Computational lithography simulation using a vector scatter model

Abstract

A non-transitory computer-readable medium stores a set of instructions that is executable by at least one processor of an apparatus to cause the apparatus to perform operations for simulating an aerial image in a simulated lithography process. The operations include obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source. The operations also include decomposing the near-field solution into a plurality of diffraction orders. The operations also include generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions. The operations also include generating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.

Description

COMPUTATIONAL LITHOGRAPHY SIMULATION USING A VECTOR SCATTER MODELCROSS REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to U.S. Application No. 63 / 765,471, filed February 28, 2025, and which is incorporated herein in its entirety by reference.FIELD

[0002] The description herein relates to lithographic apparatuses and processes. More particularly, the description herein relates computational lithographyBACKGROUND

[0003] A lithographic apparatus is a machine that applies a desired pattern onto a substrate by illuminating a mask that includes the pattern. Lithographic apparatuses are used in the manufacture of integrated circuits (ICs) having very small nanoscale features. An IC chip (e.g., a processor) can be as small as a person’s thumbnail and yet include billions of transistors. Making an IC is a complex and time-consuming process, with circuit components in different layers and including hundreds of individual steps. Errors in even one lithography step has the potential to result in problems with the final IC and can cause device failure. High process yield and high wafer throughput can be impacted by the presence of defects.

[0004] As feature sizes become significantly smaller than the wavelength of the illumination used to transfer the pattern, it is increasingly more difficult to maintain adequate process margins in the lithography process. For example, an aerial image created by a mask and exposure tool lose contrast and sharpness as the ratio of feature size to wavelength decreases. Loss of sharpness and contrast decreases accuracy of pattern transfer. Computational lithography techniques can be used to improve accuracy by simulating a lithography process and optimizing lithography process parameters. Some process parameters can relate to patterns on the mask. For example, the simulated illumination (electromagnetic field) in the vicinity of a mask (near field) can carry the pattern information from a resulting light-matter interaction between illumination and mask pattern. The simulated near field image of the patterned illumination near the mask is represented as a mask image. The simulation then propagates the patterned illumination into the far field, which interacts with a resist on a substrate. The simulated far field image is represented as an aerial image. The chemical alteration of the resist is simulated, and the simulated image that represents the altered resist is a resist image. An etch image can also be used, which represents an etch simulation result of the altered resist (or the etching of unaltered resist for a negative resist).

[0005] Abbe simulation can be used to simulate the optics of a lithography process. For simulations of two-dimensional and three-dimensional physical spaces, it is customary to dissect the continuous space into grids (discrete points) for purposes of computation. When information in between the gridpoints are needed, interpolation can be used. Averaging and interpolation techniques can carry a risk of introducing numerical errors and also impose a high computational burden.SUMMARY

[0006] Embodiments of the present disclosure provide a system and method for simulating an aerial image in a simulated lithography process.

[0007] In some embodiments, a non-transitory computer readable medium for simulating an aerial image in a simulated lithography process is provided. The non-transitory computer readable medium can store a set of instructions. The instructions are executable by at least one processor of an apparatus to cause the apparatus to perform operations for simulating the aerial image. The operations can comprise obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source. The operations can also comprise decomposing the near-field solution into a plurality of diffraction orders. The operations can also comprise generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions. The operations can also comprise generating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.

[0008] In some embodiments, a system for simulating an aerial image in a simulated lithography process is provided. The system can comprise one or more processors and one or more memory devices. The memory devices can store a set of instructions that is executable by the one or more processors to cause the system to perform operations for simulating an aerial image in a simulated lithography process. The operations can comprise The operations can comprise obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source. The operations can also comprise decomposing the near-field solution into a plurality of diffraction orders. The operations can also comprise generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions. The operations can also comprise generating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.

[0009] In some embodiments, a method for simulating an aerial image in a simulated lithography process is provided. The method can comprise obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source. The method can also comprise decomposing the near-field solution into a plurality of diffraction orders. The method can also comprise generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions. The method can also comprise generating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.BRIEF DESCRIPTION OF FIGURES

[0010] The above and other aspects of the present disclosure will become more apparent from the description of example embodiments, taken in conjunction with the accompanying drawings.

[0011] FIG. 1 is a schematic diagram illustrating example subsystems of a lithographic apparatus, consistent with embodiments of the present disclosure.

[0012] FIG. 2 is a flowchart of an example method for simulating lithography in a lithographic apparatus, consistent with embodiments of the present disclosure.

[0013] FIG. 3 is a flowchart of an example method for source or mask optimization of a patterning process, consistent with embodiments of the present disclosure.

[0014] FIG. 4 illustrates an example pupil profile, consistent with embodiments of the present disclosure.

[0015] FIG. 5 illustrates an example near-field grid of a near-field image in terms of diffraction orders, consistent with embodiments of the present disclosure

[0016] FIG. 6 illustrates an example flowchart of a method for a vector-scatter-model-based process for generating raw near-field data and an aerial image, consistent with embodiments of the present disclosure.

[0017] FIG. 7 illustrates an example flowchart of a vector scatter model generator, consistent with embodiments of the present disclosure.

[0018] FIG. 8 illustrates an example of a light-mask interaction based on its input and output illumination, consistent with embodiments of the present disclosure.

[0019] FIG. 9 illustrates example contents of a vector scatter model, consistent with embodiments of the present disclosure.

[0020] FIG. 10 illustrates an example error analysis scheme used for analyzing the impact of expansion terms used for a fit operation, consistent with embodiments of the present disclosure.

[0021] FIG. 11 illustrates a flowchart of an example method for simulating a mask image, consistent with embodiments of the present disclosure.

[0022] FIG. 12 illustrates a block diagram of an example server, consistent with some embodiments of the disclosure.DETAILED DESCRIPTION

[0023] Reference will now be made in detail to example embodiments, examples of which are illustrated in the drawings. The following description refers to the accompanying drawings in which the same numbers in different drawings represent the same or similar elements unless otherwise represented. The implementations set forth in the following description of example embodiments do not represent all implementations consistent with the invention. Instead, they are merely examples of apparatuses, systems, and methods consistent with aspects related to subject matter that may be recited in the appended claims.

[0024] To print nano-accurate features on a substrate (e.g., nano-transistors), a lithographic apparatus can illuminate a mask. The mask acts like a stencil for the illumination and the patterned illumination is projected onto the substrate, thereby achieving a pattern transfer from mask to substrate. It is desirable for the illumination used in this process to be conditioned to a high degree of accuracy in terms of wavelength, dose, uniform intensity spread, uniform wavefront, or the like. Instability of the illumination can introduce defects and reduce yield. A goal of the manufacturing process is to avoid such defects to maximize the number / yield of functional ICs made in the process.

[0025] Along an illumination path of a lithographic apparatus (e.g., from the source, to the mask, and subsequently to the substrate), the illumination can interact with a plurality of optical hardware, such as an array of discrete adjustable mirrors disposed at a pupil plane of the source. Each mirror can be toggled on and off. A given configuration of on / off mirrors defines a pupil profile. Considering that there can be numerous discrete mirrors at the pupil plane, each mirror activated / deactivated can improve or worsen the fidelity of the image of the mask pattern that is focused at the substrate (e.g., an aerial image). Hence, selecting which mirrors to activate / deactivate can be an optimization problem. Computationally optimizing the pupil profile (optimizing the source) via lithographic simulation can be referred to as source optimization (SO).

[0026] Computational lithography can be used to simulate the interaction between illumination and mask features while adjusting / optimizing the mask pattern itself (e.g., biasing the mask) in order to realize the sharp features on the substrate itself. Computationally optimizing a mask via lithographic simulation can be referred to as mask optimization (MO). Computationally co-optimizing a source and a mask via lithographic simulation can be referred to as source-mask optimization (SMO). Other optimizations directed to other parameters of lithography processes can also be implemented using embodiments of the present disclosure.

