input data. Each level learns to transform its input data into a slightly more abstract and composite representation, thus deriving low and high level knowledge from the training data. The hidden layers can have differing sizes and tasks such as convolutional or pooling layers.Using a trained machine learning model allows for an accurate and fast computationof the plausible design from the acquired aerial image. The accuracy is determined by the selected training data. The computation time is very fast, since only a forward pass of the trained machine learning model is required during inference. According to an example of the invention, solving the optimization problem in step iv.) comprises minimizing the deviation of a simulated aerial image of the plausible design from the acquired aerial image, wherein the plausible design is obtained by modifyingthe underlying design of the acquired aerial image. . Solving an optimization problemin this way to generate the plausible design increases the accuracy of the plausibledesign by minimizing the deviation of the simulated aerial image from the acquiredaerial image. In this way, a plausible design can be obtained that could be used toapproximately generate the acquired aerial image. In a preferred example, a parametric representation of the underlying design is opti- mized by solving the optimization problem. In this way, a flexible representation of the underlying design can be obtained that can be modified to obtain the plausible design. In an example, the parametric representation describes structure boundaries of theunderlying design, e.g., geometric shapes, contours, or polygons. In this way, a par-ticularly simple and flexible parametric representation of low complexity is obtainedthat is well suited for optimization. In particular, the parametric representation cancomprise contours represented by graphs containing nodes and edges. The locationof the nodes and edges can be optimized by solving the optimization problem. According to an example, the parametric representation of the underlying design that is optimized by the optimization problem to obtain the plausible design is over-param- eterized, i.e., the parametric representation comprises more parameters than neces- sary to describe the underlying design. In this way, the resolution of the plausible design can be adapted, and structures in the plausible design, in particular defects, can be modeled with higher accuracy. According to an example, the optimization problem imposes further constraints on the parametric representation of the underlying design. The further constraints can com- prise hard constraints or soft constraints. Hard constraints are constraints that strictly have to be fulfilled by the solution of the optimization problem. They can, for example, be implemented using Lagrange multipliers. Soft constraints influence the solution of the optimization problem without having to be strictly fulfilled. They can, for example, be implemented as additional terms of the objective function with a weighting factor that are optimized in parallel with the other terms. Further constraints comprise, for example, smoothness constraints, shape constraints, contour length constraints, area constraints, symmetry constraints, etc.According to an aspect of the first embodiment, the optimization problem comprisesparameters that describe a deviation from the underlying design, and the optimizationproblem imposes a sparsity constraint on these parameters. In this way, the accuracyof the optimized parameters and, thus, of the plausible design is improved. In addition,the plausible design is less susceptible to noise in the acquired aerial image. Further- more, the optimization problem is simplified. In an example, the simulated aerial image of the plausible design is obtained by ap-plying an aerial image simulation method to the plausible design that simulates theapplication of inspection system to the plausible design. In an example, the emulated aerial image is obtained by applying an aerial image simulation method to the generated plausible design that simulates the imaging pro- cess in the photolithography system or the optical mask qualification system. An aerial image simulation method simulates the generation of an aerial image of a photolithography mask from a design of the photolithography mask. The aerial image simulation method can use physics-based models, e.g., physics-based models of the photolithography mask and / or of the propagation of electromagnetic waves through the photolithography mask. The aerial image simulation method can also use non- physics-based models, e.g., machine learning models that are trained using training data. The aerial image simulation method can also use a combination of a physics- based model and a non-physis-based model such as a machine learning model. In an example, the aerial image simulation method comprises the use of a physics-based model for generating an aerial image from a design. The use of a physics-based model leads to highly accurate simulations that adhere to the laws of physics.According to an example, the aerial image simulation method comprises a machinelearning model. In this way, the aerial image simulation method is improved, sincemachine learning models directly and automatically learn important correlations fromtraining data without requiring hand-crafted, rule-based, usually error-prone pro-grams. In addition, the machine learning model can be used to improve the accuracyof other methods used in the aerial image simulation method. It, thus, allows using less complex methods of lower computation time for simulating an aerial image from a design. In a preferred example, the aerial image simulation method comprises a physics-based model and a machine learning model for generating an aerial image from adesign. The machine learning model can be applied subsequently to the physics-based model. In this way, the machine learning model can increase the accuracy ofthe output of the physics-based model. At the same time, it allows for using less com-plex physics-based models, since the result is processed further by the machinelearning model. In an example, the aerial image simulation method comprises a machine learningmodel that maps a design to an aerial image. Since machine learning models directlylearn important correlations from training data, the aerial image simulation method is highly accurate. In addition, the computation time is low as only a single forward passis required at inference time.According to a preferred example of the invention referred to as not quite rigorous(NQR) in the following, the aerial image simulation method generates an aerial imagefrom a design, in particular the plausible design, under illumination of a correspondingphotolithography mask by incident electromagnetic waves in an optical system, in particular in an inspection system, in a photolithography system or in an optical maskqualification system, and comprises: a) Approximately simulating the propagation ofthe incident electromagnetic waves within a first section of the photolithography maskthat comprises multiple structures; b) Simulating the propagation of the simulatedelectromagnetic waves from step a) within a second section of the photolithographymask analytically or numerically; c) Simulating a representation of an electromagneticnear field of the design by propagating the simulated electromagnetic waves from stepb) to a near field plane; and d) Simulating an aerial image from the design by applyinga simulation of an imaging process of the optical system to the representation of theelectromagnetic near field. For example, the first section of an EUV photolithography mask can contain absorber and non-absorber structures, whereas the second sectionof the EUV photolithography mask can contain a multilayer as further described be-low. This aerial image simulation method generates aerial images of higher accuracyand at lower computation times than standard aerial image simulation methods. Thus,the accuracy of the quality evaluation is increased and the required computation timeis reduced. The electromagnetic near field is computed in different ways within the first section of the photolithography mask and within the second section of the photolithography mask. Within the first section several assumptions described below can be made in the photolithography setting, which allow for a simplified and fast computation of the propagation of the electromagnetic waves within the first section. The propagation of the electromagnetic waves within the first section is computed by means of a wave propagation method, which takes into account the inhomogeneity of the medium within the first section of the photolithography mask. Within the second section, thepropagation of the electromagnetic waves is computed analytically or numerically. Inthis way, a highly accurate approximation of the propagation of the electromagnetic waves within the photolithography mask is obtained, requiring computation times sev- eral magnitudes below rigorous simulation methods. Thus, the simulation of electro- magnetic near fields and aerial images within industry applications becomes feasible.The design of the photolithograph mask preferably describes the photolithographymask at least partially in a dimension orthogonal to a base plane of the photolithogra-phy mask. The design of the photolithography mask can comprise one or more differ-ent sections of the photolithography mask or parts thereof, for example the first sec- tion and / or the second section. The one or more different sections can be arranged at different depths with respect to the normal of the surface. The first section of the photolithography mask comprises multiple structures. These structures can be arranged in a design that determines the patterns imprinted on the wafer during the printing process. The design can comprise structures and non-struc- tures, in particular absorber structures and non-absorber structures. The second sec- tion of the photolithography mask can contain a mask carrier that can comprise one or multiple layers of one or more materials. The structures and the non-structures can be deposited on the mask carrier. The mask carrier can comprise a substrate layer. The second section can be configured to transmit the incident electromagnetic waves (for transmission-based photolithography masks) or it can be configured to reflect the incident electromagnetic waves (for reflection-based photolithography masks). The first section can be directly adjacent to the second section of the photolithography mask. Thus, the first section and the second section can have a common boundary, e.g., a boundary plane. The mask carrier in the photolithography mask can be delim- ited by the boundary plane and a base plane. The boundary plane can be a surface plane of the mask carrier. The base plane is preferably parallel to the boundary plane. The base plane can delimit the second section from the outside. It can form an inter- face between the mask carrier and the outside of the photolithography mask through which the electromagnetic waves propagate. The structures in the first section of the photolithography mask can be delimited by the boundary plane and a structure plane. The structure plane can delimit the first section of the photolithography mask from the outside. The structure plane can contain the portion of the surface of the structures, which is facing away from the boundary plane. Preferably, the structure plane is par- allel to the boundary plane. The first section of the photolithography mask can extendbetween the structure plane and the boundary plane and can be delimited by theseplanes. The second section of the photolithography mask can extend between the boundary plane and the base plane. It can be delimited by the boundary plane and the base plane. The second section can contain a stack of homogeneous parallel layers. Homogeneous means that the material properties do not change within a layer. Other constructions of photolithography masks containing a first section and a second section can also be used. An electromagnetic near field indicates the distribution of the electromagnetic waves in a near field plane. The near field plane can be located next to a structure plane of the photolithography mask that delimits the first section of the photolithography mask from the outside. Preferably, the near field plane is parallel to the structure plane of the photolithography mask. The near field plane can be located anywhere between the structure plane and a wafer plane, for example, the near field plane can be located at a distance between 0 and 1000 nm from the structure plane, preferably at a dis- tance between 0 and 100 nm, more preferably at a distance between 0 and 50 nm, even more preferably at a distance between 0 and 20 nm and most preferably at a distance between 0 and 10 nm. In a preferred embodiment of the invention the near field plane and the structure plane are identical. The near field plane could, in princi- ple, also lie within the first section, within the second section, on the structure plane, on the base plane, or outside of the photolithography mask at the side of the base plane of the photolithography mask, for example in case the electromagnetic waves are re-propagated back into the photolithography mask after propagation through the first section. A representation of an electromagnetic (near) field can refer to the (complex) electricfield E or the (complex) scattered electric field ^^^ = ^ − ^^^^, where ^^^^ denotes theincident electric field. A complex electromagnetic field can be represented for exam- ple, in terms of the real and imaginary part, or the amplitude and phase, etc. A repre- sentation of an electromagnetic field can refer to the (complex) magnetic field H orthe (complex) scattered magnetic field ^^^ = ^ − ^^^^, where ^^^^ denotes the inci-dent magnetic field. A representation of an electromagnetic field can comprise meas-urements derived from the electromagnetic field, e.g., diffraction orders, the spectrum,the far field or the intensity field, etc. A representation of an electromagnetic fieldwithin the photolithography mask can refer to the electromagnetic field within the pho- tolithography mask, to a section of the electromagnetic field within the photolithogra- phy mask, to an electromagnetic field next to the photolithography mask, e.g., a near field, etc. A representation of an electromagnetic field can comprise representations of the electromagnetic field for different spatial directions. For example, a representa- tion of an electromagnetic field can comprise a 2D or 3D image containing one, two or more channels, such that the 2D or 3D image comprises a representation of the electromagnetic field in each spatial direction, e.g., the complex electric field in x and y or in x, y and z direction yielding a 2D or 3D image with four or six channels. According to a first example of the embodiment, the propagation of the incident elec- tromagnetic waves within the first section of the photolithography mask in step a) is approximately simulated using a Helmholtz equation. In this way, the approximation is simplified and, thus, the complexity and the computation time reduced. According to a second example of the embodiment, the propagation of the incident electromagnetic waves within the first section of the photolithography mask in step a) is approximately simulated using a machine learning model. By using a machine learning model, the computation time can be strongly reduced, as after training a sin- gle and fast forward pass is sufficient to compute the propagation of the incident elec- tromagnetic waves. According to an aspect of the first example, the Helmholtz equation is approximated using a forward Helmholtz equation. In this way, the approximation is simplified and, thus, the complexity and the computation time reduced. The forward Helmholtz equation can be solved using a beam propagation method. In this way, the approximation is simplified and, thus, the complexity and the computa- tion time reduced. In a preferred embodiment, the forward Helmholtz equation is solved using a wave propagation method that approximately describes the propagation of electromagnetic waves through an inhomogeneous medium. By using the wave propagation method, the forward Helmholtz equation is solved quickly, thereby reducing the computation time of the method. Furthermore, by taking into account the inhomogeneity of the first section of the photolithography mask, e.g., due to different materials in absorber struc- tures and non-absorber structures, the wave propagation is simulated with high accu- racy. A method for evaluating the quality of a photolithography mask for use of the photoli-thography mask in a photolithography system