WO2025087650A1

WO2025087650A1 - Monitoring fading effect based on computational lithography simulation

Info

Publication number: WO2025087650A1
Application number: PCT/EP2024/077408
Authority: WO
Inventors: Xingyue Peng; Zhan Shi; Rafael C. Howell
Original assignee: ASML Netherlands BV
Current assignee: ASML Netherlands BV
Priority date: 2023-10-25
Filing date: 2024-09-30
Publication date: 2025-05-01
Anticipated expiration: 2026-04-25
Also published as: TW202530881A

Abstract

Described herein is a method and system for simulating fading effect in lithography. Patterns or cutlines are selected based on their fading sensitivity for monitoring fading effect. The fading sensitivity may be determined based on (a) a spatial derivative of a fading image, or (b) a second order derivative of an aerial image of the patterns. A fading source representing a scanning location dependent illumination pupil profile of a lithographic apparatus is obtained and a fading image is determined using the fading source. A derivative of a fading sensitivity indicator with respect to wafer stage modulation is determined based on a spatial derivative of the fading image. Patterns are represented using a feature vector of the derivates of the fading sensitivity indicator. The feature vectors are grouped, and a set of patterns are selected based on the grouping for monitoring the fading effect.

Description

METHOD AND SYSTEM FOR SELECTING PATTERN AND CUTLINE FOR MONITORING FADING EFFECT

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority of US application 63/545,731 which was filed on October 25, 2023 and US application 63/554,359 which was filed on February 16, 2024 which are incorporated herein in its entirety by reference.

TECHNICAL FIELD

[0002] The embodiments provided herein relate to semiconductor manufacturing, and more particularly to monitoring fading effect based on computational lithography simulation.

BACKGROUND

[0003] A lithographic apparatus is a machine that applies a desired pattern onto a target portion of a substrate. The lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). For example, an IC chip in a smart phone, can be as small as a person’s thumbnail, and may include over 2 billion 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 step have 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.

BRIEF SUMMARY

[0004] In some embodiments, the techniques described herein relate to a method for simulating fading effect in lithography, the method including: obtaining an illumination pupil profile of a source of a lithographic apparatus; determining a fading source representing a scanning location dependent illumination pupil profile of the lithographic apparatus, wherein the fading source is determined based on a product of the illumination pupil profile and a location in a scanning direction of a slit in the lithographic apparatus; determining a fading image of a pattern using the fading source; and determining a fading sensitivity of the patterns based on a spatial derivative of the fading image.

[0005] In some embodiments, the techniques described herein relate to a method for simulating fading effect in lithography, the method including: obtaining an aerial image of patterns to be printed on a substrate; determining fading sensitivity of the patterns based on second order spatial derivatives of the aerial image; and selecting a set of patterns from the patterns based on the fading sensitivity for monitoring fading effect.

[0006] In some embodiments, there is provided a non-transitory computer readable medium having instructions that, when executed by a computer, cause the computer to execute a method of any of the above embodiments. [0007] In some embodiments, there is provided an apparatus includes a memory storing a set of instructions and a processor configured to execute the set of instructions to cause the apparatus to perform a method of any of the above embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Embodiments will now be described, by way of example only, with reference to the accompanying drawings in which:

[0009] Figure 1 illustrates a block diagram of various subsystems of a lithographic projection apparatus, according to an embodiment.

[0010] Figure 2 is a schematic diagram of a lithographic projection apparatus, according to an embodiment.

[0011] Figure 3 illustrates an exemplary flow chart for simulating lithography in a lithographic projection apparatus, according to an embodiment.

[0012] Figure 4 is a block diagram of an exemplary system for selecting patterns for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments. [0013] Figure 5 illustrates a dynamic pupil whose shape varies along the scanning direction, consistent with various embodiments.

[0014] Figure 6 illustrates placement of cutlines on a pattern, consistent with various embodiments.

[0015] Figure 7A is a flow diagram of an exemplary method for selecting patterns for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments. [0016] Figure 7B is a flow diagram of an exemplary method for determining derivatives of a fading sensitivity indicator for a pattern, consistent with various embodiments.