[0027] Computational techniques for lithography imaging simulation (e.g., Abbe simulation method) can employ a mask model based on an electromagnetic solver while using a grid system to define coordinates in two dimensions and three dimensions. The Abbe simulation method calculates and determines an image created by a light source, which can be used to simulate how an optical system forms an image (e.g., aerial image) based on parameters of the optical system used to create the image. It can be challenging to perform the simulations in the continuous realm of two and three dimensional spaces, and it is rare to be able to calculate analytical solutions for electromagnetic wave propagation, even for simple boundary conditions. The grid system of Abbe simulation allows for a discretized numerical approach when modeling and simulating electromagnetic wave propagation in connection with lithography imaging. However, lithography imaging simulations that use grid system can encounter issues when important electromagnetic information is to be mapped to, or referenced from, locations of the grid where a grid point does not exist (e.g., in between grid points). To address this problem, averaging or interpolation can be used to obtain or otherwise infer information between the grid points. In conventional methods, an aerial image can be simulated based on a mask imagelocated near the mask (also referred to as a near-field image in reference to the near field interaction between the illumination and the mask). In the Abbe simulation method, the electromagnetic field information of the mask image, averaged for in between grid points, can be referred to as the averaged near-field per sub-source. However, averaging and interpolation are accompanied by risk of numerical error and high computational intensity.

[0028] Embodiments of the present disclosure provide method for generating an aerial image using a vector scatter model, which mitigate at least the above noted problems of interpolation and averaging. The vector scatter model can be used to replace the mask image, thereby eliminating any issues that accompany averaging methods such as the averaged near-field per sub-source. In the vector scatter model, each near field solution is represented by using a set of diffraction Jones Pupils. Elements of Jones calculus, such as Jones pupils and Jones matrices, will be described below with respect to FIGS. 7-10. Specifically, the EM solver solution for each angle is decomposed into different diffraction orders that are each fit into Zernikes (or other orthogonal functions), thereby characterizing each diffraction order with a diffraction-based Jones pupil, or collectively a “Vector Scatter Model.” The diffraction-based Jones pupils are used directly as input to downstream optical model.Diffraction-based Jones pupils can be generated by defining a relationship between the incident EM field at the mask and the output diffracted near field. Furthermore, the diffraction-based Jones pupils describe the pupil plane as a continuum (no discrete grid points), thereby circumventing the issues that accompany averaging and interpolation techniques.

[0029] Objects and advantages of the disclosure can be realized by the elements and combinations as set forth in embodiments described herein. However, embodiments of the present disclosure are not necessarily required to achieve such example objects or advantages. Some embodiments can achieve a different feature or enhancement without necessarily achieving any expressly stated object or advantage.

[0030] As used herein, unless specifically stated otherwise, the term “or” encompasses all possible combinations, except where infeasible. For example, if it is stated that a component can comprise A or B, then, unless specifically stated otherwise or infeasible, the component can comprise A, or B, or A and B. As a second example, if it is stated that a component can comprise A, B, or C, then, unless specifically stated otherwise or infeasible, the component can comprise A, or B, or C, or A and B, or A and C, or B and C, or A and B and C.

[0031] Relative dimensions of components in drawings may be exaggerated for clarity. Within the following description of drawings, the same or like reference numbers refer to the same or like components or entities, and only the differences with respect to the individual embodiments are described.

[0032] The term “patterning device” may be considered synonymous with similar terms of art, such as “reticle” or “mask.” The term “patterning device” used herein should be broadly interpreted asreferring to any device that can be used to impart a pattern on a cross section of a radiation beam. The radiation beam then can recreate the pattern in a target portion of a substrate.

[0033] The term “projection system” used herein should be broadly interpreted as encompassing any type of projection system, including refractive, reflective, catadioptric, magnetic, electromagnetic, or electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, or for other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term “projection lens” herein may be considered as synonymous with the more general term “projection system.”

[0034] Illumination can be understood to be a form of radiation. The terms “radiation” and “illumination” can be used herein interchangeably. Embodiments described in the context of illumination are also applicable in the context of radiation in general. Furthermore, the terms “radiation” and “beam” can encompass all types of electromagnetic radiation, including ultraviolet radiation (e.g., with a wavelength of 365, 248, 193, 157 or 126 nm) and EUV (extreme ultra-violet radiation, e.g., having a wavelength in the range 5-20 nm, such as 13.5 nm).

[0035] The term “optimizing” and “optimization” as used herein can indicate adjusting a lithographic apparatus such that results or processes of lithography have more desirable characteristics, such as higher accuracy of projection of design layouts on a substrate, larger process windows, etc.

[0036] FIG. 1 illustrates an example lithographic apparatus 100, consistent with embodiments of the present disclosure. In some embodiments, lithographic apparatus 100 comprises a radiation source 102, which can be a deep-ultraviolet excimer laser source or other type of source including an extreme ultra violet (EUV) source (the lithographic apparatus itself need not have the radiation source), illumination optics which define the partial coherence (denoted as sigma) and which can include optic components 104, 106a, and 106b that shape radiation from source 102; a patterning device 108; and transmission optics 106c that project an image of the patterning device pattern onto a substrate plane 109. An adjustable filter or aperture 107 at disposed in among the optics can restrict the range of beam angles that impinge on the substrate plane 109. A largest possible angle 0max can define the numerical aperture NA of the projection optics as NA = n sin(0max), where n is the index of refraction of the medium in which the final lens element is working (e.g., a lens closest to the substrate).

[0037] In an optimization process of a lithographic projection system, a figure of merit of the system can be represented as a cost function. The optimization process can determine a set of parameters (design variables) of the system that minimizes the cost function. The cost function can have any suitable form depending on the goal of the optimization. For example, the cost function can be a weighted root mean square (RMS) of deviations of certain characteristics (evaluation points) of the system with respect to the intended values (e.g., ideal values) of these characteristics. The cost function can be the maximum of these deviations (e.g., worst deviation). The term “evaluation points” herein should be interpreted broadly to include any characteristics of the system. The design variables of the system can be confined to finite ranges or be interdependent due to practicalities ofimplementations of the system. In case of a lithographic apparatus, the constraints are often associated with physical properties and characteristics of the hardware such as tunable ranges or patterning device manufacturability design rules, and the evaluation points can include physical points on a resist image on a substrate, as well as non-physical characteristics such as dose and focus of the illumination used.

[0038] In a lithographic apparatus, a source can provide illumination (e.g., light). Projection optics can direct and shape the illumination via a patterning device and onto a substrate. The term “projection optics” is broadly defined to include any optical component that can alter the wavefront of the radiation beam. For example, projection optics can include at least some of components 104, 106a, 106b, and 106c. An aerial image is the radiation intensity distribution at substrate level. A resist layer on the substrate is exposed and the aerial image is transferred to the resist layer as a latent “resist image” therein. The resist image can be defined as a spatial distribution of solubility of the resist in the resist layer. A resist model can be used to calculate the resist image from the aerial image. An example of a resist model can be found in U.S. Patent No. 8,200,468, the contents of which are incorporated herein by reference in their entirety. The resist model is related to properties of the resist layer (e.g., effects of chemical processes which occur during exposure, post-exposure bake (PEB), and development). Optical properties of the lithographic apparatus (e.g., properties of the source, the patterning device, and the projection optics) dictate the aerial image. Since the patterning device used in the lithographic apparatus can be changed, it is desirable to separate the optical properties of the patterning device from the optical properties of the rest of the lithographic apparatus including at least the source and the projection optics.

[0039] While FIG. 1 illustrates a transmission-based lithographic apparatus (e.g., the mask or patterning device 108 allows illumination to be transmitted through), embodiments of the present disclosure can be implemented with any suitable lithographic apparatus, for example, a reflectionbased lithographic apparatus (e.g., the mask reflects illumination).

[0040] FIG. 2 illustrates a flowchart of an example method 200 for simulating lithography in a lithographic apparatus, consistent with embodiments of the present disclosure. In some embodiments, a source model 202 represents optical characteristics of the source (e.g., including radiation intensity distribution, phase distribution, or the like). A projection optics model 204 can represent optical characteristics of the projection optics (e.g., including changes to radiation intensity / phase distribution caused by the projection optics). A mask model 206 can represent optical characteristics of a design layout (e.g., including changes to radiation intensity / phase distribution caused by a given design layout), which is the representation of an arrangement of features on, or formed by, a patterning device. An aerial image 208 can be simulated from source model 202, projection optics model 204, and mask model 206. A resist image 212 can be simulated from aerial image 208 using a resist model 210. Simulation of lithography can, for example, predict lithographic pattern transfer results, whichcan include feature contours, edge placement errors (EPE), critical dimensions (CDs), or the like, in the resist image.