or optical mask qualification systemaccording to a second embodiment of the invention comprises: emulating an aerialimage of the photolithography mask obtained by the photolithography system or opti- cal mask qualification system according to a method of the first embodiment of the invention; and evaluating the quality of the photolithography mask using the emulated aerial image. The quality of a photolithography mask can, for example, be evaluated by detecting defects in the emulated aerial image, by examining if a potential defect actually prints on a wafer when using the photolithography mask in a photolithographysystem, i.e., if it appears in the emulated aerial image. Optionally, the criticality of adefect that prints on a wafer can also be examined, i.e., if it is critical or not. This canbe accomplished by classifying potential defects in the emulated aerial image withrespect to the likelihood that a defect actually prints on a wafer and, potentially, withrespect to the likelihood that a printing defect is critical, or by evaluating the quality of the emulated aerial image, for example, by measuring properties of the emulated aer- ial image, e.g., the magnitude of variation in a critical dimension caused by the defect. The term “defect” refers to a localized deviation of an integrated circuit pattern in a photolithography mask or on a wafer from an a priori defined norm of the integrated circuit pattern. For instance, a defect of an integrated circuit pattern, e.g., of a semi- conductor structure, can result in malfunctioning of an associated semiconductor de- vice. Depending on the detected defect, for example, the photolithography process can be improved, or photolithography masks or wafers can be repaired or discarded. The norm of the integrated circuit pattern can be defined by one or more correspond- ing reference objects or reference datasets, e.g., by design datasets, simulated da- tasets or acquired defect-free datasets. In an example, the quality of the photolithography mask is evaluated by obtaining one or more potential defect locations in the photolithography mask and analyzing the potential defect locations in the emulated aerial image. If a potential defect also occurs in the emulated aerial image, it will also print on a wafer during the photolithography process. If such a defect is critical, it should be repaired in the photolithography mask,or the photolithography mask should be discarded, e.g., in case the number of defectsexceeds a certain limit. Otherwise, the defect does not print on a wafer during the photolithography process and can be ignored. In this way, time and resources can besaved as unnecessary repairs of defects are prevented.In an example, the quality of the photolithography mask is evaluated by classifying potential defects in the photolithography mask according to a defect likelihood using the emulated aerial image. The classification can be less exact than a defect detec- tion, e.g., using classes such as “printing defect”, “not printing defect”, “requires re- view”, etc. In this way, the method can be used as a pre-processing method. Subse- quently, potential defects that require review can be examined further, e.g., using anoptical mask qualification system or user feedback. Using a two-step procedure, thecomputation time and processing time can be reduced as only few potential defectsrequire review, and the accuracy of the quality evaluation can be improved.A computer implemented method for training a machine learning model according toa third embodiment of the invention comprises training any of the machine learningmodels configured to be used in a method according to an example or aspect of thefirst embodiment of the invention.An inspection system for evaluating the quality of a photolithography mask for use ina photolithography system or optical mask qualification system according to a fourthembodiment of the invention comprises: a subsystem for acquiring an aerial image ofthe photolithography mask; and a data analysis device comprising at least onememory and at least one processor configured to perform the steps of the method for detecting defects in a photolithography mask according to any of the examples oraspects of the second embodiment of the invention.The invention described by embodiments, examples and aspects is not limited to theembodiments, examples and aspects, but can be implemented by those skilled in theart by various combinations or modifications thereof. Brief Description of the Drawings Fig.1 illustrates an exemplary transmission-based photolithography system, e.g., a deep ultraviolet (DUV) photolithography system;Fig. 2 illustrates an exemplary reflection-based photolithography system,e.g., an extreme ultra-violet light (EUV) photolithography system;Fig. 3 illustrates a method for evaluating the quality of a photolithographymask using an inspection system and an optical mask qualification sys-tem for emulating the photolithography system;Fig. 4 shows a flowchart of a method for emulating an aerial image of a pho-tolithography mask obtained by a photolithography system or optical mask qualification system according to a first embodiment of the inven- tion;Fig. 5 illustrates an exemplary application of the method in Fig.4;Fig. 6a, b illustrate solving an optimization problem by applying a machine learn-ing model that is trained to map an acquired aerial image to a plausible design of the acquired aerial image;Fig. 7a – c illustrate solving an optimization problem that comprises minimizing thedeviation of a simulated aerial image of the plausible design from the acquired aerial image, wherein the plausible design is obtained by modifying the underlying design of the acquired aerial image.Fig. 8 illustrates the adaptation of a parametric representation of a design toincrease its flexibility for defect detection;Fig. 9a shows a flowchart of a not quite rigorous method for simulating an aer-ial image of a photolithography mask;Fig. 9b illustrates the propagation of incoming electromagnetic waves througha transmission-based photolithography mask;Fig. 9c shows a flowchart of the not quite rigorous method for simulating anaerial image of a transmission-based photolithography mask;Fig. 9d illustrates the propagation of incoming electromagnetic waves througha reflection-based photolithography mask;Fig. 9e shows a flowchart of the not quite rigorous method for simulating anaerial image of a reflection-based photolithography mask;Fig. 9f shows a flowchart of an example of the not quite rigorous method forsimulating an aerial image of a photolithography mask including an ad- ditional characteristic function step;Fig. 9g illustrates the dependency of the phase shift vector α on the angle ϕ ofthe incoming electromagnetic waves;Fig. 9h a)-d) illustrate the steps of the not quite rigorous method for simulating anaerial image of a photolithography mask according to an example;Fig. 10 illustrates an aerial image simulation method that comprises a machinelearning model;Fig.11 shows a flowchart of a method for evaluating the quality of a photoli-thography mask for use in a photolithography system or optical maskqualification system according to a second embodiment of the inven-tion;Fig. 12 illustrates an exemplary application of the method in Fig. 11;Fig. 13 illustrates an inspection system for evaluating the quality of a photoli-thography mask according to a fourth embodiment of the invention.Detailed Description In the following, advantageous exemplary embodiments of the invention are de- scribed and schematically shown in the figures. Throughout the figures and the de- scription, same reference numbers are used to describe same features or compo- nents. Dashed lines indicate optional features. The methods and systems herein can be used with a variety of photolithography sys- tems, e.g., transmission-based photolithography systems 10 or reflection-based pho- tolithography systems 10’. Fig.1 illustrates an exemplary transmission-based photolithography system 10, e.g., a DUV photolithography system. Major components are a radiation source 12, which may be a deep-ultraviolet (DUV) excimer laser source, imaging optics which, for ex- ample, define the partial coherence and which may include optics that shape radiation from the radiation source 12, a photolithography mask 14, illumination optics 16 that illuminate the photolithography mask 14 and projection optics 17 that project an imageof the photolithography mask design 292 onto a wafer plane 18. An adjustable filteror aperture at the pupil plane of the projection optics 17 may restrict the range of beam angles that impinge on the wafer plane 18, where the largest possible angle defines the numerical aperture of the projection optics NA= n sin(Gmax), wherein n is the refractive index of the media between the substrate and the last element of the pro- jection optics 17, and Gmax is the largest angle of the beam exiting from the projection optics 17 that can still impinge on the wafer plane 18. In the present document, the terms "radiation" or “beam” are used to encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g. with a wave- length of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g. having a wavelength in the range of about 3-100 nm). Illumination optics 16 may include optical components for shaping, adjusting and / or projecting radiation from the radiation source 12 before the radiation passes the pho- tolithography mask 14. Projection optics 17 may include optical components for shap- ing, adjusting and / or projecting the radiation after the radiation passes the photoli- thography mask 14. The illumination optics 16 exclude the light source 12, the pro- jection optics exclude the photolithography mask 14. Illumination optics 16 and projection optics 17 may comprise various types of optical systems, including refractive optics, reflective optics, apertures and catadioptric op- tics, for example. Illumination optics 16 and projection optics 17 may also include components operating according to any of these design types for directing, shaping or controlling the projection beam of radiation, collectively or singularly.Fig. 2 illustrates an exemplary reflection-based photolithography system 10’, e.g., anextreme ultraviolet light (EUV) lithography system. Major components are a radiation source 12, which may be a laser plasma light source, illumination optics 16 which, for example, define the partial coherence and which may include optics that shape radi- ation from the radiation source 12, a photolithography mask 14, and projection optics17 that project an image of the photolithography mask design 292 onto a wafer plane18. An adjustable filter or aperture at the pupil plane of the projection optics 17 may restrict the range of beam angles that impinge on the wafer plane 18, where the larg- est possible angle defines the numerical aperture of the projection optics NA= n sin(Gmax), wherein n is the refractive index of the media between the substrate and the last element of the projection optics 17, and Gmax is the largest angle of the beam exiting from the projection optics 17 that can still impinge on the wafer plane 18.Fig.3 illustrates a method for evaluating the quality of a photolithography mask usingan inspection system 22 and an optical mask qualification system 26. The opticalmask qualification system 26 emulates the photolithography system that is to be usedfor printing wafers with the photolithography mask. A photolithography mask with thedesign 20 is inspected for defects using an inspection system 22. The inspection sys-tem 22 acquires an aerial image 24 and detects defects 30 in the photolithographymask. The acquired aerial image 24, however, can differ from the aerial image gen-erated by the photolithography system due to various factors, e.g., a different illumi- nation, due to the imaging optics of the photolithography system, due to a differentwavelength, etc. Therefore, a further aerial image 28 is obtained at the potential defectlocations identified by the optical inspection system using the optical mask qualifica-tion system 26 in order to examine the potential defect locations in detail. Using theaerial image 24 acquired by the inspection system, the detected defects 30 and thefurther aerial image 28 obtained using the optical mask qualification system 26, it canbe examined if the detected defects 30 print on a wafer or not. In case the detected defects also occur in the further aerial image 28, the defects 30 print on the wafer and,depending on their criticality, may need to be repaired. They are printing defects 32.Otherwise, the detected defects 30 do not print on the wafer and can be ignored.However, using an inspection system 22 and an optical mask qualification system 26for examining if a defect prints on a wafer or not requires resources and time. Fig.4 shows a flowchart of a method for emulating an aerial image of a photolithog-raphy mask obtained by a photolithography system or optical mask qualification sys-tem according to a first embodiment of the invention, the method comprising: i.) ob-taining one or more parameters describing the photolithography process in the pho-tolithography system or optical mask qualification system in a step M1; ii.) acquiringan aerial image of the photolithography mask using an inspection system in a stepM2; iii.) obtaining an underlying design of the photolithography mask in a step M3; iv.)generating a plausible design of the acquired aerial image using the underlying designby solving an optimization problem in a step M4; and v.) emulating an aerial image ofthe photolithography mask using the generated plausible design of the acquired aerialimage and the one or more parameters describing the photolithography process inthe photolithography system or optical mask qualification system in a step M5.Fig.5 illustrates an exemplary application of the method 34 for emulating an aerialimage 40 of a photolithography mask obtained by a photolithography system or opticalmask qualification system, illustrated in Fig.4.An aerial image 24 of the photolithography mask is acquired using an inspection sys-tem. An underlying design 20 of the photolithography mask is obtained. The underly-ing design 20 can, for example, be loaded from a database, obtained from a designerof the photolithography mask, or derived from an aerial image of the photolithographymask using image processing methods as described below. It can, for example, be aCAD design used to manufacture the photolithography mask.Using the underlying design 20 a plausible design 38 of the acquired aerial image 24is generated by solving an optimization problem. The optimization problem can usethe underlying design 20 as a starting point that is subsequently adapted until a sim-ulated aerial image of the adapted design sufficiently resembles the acquired aerialimage 24 of the photolithography mask. Alternatively or in addition, machine learningcan be used to adapt the underlying design 20 to the acquired aerial image 24.To emulate the photolithography process in a photolithography system, parameters q describing the photolithography process need to be obtained. Parameters describing the photolithography process comprise, for example, illumination parameters describ-ing the illumination setting of the photolithography system, imaging parameters or de-sign parameters. These parameters can, for example, be indicated by a user or they can be loaded from a database or memory. An aerial image 40 is emulated from theplausible design 38 using the one or more parameters q describing the photolithogra-phy process in the photolithography system. The aerial image 40 can, for example, be emulated using the Hopkins approach, the Abbe approach or the local Hopkins approach described further below. The mask spectrum ^^^in the formulas can, forexample, depend on the plausible design 38 and, thus, on design parameters in q,while the illumination pupil J and the imaging pupil P depend other parameters con- tained in q. The emulated aerial image 40 can then be used to evaluate the quality of the photo- lithography mask based on the photolithography system that will be used for printing wafers from the photolithography mask. For example, for defects detected by the in- spection system it can be examined if they actually print on a wafer in the photolithog- raphy system. Thus, time and resources can be saved if defects do not print on a wafer in the photolithography system are ignored.In a preferred example of the first embodiment, the underlying design 20 and theplausible design 38 are represented in the same way, e.g., they share the same for-mat, type of representation, color or grey value space, etc. By using the same repre- sentation, the comparability of the designs is ensured.For example, the underlying design 20 and the plausible design 38 can