[0017] Figure 8 is a block diagram of an exemplary system for selecting patterns based on second order spatial derivatives of an aerial image for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments.

[0018] Figure 9 is a flow diagram of an exemplary method for selecting patterns based on second order derivatives of an aerial image for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments.

[0019] Figure 10 is a flow diagram of a process for reducing fading sensitivity of the patterns by performing a source mask optimization (SMO) process, consistent with various embodiments.

[0020] Figure 11 A is a flow diagram of a method for modifying a cost function of an SMO process to a fading-aware cost function based on second order spatial derivates of an aerial image of a pattern, consistent with various embodiments.

[0021] Figure 1 IB is a flow diagram of another method for modifying a cost function of an SMO process to a fading-aware cost function based on a stage modulation of a wafer stage of a lithographic apparatus, consistent with various embodiments. [0022] Figure 12 is a block diagram illustrating a comparison of results between an SMO process and an enhanced SMO process with fading-aware cost function, consistent with various embodiments. [0023] Figure 13 is another block diagram illustrating a comparison of results between an SMO process and an enhanced SMO process with fading-aware cost function, consistent with various embodiments.

[0024] Figure 14 is a block diagram of an example computer system, according to an embodiment.

[0025] Embodiments will now be described in detail with reference to the drawings, which are provided as illustrative examples so as to enable those skilled in the art to practice the embodiments. Notably, the figures and examples below are not meant to limit the scope to a single embodiment, but other embodiments are possible by way of interchange of some or all of the described or illustrated elements. Wherever convenient, the same reference numbers will be used throughout the drawings to refer to same or like parts. Where certain elements of these embodiments can be partially or fully implemented using known components, only those portions of such known components that are necessary for an understanding of the embodiments will be described, and detailed descriptions of other portions of such known components will be omitted so as not to obscure the description of the embodiments. In the present specification, an embodiment showing a singular component should not be considered limiting; rather, the scope is intended to encompass other embodiments including a plurality of the same component, and vice-versa, unless explicitly stated otherwise herein. Moreover, applicants do not intend for any term in the specification or claims to be ascribed an uncommon or special meaning unless explicitly set forth as such. Further, the scope encompasses present and future known equivalents to the components referred to herein by way of illustration.

DETAILED DESCRIPTION

[0026] A lithographic apparatus is a machine that applies a designed pattern onto a target portion of a substrate. This process of transferring the designed pattern to the substrate is called a patterning process. The patterning process can include a patterning step to transfer a pattern from a patterning device (such as a mask) to the substrate. Various variations (e.g., variations in the patterning process or the lithographic apparatus) can potentially limit lithography implementation for semiconductor high volume manufacturing (HVM). In some embodiments, an “overlay” is determined as a layer-to- layer placement error between two features of two layers (e.g., adjacent layers) which are designed to align or have a known relationship. An overlay correction may be performed to reduce overlay. The overlay correction may be implemented by coordinating the movements of reticle stage, wafer stage, or an optical column of the lithographic apparatus (e.g., lens or mirrors) during processing. However, such overlay correction operations may introduce certain imaging errors such as fading (e.g., blurring). The active overlay correction induced image fading is the imaging degradation due to a deviation of a specified wafer or reticle stage movement from a requested scanner set point of wafer and reticle stage movement. Image fading can be the degradation of an imaged feature's fidelity due to mismatch between wafer and reticle stage synchronization. Such imaging degradation has an impact on one or more patterning parameters such as critical dimension (CD), CD uniformity (CDU), image log slope (ILS), normalized image log slope (NILS), edge placement error (EPE) distribution, etc. that can cause a pattern to be defective. Conventional techniques simulate fading effect from overlay correction. The conventional techniques perform the simulation at a full-chip level, which is computing resource intensive. For example, to simulate the fading effect, a stage modulation of the wafer stage (e.g., moving standard deviation (MSD) of a moving trajectory profile of the wafer stage) is perturbed in many directions (e.g., 8 directions) and the simulation is done via wavefront injection and determination of a transmission cross-coefficient (TCC) of a source of a lithographic apparatus multiple times (e.g., 9 times), all of which are computing resource intensive. The conventional simulation techniques do not identify the most fading-sensitive patterns from the full-chip layout to monitor the fading effect. Further, the conventional techniques also do not identify appropriate cutline placements on the pattern to obtain the most accurate measurement of pattern parameters for monitoring the fading effect. Furthermore, the conventional techniques don’t consider an effect of a dynamic pupil profile on the image fading.