[0041] It is noted that the source model 202 can represent optical characteristics of the source that include, but are not limited to, NA-sigma (o) settings as well as any particular illumination source shape (e.g., off-axis radiation sources such as annular, quadrupole, and dipole, etc.). Projection optics model 204 can represent the optical characteristics of the projection optics that include, but are not limited to, aberration, distortion, refractive indexes, physical sizes, physical dimensions, or the like. Mask model 206 can represent physical properties of a physical patterning device. An example of a mask model can be found in U.S. Patent No. 7,587,704, the contents of which are incorporated herein by reference in their entirety. A goal of the simulation is to accurately predict feature contours, edge placement errors (EPE), critical dimensions (CDs), or the like, which can then be compared against an intended design for a device (e.g., a simulation to determine whether a mass fabrication of a new CPU architecture is feasible). The intended design is generally defined as a pre-optical proximity correction (OPC) design layout (OPC is sometimes also referred to as “optical and process correction”), which can be provided in a standardized digital file format. The layout file can be in a Graphic Database System (GDS) format, Graphic Database System II (GDS II) format, an Open Artwork System Interchange Standard (OASIS) format, a Caltech Intermediate Format (CIF), or the like. The intended design layout can include patterns or structures for transferring onto a wafer. The patterns or structures can be mask patterns used to transfer features from photolithography masks or reticles to a wafer. In some embodiments, a layout in GDS or OASIS format, among others, can include feature information stored in a binary file format representing planar geometric shapes, text, and other information related to the wafer design.

[0042] From the design layout, one or more portions can be identified, which are referred to as “clips.” In some embodiments, a set of clips is extracted, which represents the complicated patterns in the design layout (typically about 50 to 1000 clips, although any number of clips can be used). It is to be appreciated that these patterns or clips represent small portions (e.g., circuits, cells, or patterns) of the design and especially the clips represent small portions for which particular attention or verification is desirable. In other words, clips can be the portions of the design layout or can be similar or have a similar behavior of portions of the design layout where critical features are identified either by experience (including clips provided by a customer), by trial and error, or by running a fullchip simulation. Clips can contain one or more test patterns or gauge patterns.

[0043] An initial larger set of clips can be provided a priori by a customer based on known critical feature areas in a design layout that could benefit from image optimization. Alternatively, in some embodiments, the initial larger set of clips is extracted from the entire design layout by using some kind of automated algorithm (e.g., machine vision) or manual algorithm that identifies the critical feature areas.

[0044] In some embodiments, an optimization process (e.g., source mask optimization (SMO)) relates to one or more of a patterning process that employs process models (e.g., an optics model, a mask model, a resist model, etc. of FIG. 2). The optimization process can involve execution of the one or more process models and computing a cost function that can be reduced by modifying one or more characteristics (e.g., source, mask pattern, etc.) of the patterning process. In some embodiments, the one or more characteristics is described by design variables. Hence, an optimized characteristic can also be referred to as an optimized design variable, where a design variable is optimized based on a cost function.

[0045] In some embodiments, modifying the one or more characteristics is based on a gradient of the cost function that guides how the characteristic should be modified to reduce the cost function. A cost function can be a function of a certain continuous metric such as an edge placement error (e.g., a difference between contours of printed pattern and a target pattern). Using a continuous metric or a cost function of a continuous nature allows use of gradient-based optimizing algorithms that have acceptable runtime performance of an optimization process.

[0046] Details of example techniques and models used to transform a patterning device pattern into various lithographic images (e.g., an aerial image, a resist image, an etch image, etc.), apply OPC (e.g., using models) and evaluate performance (e.g., in terms of process window) can be found in U.S. Patent Nos. 7,695,876; 7,707,538; 7,747,978; 7,882,480; 8,413,081; 8,438,508; and 9,360,766, the contents of which are incorporated herein by reference in their entirety.

[0047] FIG. 3 illustrates a flowchart of an example method 300 of source or mask optimization of a patterning process, consistent with embodiments of the present disclosure. In a typical high-end design, almost every feature edge can benefit from some modification to achieve printed patterns that come sufficiently close to the target design. These modifications can include shifting or biasing of edge positions or line widths as well as application of “assist” features that are not intended to print themselves, but can affect the properties of an associated primary feature. Furthermore, optimization techniques applied to the source of illumination can have different effects on different edges and features. Optimization of illumination sources can include the use of pupils to restrict source illumination to a selected pattern of light. Embodiments of the present disclosure provide optimization methods that can be applied to both source and mask configurations.

[0048] A method of performing source and mask optimization (SMO) can allow full chip pattern coverage while lowering the computation cost by intelligently selecting a small set of critical design patterns from the full set of clips to be used in SMO. SMO can be performed on these selected patterns to obtain an optimized source. The optimized source can then be used to optimize the mask (e.g., using OPC and local mechanical-stress control) for the full chip, and the results can be compared. Various methods are provided for iteratively converging on an optimal result.

[0049] A target design 301 (e.g., comprising a layout in a standard digital format such as OAIS, GDSII, etc.) for which a lithographic process is to be optimized can include memory, test patterns,and logic. From this design, a full set of clips 302 can be extracted, which represents complex patterns in design 301 (e.g., about 50 to 1000 clips). It is to be appreciated that these clips represent small portions (i.e., circuits, cells, or patterns) of the design for which particular attention and / or verification is of interest. At operation 304, a small subset of clips 306 (e.g., 15 to 50 clips) can be selected from full set of clips 302. As will be explained in more detail below, the selection of clips can be performed such that the process window of the selected patterns matches the process window for the full set of critical patterns as close as possible. The effectiveness of the selection can be measured by the total run time (pattern selection and SMO) reduction.

[0050] At operation 308, SMO can be performed with the selected patterns (15 to 50 patterns) of subset of clips 306. In particularly, an illumination source can be optimized for the selected patterns of subset of clips 306. Examples of other source optimization methods can be found in, for example, U.S. Patent Application Publication No. 2004 / 0265707, the contents of which are incorporated herein by reference in their entirety.

[0051] At operation 310, manufacturability verification of the selected patterns of subset of clips 306 can be performed with the source obtained in operation 308. In particular, verification can include performing an aerial image simulation of the selected patterns of subset of clips 306 and the optimized source and verifying that the patterns will print across a sufficiently wide process window. An example verification process can be found in U.S. Patent No. 7,342,646, the contents of which are incorporated herein by reference in their entirety. If the verification at operation 310 is satisfactory, as determined in operation 312, then processing can advance to full chip optimization (e.g., advanced to operations using optimized source 314). Otherwise, processing can return to operation 308, where SMO is performed again but with a different source or set of patterns. For example, the process performance as estimated by the verification tool can be compared against thresholds for certain process window parameters such as exposure latitude and depth of focus. These thresholds can be predetermined or set by a user.

[0052] After the selected patterns meet lithography performance specification as determined in step 312, the optimized source 314 can be used for optimization of the full set of clips 316 (e.g., originating from full set of clips 302).

[0053] At operation 318, model-based sub-resolution assist feature placement (MB-SRAF) and optical proximity correction (OPC) for all the patterns in the full set of clips 316 can be performed. Examples of MB-SRAF and OPC can be found in U.S. Patent Nos. 5,663,893; 5,821,014; 6,541,167; and 6,670,081, the contents of which are incorporated herein by reference in their entirety.

[0054] At operation 320, using processes similar to step 310, full pattern simulation based manufacturability verification can be performed with the optimized source 314 and the full set of clips 316 as corrected in step 318.

[0055] At operation 322, the performance (e.g., process window parameters such as exposure latitude and depth of focus) of the full set of clips 316 can be compared against subset of clips 306. Forexample, the pattern selection can be considered complete and / or the source is fully qualified for the full chip when the similar (< 10%) lithography performances are obtained for both selected patterns of subset of clips 306 and critical patterns of full set of clips 316.

[0056] Otherwise, at operation 324, hotspots can be extracted. At operation 326, the hotspots can be added to subset of clips 306 and the process starts over. For example, hotspots (e.g., features among the full set of clips 316 that limit process window performance) identified during verification step 320 can be used for further source tuning or to run SMO of operation 308 again. The source can be considered fully converged when the process window of the full set of clips 316 are the same between the last run and the run before the last run of operation 322.