be repre-sented in a vector format. The vector format is based on the mathematics of coordi-nate geometry, in which shapes are defined as a set of points in a two- or three-dimensional cartesian coordinate system. Because almost all shapes consist of an infinite number of points, the vector format defines a limited set of geometric primitives that can be specified using a finite sample of salient points called vertices. For exam- ple, a square can be unambiguously defined by the locations of three of its four cor- ners, from which the software can interpolate the connecting boundary lines and the interior space. Because it is a regular shape, a square could also be defined by the location of one corner, a size, and a rotation angle. The fundamental geometric prim- itives of the vector format comprise points, line segments, polygons, parametric shapes in two or three dimensions such as circles, ellipses, squares, spheres, super- ellipses, etc., parametric curves, in which polylines or polygons are augmented with parameters to define a non-linear interpolation between vertices, such as circular arcs, cubic Splines, Bézier curves, etc., and three-dimensional surfaces usually de- fined as a connected set of polygons or as parametric surfaces, e.g., polygon meshes or non-uniform rational basis splines (NURBS).In another example, the underlying design 20 and the plausible design 38 are repre-sented by non-binary images. The underlying design 20 and the plausible design 38can also be represented by images that share the same color or grey value space orcolor or grey value range. In another example, the underlying design 20 and the plau-sible design 38 are represented in a CAD format, e.g., gdsll, oasis, svg, dxf, etc. Theunderlying design 20 and the plausible design 38 can be represented by geometricalstructures, e.g., polygons, circles, ellipses, contours, etc. The geometrical structures can be described using coordinates such as corner points or center points, lengths,angles, directions, axes, etc. The underlying design 20 and the plausible design 38can be represented by contours delineating the structures in the design, e.g., by lines, curves, or graphs containing nodes and edges. In another example, the underlyingdesign 20 and the plausible design 38 are represented by Fourier descriptors.In the following, obtaining the underlying design and the plausible design from aerial images will be described in detail. The underlying design is, preferably, provided along with the photolithography mask. According to an example of the first embodiment, the underlying design of the photo- lithography mask can also be generated from an aerial image, in particular from a defect-free aerial image, of the photolithography mask, or from a golden reference aerial image of the photolithography mask. To this end, image processing methods can be used, e.g., image segmentation, pattern matching, thresholding, contour ex- traction, edge detection, etc. These methods can, for example, be used to find struc- tures or patterns in the aerial image, that correspond to structures in the underlying design, e.g., repetitive structures. These structures can, for example, be represented by polygons in the underlying design, e.g., circles can represent memory holes, etc. Apart from or in addition to image processing methods, machine learning methods can be used to map an aerial image to an underlying design. The machine learning methods can be trained using pairs of aerial images of photolithography masks andcorresponding underlying designs in the required representation as training data.Deep learning methods such as CNNs, U-Nets, GANs, models including attention mechanisms such as transformers, diffusion models, etc. yield particularly good re- sults. A plausible design of an acquired aerial image can be obtained in different ways. According to an example illustrated in Fig.6a and b, solving the optimization problemin step iv.) comprises applying a machine learning model 44 to the acquired aerialimage 24, wherein the machine learning model 44 is trained to map an acquired aerialimage 24 to a plausible design 38 of the acquired aerial image 24. The plausible de-sign 38, as in this case, can be represented as a non-binary image.Solving the optimization problem here comprises applying a mathematical optimiza- tion method that comprises computing the output of a model, which was trained tooptimize an objective function for a given input. The machine learning model 44 canbe trained, as shown in Fig. 6a, using training data 42 comprising pairs of acquiredaerial images 24 of photolithography masks and corresponding designs, in particularunderlying designs 20, of the photolithography masks. The machine learning model44 could also be trained using training data 42 comprising pairs of defective acquiredaerial images 24 of photolithography masks and corresponding defective designs.The objective function optimized during training can comprise the deviation of thepredicted plausible design from the corresponding design in the training data 42. Theobjective function can also comprise the deviation of the acquired aerial images 24 from simulated aerial images obtained by applying an aerial image simulation methodto the corresponding designs in the training data 42. The machine learning modelcould use the underlying design 20 as additional input. In this case, the machine learn-ing model could be trained using aerial images and perturbed underlying designs as input and the underlying designs without perturbation as output. The perturbations could model, e.g., line edge roughness or localized defects. The machine learning model can comprise a neural network, in particular a deep neu- ral network, e.g., a CNN or U-Net. The neural network can comprise one or more attention mechanisms that allow for learning connections between different regions of an acquired aerial image 24, thereby improving the results. The machine learning model can also comprise a random forest, a support vector machine, a decision tree, a clustering method, etc.During inference, as shown in Fig. 6b, the machine learning model 44 directly mapsan acquired aerial image 24 to a plausible design 38 of the acquired aerial image 24.As this only requires a single forward pass, this mapping can be carried out veryquickly. By implementing the machine learning model 44 using, for example, GPUsor TPUs that allow for parallelization, the runtime during learning and inference can be strongly reduced. According to an example illustrated in Fig.7a to c, solving the optimization problemin step iv.) comprises minimizing the deviation of a simulated aerial image 46 of theplausible design 62 from the acquired aerial image 54, wherein the plausible design 62 is obtained by modifying the underlying design 64 of the acquired aerial image 54.According to an aspect, a parametric representation 50 of the underlying design 20,which is illustrated in Fig.7a, is optimized by the optimization problem. The parametricrepresentation 50 of the underlying design 20 can be modified until the optimizationproblem is solved. The solution of the optimization problem then corresponds to theplausible design 38. The parametric representation 50 can, for example, describestructure boundaries of the underlying design 20, e.g., a parametric representation 50 can comprise lines, edges, contours or geometric shapes such as polygons, circles,ellipses, etc. These parametric representations 50 can, for example, contain the loca-tions of control points that are optimized. In an example, as shown in Fig. 7a, theparametric representation 50 comprises contours 52 represented by graphs contain-ing nodes and edges. The contours, or the nodes and edges, can be derived, for example, from an underlying CAD design. The location of the nodes can be optimized by solving the optimization problem. Contours can also be represented by analytical functions describing the curve. The corresponding optimization problem could com- prise an Active Contour or Snake objective function term to align the contours of theparametric representation 50 to the contours in the acquired aerial image 24.The parameters of the parametric representation 50 of the underlying design 20 aremodified in the optimization problem as illustrated in Fig.7b. An initial parametric rep-resentation 55 is a parametric representation 50 that corresponds to the underlyingdesign 20. This initial parametric representation 55 is optimized 54 in order to mini-mize the deviation of the acquired aerial image 24 from a simulated aerial image 46of the optimized parametric representation 56. The optimized parametric representa-tion 56 is a parametric representation 76., During an iterative optimization, a simulatedaerial image 46 is obtained from the modified parametric representation of the under-lying design 20 using an aerial image simulation method 48. The optimized parametricrepresentation 56 then corresponds to a plausible design 38 of the acquired aerialimage 24. Optionally, the optimized parametric representation 56 can be transformedinto a different representation of the plausible design 38, e.g., a binary or non-binaryimage, as shown in Fig. 7c.In an example, the optimization problem comprises parameters that describe a devi-ation from the underlying design 20. Thus, only the deviation of the parameters isoptimized by solving the optimization problem, e.g., a shift of nodes, edges or con- tours, a contour length modification, an area modification, a size modification of some line or structure, etc. The optimization problem can impose a sparsity constraint onthese parameters that describe a deviation of the underlying design 20. In this way,the solution of the optimization problem, i.e., the plausible design 38, tends to only contain few deviations of the parameters, e.g., only local modifications of points, nodes or edges. According to an aspect of the first embodiment illustrated in Fig. 8, the parametricrepresentation 50 of the underlying design 20 is adapted, e.g., it is made more flexibleto allow for modifications by means of over-parameterization. The parametric repre-sentation 50 of the underlying design 20 is over-parameterized. To this end, the con-tours can, for example, be subsampled to place additional nodes along the contours that can be shifted during optimization. Over-parametrization can increase the com- putation time, but at the same time the accuracy of the plausible design. Fig.8 showsan initial parametric representation 50 that is derived from a CAD design by placingnodes at the corner points of the structures and connecting these by contour edges. To increase the flexibility of the parametric representation 50, further nodes are added along the contour edges yielding an over-parameterized parametric representation 60. DeepSnake-like approaches as, for example, described in the conference article “Peng, Sida, et al. "Deep snake for real-time instance segmentation. "Proceedings ofthe IEEE / CVF conference on computer vision and pattern recognition.2020”, can, forexample, be used to obtain an over-parameterized parametric representations 60from a parametric representation 50. The over-parametrized parametric representa-tion 60 is optimized to fit the acquired aerial image 24 by solving the optimizationproblem, thereby yielding an optimized parametric representation 56, from which theplausible design 38 can be derived. The parametric representations 50 and the opti-mized parametric representations 56 in Fig. 7 are also over-parametrized. Over-pa-rameterization can be used to obtain a specific resolution of the plausible design 38.There are various aerial image simulation methods 48 that can be used for obtaininga simulated aerial image 46 from a design. In a preferred example, the aerial image simulation method comprises the use of aphysics-based model for generating an aerial image from a design. This leads to ac-curate results but is often time consuming. Among these methods, there are rigorous simulation methods such as finite difference time domain (FDTD) or rigorous coupled wave analysis (RCWA) that are known to a person skilled in the art. Since they require long computation times, fast approximations such as the thin element approximation (TEA) can be used. The thin element approximation (TEA) assumes that the thickness of the structures on the photolithography mask is very small compared to the wave- length, and that the widths of the structures on the photolithography mask are very large compared to the wavelength. However, as photolithographic processes use ra- diation of shorter and shorter wavelengths, and the structures on the patterning device become smaller and smaller and grow into the vertical dimension, these assumptions do not hold anymore, and mask 3D effects must be taken into account. Therefore, the results of the TEA method are less accurate but much faster to obtain than rigorous simulation results.Such methods for simulating electromagnetic near fields or aerial images often eitherrequire too much computation time or are not sufficiently accurate. Therefore, a fastand accurate method for simulating an aerial image is described in the following. According to an example, a not quite rigorous aerial image simulation method (NQR)200 can be used to simulate an aerial image of the plausible design 38. This method200 simulates an aerial image from a design under illumination of the corresponding photolithography mask 14 by incident electromagnetic waves accurately and quickly. For simulating the interaction of electromagnetic waves 222 with a photolithography mask 14 the propagation of the electromagnetic waves 222 within the different layers of the photolithography mask 14 comprising different materials with different refractive indices has to be taken into account. The not quite rigorous aerial image simulation method 200 for generating an aerialimage of a design under illumination of a corresponding photolithography mask byincident electromagnetic waves in an optical system (e.g., an inspection system, an optical mask qualification system, a photolithographs system, etc.) is illustrated in Fig.9a and comprises: a) approximately simulating the propagation of the incident elec-tromagnetic waves within a first section of the photolithography mask that comprisesmultiple structures in a step N1; b) simulating the propagation of the simulated elec-tromagnetic waves from step a) within a second section of the photolithography maskanalytically or numerically in a step N2; c) simulating a representation of an electro-magnetic near field within the photolithography mask by propagating the simulatedelectromagnetic waves from step b) to a near field plane in a step N3; and d) gener-ating an aerial image of the photolithography mask by applying a simulation of animaging process of the optical system to the representation of the electromagneticnear field in a step N4. The not quite rigorous method 200 for simulating an aerial image can be applied totransmission-based photolithography masks 14’ as illustrated in Fig.9b and reflection-based photolithography masks 14’’ as illustrated in Fig. 9d.An electromagnetic near field indicates the distribution of the electromagnetic waves 222 in a near field plane 252. The near field plane can be located next to the structure plane 230 of the photolithography mask. Preferably, the near field plane 252 is parallel to the structure plane 230 or the base plane 234 of the photolithography mask 14. The near field plane 252 can, in general, be located anywhere between the structure plane 230 and the wafer plane 18, for example, at a distance between 0 and 1000 nm from the structure plane 230, preferably at a distance between 0 and 100nm, more preferably at a distance between 0 and 50nm, even more preferably at a distance between 0 and 20nm and most preferably at a distance between 0 and 10nm. In a preferred embodiment of the invention the near field plane 252 and the structure plane 230 are identical. According to an embodiment, the photolithography mask 14 comprises a mask carrier 248 and a grating 224, the grating 224 comprises absorber structures 226 and non-absorber structures 228 forming a design 292 on at least a portion of the mask carrier248. The photolithography mask 14 comprises a first section 225 extending between a structure plane 230 and a boundary plane 232 of the photolithography mask 14 and a second section 227 extending between the boundary plane 232 and a base plane 234 of the photolithography mask 14. The first section 225 comprises the grating 224, and the second section 227 comprises the mask carrier 248. Fig.9b illustrates the propagation of incoming electromagnetic waves 222 through a transmission-based photolithography mask 14’, e.g., a DUV photolithography mask.The photolithography mask 14’ comprises a first section 225 and a second