[0027] Disclosed are embodiments for selecting patterns or cutlines based on a fading sensitivity indicator for monitoring a fading effect on the patterns. In a first example, the fading sensitivity indicator is indicative of an edge placement error (EPE), and a derivative of the fading sensitivity indicator to a stage modulation of the wafer stage of a lithographic apparatus (e.g., moving standard deviation (MSD) of a moving trajectory profile of the wafer stage) is determined using a spatial derivative of a fading image. The fading image, whose spatial derivative is indicative of a sensitivity of an aerial image to the MSD, is obtained using a first order and second order TCC of a fading source (“fading TCC”). The fading source represents a scanning location dependent illumination pupil profile of the lithographic apparatus, which is determined based on a product of the illumination pupil profile and a location in a scanning direction of a slit in the lithographic apparatus. In some embodiments, each of the patterns is represented as a feature vector including elements that are derived using a magnitude of MSD of the lithographic apparatus and the derivatives of the fading sensitivity indicator. The feature vectors can be grouped or ranked using any of a number of grouping techniques (e.g., K-means), and one or more patterns are selected based on the grouping or ranking for monitoring the fading effect. Similarly, feature vectors representative of candidate cutlines can be grouped or ranked, and one or more cutlines are selected for placement on the patterns based on the grouping. By selecting the patterns that are most sensitive to the fading effect, the amount of computing resources consumed for monitoring the fading effect is reduced significantly. For example, selecting the patterns using a fading image reduces the number of times a TCC is determined significantly (e.g., to 3 times), thereby reducing the consumption of computing resources. Additionally, by using the fading source (e.g., a first and second order fading source) in the selection of the patterns, an effect of a dynamic pupil profile along the scanning direction of the lithographic apparatus on the imaging of the patterns is also considered for selecting the patterns.

[0028] In a second example, a fading sensitivity indicator is determined based on second order spatial derivatives of an aerial image of the patterns. For example, a Hessian matrix of the second order spatial derivatives of the aerial image is used to indicate sensitivity of the pattern to image fading, and is determined for each evaluation point on a pattern. The fading sensitivity indicator may be determined based on a magnitude of the Hessian matrix (e.g., Eigen value, /.) and a metric associated with the aerial image (e.g., image log slope (ILS)). The magnitude may be determined in a number of ways, e.g., as the largest singular value of the Hessian matrix - Eigen value, 2₀, or as an average of Eigen values, 2₀

or in other ways. The magnitude of the Hessian matrix represents the sensitivity of the patterns to image fading. The patterns are selected based on the fading sensitivity indicator values. For example, those patterns that contain evaluation points with the largest fading sensitivity indicator values (e.g., (2₀//LS’) value) are selected for monitoring the fading effect. Similarly, the cutlines are chosen based on the fading sensitivity indicator values associated with a pair of evaluation points corresponding to its endpoints. For example, those cutlines with the largest sum of fading sensitivity indicator values for the pair of evaluation points corresponding to its endpoints are selected for placement on the patterns. As in the first example, by selecting the patterns that are most sensitive to the fading effect, the amount of computing resources consumed for monitoring the fading effect is reduced significantly. By using the second order spatial derivatives of the aerial image to select the patterns, the need for determining a TCC is eliminated if the aerial image is already available, or the number of times a TCC is determined is reduced significantly (e.g., to one time - to determine the aerial image), thereby reducing the consumption of computing resources significantly.