[0057] OPC calibration can be performed by modeling or simulation. For example, for the desired yield, the total number of features, and their respective probabilities of failure, simulation can be performed to optimize OPC for a lowest yielding feature. OPC addresses the fact that, in addition to any demagnification by the lithographic projection apparatus, the final size and placement of an image of the patterning device pattern projected on the substrate will not be identical to, or simply depend only on the size and placement of, the corresponding patterning device pattern features on the patterning device.

[0058] In some embodiments, the measurement data (e.g., stochastic variations) related to the printed pattern can be employed in optimizing the patterning process or adjusting parameters of the patterning process. For small feature sizes and high feature densities present on some design layouts, the position of a particular edge of a given feature can be influenced to a certain extent by the presence or absence of other adjacent features. These proximity effects arise from minute amounts of radiation coupled from one feature to another or non-geometrical optical effects such as diffraction and interference. Similarly, proximity effects can arise from diffusion and other chemical effects during post-exposure bake (PEB), resist development, and etching that generally follow lithography.

[0059] To ensure that the projected image of the patterning device pattern is in accordance with tolerances of a given target design, proximity effects should be predicted and compensated for using sophisticated numerical models, corrections, or pre-distortions of the patterning device pattern. The article “Full-Chip Lithography Simulation and Design Analysis — How OPC Is Changing IC Design,” C. Spence, Proc. SPIE, Vol. 5751, pp 1-14 (2005) provides an overview of “model-based” optical proximity correction processes, the contents of which are incorporated herein by reference in their entirety. In a typical high-end design, almost every feature of the patterning device pattern has some modification to achieve high fidelity of the projected image to the target design. These OPC modifications can include shifting or biasing of edge positions or line widths and / or application of “assist” features that are intended to assist projection of other features.

[0060] Application of model-based OPC to a target design can involve good process models and considerable computational resources, given the many millions of features typically present in a device design. However, applying OPC is generally an empirical, iterative process that does notalways compensate for all possible proximity effects. Therefore, the effect of OPC, e.g., patterning device patterns after application of OPC and any other resolution enhancement technique (RET), should be verified by design inspection, e.g., intensive full-chip simulation using calibrated numerical process models, to reduce or minimize the possibility of design flaws being built into the patterning device pattern. This is driven by the enormous cost of making high-end patterning devices, as well as by the impact on turn-around time by reworking or repairing existing patterning devices once they have been manufactured. OPC and full-chip RET verification can be based on numerical modelling systems and methods. Examples of such methods can be found in U.S. Pat. No. 7,003,758 and an article titled “Optimized Hardware and Software For Fast, Full Chip Simulation”, by Y. Cao et al., Proc. SPIE, Vol. 5754, 405 (2005), the contents of which are incorporated herein by reference in their entirety.

[0061] The illumination source can also be optimized, either jointly with patterning device optimization or separately, to improve the overall lithography fidelity. The terms “illumination source” and “source” can be used interchangeably in this disclosure. Off-axis illumination (e.g., annular, quadrupole, dipole, or the like) can be used to resolve fine structures (e.g., target features) contained in the patterning device. However, when compared to a traditional illumination source, an off-axis illumination source usually provides less radiation intensity for the aerial image. Thus, it is can be desirable to optimize the illumination source to achieve balance between finer resolution (relevant to yield) and reduced radiation intensity (relevant to throughput).

[0062] FIG. 4 illustrates an example pupil profile 400, consistent with embodiments of the present disclosure. In some embodiments, pupil profile 400 is a two-dimensional representation of how illumination intensities are distributed across a pupil plane 402 (the pupil plane can be defined by an xy-coordinate system). Pupil plane 402 can be a pupil plane of source 102 (FIG. 1). The pupil plane of source 102 can be a conjugate patterning device 108 (FIG. 1) (e.g., as dictated by Fourier optics).

[0063] A mask 404 can be used as patterning device 108. Hence, the positions of pupil profile 400 can be associated with different incidence angles o that impinge on mask 404. To better illustrate the incidence angle relationship, pupil profile 400 and pupil plane 402 are re-drawn at a different perspective (denoted as pupil profile 400' and pupil plane 402'). Upon diffractive interaction with mask 404, the incident input illumination can emerge from mask 404 as a plurality of diffraction waves as the output. The output diffraction waves can form a near-field image 406 (also referred to as a mask image).

[0064] Computationally processing different optical planes (such as pupil profile 400) can comprise dissecting the optical planes into pixel grids. The spatial sampling (coarseness or fineness) of the pixel grid can be selectable. Using a fine pixel grid can be helpful where fine resolution is conducive to improving accuracy (e.g., improving fidelity between lithography simulation and actual lithographic fabrication). However, selecting a high pixel density can be computationally intensive and slow.Hence, in some embodiments, a coarse pixel grid 408 is used. Each pixel position is denoted by a cross (e.g., the pixel position is at the center of the cross).

[0065] When coarse pixel grid 408 is used for simulation processes of methods 200 and 300 (FIGS.2 and 3) much of the intensity of pupil profile information can find itself off-pixel or off-center. During simulation, some processes may need off-pixel information. This can be a problem for Abbe simulation, which employs a mask model based on an electromagnetic (EM) solver (e.g., represented in FIG. 2 as mask model 206). One way to address this gap is to interpolate the intensity information in between the pixels. A similar process can be used to interpolate a grid defined near-field 406. When near-field 406 is also discretized using a coarse grid, obtaining an interpolated near-field solution can comprise analyzing nearest neighbor pixels of the near field grid (e.g., weighted averaging). However, this process re-inherits the problem of high computational intensity, as well as introducing new problems, such as averaging errors and an undesirable dependence on pupil profile 400 (e.g., required as an input by the interpolation or averaging software). The quality of near-field image simulation affects accurate production of an aerial image 410 (e.g., can be aerial image 208 (FIG.2)).

[0066] Some embodiments of the present disclosure can advantageously eliminate the dependence on the source shape (e.g., eliminate the pupil profile as an input to a process) by replacing the interpolation or averaging operations with a vector scatter model (VSM). In a VSM, each near field solution is represented by using a set of diffraction Jones pupils. Specifically, the EM solver solution for each angle is decomposed into different diffraction orders that are each fit into Zernikes (or other orthogonal functions), thereby characterizing each diffraction order with a diffraction-based Jones pupil, or collectively a “Vector Scatter Model.” As described below in reference to FIG.8, diffraction-based Jones pupils can be generated by defining a relationship between the incident EM field at the mask and the output diffracted near field. Furthermore, the diffraction-based Jones pupils describe the pupil plane as a continuum (no discrete grid points), thereby circumventing the issues that accompany averaging and interpolation techniques.

[0067] FIG. 5 illustrates an example near-field grid 500 of near-field image 406 (FIG.4) in terms of diffraction orders 502, consistent with embodiments of the present disclosure. Reference to elements of FIG.4 can be made in the description of FIG. 5. In some embodiments, the mask image plane is dissected into discrete pixel elements using near-field grid 500, similar to how pupil profile 400 can be apportioned via coarse grid 408. Diamond symbols are used in near-field grid 500 so as not to confuse the cross symbols of coarse pixel grid 408.

[0068] Mask patterns of mask 404 can comprise diffractive structures. Hence, different positions of the near-field image plane can comprise a one or more diffraction orders. For example, a grid position 504 of near-field grid 500 can be decomposed into diffraction orders 502. The specific diffraction order illustrated in FIG.5 is a non-limiting example. More or fewer diffraction orders can be used (e.g., two or more orders, three or more orders, four or more orders, or the like). The diffraction orders can be two dimensional, e.g., corresponding to the diffractions in X and Y directions respectively. Anon-limiting nomenclature (n,n') for diffraction orders can be used in the present disclosure. The example diffraction orders that are illustrated in FIG. 5 are a 0thorder or central wave (0,0), a (-1,1) order, a (0,1) order, a (1,1) order, a (-1,0) order, a (1,0) order, and unlabeled (-1,-1), (0-1), and (1,-1) orders.