section227. The first section 225 contains a grating 224, and the second section 227 contains a mask carrier 248. The grating 224 is formed by a combination of absorber structures226 and non-absorber structures 228. The absorber structures 226 are made of oneor more materials which absorb electromagnetic waves 222, e.g. titanium nitride or tantalum nitride, etc. The non-absorber structures 228 are made of one or more ma- terials which absorb electromagnetic waves 222 to a lower degree than the absorbermaterial. For example, the non-absorber structures 228 can comprise vacuum. Thus,the grating 224 is an inhomogeneous medium. The absorber structures 226 and the non-absorber structures 228 are deposited on a mask carrier 248. The mask carrier 248 can comprise a substrate layer 246. The mask carrier 248 in the photolithographymask 14’ is delimited by a boundary plane 232 and a base plane 234 which is prefer-ably parallel to the boundary plane 232. The boundary plane 232 is a surface plane of the mask carrier 248. The base plane 234 is a boundary plane through which the electromagnetic waves 222 enter the grating 224. The incoming electromagnetic wave 222 impinge on the base plane 234. The base plane 234 is forming an interface between the mask carrier 248 and the outside of the photolithography mask 14’ through which the electromagnetic waves 222 propagate. The absorber structures226 in the grating 224 of the photolithography mask 14’ are delimited by the boundaryplane 232 and a structure plane 230. The structure plane 230 is a boundary plane which contains the portion of the surface of the absorber structures 226, which is facing away from the boundary plane 232. Preferably, the structure plane 230 is par- allel to the boundary plane 232. The first section 225 of the photolithography mask14’ extends between the structure plane 230 and the boundary plane 232 and is de-limited by these planes. The second section 227 of the photolithography mask 14’ extends between the boundary plane 232 and the base plane 234 and is delimited by the boundary plane 232 and the base plane 234. For transmission-based photolithography masks 14’, according to an example, the simulated electromagnetic waves 222 are incident on the base plane 34, propagatedwithin the second section 227 of the photolithography mask 14’ from the base plane234 to the boundary plane 232, and within the first section 225 of the photolithographymask 14’ from the boundary plane 232 to the structure plane 230.Fig.9c shows a flowchart of the not quite rigorous method 200 for simulating an aerialimage in case of a transmission-based photolithography mask 14’ as shown in Fig.9b. The simulated electromagnetic waves 222 are incident on the photolithography mask, e.g., on the base plane 234, propagated within the second section 227 of the photolithography mask, e.g., from the base plane 234 to the boundary plane 232, ina step P1, and within the first section 225 of the photolithography mask 14’, e.g., fromthe boundary plane 232 to the structure plane 230, in a step P2. Then a representationof the electromagnetic near field of the photolithography mask 14’ in a near field plane252 is obtained in a step P3. Finally, an aerial image is generated from the represen- tation of the near field by applying a simulation of an imaging process of an opticalsystem to the representation of the electromagnetic near field in a step P4.For reflection-based photolithography masks 14’’, according to an example illustrated in Fig.9d, the mask carrier 248 comprises a multilayer 238 in the form of a stack of optical thin films 240 for reflecting the electromagnetic waves 222, and the simulated electromagnetic waves 222 are incident on the structure plane 230, propagated withinthe first section 225 of the photolithography mask 14’’ from the structure plane 230 tothe boundary plane 232, reflected within the multilayer 238 in the second section 227of the photolithography mask 14’’ and propagated within the first section 225 of thephotolithography mask 14’’ from the boundary plane 232 to the structure plane 230.In this way, the not quite rigorous method 200 for simulating an aerial image can be applied to reflection-based photolithography masks, e.g., EUV photolithography masks. Fig.9d illustrates the propagation of incoming electromagnetic waves 222 through a reflection-based photolithography mask 14’’, e.g., an EUV photolithography mask.The photolithography mask 14’’ comprises a first section 225 and a second section227. The first section 225 contains a grating 224, and the second section 227 contains a mask carrier 248. The grating 224 contains absorber structures 226 and non-ab-sorber structures 228 forming a design on at least a portion of the mask carrier 248to be printed onto a wafer. The absorber structures 226 are made of one or more materials which absorb electromagnetic waves 222, e.g., titanium nitride or tantalum nitride, etc. The non-absorber structures 228 are made of one or more materials which absorb electromagnetic waves 222 to a lower degree than the absorber material. For example, the non-absorber structures 228 can comprise vacuum. Thus, the absorber structures 226 and the non-absorber structures 228 form an inhomogeneous medium. The absorber structures 226 and the non-absorber structures 228 are deposited on a mask carrier 248. The mask carrier 248 comprises a multilayer 238 in the form of a stack of optical thin films 240 for reflecting the electromagnetic waves 222. The maskcarrier 248 can comprise a capping layer 242 and / or a substrate layer 246. The re-flection of the electromagnetic waves 222 by the stack of optical thin films 240 corre- sponds to a reflection of the electromagnetic waves 222 at the effective mirror plane244. The mask carrier 248 in the photolithography mask 14’’ is delimited by a bound-ary plane 232 and a base plane 234 which is preferably parallel to the boundary plane 232. The boundary plane 232 is a surface plane of the mask carrier 248. The absorberstructures 228 in the grating 224 of the photolithography mask 14’’ are delimited bythe boundary plane 232 and a structure plane 230. The structure plane 230 is a boundary plane which contains the portion of the surface of the absorber structures 226, which is facing away from the boundary plane 232. Preferably, the structure plane 230 is parallel to the boundary plane 232. The structure plane 230 is a boundary plane through which the electromagnetic waves 222 enter the first section 225, e.g., the grating 224. The incoming electromagnetic waves 222 impinge on the structure plane 230. The structure plane 230 is forming aninterface between the photolithpgraphy mask 14’’ and the outside of the photolithog-raphy mask 14’’ through which the electromagnetic waves 222 propagate. The32ec-tionn 225 of the photolithography mask 14’’ extends between the structure plane 230and the boundary plane 232 and is delimited by these planes. The second section227 of the photolithography mask 14’’ extends between the boundary plane 232 andthe base plane 234 and is delimited by the boundary plane 232 and the base plane 234. Fig. 9e shows a flowchart of an example of the not quite rigorous method 200 forgenerating an aerial image of a photolithography mask 14 in case of a reflection-based photolithography mask 14’’ as shown in Fig. 9d. The mask carrier 248 com-prises a multilayer 238 in the form of a stack of optical thin films 240 for reflecting the electromagnetic waves 222, and the simulated electromagnetic waves 222 are inci- dent on the photolithography mask, e.g., on the structure plane 230, propagated within the first section 225 of the photolithography mask 14’’, e.g., from the structure plane 230 to the boundary plane 232, in a step Q1, reflected within the multilayer 238 in thesecond section 227 of the photolithography mask 14’’ in a step Q2 and propagatedwithin the first section 225 of the photolithography mask 14’’, e.g., from the boundary plane 232 to the structure plane 230, in a step Q3. Then a representation of the elec-tromagnetic near field of the photolithography mask 14’’ in a near field plane 252 isobtained in a step Q4. Finally, an aerial image is generated from the representation of the near field by applying a simulation of an imaging process of an optical system such as an inspection system, an optical mask qualification system or a photolithog-raphy system, etc. to the representation of the electromagnetic near field in a step Q5.Instead of solving the Maxwell equations directly in the first section 225, different ap- proximations can be used to reduce the computation time of the method. According to an example, the propagation of the incident electromagnetic waves within the firstsection 225 of the photolithography mask 14 in step a) is approximately simulatedusing a Helmholtz equation, in particular a forward Helmholtz equation. In the photolithography setting, the following assumptions can be made: 1) the refrac- tive index is similar for the different materials of the photolithography mask 14, e.g.,the refractive index of the structures 226, in particular the absorber structures, is closeto the refractive index outside the structures 226, in particular the non-absorber struc- tures, e.g., vacuum.2) The refractive index distribution in the first section 225 is piece- wise constant without requiring a transition to be modeled.3) The main propagation direction 250 of the incoming electromagnetic waves 222 is near vertical with respect to a main surface of the photolithography mask, in particular to the base plane 234. These assumptions allow for a simplified approximation of the propagation of the elec- tromagnetic waves 222 within the first section 225.Based on the time-harmonic Maxwell equations, the following equation can be derivedfor the electric field E of an electromagnetic wave 222: where ω is the angular frequency, c the speed of light and ^(r,ω) the dielectric functioncharacterizing the specific material. These relations are connected to the refractiveindex n(r, ω) of a material via ^(^, ^) = ^(^, ^)^. The right-hand side couples the elec-tric field components, which makes it hard to find solutions to this equation. Therefore, the right-hand side is preferably neglected. The neglection of the right-hand side re- mains valid if the following two assumptions are fulfilled: the considered optical sys- tem does not show a distinctive response depending upon the incident polarization, and there is no cross coupling between individual polarization components. For the lithography setting at short wavelengths, e.g., for DUV or EUV photolithography masks, there are two reasons for neglecting polarization and phononic effects, so these assumptions are valid. Firstly, the contrasts in the refractive index are low with respect to the different materials within the structures 226 and outside the structures in the first section 225. Secondly, the height a of the structures 226 is larger than the wavelength λ, i.e. ≥ 2. Therefore, the right-hand side of equation (1) can be ne-glected resulting in a Helmholtz equation △^(^, ^) ^) = 0The Helmholtz equation can be simplified further. Using the following relations con- cerning the magnitude of the wave number|^|ω)  and its connection to the wavelength λ |^| = ^^n(r, ω) =2π ()λn r, ω ,^where ^^and λ^are respectively the wave vector and wavelength in vacuum, the Helmholtz equation can be rewritten as △^(^, ^) + ^^^^^(^, ^)^(^, ^) = 0.This equation can be rewritten using the transverse Helmholtz operator as follows: (^^^ + ℋ^)^^,^ = 0,where ℋ^ = ^^^ + ^^^ + ^ ^^^^(^, ^, ^).This equation can be rewritten as Here, the square root Helmholtz operator is introduced, being formally defined in terms of a power-series. Moreover, it is assumed that the commutator δ^^ℋ^can be neglected, which physically implies that back reflections within the inhomogeneous medium are ignored. Then, the forward Helmholtz equation is identified as The ordinary partial differential equation can be solved using multiplication with an integrating factor: ^^,^(^, ^, ^^ + ^) = ^^^^^^ℋ^ ^^,^(^, ^, ^^) .The exponential operator can be approximated by an integral operator as shown in the appendix A of the PhD thesis “Efficient wave-optical simulations for the modeling of micro-optical elements” by Soeren Schmidt at the University of Jena. Reference is hereby made in full to the aforementioned PhD thesis, and its disclosure content is included in the description of this invention. The approximation by the integral opera-tor yields: This approach is referred to as the angular spectrum of plane wave decomposition (ASPW) as shown in equation 1.8 of the aforementioned PhD thesis. It assumes that the electromagnetic waves are propagated within a homogeneous medium with re- fractive index n. However, this does not hold for the first section 225 of the photoli- thography mask 14 comprising structures 226 and non- structures. Therefore, an extension of the ASPW to inhomogeneous media is required to describe the propagation of electromagnetic waves 222 within the first section 225 of the pho- tolithography mask 14. In order to account for inhomogeneous media, the propagation constant in a subse- quent plane z to a given plane ^^is computed according to the refractive index distri- bution as described in section 1.4 of the aforementioned PhD thesis Therefore, according to an example, the forward Helmholtz equation can be solved using a wave propagation method. The wave propagation method is a generalization of the ASPW to inhomogeneous media and describes a wave propagation step in a plane ^^along the z-direction perpendicular to the base plane by where E denotes the electric field component of the electromagnetic field and^^^, ^^ , ^^^^ the wave vector, which locally obeys the dispersion relation ^^ where ^^=^^denotes the wavenumber of light with a wavelength λ^in vacuum,^(^, ^, ^) the refractive index distribution and ℱ the spatial Fourier Transform. Themagnitude of the wave vector k is inversely proportional to the wavelength λ, and the direction of the wave vector is perpendicular to the wave front. By using this wave propagation method, the propagation of the electromagnetic waves within an inhomo- geneous medium can be modeled leading to an accurate approximation of the prop- agation of the electromagnetic waves within the first section of the photolithography mask. In an embodiment, the first section 225 of the photolithography mask 14 comprises structures 226 and non-structures 228 forming an inhomogeneous medium, e.g., the grating 224 comprises absorber structures and non-absorber structures. The simula- tion of the propagation of the electromagnetic waves 222 within the first section 225 takes into account this inhomogeneity of the material within the first section 225. At the same time, several simplifying assumptions can be exploited in the photolithogra- phy setting. In addition, the simulation of the propagation of the electromagnetic waves 222 within the second section 227 is computed analytically or numerically. In this way, an accurate and fast simulation of the propagation of the electromagnetic waves 222 within the photolithography mask 14 is obtained. Alternatively, the forward Helmholtz equation can be solved using a beam propaga- tion method. The beam propagation method is described, for example, in chapter 1.3 of the above-mentioned PhD thesis “Efficient wave-optical simulations for the model- ing of micro-optical elements” by Soeren Schmidt. In an example, the propagation of the incident electromagnetic waves within the firstsection of the photolithography mask in step a) is approximately simulated using amachine learning model. The machine learning model can, for example, comprise a neural network, e.g., a deep learning model. For example, the machine learning model can comprise a U-Net or a neural network with at least one attention mechanism, e.g., a Transformer machine learning model. The machine learning model can use a model of the photolithography mask, e.g., a design pattern, as input and map the input to an electromagnetic field as output. The machine learning model can be trained using training data obtained, e.g., from simulations described above. By using a machine learning model, the computation time can be strongly reduced, as after training a sin- gle and fast forward pass is sufficient to compute the propagation of the incident elec- tromagnetic waves. Due to the dependence of the dispersion relation in (3) on the spatial variables (x,y) the wave propagation method in (2) cannot be implemented using Fast Fourier Trans- forms (FFT). In order to use FFTs and reduce the computation time the wave propa- gation method in (2) can be reformulated using characteristic functions. In an example, the first section 225 of the photolithography mask 14 is decomposed into different materials by defining a