[0029] As described above, different patterns may have different sensitivities to image fading, and the patterns may be selected for monitoring the fading effect based on their fading sensitivity indicator values. The fading sensitivity of the patterns may be reduced by adjusting at least one of the source parameters (e.g., illumination pupil profile), mask parameters (e.g., mask pattern), or wavefront. A source mask optimization (SMO) process is a process that optimizes at least one of source parameters, mask parameters, or wavefront in order to increase imaging quality (e.g., one or more parameters such as image contrast, edge placement error, CD uniformity, resist contours, depth of focus, throughput, etc.) of the patterning process. In some embodiments, the SMO process may be enhanced or adjusted to reduce the fading sensitivity of the patterns as well.

[0030] Disclosed are embodiments for reducing the fading sensitivity of the patterns using an enhanced SMO process. In some embodiments, a cost function of the SMO may be modified to include parameters that have an impact on the fading sensitivity of the patterns. For example, second order spatial derivatives of an aerial image of a pattern (e.g., Hessian matrix), which have an impact on the fading effect of a pattern as described above, may be added as a cost term to the cost function of the SMO process to generate a “fading-aware” cost function. In some embodiments, as described above, a stage modulation of a wafer stage of the lithographic apparatus (e.g., MSD) has an impact on the fading effect of a pattern. Accordingly, the MSD may be incorporated in the cost function of the SMO process to generate a fading aware cost function. For example, the cost function of the SMO, which is computed for a process window condition for a set of parameters (e.g., dose, focus, etc.) may be adjusted to include MSD as an additional parameter in the process window parameters to generate the fading aware cost function. The enhanced SMO process with the fading-aware cost function generates output parameters, which includes at least one of source parameters, mask parameters, or wavefront, that are optimized to reduce the fading sensitivity of the patterns. In some embodiments, by enhancing the SMO process, the fading sensitivity of patterns with different fading sensitivities may be simultaneously reduced. Additionally, since the existing SMO process is enhanced for reducing the fading sensitivity, it eliminates the need for implementing an additional process to reduce the fading sensitivity thereby reducing the computing resources that may otherwise have been consumed in reducing the fading sensitivity.

[0031] In the present disclosure, although specific reference may be made to the manufacture of ICs, it should be explicitly understood that the description herein has many other possible applications. For example, it may be employed in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, liquid crystal display panels, thin film magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms “reticle”, “wafer” or “die” in this text should be considered as interchangeable with the more general terms “mask”, “substrate” and “target portion”, respectively. [0032] In the present document, the terms “radiation” and “beam” are used to 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 of about 5-100 nm). In the present document, the term “radiation source” or “source” is used to encompass all types of sources of radiation, including laser sources, incandescent sources, etc. which may include treatment of the radiation between the radiation source and the target or other parts of the optics, including filtering, collimating, focusing, etc.

[0033] A patterning device can comprise, or can form, one or more design layouts. The design layout can be generated utilizing CAD (computer-aided design) programs. This process is often referred to as EDA (electronic design automation). Most CAD programs follow a set of predetermined design rules in order to create functional design layouts/patterning devices. These rules are set based processing and design limitations. For example, design rules define the space tolerance between devices (such as gates, capacitors, etc.) or interconnect lines, to ensure that the devices or lines do not interact with one another in an undesirable way. One or more of the design rule limitations may be referred to as a “critical dimension” (CD). A critical dimension of a device can be defined as the smallest width of a line or hole, or the smallest space between two lines or two holes. Thus, the CD regulates the overall size and density of the designed device. One of the goals in device fabrication is to faithfully reproduce the original design intent on the substrate (via the patterning device).