[0069] The diffraction order output from mask 404 can be a function of the interaction between illumination from pupil profile 400 and patterns on mask 404. The interaction can be modeled as an EM problem to be solved via simulation, which is encompassed in the described source model 202 and mask model 206 (e.g., mask model) of FIG. 2. As a precursor to generating aerial image 208 (FIG. 2), a near-field image is obtained by solving the EM interaction of the source and mask. As explained earlier, a problem with conventional methods to interpolate the information in between the grid positions of near-field grid 500 is intense computational burden, introduction of errors in the averaging process, and a dependence on having pupil profile information available. Some embodiments of the present disclosure can overcome at least these problems by implementing a vector scatter model (VSM).

[0070] FIG. 6 illustrates a flowchart of an example method 600 for a VSM-based process for generating raw near-field data and accordingly an aerial image, consistent with embodiments of the present disclosure. In some embodiments, mask model 206 (FIG. 2) can comprise physical information about the mask (e.g., pattern shapes, refractive index, or the like). For computational simulation, the physical information can be provided as information about mask pattern 602 and mask film stack 604. Mask pattern 602 and mask film stack 604 pertain to geometric and physical information about mask 404 (FIG.4) that is provided to an EM solver 606 as input. In FIG. 6, whitebox elements can represent input / output entities of black-box elements (processes). EM solver 606 can simulate the possible light-matter interactions (e.g., via Maxwell’s equations) to output raw near-field data 608. Raw near-field data 608 can correspond to near-field image 406 (FIG. 4) and comprises information about diffraction orders 502. The qualifier “raw” can refer to data that is yet to be transformed into another form or otherwise modified via averaging, interpolation, or fitting.

[0071] According to embodiments of the present disclosure, raw near-field data 608 can be provided as input to a VSM generator 610. The output of VSM generator 610 is a vector scatter model 612. Further details of the operations of VSM generator 610 are described below with respect to FIGS. 7-11. Vector scatter model 612 can represent characteristics of interaction between input illumination and a source of illumination (e.g., diffraction behavior). Specifically, in vector scatter model 612, each near field solution can be represented by decomposing raw near-field data 608 into different diffraction orders that are each fit into Zernikes (or other orthogonal basis functions). The orthogonal basis functions solve are continuous over the mask image plane, thereby providing electromagnetic information for even the grid positions that are in between grid points. Hence, vector scatter model 612 circumvents the issues associated with averaging and interpolating electromagnetic information in between grid points. Vector scatter model 612 can be used to predict the details of the near-fieldimage based on the shape (spatial distribution) of illumination from a source 614 (e.g., can be a pupil profile). Projection simulation process 616 can receive vector scatter model 612 and source 614 as input. Projection simulation process 616 can generate aerial image 618 (e.g., can correspond to aerial images 208 or 410 (FIGS.2 and 4)).

[0072] A problem of conventional methods relates to the apportionment of raw near-field data 608 according to near-field grid 500 (FIG. 5) (e.g., risk of generating errors). Also, in the flow of method 600, a conventional averaging or interpolation process would be introduced at about where VSM generator 610 is located. Since conventional averaging and interpolation requires information about the exact shape of the source, such averaging and interpolation processes can require source 614' as input (source 614' can represent a pupil profile used in a conventional averaging / interpolation process flow). A technical significance of vector scatter model 612 is that it is independent of any source shape. Hence, it is not necessary to directly input source 614' into VSM generator 610. For example, a specific source shape is not needed for VSM generator 610, computations can be facilitated by initially assuming a plain pupil profile with uniform intensity across the entire pupil plane (no details in the source shape) to compute vector scatter model 612 without adverse consequence, whereas a conventional averaging or interpolation technique relies on knowing the detailed source shape in order to output an accurate aerial image 618.

[0073] FIG. 7 illustrates a flowchart of an example VSM generator 700 operable to generate a VSM, consistent with embodiments of the present disclosure. In some embodiments, VSM generator 700 represents a more detailed view of operations within VSM generator 610 (FIG.6). As discussed above in reference to FIG.6, raw near-field data 704 (e.g. raw near-field data 608 in FIG.6) can be provided to VSM generator 700 as input. In FIG. 7, white -box elements can represent input / output entities of black-box elements (processes).

[0074] As explained above, raw near-field data 704 can be generate by using an EM solver and can comprise information about diffractive behavior of mask 404 (FIG.4). A decomposition process 706 can be executed on the diffraction information represented by the raw near-field data 704. The output can be the constituent diffraction orders 708 of raw near-field data 704. Diffraction orders 708 can be represented as orthogonal basis functions using fitting processes 710. In this example, fitting processes 710 can comprise, for example, operations to fit diffraction orders 708 into Zernike polynomials (Zernikes). The output of fitting processes 710 can be diffraction-based Jones pupils (DJP) 712. However, the present disclosure is not limited to Zernikes. Any suitable set of basis functions can be used without departing from the scope of the present disclosure (e.g., Tatian polynomials).

[0075] In optics, polarized illumination can be described using Jones calculus. In Jones calculus, polarized illumination can be represented as a vector (e.g., a Jones vector) and an optical element or system, such as lenses, can be represented as a matrix (e.g., a Jones matrix). When illumination interacts with the optical element, a solution for the output illumination from the optical element canbe found by calculating the product of the Jones matrix of the optical element and the Jones vector of the incident illumination. When Jones calculus is implemented in the context of a pupil-based optical element of a lithographic apparatus (e.g., a source, a projection system, or the like), information of the Jones matrix can be framed as Jones pupil that describes the imaging properties of the pupil-based optical element. Diffraction-based Jones pupils 712 is an enhancement over conventional implementations of a Jones pupil. Diffraction-based Jones pupils 712 is a plurality of Jones pupils that are distinguishable from one another according to one or more different associations, such as diffraction order, angle of incidence on a mask (which maps to a location at pupil plane of a source), polarization, real or imaginary component, or the like.

[0076] A grouping operation 714 can be performed on diffraction-based Jones pupils 712. Grouping operation 714 can aggregate the different diffraction-based Jones pupils 712 into a single collection or grouping (e.g., table, library, file, or the like) that a downstream process can then reference with specificity to association or classification (e.g., diffraction order, angle of incidence on a mask, polarization, real or imaginary component, or the like). The collection or grouping is vector scatter model 716. Vector scatter model 716 can correspond to vector scatter model 612 of FIG. 6 (e.g., see FIG. 9 for a visual representation of a grouping of different DJPs according to diffraction order, angle of incidence, polarization, and the like).

[0077] FIG. 8 illustrates an example of a light-mask interaction based on its input and output illumination, consistent with embodiments of the present disclosure. In some embodiments, the lightmatter interaction illustrated in FIG. 8 corresponds to the physical interaction solved by EM solver 606 (FIG. 6). As explained below, embodiments of VSMs are not limited by discrete grid points and instead provide a continuum of spatial information to fully describe pupil illumination properties. VSMs can be used to replace the mask image, thereby eliminating any issues that accompany averaging and interpolation methods that attempt to cope with the discrete nature of a grid-based mask image. A computational lithography process can simulate an interaction between a sourced beam of illumination from source 102 (FIG. 1) (e.g., input illumination) and a mask 802. Mask 802 can be used as patterning device 108 or mask 404 (FIGS. 1 and 4). Using optical elements of source 102 (FIG. 1), the illumination incident on mask 802 can have a plurality of incidence angles. The plurality of incidence angles can be represented as a pupil profile image of source 102 (e.g., on the basis of a Fourier conjugate relation between the pupil plane of source 102 and the input plane of mask 802) (see pupil profile 400 (FIG. 4)). In FIG.8, an example incidence angle 6o is illustrated by an input electric field (Ein) 804. For drawing clarity, other incidence angles are not shown. The symbol n can be used herein to denote an incidence angle variable, which can also be marked with suitable subscripts or superscripts as described below.

[0078] Mask patterns of mask 802 can cause Ein804 to diffract into a plurality of output electric field (Eout) wavefronts. The aggregate output fields can be referred to as a near-field image (or mask image)806. A few example output fields are shown (e.g., E° o)(°b) 808, ^(^o) 810, E('“(tl)(o'o) 812). For convenience, the nomenclature E°^nr^ (<r) can be used to identify the various associations of the output field, such as n denoting diffraction orders along an arbitrary horizontal direction, n' denoting diffraction orders along a direction that is not parallel to the horizontal direction (e.g., along a vertical direction), and a being a single index to represent angle of incidence (can be split into sub-indices when differentiating between x- and y-directions). Depending on the mask patterns, more or fewer diffraction orders, or select ones thereof, can be of interest for lithography simulation. Inset 814 provides example diffraction orders that can be selected for simulation (can correspond to diffraction orders 708 (FIG.7)). Each diffraction order can have one or more attributes (e.g., properties or characteristics) that are useful for lithography simulation, such as field direction (e.g., polarization), magnitude (e.g., intensity or amplitude), timing (e.g., delay or phase), field type (e.g., electric or magnetic), real or imaginary component, or the like.