characteristic function for each material that indi- cates the presence of the material within different locations in the first section 225 of the photolithography mask 14, wherein at least one characteristic function is non-bi- nary. The first section 225 of the photolithography mask 14 can be decomposed into a finite number M of pairwise disjoint and homogeneous subregions with refractive index ^^.Then, the refractive index distribution ^(^, ^, ^^) within a given layer ^^ can be rewrit-ten using characteristic functions. A characteristic function X × Y → ^ for a ma-terial m is a mapping from a spatial domain X  × ^  ⊆ ℝ  × ℝ to a value range ^, whichrepresents the presence of the material m for each location (x,y) of the spatial domain.For example indicates a binary characteristic function with a value range ^ = {0,1}, where ^^ indi-cates the refractive index of material m. ^ can, for example, be a subset of the realnumbers ℝ (^ ⊆ ℝ) or of the complex numbers ℂ (^ ⊆ ℂ).Fig. 9f shows a flowchart of the not quite rigorous method for simulating an aerialimage according to an example, comprising an additional characteristic function stepR1 followed by simulating a representation of an electromagnetic near field in a stepR2 and by applying a simulation of an imaging process of an optical system to therepresentation of the resulting electromagnetic near field in a step R3. The step R1 comprises: identifying a number M of materials of the structures 226 inthe first section 225 forming the design 292, e.g., the design pattern, of the photoli-thography mask 14; defining a characteristic function ^^^^ : ^ × Y → ^ for each materialm ∈ {1, .. , ^} indicating the presence of the material for locations (x,y) of the photoli-thography mask 14 within a subset X  × ^  ⊆ ℝ  × ℝ of an x / y-plane at ^ = ^^, whereinthe x / y-plane is orthogonal to the z-direction, which is perpendicular to the base plane 34; simulating the propagation of the electromagnetic waves 222 as a weighted sum over a propagation step within each of the identified materials: where ℱ^^ indicates the inverse Fourier Transform. The use of characteristic func-tions allows for an FFT based implementation of the wave propagation method in (2), thus saving computation time. The integrator in (4) converges linearly with the step size. However, the discretization of the commonly used binary characteristic functions is problematic. Since binary characteristic functions are discontinuous, the Shannon- Nyquist theorem requires a very high sampling frequency (at least twice the maximum frequency of the signal) and, thus, a very high resolution of the sampling grid. In par- ticular, if the edges of the structures 226 do not align with the sampling grid, the sam- pling is inaccurate. In addition, the resolution of the sampling grid depends on the size of the smallest feature. The high resolution of the sampling grid in turn leads to high computation times for generating the aerial image. Therefore, according to an aspect of the example the characteristic functions are band-limited. A band-limited characteristic function is a characteristic function for which a finite frequency ω^exists such that ℱ(ω) = 0 ^^^ |ω| > ω^.According to the Shannon-Nyquist theorem, on the one hand the required sampling frequency of the discretization of a band-limited characteristic function depends on its maximum frequency. On the other hand, a given sampling frequency of a discretiza- tion of a band-limited characteristic function directly implies its maximum frequency. By using band-limited characteristic functions, the maximum frequency of the charac- teristic functions can be limited. In this way, according to the Shannon-Nyquist theo- rem, the required sampling frequency is reduced, so a sampling grid of lower resolu- tion can be used for discretizing the characteristic functions (than in case of binary characteristic functions). In this way, the required computation times for generating the aerial image can be reduced. In addition, the resolution of the sampling grid is independent from the feature size of the features in the design, e.g., the design pat- tern, of the photolithography mask. In contrast, for binary characteristic functions, thesampling grid resolution depends on the smallest feature of the design of the photoli-thography mask. A justification of using discretized band-limited characteristic functions is given in thefollowing: Assuming that the electromagnetic field E only contains energy at longwavelengths in the x / y-plane perpendicular to the base plane 234 of the photolithog-raphy mask 14, a linear space invariant low-pass filter P has no effect when appliedto the electromagnetic field E, that is: ^(^) ≈ ^.Equivalently, P can be written as a convolution in time domain, and the above implies:^(^) If the filter P is applied to the product of E with a function Θ having energy at shorterwavelengths, it follows: P(^ ∙ Θ) =E ∙ P(Θ).Thus, if a low pass filter is applied to the product of a slowly varying function E and afastly varying function Θ, then the result is approximately the product of the slowlyvarying function E and the filtered fastly varying function ^(Θ).Applying this result to the propagator of the wave propagation method in (4) where O denotes the linear ASPW propagator ℱ{^(^^)}^ (5)and assuming that the electromagnetic field ^(^^) varies on a longer scale than thecharacteristic functions ^^^^, it follows: P[^(^^ + Δ^)] ≈ ∑^ ^^^ ^^^^^^^O[^(^^)] .Thus, the propagator for the low frequency part of the field E in the wave propagationmethod in (4) is obtained by applying the filter P to the characteristic functions. By generalizing the concept of characteristic functions to non-binary characteristic functions sub-pixel design features can be resolved, and a speedup factor of about 100 can be achieved. Apart from band-limited characteristic functions, it is also advantageous to use other non-binary characteristic functions to describe the presence of specific materials indifferent locations (^, y) ∈ ^ × ^ of the photolithography mask 14 at ^ = ^^.For example, it is advantageous to use continuous characteristic functions or complex valued characteristic functions. In this way, the material distribution within the photo- lithography mask can be described in a more flexible way leading to approximations of higher accuracy.According to an aspect of the example, the value range ^ of at least one characteristicfunction comprises at least one value (^, ^) ∉ {0,1}. Thus, at least one character-istic function is not a binary characteristic function, since it maps to at least one non- binary value. In this way, different materials m can be present in the same location (x,y) allowing for a more flexible modeling of the refractive index distribution in the photolithography mask 14, thereby obtaining a more general description of the mate- rial distribution in the photolithography mask. By using characteristic functions having overlapping support the accuracy of the wave propagation method can be improved. The support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. On the one hand, the presence of differ- ent materials in the same location of the photolithography mask can be used to model the distribution of materials in case that different materials are present in the same location. On the other hand, assuming the presence of different materials in the same location can be used as a mathematical means to improve the accuracy of the elec- tromagnetic near field and aerial image even if this material distribution does not cor- respond to the true material distribution. In this way, more accurate electromagnetic near fields and aerial images can be computed. According to an aspect of the example, the characteristic functions form an affine combination at each location in the first section of the photolithography mask. Thatmeans that at ^ = ^^: ∑^ ^^^^^ ^^ (^, ^) = 1 ∀ (^, y) ∈ ^ × ^. In particular, the character-istic functions can form a convex combination at each location of the photolithographymask at ^ = ^^. This constraint ensures that the amount of material present in each location of the domain of the characteristic functions is the same and amounts to 1. Thus, an accurate description of the material distribution within the photolithography mask 14 is obtained leading to an accurate approximation of the propagation of the electromagnetic waves 222 within the photolithography mask 14. According to an aspect of the example, obtaining the characteristic functions com-prises decomposing the design of the photolithography mask 14 into elements (e.g.,using mathematical functions that describe the contours or aera of the structures 226 such as polygons, Splines, curvilinear elements, etc.), representing the elements 294 by characteristic functions, in particular by binary characteristic functions, and apply- ing a low pass filter to the characteristic functions. The elements 294 can, for example, be represented by characteristic functions taking on a non-zero value, for example 1, inside the element 94 and 0 outside the element 294. For example, each element 294 can be decomposed into one or more triangles, and the triangles can be represented by characteristic functions. The Fourier Transform of polygons can be obtained as described in appendix A of the PhD thesis “Photolithography Simulation by Heinrich Kirchauer at the Technical University of Wien”. Reference is hereby made in full to the aforementioned PhD thesis, and its disclosure content is included in the descrip- tion of this invention. By applying a low pass filter to the characteristic functions band- limited characteristic functions 68 are obtained. Thus, the wave propagation method in (4) can be simulated using a coarse sampling grid as described above, thereby reducing the computation time. In an example, a low pass filter is applied to the characteristic functions. In particular, applying a low pass filter to the characteristic functions can comprise applying a spa- tial analytical Fourier Transform to the characteristic functions followed by an inverse Fourier Transform. The analytical Fourier transform can be computed only for the spatial frequencies of the discretized domain of the inverse FFT. This subsampling of the spatial domain limits the maximum frequency of the characteristic functions ac- cording to the Shannon-Nyquist theorem. Thus, the discretization corresponds to a low pass filter of the characteristic functions. The result is a representation of the de-sign of the photolithography mask by means of band-limited characteristic functions,which can be discretized using a sampling grid of a resolution much lower than for binary characteristic functions, thereby reducing the computation time. According to an example, the analytical Fourier Transform used in the wave propa- gation method in equation (4) is approximated by a Fast Fourier Transform (FFT) and / or an analytical inverse Fourier Transform by a Fast Inverse Fourier Transform.In this way, the computation time is reduced.The FFT implies periodic boundary conditions. However, due to the arbitrary angle of the incident electromagnetic waves, this assumption does not hold anymore. This in-accuracy is often ignored by approximation methods. Even if the design 292 is as-sumed to be periodic, the arbitrary illumination angle of the incident electromagnetic waves 222, e.g., with respect to the normal 254 of the structure plane 230, implies that the solution of equation (4) is only quasi periodic according to the Floquet Theo- rem, that means periodic with an additional phase shift α: ^(^ + ^Δ^) = ^(^)^^^^^^ .Therefore, according to an example, the wave propagation method approximates an analytical Fourier Transform by a Fast Fourier Transform, and the wave propagation method takes into account the angle ϕ of the incident electromagnetic waves 222, e.g., the angle with respect to the normal 254 of the structure plane 230, by assuming quasiperiodic boundary conditions in the propagator step in equation (4) at one or more pairs of opposite boundaries perpendicular to a base plane 234 of the photoli- thography mask 14, that is in the x / y-plane. By assuming quasiperiodic boundary con- ditions, the accuracy of the simulated electromagnetic near field is improved.Let E(x, y, z^) be quasi-periodic in the x and y coordinates. Then, according to theFloquet theorem, E can be rewritten as a part E’ that is periodic in x and y multipliedwith a non-periodic phase shift α = as follows: Then the Fourier transform of the periodic part E’ can be written as It follows that Using we obtain From this it can be concluded that a phase shift α in the input field that is linear in the x and y coordinates can be accommodated by reformulating the dispersion relation in equation (3) as follows: Therefore, according to an example, the dispersion relation in (3) can be reformulated using the Floquet theorem. The term within the inverse Fourier Transform is then pe- riodic and can be computed using standard FFT. In particular, the dispersion relation of the electromagnetic waves 222 within the first section 225 depends on the angle ϕ of the incident electromagnetic waves 222. In particular, the dispersion relation within the first section 225 is modified by a phase shift in the coordinates parallel to the base plane 234 of the photolithography mask 14. Fig.9g illustrates the dependency of the phase shift vector α on the angle ϕ of the incoming electromagnetic waves 222. The angle ϕ can be measured with respect tothe normal 254 of the structure plane ^^. The electromagnetic waves 222 are propa-gated in the direction of the wave vector 256. Let ^^and ^^indicate the boundaries of the unit cell in the x-direction, that is the smallest non-periodic subset of the periodicdesign 292. Then, using the relation^^^ ϕ =^ ^^ − ^^the phase difference between ^^and ^^can be expressed in terms of ϕ as follows: The dependence of ^^on the angle of the incoming electromagnetic waves 222 can be computed analogously. In an example, the photolithography mask 14 is a transmission-based photolithogra- phy mask. In another example, the photolithography mask 14 is a reflection-based photolithog- raphy mask, and the second section 227 comprises a multilayer 238 in the form of a stack of optical thin films 240 for reflecting the electromagnetic waves 222. For reflection-based photolithography masks 14, simulating the propagation of thesimulated electromagnetic waves 222 from step a) within the second section 227 ofthe photolithography mask 14 analytically or numerically can comprise using an ana- lytical description of the electromagnetic wave propagation within the mask carrier 248 and analytically computing the reflection of the electromagnetic waves 222 at the multilayer 238. Therefore, according to an example, simulating the reflection of the simulated elec-tromagnetic waves 222 from step a) within the multilayer 238 comprises the analyticalcomputation of reflection coefficients at a boundary, e.g., at the boundary plane 232, between the second section 227 and the first section 225 of the photolithographymask 14, the reflection coefficients describing the propagation of the electromagneticwaves 222 within the stack of optical thin films 240 of the multilayer 238. The propa- gation within the stack of optical thin films 240 of the multilayer 238 corresponds to a reflection at an effective mirror plane 244 at a specific distance from the boundary plane 232. In particular, the reflection coefficients at the boundary 232 can be computed sepa- rately within the structures 26 and outside the structures 226 in the first section 225 of the photolithography mask 14. For example, the reflection coefficients can be com- puted separately for each medium of the absorber structures and the non-absorber structures of the grating 224 at the location of the boundary plane 232. In this way, the accuracy of the generated aerial image is improved. In an example, simulating the propagation of the simulated electromagnetic waves 222 within the second section 227 of the photolithography mask 14 comprises apply- ing the reflection coefficients to the electromagnetic waves 222 incident on the bound- ary 232. In particular, simulating the reflection of the electromagnetic waves 222 within the multilayer 238 comprises replacing the phase term e^^^^ ^^^,^^^^^in (4) by analytical reflection coefficients ^^at the boundary plane ^^: where ^^^indicates the scalar electric field at the boundary plane ^^directed towards the structure plane 230 of the photolithography mask 14, and ^^^^^indicates the scalar electric field at the boundary plane ^^directed towards the base plane 234 of the photolithography mask 14. In this way, the computer implemented method forgenerating an aerial image of a design of a photolithography mask can be applied toreflection-based photolithography masks. In addition, the accuracy of the method is improved. As shown in the article “Optical properties of a thin-film stack illuminated by a focused field” by S. Kim, Y. Kim and I. Park, Journal of the Optical Society of America A, Vol. 17, No.8, August 2000, equations 33 to 41, the analytical reflection coefficients ^^for each of the N optical thin films 240 of the multilayer 238 can be computed for s-polar- ized waves and p-polarized waves as follows: where ^^^are the elements of the characteristic matrix A Here, ^^^^is given by where ℎ^^^ = ^^^^^^^^ ^^^ θ^^^polarized waveswaves. Here ^^denotes the vacuum permittivity, μ^the vacuum magnetic permeability, ^^^^the refractive index of the j+1-th optical thin film 240 and ^^^^the thickness of the j+1- th optical thin film 240. Reference is hereby made in full to the aforementioned article, and its disclosure content is included in the description of this invention. In another example, the reflection of the electromagnetic waves by the multilayer 238 could be computed numerically as follows: in a first step, the electric field at the bound- ary plane 232 is decomposed in its Fourier Modes. In a second step, for each Fourier mode, the reflected electromagnetic field can be computed using, for example, the transfer matrix method (described in Section 2.2 of the article “Domain Decomposition Method for Maxwell’s Equations: Scattering off Periodic Structures, Achim Schädle, Lin Zschiedrich, Sven Burger, Roland Klose, Frank Schmidt, in arXiv:math / 0602179v1”). In a third step, the superposition of the reflected Fourier modes yields the reflected electromagnetic waves. Alternatively, a machine learning model can be trained for numerically simulating the propagation of the electromag- netic waves within the second section of the photolithography mask.Fig.9h a) to d) illustrate the steps of an example of the not quite rigorous method 200for simulating an aerial image. The design 292 of the photolithography mask 14 com-prises elements 294 consisting of polygons in the form of rectangles shown in Fig.9h a). In a characteristic function step R1, the elements 294 are represented by charac- teristic functions, e.g., by binary characteristic functions, obtained by any of the meth- ods described above. For example, the elements 294 are represented by binary char- acteristic functions having the value 1 within the elements 294 and the value 0 outside. Then a spatial analytical Fourier transform is applied to the characteristic functions followed by an inverse FFT for back transformation resulting in band-limited charac- teristic functions 268. Here, the analytical Fourier transform is only computed for the spatial frequencies of the discretized domain of the inverse FFT. This subsampling of the spatial domain limits the maximum frequency of the characteristic functions ac- cording to the Shannon-Nyquist theorem. Thus, the discretization corresponds to a low pass filter of the characteristic functions. The result is a band-limited discretizedrepresentation of the design 292 of the photolithography mask 14, i.e., band-limitedcharacteristic functions 268 sampled on a sampling grid of low resolution shown in Fig.9h b). Based on the band-limited characteristic functions 268 a representation of an electromagnetic near field 220 in the form of its amplitude is shown in Fig.9h c), which is simulated by propagating the simulated electromagnetic waves to a near fieldplane. Finally, an aerial image 264 shown in Fig. 9h d) is computed by applying asimulation 290 of the imaging process of the photolithography system 10, 10’ within a projection section 19 between the near field plane 252 and a wafer plane 18 to the representation of the electromagnetic near field 220. The imaging process can include resampling of the electromagnetic near field 220 to a grid of higher resolution. By computing the aerial image 264 by applying step R1 and step R3 an accurate aerialimage 264 can be simulated for the design 292 of the photolithography mask 14 atlow computation times due to the low resolution of the sampling grid. Thus, the com- putation time for obtaining the aerial image 264 is reduced compared to the simulation of an aerial image 264 by applying a rigorous simulation method (such as RCWA) tothe design 292 of the photolithography mask 14 by means of rigorous simulation 295,which requires a sampling grid of high resolution. Further details of the not quite rigorous method for simulating an aerial image are described in the international patent application PCT / EP2023 / 087651 and in the Ger- man patent application 102022135019.3 which are hereby incorporated by reference in their entirety. According to an example, the aerial image simulation method 48 comprises a machine learning model. The aerial image simulation method 48 can, for example, be config- ured as shown in Fig.10. The input is a design, in particular a parametric representa- tion 50 of the design. The output is a simulated aerial image 46. A transformation method 62 can, optionally, be used to transform the parametric representation 50 to a standardized representation that is used as input for the following methods 64, 66. For example, a parametric representation 50 containing contours, or nodes and edges, or geometrical shapes, etc. can be transformed, for example, into an image or into a vector format, etc. A simulation method 64 is used for simulating the aerial image generation from the input design or from the standardized representation of the design. The simulation method 64 can, for example, comprise a physics-based model of the electromagnetic wave propagation within the photolithography mask, or it can comprise a machine learning model simulating the electromagnetic wave propagation within the photoli- thography mask. The simulation method 64 can, for example, comprise the TEA method, a rigorous simulation method, a physics-based machine learning model or any other simulation method that can be used to compute an aerial image from a design of a photolithography mask. In an example, the aerial image simulation method 48 comprises a machine learning model 66. The machine learning model 66 comprises at least one parameter, prefer- ably multiple parameters, e.g., 100.000 parameters. The values of the at least one parameter are determined during training of the machine learning model 66 on train- ing data. The machine learning model 66 can be applied to the result of the simulation method 64. In this way, the machine learning model 66 can improve the accuracy of the simulation method 64. Thus, less accurate or less complex simulation methods 64 can be used, since these are followed by a machine learning model 66 that im- proves the accuracy of the simulation result. In addition, a less complex machine learning model 66 can be used, since the input to the machine learning model 66 is already obtained from a physics-based simulation. In this way, the computation time and the accuracy can be improved. The training of the aerial image simulation method 48 can be carried out in different ways. In a first example, the simulation method 64 is adapted or trained first. In case, the simulation method 64 does not contain a machine learning model, parameters of the simulation method 64 can be adjusted, e.g., using training data or prior knowledge. In case, the simulation method contains a machine learning model, the machine learn- ing model can be trained using training data, e.g., comprising pairs of designs and corresponding aerial images. The following machine learning model 66 can be trained in a following step using pairs comprising the output of the simulation method 64 and corresponding aerial images 24 as training data. In a second example, the simulation method 64 and the machine learning model 66 are trained jointly. In this case, weighting the influence of the parameters of the sim- ulation method 64 and of the parameters of the machine learning model 66 in theobjective function is beneficial. In case, the optional transformation method 62 con-tains a machine learning model, this machine learning model can also be trained jointly. Due to the joint training a higher accuracy of the aerial image simulation method 48 can be achieved. In an example, the aerial image simulation method 48 comprises a machine learning model that maps a design to an aerial image. The design can be optimized by solving the optimization problem, e.g., in an iterative way. The deviation of the simulated aer- ial image from the acquired aerial image can be minimized in the objective function of the optimization problem. An optimization problem that minimizes the deviation of the acquired aerial imagefrom a simulated aerial image, wherein the simulated aerial image is obtained by ap-plying an aerial image simulation method to a design, could be formulated as follows.Let p denote a set of parameters of a parametric representation of the plausible de-sign. p can also include design parameters such as a mask thickness, refractive indi-ces, etc. The parameters can, for example, be optimized in an iterative way. To this end, an optimization problem such as the following can be solved: ^^^^ is the target parameter vector that minimizes a difference measure χ between anacquired aerial image ^^^^ and a simulated aerial image ^^^^(^, ^). χ is an objectivefunction or loss function that defines the optimality condition. It can be linked to a noise model of the acquired aerial image, e.g., for Gaussian i.i.d. noise the ^^-norm can be used. q is a parameter vector containing the photolithography process param- eters such as illumination parameters, imaging parameters and / or design parameters as described above. Different approaches are known for computing the intensity of the simulated aerial image ^^^^using partially coherent imaging from an incoming electromagnetic near field corresponding to different illumination angles: for example, the Hopkins ap- proach, the Abbe approach and the local Hopkins approach. The incoming electro- magnetic near field can be computed using a near field simulation method such asRCWA, FDTD,TEA or NQR. The Hopkins approach relies on the observation that for small variations of the inci- dence angles of the light waves only very small deviations of the intensity, phase and polarization of the light waves can be expected. Thus, a change in the illumination angle approximately only results in a frequency shift of the respective diffraction spec-trum of the photolithography mask. The same mask spectrum ℱ{^^^(^, ^, ^, ^)} of anincoming electromagnetic near field ^^^(^, ^, ^, ^) is, therefore, used for all illuminationangles with a shift according to the illumination angle: ^^^^(^^ , ^^, ^, ^) where I^^^(x’, y’, p, q) indicates the intensity of the simulated aerial image ^^^^(^, ^),^^(^^ , ^^ , ^) a complex imaging pupil function, ^^^^ ^^^^^,^ , ^ ^^^^^,^ , ^^ an illumination angleweighting distribution (e.g., with respect to the illumination intensity), ^^the electric permittivity, ^^the speed of light assuming vacuum, ^^^^^^,^ , ^^^^,^^^the illumination angles and N^^^^the number of illumination angles in the illumination angle distribution in the pupil plane. This approach is simple and fast. For simulations using the thin mask or Kirchhoff approach such as the TEA this assumption is always fulfilled. However, in case that the thickness of the structures on the photolithography mask cannot be neglected anymore and require rigorous electromagnetic field simulations of mask diffraction for varying illumination angles, the Hopkins approach is not sufficiently accurate. In this case the Abbe approach may be used to accommodate for the non-constant diffraction spectra of the photolithography mask, since the Abbe approach assumesillumination angle dependent diffraction spectra ℱ{^^^,^}, i = 1, .. N^^^^: where I^^^(x’, y’, p, q) indicates the intensity of the simulated aerial image ^^^^(^, ^).However, the Abbe approach is highly computationally expensive, since an electro- magnetic near field has to be simulated for every single illumination angle. Thus, the Abbe approach may not be suitable for use in, e.g., a full chip die-to-die or die-to- database defect detection method. In order to obtain a fast and accurate simulation method for aerial images of photoli- thography masks, the local Hopkins approach can be used as disclosed, for example, in US 2007 / 0253637 A1. The local Hopkins approach is a combination of the Hopkins approach and the Abbe approach based on locally assuming constant diffraction spectra of the photolithography mask. To this end, the source maps are partitioned into a number of segments. For each segment the diffraction spectra are assumed constant, such that only a single diffraction spectrum for each segment has to be simulated. Hence, with the local Hopkins approach, a smaller number of spectra =1, .. N^^^, ^^^^ ≪ ^^^^^, for a subset of selected illumination angles issimulated. For the remaining illumination angles the simulated spectra are shifted ac- cording to the illumination angle: where I^^^(x’, y’, p, q) indicates the intensity of the simulated aerial image ^^^^(^, ^).The local Hopkins approach requires a careful selection of segments and illumination angles within the segments, for which the diffraction spectra are simulated, as, for example, described in US 2007 / 0253637 A1. The Hopkins, Abbe or local Hopkins approach allow to compute the gradient of the objective function with respect to the parameter vector p. Thus, by using one of theseapproaches to compute the simulated aerial image ^) from the electromagneticnear field in the objective function χ above, the parameters p can be optimized in an iterative way, e.g., by means of gradient descent. Note that, contrary to design pa- rameters (e.g. bias, corner-rounding, etc.), the optimization of optical parameters (e.g. Zernike aberrations) which can be modelled by changes in the pupil function, do not require new simulations of the electromagnetic near field but just a re-evaluation ofthe Hopkins, Abbe or local Hopkins imaging equations above, thereby simplifying andspeeding up the optimization of the parameter vector p. The optimization problem can comprise a sparsity constraint imposed on one or more parameters of the parameter vector p. A sparsity constraint can be used to penalize deviations of parameters from initial parameters, i.e., a deviation of a parametric rep- resentation 50 of the plausible design 38 from a parametric representation 50 of theunderlying design 20. In this case many of the parameters in the solution of the opti-mization problem are usually 0 as only few defects are expected in the plausible de-sign. A sparsity constraint can be implemented using, e.g., ^^-norm regularization. In order to preserve differentiability of the objective function, the ^^-norm can, for exam-ple, be approximated using a Huber loss function, which penalizes small parameterdeviations quadratically and larger parameter deviations linearly and is differentiable,or a different optimizer such as the fast iterative shrinkage-thresholding algorithm(FISTA). Sparsity constraints can also be implemented using, e.g., ^^-regularizationfor 0 ≤ ^ ≤ 1.The objective function may also include additional regularization terms as functions of p, in particular to obtain a well-posed objective function, e.g., smoothness priors, convexity priors, length priors, shape priors, etc. Smoothness priors can, for example, be imposed by penalizing distance changes between neighboring polygon nodes or by penalizing contour length. Alternatively, parameterizations based on smooth basis functions such as Splines or Fourier functions can be used to impose smoothness. Shape priors can, for example, be used to prohibit line crossings, e.g., of contouredges. Contour length priors can be used to promote shorter circumferences ofshapes and, thus, simpler or convex shapes. The resulting optimization problem can be solved in different ways. For example, a downhill simplex approach or a gradient descent approach can be used. Preferably conjugate gradients are used or the Limited-memory Broyden-Fletcher-Goldfarb- Shanno (LBFGS) algorithm. Furthermore, evolutionary algorithms such as simulated annealing could also be used to solve the optimization problem.Fig. 11 illustrates a method 67 for evaluating the quality of a photolithography mask14 that is to be used in a photolithography system 10, 10’ or optical mask qualificationsystem 26 according to the second embodiment of the invention. The method com-prises: a) obtaining one or more parameters describing the photolithography processin the photolithography system or in the optical mask qualification system in a stepS1; b) acquiring an aerial image of the photolithography mask using an inspectionsystem in a step S2; c) obtaining an underlying design of the