[0079] As explained above, the intensity distribution at the pupil plane of source 102 (FIG. 1) can represent the numerous angles of incidence a that impinge on mask 802. Hence, a coordinate nomenclature, such as a = (ox,oy), can denote locations of the pupil plane in view of their correspondence to angles of incidence on mask 802.

[0080] With this understanding, just as Ein804 (with incidence angle Go) can generate a two dimensional diffraction order of (0,0) (n = 0 and n' = 0), it follows that each incidence angle input can have a (0,0) output, a (-1,0) output, a (1,0) output, a (1,1) output, and so on. The example that follows next will focus on the 0thorder (0,0). One or more attributes of the 0thorders can be mapped to pixel positions of the pupil profile (e.g., positions of the pupil plane). The mapping can be achieved by associating each of the 0thorders to the incidence angle that generated it. The mapping can be represented as a diffraction-based Jones pupil (can correspond to one of diffraction-based Jones pupils 712). An example of a 0thorder mapping is illustrated as DJP intensity map 816. In the example of FIG. 8, the intensity metric represents the E-field intensity (in arbitrary units), but any suitable attribute, of electromagnetic (EM) radiation can be represented in this manner. The intensity scale goes from lower intensity (dark / black) to higher intensity (bright / white).

[0081] To generate the values of the attribute (e.g., intensity for DJP intensity map 816), a relationship can be defined between the input Einand the output Eout. Equation 1 provides an example relationship that uses a Jones matrix j(nx’ny which can be a 2x2 complex valued matrix.>

[0082] One of the functions of VSM generator 610 (FIG. 6) is to perform equation fitting to obtain orthonormal basis set for the Jones matrix J(n’n'>. The output vector scatter model 612 can comprisethe contents of the computed Jones matrix Equation 2 is an example of an orthonormal basis based on Zernike polynomial expansion.(Eq-2

[0083] The coefficients c; are Zernike expansion coefficients. The coefficients c; are a function of diffraction order as indicated by the superscript (n,nT). Z; are orthonormal Zernike polynomials as a function of incidence angle (denoted by o). In one example, generating DJP intensity map 816 (0thdiffraction order) can comprise solving for the Zernike coefficients c-0'°\ thereby obtaining the orthonormal basis set for the Jones matrix / (0,0,(o'). The processes described in reference to FIG.8 can be performed while maintaining a smooth function of the incidence angle n. Hence the output DJP intensity map 816 can also be a smooth function over incidence angle n. The number of incidence angles a considered in the calculation process can be reduced compared to conventional methods of generating aerial images that use averaging or interpolation of a grid. Reduction o-samplcs using the smooth function of the present disclosure can reduce computational burden and accelerate generation of an aerial image while maintaining comparable accuracy.

[0084] FIG. 9 illustrates example contents of a vector scatter model 900, consistent with embodiments of the present disclosure. In some embodiments, vector scatter model 900 comprises a plurality of DJP representations. The representations of DJPs may be rendered graphically, as illustrated in FIG.9, or otherwise (e.g., represented as Zernike coefficients, which can be used to generate the graphical representations).

[0085] The example described in reference to FIG. 8 (focused on the 0thdiffraction order (0,0)) can be implemented for other diffraction orders so as to generate the DJPs of FIG.9. The intensity scale 902 goes from lower intensity (dark / black) to higher intensity (bright / white) (intensity scale is replicated at each row). The contents of vector scatter model 900 are organized according to Jones matrix element (real (Re), imaginary (Im), xx, yx, xy, and yy) and a few choice diffraction orders (n,n'). A technical significance of generating the DJP representations is that vector scatter model 900 can physically model what happens to an input wave Einas it interacts with the mask pattern of mask 802 (FIG. 8). The model can predict the form ofbased on the shape of illumination used (e.g., pupil profile 400 (FIG. 4)). The resulting near-field image can be processed by projection optics simulation to generate an aerial image (e.g., see projection simulation process 616 and aerial image 618 (FIG. 6)).

[0086] A Zernikes fit (or any other suitable fit) offers a high-fidelity representation of the original diffraction order data prior to the fitting (e.g., analogous to evaluating standard deviation of an equation line fit) can be verified according to embodiments of the present disclosure. A polynomial expansion (e.g., Zernikes, Taylor, or the like) can become more accurate by allowing the use ofadditional higher order terms of the expansion. In some embodiments, an optimal number of expansion terms can be determined through an optimization process (e.g., finding the minimum number of terms that meet an error threshold condition).

[0087] FIG. 10 illustrates an example error analysis scheme 1000 used for analyzing the impact of expansion terms used for a fit operation, consistent with embodiments of the present disclosure. In some embodiments, a vector scatter model generator generates raw diffraction order data 1002 (e.g., VSM generator 700 generates diffraction orders 708 of raw near-field data 704 (FIG. 7)). Raw diffraction order data 1002 can be represented as an intensity map using pupil plane coordinates (e.g., incident angles 6 = (ox,oy)). The intensity scale of the various elements in FIG. 10 goes from lower intensity (dark / black) to higher intensity (bright / white).

[0088] By applying the fitting processes described in reference to FIG.8 to raw diffraction order data 900, diffraction-based Jones pupils 1004 and 1006 can be generated. Diffraction-based Jones pupil 1004 can correspond to a Zernikes fit using 20 or fewer terms. Diffraction-based Jones pupil 1006 can correspond to a Zernikes fit using 60 to 70 terms. A residual 1008 can be computed by evaluating a difference between raw diffraction order data 1002 and diffraction-based Jones pupil 1004. A residual 1010 can be computed by evaluating a difference between raw diffraction order data 1002 and diffraction-based Jones pupil 1006. The residuals are computed at pixel centers and disregard the off-pixel information of the DJPs (hence, residuals are illustrated as a discretized pixel grid).

[0089] For clarity, the intensity scale of residual 1010 is at a xlO multiplier as compared to the intensity scale of residual 1008 (otherwise the graphical representation of residual 1008 would be uninformative, as the intensity map would be completely dark due to the much smaller residual values). Any suitable error evaluation can be applied. For example, an average of the entire residual field can be assessed, a maximum residual value can be a point of focus, or the like. From a maximum residual value perspective, a maximum residual value of residual 1010 can be smaller than a maximum residual value of residual 1008 by approximately a factor of 10. Based on the residual data and a selected threshold condition, a number of expansion terms can be selected for the fitting process described in reference to FIGS. 7-9. For example, using error analysis scheme 1000, it can be determined that a suitable number of Zernike terms for a fitting operation is about 70 or fewer expansion terms, 60 or fewer expansion terms, 50 or fewer expansion terms, 40 or fewer expansion terms, 40 to 70 expansion terms, 50 to 70 expansion terms 60 to 70 expansion terms, 40 to 60 expansion terms, 40 to 50 expansion terms, or the like.

[0090] FIG. 11 illustrates a flowchart of an example method 1100 for simulating a mask image, consistent with embodiments of the present disclosure. In some embodiments, method 1100 is performed using devices and functions described in reference to FIGS. 1-10 and 12. At operation 1102, a near-field solution is obtained (raw near-field data 608 or 704 (FIGS. 6 and 7)). The raw near-field data can be generated by EM solver 606 based on the inputs mask pattern 602 and maskfilm stack 604 (FIG. 6). The near-field solution can be associated with an interaction between a mask pattern of a mask and an electromagnetic (EM) field from a source (e.g., mask pattern of mask patterning device 108 or mask 404, Ein804 from source 102 (FIGS. 1, 4, and 8)).

[0091] At operation 1104, the near-field solution can be decomposed into a plurality of diffraction orders (e.g., diffraction orders 708, F^0)(<r0) 808, E(O“o)(CTo) 810, F™o)(ffo) 812) (FIGS. 7 and 8)). The decomposition can be performed via decomposition process 706 (FIG.7).