photolithography maskin a step S3; d) generating, in particular, receiving a plausible design of the acquiredaerial image using the underlying design by solving an optimization problem in a stepS4; e) emulating an aerial image of the photolithography mask using the generatedplausible design of the acquired aerial image and the one or more parameters de- scribing the photolithography process in the photolithography system or optical maskqualification system in a step S5; and f) evaluating the quality of the photolithographymask 14 using the emulated aerial image 40 in a step S6.Using the emulated aerial image 40 that emulates the photolithography process in thephotolithography system or optical mask qualification system for the plausible design38, the quality of the examined photolithography mask can be evaluated. The qualitycan be evaluated, e.g., by detecting defects in the emulated aerial image 40, or byexamining if a defect detected in the acquired aerial image 24 actually prints in the emulated aerial image 40, or by measuring image properties such as brightness, con-trast, critical dimension (CD), CD uniformity (CDU), normalized image log-slope(NILS), or by pre-classifying defects, for example based on CD variations or place-ment errors, etc. Methods for the automatic detection of defects in photolithography masks include de- fect detection algorithms, which are often based on a die-to-die or die-to-database principle. The die-to-die principle compares an imaging dataset of a photolithography mask to a reference dataset comprising the same structures obtained from a different portion of the same photolithography mask or from a different photolithography mask. The discovered deviations are treated as defects. However, this method requires the availability and time-consuming scanning of two corresponding portions of photoli- thography masks and exact knowledge about their relative position. In addition, it fails in case of repeater defects. The die-to-database principle compares an imaging dataset of a photolithography mask with a reference dataset from a database, e.g., a simulated image, a design, a CAD file or a model of the photolithography mask, thereby discovering deviations from the ideal data. Unexpected patterns in the imaging dataset are detected due to large differences. Repeater defects can be handled. However, the comparison of an aerial image to a reference dataset is difficult due to different acquisition parameters, styles and qualities of the images. Properties of the emulated aerial image 40 can be measured to evaluate the qualityof the photolithography mask. The critical dimension (CD) refers to the smallest sizeof a feature that is printed on a wafer. The normalized image log-slope (NILS) is anessential metric for describing photolithographic image quality. It is defined as theslope of the log of intensity ^ of the emulated aerial image, multiplied by the linewidth^: ^^^^^^(^)^^^^^ =^ ^^ Essentially, NILS indicates the % change in width for a given % change in dose. This is particularly critical for EUV lithography, where stochastic variations in dose are nat- urally occurring.To examine if a defect actually prints in the emulated aerial image 40, one or morepotential defect locations in the photolithography mask can be obtained and analyzedin the emulated aerial image as, for example, illustrated in Fig.12. The potential defectlocations can be obtained from the inspection system 22 that generates a list of de- fects 30 from the acquired aerial image 24 of the photolithography mask. However, it is not clear if the defects 30 actually print on the wafer during the photolithography process. Therefore, an emulated aerial image 40 is generated from the plausible de- sign using photolithography process describing parameters q. The potential defectlocations 68 are analyzed in the emulated aerial image 40. The analysis can alsocomprise a modelling or simulation of the (nonlinear) resist and etching behavior inthe photolithography process in order to accurately predict the final patterns on thewafer as well as the respective assessment of the impact of the defect. The emulatedaerial image 40 also contains the defect 30. Thus, the defect 30 is a printing defect 32. Since the defect prints on the wafer the photolithography mask should be repairedor discarded. The resist and etching behavior could be described in the Abbe ap-proach, Hopkins approach or local Hopkins approach above using a non-linear func-tion g depending on the simulated aerial image and the parameters q as follows:g(^^^^(^^ , ^^, ^, ^), ^).The quality of the photolithography mask can also be evaluated by classifying poten-tial defects in the photolithography mask according to a defect likelihood using theemulated aerial image 40. A classification can, for example, classify potential defects68 as a defect that definitely prints on the wafer, that definitely does not print on thewafer, or that requires review by a user or another method or system. In this way, apre-evaluation of the potential defect locations can be carried out. Only the potential defects requiring review need to be analyzed further, thereby saving computation time, effort and costs. A second embodiment of the invention relates to a computer implemented method for training a machine learning model according to any of the examples in the first em- bodiment, i.e., a machine learning model for mapping an acquired aerial image to a plausible design of the acquired aerial image such that the acquired aerial image is a plausible result of an aerial image simulation method applied to the plausible design, or a machine learning model that is part of an aerial image simulation method as described above.A inspection system 70 for evaluating the quality of a photolithography mask 14 ac-cording to a third embodiment of the invention illustrated in Fig. 13 comprises: a sub-system 72 for acquiring an aerial image 24 of the photolithography mask 14; and adata analysis device 74 comprising at least one memory 76 and at least one processor78 configured to perform the steps of the method 34 for emulating an aerial image 40of a photolithography mask 14 obtained by a photolithography system or optical maskqualification system 26 according to any of the examples or aspects of the first em- bodiment.The subsystem 72 for acquiring an aerial image 24 of the photolithography mask 14can, for example, comprise an aerial image measurement system. The subsystem 72for obtaining an aerial image 24 of the photolithography mask 14 provides the aerialimage 24 to the data analysis device 74. The data analysis device 74 includes a pro-cessor 78, e.g., implemented as a CPU or GPU. The processor 78 can receive theaerial image 24 via an interface 80. The processor 78 can load program code from amemory 76, e.g., program code for evaluating the quality of a photolithography mask14 according to the first embodiment described above. The processor 78 can executethe program code. Reference throughout this specification to “an embodiment” or “an example” or “an aspect” means that a particular feature, structure or characteristic described in con- nection with the embodiment, example or aspect is included in at least one embodi- ment, example or aspect. Thus, appearances of the phrases “according to an embod- iment”, “according to an example” or “according to an aspect” in various places throughout this specification are not necessarily all referring to the same embodiment, example or aspect, but may. Furthermore, the particular features or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments. Furthermore, while some embodiments, examples or aspects described herein in- clude some but not other features included in other embodiments, examples or as- pects combinations of features of different embodiments, examples or aspects are meant to be within the scope of the claims, and form different embodiments, as would be understood by those skilled in the art. The invention can be described by the following clauses:1. A method 34 for emulating an aerial image 40 of a photolithography mask 14 ob-tained by a photolithography system 10, 10’ or an optical mask qualification sys- tem 26, the method comprising:i.) Obtaining one or more parameters q describing the photolithography pro-cess in the photolithography system 10, 10’ or the optical mask qualification system 26;ii.) Acquiring an aerial image 24 of the photolithography mask 14 using an in-spection system 22; iii.) Obtaining an underlying design 20 of the photolithography mask 14;iv.) Generating a plausible design 38 of the acquired aerial image 24 using theunderlying design 20 by solving an optimization problem that minimizes the deviation of a simulated aerial image 46 of the plausible design 38 from the acquired aerial image 24;v.) Emulating an aerial image 40 of the photolithography mask 14 using thegenerated plausible design 38 of the acquired aerial image 24 and the one or more parameters q describing the photolithography process in the pho- tolithography system 10, 10’ or optical mask qualification system 26.The method of clause 1, wherein the underlying design 20 and the plausible de-sign 38 are represented in a vector format.The method of any one of the preceding clauses, wherein the underlying design20 and the plausible design 38 are represented by non-binary images.The method of any one of the preceding clauses, wherein the underlying design20 of the photolithography mask 14 is generated from the acquired aerial image 24.The method of any one of the preceding clauses, wherein solving the optimizationproblem in step iv comprises applying a machine learning model 44 to the ac-quired aerial image 24, wherein the machine learning model 44 is trained to map an acquired aerial image 24 to a plausible design 38 of the acquired aerial image 24.The method of any one of the preceding clauses, wherein solving the optimizationproblem in step iv comprises minimizing the deviation of a simulated aerial image46 of the plausible design 38 from the acquired aerial image 24, wherein theplausible design 38 is obtained by modifying the underlying design 20 of the ac-quired aerial image 24.The method of any one of the preceding clauses, wherein the simulated aerialimage 46 of the plausible design 38 in step iv is obtained by applying an aerialimage simulation method 48 to the plausible design 38 that simulates the imagingprocess in the inspection system 22.The method of any one of the preceding clauses, wherein the emulated aerialimage 40 in step v is obtained by applying an aerial image simulation method 48 to the generated plausible design 38 that simulates the imaging process in the photolithography system 10, 10’ or the optical mask qualification system 26.9. The method of clause 7 or 8, wherein the aerial image simulation method 48comprises the use of a physics-based model for simulating an aerial image 46from a design.10. The method of any one of clauses 7 to 9, wherein the aerial image simulationmethod 48 comprises a machine learning model 66.11. The method of clause 10, wherein the machine learning model 66 maps a designto an aerial image 46.12. The method of any one of clauses 7 to 9, wherein the aerial image simulationmethod 48 comprises the use of a physics-based model for simulating an aerialimage 46 from a design, and wherein a machine learning model 64 is subse- quently applied to the simulated aerial image to improve its accuracy.13. The method of any one of clauses 7 to 12, wherein the aerial image simulationmethod 48 simulates an aerial image 46 from the plausible design 38 under illu-mination of a corresponding photolithography mask 14 by incident electromag-netic waves 222 in an optical system, in particular in an inspection system 22, a photolithography system 10, 10’ or an optical mask qualification system 26, and comprises: a) Approximately simulating the propagation of the incident electromagneticwaves 222 within a first section 225 of the photolithography mask 14 that comprises multiple structures 226; b) Simulating the propagation of the simulated electromagnetic waves 222 fromstep a within a second section 227 of the photolithography mask 14 analyti-cally or numerically; c) Simulating a representation of an electromagnetic near field 220 of the de-sign by propagating the simulated electromagnetic waves from step b to anear field plane 252; and d) Simulating an aerial image from the plausible design 38 by applying a simu-lation of an imaging process of the optical system to the representation of theelectromagnetic near field 220.14. The method of clause 13, wherein the propagation of the incident electromagneticwaves within the first section 225 of the photolithography mask 14 in step a isapproximately simulated using a Helmholtz equation.15. The method of clause 13, wherein the propagation of the incident electromagneticwaves within the first section 225 of the photolithography mask 14 in step a isapproximately simulated using a machine learning model.16. The method of clause 14, wherein the Helmholtz equation is approximated usinga forward Helmholtz equation.17. The method of clause 16, wherein the forward Helmholtz equation is solved usinga beam propagation method.18. The method of clause 16, wherein the forward Helmholtz equation is solved usinga wave propagation method that approximately describes the propagation of electromagnetic waves 222 through an inhomogeneous medium.19. The method of any one of the preceding clauses, wherein a parametric represen-tation 50 of the underlying design 20 is optimized by the optimization problem.20. The method of clause 19, wherein the parametric representation 50 describesstructure boundaries of the underlying design 20.21. The method of clause 19 or 20, wherein the parametric representation 50 com-prises contours 52 represented by graphs containing nodes and edges, whose location is optimized by solving the optimization problem.22. The method of any one of clauses 19 to 21, wherein the optimization problemimposes further constraints on the parametric representation 50 of the underlying design 20.23. The method of any one of the preceding clauses, wherein the optimization prob-lem comprises parameters that describe a deviation from the underlying design 20, and wherein the optimization problem imposes a sparsity constraint on these parameters.24. A method 67 for evaluating the quality of a photolithography mask 14 for use ofthe photolithography mask 14 in a photolithography system 10, 10’ or in an opticalmask qualification system 26, the method comprising:- Emulating an aerial image 40 of the photolithography mask 14 obtained bythe photolithography system 10, 10’ or the optical mask qualification system26 according to any one of the preceding clauses; -Evaluating the quality of the photolithography mask 14 using the emulatedaerial image 40.25. The method of clause 24, wherein the quality of the photolithography mask 14 isevaluated by obtaining one or more potential defect locations in the photolithog- raphy mask 14 and analyzing the potential defect locations in the emulated aerial image 40.26. The method of clause 24, wherein the quality of the photolithography mask 14 isevaluated by classifying potential defects in the photolithography mask according to a defect likelihood using the emulated aerial image 40.27. A computer implemented method for training a machine learning model 44, 66according to any one of clauses 5, 9, 10 or 11.28. An inspection system 70 for evaluating the quality of a photolithography mask 14for use of the photolithography mask 14 in a photolithography system 10, 10’ orin an optical mask qualification system 26, the inspection system 70 comprising:a. a subsystem 72 for acquiring an aerial image 24 of the photolithographymask 14; b. a data analysis device 74 comprising at least one memory 76 and at leastone processor 78 configured to perform the steps of the method 67 for eval- uating the quality of the photolithography mask 14 according to any one of clauses 24 to 26.In summary, the invention relates to a method for emulating an aerial image 40 of aphotolithography mask obtained by a photolithography system, the method compris- ing: a) obtaining one or more parameters q describing the photolithography process in the photolithography system; b) acquiring an aerial image 24 of the photolithogra- phy mask using an inspection system; c) obtaining an underlying design 20 of the photolithography mask; d) generating a plausible design 38 of the acquired aerial im- age 24 using the underlying design 20 by solving an optimization problem; and e) emulating an aerial image 40 of the photolithography mask using the plausible design 38 and the one or more parameters q describing the photolithography process. The invention also relates to a method for evaluating the quality of a photolithography mask and a corresponding system.