[0092] At operation 1106, a vector scatter model can be generated (e.g., vector scatter models 612, 716, 900 (FIGS. 6, 7, and 9)). The vector scatter model can comprise a representation of each diffraction order of the plurality of diffraction orders using a set of basis functions (e.g., Zernikes, equation 2), thereby characterizing each diffraction order with a DJP (e.g., graphical representations illustrated in FIG.9). The vector scatter model can be generated using VSM generator 610 or 700 (FIGS. 6 and 7). The set of orthogonal basis functions are defined in the continuum of incidence angles (continuous in the pupil plane of the source). Hence, DJPs are not limited by the issues and constraints of grid-based computations such as averaging and interpolation. Generating the scatter model can be performed independent of a shape of the source illumination. Instead, the inputting of the source shape into the simulation system can be delayed until the execution of projection simulation process 616 (FIG.6), where the vector scatter model and source 614 are provided to projection simulation process 616 for generating an aerial image.

[0093] At operation 1108, an aerial image can be generated based on the vector scatter model (e.g., aerial image 208 or 618 (FIGS. 2 and 6)). The aerial image can be a projection onto a substrate in a simulated lithography process. The aerial image can be generated using projection simulation process 616 (FIG. 6).

[0094] Some embodiments of the present disclosure can comprise additional or alternative operations based on the description of FIGS. 1-10. For example, operations can comprise generating the representation by fitting each diffraction order of the plurality diffraction orders into the set of basis functions by using a Jones pupil matrix. Operations can comprise generating a diffraction Jones pupil representation for each diffraction order of the plurality of diffraction orders using the Jones pupil matrix. The interaction between the mask pattern and the EM field can be based on a plurality of incidence angles of the EM field as input to the mask and the plurality of diffraction orders as output. Each incidence angle input of the plurality of incidence angles can have a corresponding one or more diffraction order outputs of the plurality of diffraction orders. The set of basis functions can comprise Zernike polynomials to represent the plurality of diffraction orders. The operations can comprise selecting a number of terms of the Zernike polynomials to use for generating the aerial image. The number of terms the Zernike polynomials can be determined by an error analysis. Performing the error analysis can comprise determining one or more residuals between the near-field solution and the vector scatter model. Generating the vector scatter model can be performed using a vector scattermodel generator that is operable based on the near-field solution as input and without using an illumination geometry of the source as additional input to the vector scatter model.

[0095] FIG. 12 shows a block diagram of an example server 1200, consistent with embodiments of the disclosure. In some embodiments, server 1200 can comprise processor 1202. When processor 1202 executes instructions described herein, server 1200 can become a specialized machine. It is appreciated that server 1200 can be involved with computationally intensive tasks, such as generating a vector scatter model or using a vector scatter model to generate an aerial image.

[0096] Processor 1202 can be any type of circuitry capable of manipulating or processing information. For example, processor 1202 can comprise any combination of any number of a central processing unit (“CPU”), a graphics processing unit (“GPU”), a neural processing unit (“NPU”), a microcontroller unit (“MCU”), an optical processor, a programmable logic controller, a microcontroller, a microprocessor, a digital signal processor, an intellectual property (IP) core, a Programmable Logic Array (PLA), a Programmable Array Logic (PAL), a Generic Array Logic (GAL), a Complex Programmable Logic Device (CPLD), a Field-Programmable Gate Array (FPGA), a System On Chip (SoC), an Application- Specific Integrated Circuit (ASIC), or the like. In some embodiments, processor 1202 can also be a set of processors grouped as a single logical component. Processor 1202 can comprise multiple processors, including processor 1202a, processor 1202b, and so on up to processor 1202n.

[0097] Server 1200 can further comprise memory 1204 configured to store data (e.g., a set of instructions, computer codes, intermediate data, or the like). The stored data can comprise program instructions and data for processing. Processor 1202 can access the program instructions and data for processing (e.g., via bus 1210), and execute the program instructions to perform an operation or manipulation on the data for processing. Memory 1204 can comprise a high-speed random-access storage device or a non-volatile storage device. In some embodiments, memory 1204 comprises any combination of any number of non-transitory computer-readable media. Memory 1204 can also be a group of memories grouped as a single logical component.

[0098] Bus 1210 can be a communication device that transfers data between components inside server 1200, such as an internal bus (e.g., a CPU-memory bus), an external bus (e.g., a universal serial bus port, a peripheral component interconnect express port), or the like.

[0099] Processor 1202 and other data processing circuits can collectively be referred to as a “data processing circuit” in the present disclosure (intended to reduce ambiguity and not to limit). The data processing circuit can be implemented entirely as hardware, or as a combination of software, hardware, or firmware. In addition, the data processing circuit can be a single independent module or can be combined entirely or partially into any other component of server 1200.

[0100] Server 1200 can further comprise network interface 1206 to provide wired or wireless communication with a network (e.g., the Internet, an intranet, a local area network, a mobile communications network, or the like). In some embodiments, network interface 1206 can compriseany combination of any number of a network interface controller (NIC), a radio frequency (RF) module, a transponder, a transceiver, a modem, a router, a gateway, a wired network adapter, a wireless network adapter, a Bluetooth adapter, an infrared adapter, a near-field communication (“NFC”) adapter, a cellular network chip, or the like.

[0101] In some embodiments, optionally, server 1200 can further comprise a peripheral interface 1208 to provide a connection to one or more peripheral devices. The peripheral device(s) can include, but are not limited to, a cursor control device (e.g., a mouse, a touchpad, or a touchscreen), a keyboard, a display (e.g., a cathode-ray tube display, a liquid crystal display, or a light-emitting diode display), a video input device (e.g., a camera or an input interface coupled to a video archive), or the like.

[0102] In some embodiments, vector scatter model generation and aerial image generation can be performed using server 1200 in instances where server 1200 can provide higher computational capacity compared to a smaller computational system, such as a personal computer.

[0103] A non-transitory computer-readable medium can be provided that stores instructions for a processor of a controller for generating vector scatter models according to operations described in reference to FIGS.6-11, consistent with embodiments in the present disclosure. For example, the instructions stored in the non-transitory computer-readable medium can be executed by the circuitry of the controller for performing method 600 in part or entirely. Common forms of non-transitory media include, for example, a floppy disk, a flexible disk, hard disk, solid-state drive, magnetic tape, or any other magnetic data storage medium, a Compact Disc Read-Only Memory (CD-ROM), any other optical data storage medium, any physical medium with patterns of holes, a Random Access Memory (RAM), a Programmable Read-Only Memory (PROM), a Field Programmable Gate Array (FPGA), and Erasable Programmable Read-Only Memory (EPROM), a FLASH-EPROM or any other flash memory, Non-Volatile Random Access Memory (NVRAM), a cache, a register, any other memory chip or cartridge, and networked versions of the same.