[0002] Reference number list 10, 10’ Photolithography system12 Radiation source14 Photolithography mask14’ Transmission-based photolithography mask14’’ Reflection-based photolithography mask16 Illumination optics17 Projection optics18 Wafer plane19 Projection section20 Underlying design22 Inspection system24 Acquired aerial image26 Optical mask qualification system28 Further aerial image30 Defect32 Printing defect34 Method38 Plausible design40 Emulated aerial image42 Training data44 Machine learning model46 Simulated aerial image48 Aerial image simulation method50 Parametric representation52 Contour54 Optimization55 Initial parametric representation56 Optimized parametric representation58 Deviation60 Over-parameterized parametric representation62 Transformation method64 Simulation method66 Machine learning model67 MethodPotential defect locationInspection systemSubsystemNot quite rigorous aerial image simulation methodNear fieldElectromagnetic waveGratingFirst sectionStructuresSecond sectionNon-structuresStructure planeBoundary planeBase planeMultilayerOptical thin filmCapping layerEffective mirror planeSubstrate layerMask carrierMain propagation directionNear field planeNormalWave vectorAerial imageBand-limited characteristic functionDesignElementsRigorous simulation

Claims

Claims1. A method (67) for evaluating the quality of a photolithography mask (14) for useof the photolithography mask (14) in a photolithography system (10, 10’) or in anoptical mask qualification system (26), the method comprising:i.) Obtaining one or more parameters (q) describing a photolithography pro-cess in the photolithography system (10, 10’) or in the optical mask qualifi-cation system (26);ii.) Acquiring an aerial image (24) of the photolithography mask (14) using aninspection system (22); iii.) Obtaining an underlying design (20) of the photolithography mask (14);iv.) Generating a plausible design (38) of the acquired aerial image (24) usingthe underlying design (20) by solving an optimization problem that mini-mizes the deviation of a simulated aerial image (46) of the plausible design (38) from the acquired aerial image (24); v.) Emulating an aerial image (40) of the photolithography mask (14) using thegenerated plausible design (38) of the acquired aerial image (24) and the one or more parameters (q) describing the photolithography process in the photolithography system (10, 10’) or optical mask qualification system (26);vi.) Evaluating the quality of the photolithography mask (14) using the emulatedaerial image (40).

2. The method of claim 1, wherein the underlying design (20) and the plausible de-sign (38) are represented in a vector format.

3. The method of any one of the preceding claims, wherein the underlying design(20) and the plausible design (38) are represented by non-binary images.

4. The method of any one of the preceding claims, wherein the underlying design(20) of the photolithography mask (14) is generated from the acquired aerial im- age (24).

5. The method of any one of the preceding claims, wherein solving the optimizationproblem in step iv) comprises applying a machine learning model (44) to the ac- quired aerial image (24), wherein the machine learning model (44) is trained tomap an acquired aerial image (24) to a plausible design (38) of the acquired aerialimage (24).

6. The method of any one of the preceding claims, wherein solving the optimizationproblem in step iv) comprises minimizing the deviation of a simulated aerial image (46) of the plausible design (38) from the acquired aerial image (24), wherein the plausible design (38) is obtained by modifying the underlying design (20) of the acquired aerial image (24).

7. The method of any one of the preceding claims, wherein the simulated aerialimage (46) of the plausible design (38) in step iv) is obtained by applying an aerial image simulation method (48) to the plausible design (38) that simulates the im-aging process in the inspection system (22).

8. The method of any one of the preceding claims, wherein the emulated aerial im-age (40) in step v) is obtained by applying an aerial image simulation method (48)to the generated plausible design (38) that simulates the imaging process in the photolithography system (10, 10’) or the optical mask qualification system (26).

9. The method of claim 7 or 8, wherein the aerial image simulation method (48)comprises the use of a physics-based model for simulating an aerial image (46)from a design.

10. The method of any one of claims 7 to 9, wherein the aerial image simulationmethod (48) comprises a machine learning model (66).

11. The method of claim 10, wherein the machine learning model (66) maps a designto an aerial image (46).

12. The method of any one of claims 7 to 9, wherein the aerial image simulationmethod (48) comprises the use of a physics-based model for simulating an aerialimage (46) from a design, and wherein a machine learning model (64) is subse-quently applied to the simulated aerial image to improve its accuracy.

13. The method of any one of claims 7 to 12, wherein the aerial image simulationmethod (48) simulates an aerial image (46) from the plausible design (38) underillumination of a corresponding photolithography mask (14) by incident electro-magnetic waves (222) in an optical system, in particular in an inspection system(22), a photolithography system (10, 10’) or an optical mask qualification system (26), and comprises: a) Approximately simulating the propagation of the incident electromagneticwaves (222) within a first section (225) of the photolithography mask (14) that comprises multiple structures (226); b) Simulating the propagation of the simulated electromagnetic waves (222)from step a) within a second section (227) of the photolithography mask (14)analytically or numerically; c) Simulating a representation of an electromagnetic near field (220) of the de-sign by propagating the simulated electromagnetic waves from step b) to anear field plane (252); and d) Simulating an aerial image from the plausible design (38) by applying a sim-ulation of an imaging process of the optical system to the representation ofthe electromagnetic near field (220).

14. The method of claim 13, wherein the propagation of the incident electromagneticwaves within the first section (225) of the photolithography mask (14) in step a) isapproximately simulated using a Helmholtz equation.

15. The method of claim 13, wherein the propagation of the incident electromagneticwaves within the first section (225) of the photolithography mask (14) in step a) isapproximately simulated using a machine learning model.

16. The method of claim 14, wherein the Helmholtz equation is approximated using aforward Helmholtz equation.

17. The method of claim 16, wherein the forward Helmholtz equation is solved usinga beam propagation method.

18. The method of claim 16, wherein the forward Helmholtz equation is solved usinga wave propagation method that approximately describes the propagation of electromagnetic waves (222) through an inhomogeneous medium.

19. The method of any one of the preceding claims, wherein a parametric represen-tation (50) of the underlying design (20) is optimized by the optimization problem.

20. The method of claim 19, wherein the parametric representation (50) describesstructure boundaries of the underlying design (20).

21. The method of claim 19 or 20, wherein the parametric representation (50) com-prises contours (52) represented by graphs containing nodes and edges, whose location is optimized by solving the optimization problem.

22. The method of any one of claims 19 to 21, wherein the optimization problem im-poses further constraints on the parametric representation (50) of the underlying design (20).

23. The method of any one of the preceding claims, wherein the optimization problemcomprises parameters that describe a deviation from the underlying design (20), and wherein the optimization problem imposes a sparsity constraint on these pa- rameters.

24. The method of any one of the preceding claims, wherein the quality of the photo-lithography mask (14) is evaluated by obtaining one or more potential defect loca- tions in the photolithography mask (14) and analyzing the potential defect loca- tions in the emulated aerial image (40).

25. The method of any one of the preceding claims, wherein the quality of the photo-lithography mask (14) is evaluated by classifying potential defects in the photoli- thography mask according to a defect likelihood using the emulated aerial image (40).

26. A computer implemented method for training a machine learning model (44, 66)configured for being used in a method according to any one of claims 5, 9, 10 or11.

27. An inspection system (70) for evaluating the quality of a photolithography mask(14) for use in a photolithography system (10, 10’) or in an optical mask qualifi-cation system (26), the inspection system (70) comprising:a. a subsystem (72) for acquiring an aerial image (24) of the photolithographymask (14); b. a data analysis device (74) comprising at least one memory (76) and at leastone processor (78) configured to perform the steps of the method (67) forevaluating the quality of the photolithography mask (14) according to any oneof claims 1 to 25.

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