[0104] Embodiments of the present disclosure can be further described by the following clauses. 1. A non-transitory computer-readable medium that stores a set of instructions that is executable by at least one processor of an apparatus to cause the apparatus to perform operations for simulating an aerial image in a simulated lithography process, the operations comprising:obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source;decomposing the near-field solution into a plurality of diffraction orders;generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions; andgenerating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.2. The non-transitory computer-readable medium of claim 1, wherein the representation is generated by fitting data from each diffraction order of the plurality diffraction orders using the set of basis functions and a Jones pupil matrix.3. The non-transitory computer-readable medium of claim 2, wherein the operations further comprise:generating a diffraction Jones pupil representation for each diffraction order of the plurality of diffraction orders using the Jones pupil matrix.4. The non-transitory computer-readable medium of claim 1, wherein:the interaction is based on a plurality of incidence angles of the EM field at the mask;a first set of one or more diffraction orders of the plurality of diffraction orders is associated with a first incidence angle of the plurality of incidence angles; anda second set of one or more diffraction orders of the plurality of diffraction orders is associated with a second incidence angle of the plurality of incidence angles.5. The non-transitory computer-readable medium of claim 1, wherein the set of basis functions comprises Zernike polynomials to represent the plurality of diffraction orders.6. The non-transitory computer-readable medium of claim 5, wherein the operations further comprise:selecting a number of terms of the Zernike polynomials to use for generating the aerial image.7. The non-transitory computer-readable medium of claim 6, wherein the number of terms the Zernike polynomials is determined by an error analysis.8. The non-transitory computer-readable medium of claim 7, wherein the operations further comprise:performing the error analysis by determining one or more residuals between the near-field solution and the vector scatter model.9. The non-transitory computer-readable medium of claim 1, wherein generating the vector scatter model is performed using a vector scatter model generator that is operable based on the near-field solution as input and is configured to generate the vector scatter model independent of an illumination geometry of the source.10. The non-transitory computer-readable medium of claim 1, wherein the basis functions are continuous as a function of incidence angle of the EM field at the mask.11. The non-transitory computer-readable medium of claim 1, wherein generating the aerial image comprises using the vector scatter model and an illumination geometry of the source as input to an optical model that simulates a projection optics system used in the simulated lithography process. 12. A system comprising:one or more processors; andone or more memory devices configured to store a set of instructions that is executable by the one or more processors to cause the system to perform operations for simulating an aerial image in a simulated lithography process, the operations comprising:obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source;decomposing the near-field solution into a plurality of diffraction orders; generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions; andgenerating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.13. The system of claim 12, wherein the representation is generated by fitting data from each diffraction order of the plurality diffraction orders using the set of basis functions and a Jones pupil matrix.14. The system of claim 13, wherein the operations further comprise:generating a diffraction Jones pupil representation for each diffraction order of the plurality of diffraction orders using the Jones pupil matrix.15. The system of claim 12, wherein:the interaction is based on a plurality of incidence angles of the EM field at the mask;a first set of one or more diffraction orders of the plurality of diffraction orders is associated with a first incidence angle of the plurality of incidence angles; anda second set of one or more diffraction orders of the plurality of diffraction orders is associated with a second incidence angle of the plurality of incidence angles.16. The system of claim 12, wherein the set of basis functions comprises Zernike polynomials to represent the plurality of diffraction orders.17. The system of claim 16, wherein the operations further comprise:selecting a number of terms of the Zernike polynomials to use for generating the aerial image 18. The system of claim 17, wherein the number of terms the Zernike polynomials is determined by an error analysis.19. The system of claim 18, wherein the operations further comprise:performing the error analysis by determining one or more residuals between the near-field solution and the vector scatter model.20. The system of claim 12, wherein generating the vector scatter model is performed using a vector scatter model generator that is operable based on the near-field solution as input and is configured to generate the vector scatter model independent of an illumination geometry of the source.21. The system of claim 12, wherein each the basis functions are continuous as a function of incidence angle of the EM field at the mask.22. The system of claim 12, wherein generating the aerial image comprises using the vector scatter model and an illumination geometry of the source as input to an optical model that simulates a projection optics system used in the simulated lithography process.23. A method for simulating an aerial image in a simulated lithography process, the method comprising:obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source;decomposing the near-field solution into a plurality of diffraction orders;generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions; andgenerating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.24. The method of claim 23, wherein the representation is generated by fitting data from each diffraction order of the plurality diffraction orders using the set of basis functions and a Jones pupil matrix.25. The method of claim 24, further comprising:generating a diffraction Jones pupil representation for each diffraction order of the plurality of diffraction orders using the Jones pupil matrix.26. The method of claim 23, wherein:the interaction is based on a plurality of incidence angles of the EM field at the mask;a first set of one or more diffraction orders of the plurality of diffraction orders is associated with a first incidence angle of the plurality of incidence angles; anda second set of one or more diffraction orders of the plurality of diffraction orders is associated with a second incidence angle of the plurality of incidence angles.27. The method of claim 23, wherein the set of basis functions comprises Zernike polynomials to represent the plurality of diffraction orders.28. The method of claim 27, further comprising:selecting a number of terms of the Zernike polynomials to use for generating the aerial image.29. The method of claim 28, wherein the number of terms the Zernike polynomials is determined by an error analysis.30. The method of claim 29, further comprising:performing the error analysis by determining one or more residuals between the near-field solution and the vector scatter model.31. The method of claim 23, wherein generating the vector scatter model is performed using a vector scatter model generator that is operable based on the near-field solution as input and is configured to generate the vector scatter model independent of an illumination geometry of the source.32. The method of claim 23, wherein each the basis functions are continuous as a function of incidence angle of the EM field at the mask.33. The method of claim 23, wherein generating the aerial image comprises using the vector scatter model and an illumination geometry of the source as input to an optical model that simulates a projection optics system used in the simulated lithography process.

[0105] It will be appreciated that the embodiments of the present disclosure are not limited to the exact construction that has been described above and illustrated in the accompanying drawings and that various modifications and changes can be made without departing from the scope thereof.

Claims

CLAIMS1. A non-transitory computer-readable medium that stores a set of instructions that is executable by at least one processor of an apparatus to cause the apparatus to perform operations for simulating an aerial image in a simulated lithography process, the operations comprising:obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source;decomposing the near-field solution into a plurality of diffraction orders;generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions; andgenerating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.

2. The non-transitory computer-readable medium of claim 1, wherein the representation is generated by fitting data from each diffraction order of the plurality diffraction orders using the set of basis functions and a Jones pupil matrix.

3. The non-transitory computer-readable medium of claim 2, wherein the operations further comprise:generating a diffraction Jones pupil representation for each diffraction order of the plurality of diffraction orders using the Jones pupil matrix.

4. The non-transitory computer-readable medium of claim 1, wherein:the interaction is based on a plurality of incidence angles of the EM field at the mask;a first set of one or more diffraction orders of the plurality of diffraction orders is associated with a first incidence angle of the plurality of incidence angles; anda second set of one or more diffraction orders of the plurality of diffraction orders is associated with a second incidence angle of the plurality of incidence angles.

5. The non-transitory computer-readable medium of claim 1, wherein the set of basis functions comprises Zernike polynomials to represent the plurality of diffraction orders.

6. The non-transitory computer-readable medium of claim 5, wherein the operations further comprise:selecting a number of terms of the Zernike polynomials to use for generating the aerial image.

7. The non-transitory computer-readable medium of claim 6, wherein the number of terms the Zernike polynomials is determined by an error analysis.

8. The non-transitory computer-readable medium of claim 7, wherein the operations further comprise:performing the error analysis by determining one or more residuals between the near-field solution and the vector scatter model.

9. The non-transitory computer-readable medium of claim 1, wherein generating the vector scatter model is performed using a vector scatter model generator that is operable based on the near-field solution as input and is configured to generate the vector scatter model independent of an illumination geometry of the source.

10. The non-transitory computer-readable medium of claim 1, wherein the basis functions are continuous as a function of incidence angle of the EM field at the mask.

11. The non-transitory computer-readable medium of claim 1, wherein generating the aerial image comprises using the vector scatter model and an illumination geometry of the source as input to an optical model that simulates a projection optics system used in the simulated lithography process.

12. A method for simulating an aerial image in a simulated lithography process, the method comprising:obtaining a near-field solution associated with an interaction between a mask and an electromagnetic (EM) field from a source;decomposing the near-field solution into a plurality of diffraction orders;generating a vector scatter model comprising a representation of each diffraction order of the plurality diffraction orders by using a set of basis functions; andgenerating the aerial image based on the vector scatter model, wherein the aerial image is projected onto a substrate in the simulated lithography process.

13. The method of claim 12, wherein the representation is generated by fitting data from each diffraction order of the plurality diffraction orders using the set of basis functions and a Jones pupil matrix, and further comprising:generating a diffraction Jones pupil representation for each diffraction order of the plurality of diffraction orders using the Jones pupil matrix, wherein each the basis functions are continuous as a function of incidence angle of the EM field at the mask.

14. The method of claim 12, wherein:the interaction is based on a plurality of incidence angles of the EM field at the mask; a first set of one or more diffraction orders of the plurality of diffraction orders is associated with a first incidence angle of the plurality of incidence angles; anda second set of one or more diffraction orders of the plurality of diffraction orders is associated with a second incidence angle of the plurality of incidence angles.

15. The method of claim 12, wherein the set of basis functions comprises Zernike polynomials to represent the plurality of diffraction orders, further comprising: selecting a number of terms of the Zernike polynomials to use for generating the aerial image, wherein the number of terms the Zernike polynomials is determined by an error analysis; and performing the error analysis by determining one or more residuals between the near-field solution and the vector scatter model.

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