[0034] The term “mask” or “patterning device” as employed in this text may be broadly interpreted as referring to a generic patterning device that can be used to endow an incoming radiation beam with a patterned cross-section, corresponding to a pattern that is to be created in a target portion of the substrate. The term “light valve” can also be used in this context. Besides the classic mask (transmissive or reflective; binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include a programmable mirror array. An example of such a device is a matrix-addressable surface having a viscoelastic control layer and a reflective surface. The basic principle behind such an apparatus is that (for example) addressed areas of the reflective surface reflect incident radiation as diffracted radiation, whereas unaddressed areas reflect incident radiation as undiffracted radiation. Using an appropriate filter, the said undiffracted radiation can be filtered out of the reflected beam, leaving only the diffracted radiation behind; in this manner, the beam becomes patterned according to the addressing pattern of the matrix-addressable surface. The required matrix addressing can be performed using suitable electronic means. Examples of other such patterning devices also include a programmable LCD array. An example of such a construction is given in U.S. Patent No. 5,229,872, which is incorporated herein by reference.

[0035] The term “projection optics” as used herein should be broadly interpreted as encompassing various types of optical systems, including refractive optics, reflective optics, apertures and catadioptric optics, for example. The term “projection optics” 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. The term “projection optics” may include any optical component in the lithographic projection apparatus, no matter where the optical component is located on an optical path of the lithographic projection apparatus. Projection optics may include optical components for shaping, adjusting and/or projecting radiation from the source before the radiation passes the patterning device, and/or optical components for shaping, adjusting and/or projecting the radiation after the radiation passes the patterning device. The projection optics generally exclude the source and the patterning device.

[0036] Figure 1 illustrates a block diagram of various subsystems of a lithographic projection apparatus 10A, according to an embodiment. Major components are a radiation source 12A, which may be a deep-ultraviolet excimer laser source or other type of source including an extreme ultra violet (EUV) source (the lithographic projection apparatus itself need not have the radiation source), illumination optics which, e.g., define the partial coherence (denoted as sigma) and which may include optics 14 A, 16Aa and 16 Ab that shape radiation from the source 12 A; a patterning device (or mask) 18A; and transmission optics 16Ac that project an image of the patterning device pattern onto a substrate plane 22A. [0037] A pupil 20A can be included with transmission optics 16 Ac. In some embodiments, there can be one or more pupils before and/or after mask 18 A. As described in further detail herein, pupil 20A can provide patterning of the light that ultimately reaches substrate plane 22A. An adjustable filter or aperture at the pupil plane of the projection optics may restrict the range of beam angles that impinge on the substrate plane 22A, where the largest possible angle defines the numerical aperture of the projection optics NA= n sin(0max), wherein n is the refractive index of the media between the substrate and the last element of the projection optics, and ©max is the largest angle of the beam exiting from the projection optics that can still impinge on the substrate plane 22 A.

[0038] In a lithographic projection apparatus, a source provides illumination (i.e., radiation) to a patterning device and projection optics direct and shape the illumination, via the patterning device, onto a substrate. This is not to disclaim that the source does not itself provide patterning, directing, or shaping to the radiation or that patterning, directing, or shaping does not occur between the source and the projection optics. The projection optics may include at least some of the components 14A, 16Aa, 16Ab and 16Ac. An aerial image (Al) is the radiation intensity distribution at substrate level. A resist model can be used to calculate the resist image from the aerial image, an example of which can be found in U.S. Patent Application Publication No. US 2009-0157360, the disclosure of which is hereby incorporated by reference in its 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 projection apparatus (e.g., properties of the illumination, the patterning device and the projection optics) dictate the aerial image and can be defined in an optical model. Since the patterning device used in the lithographic projection 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 projection apparatus including at least the source and the projection optics. Details of techniques and models used to transform a design layout into various lithographic images (e.g., an aerial image, a resist image, etc.), apply OPC using those techniques and models and evaluate performance (e.g., in terms of process window) are described in U.S. Patent Application Publication Nos. US 2008-0301620, 2007-0050749, 2007-0031745, 2008-0309897, 2010-0162197, and 2010-0180251, the disclosure of each which is hereby incorporated by reference in its entirety.

[0039] One aspect of understanding a lithographic process is understanding the interaction of the radiation and the patterning device. The electromagnetic field of the radiation after the radiation passes the patterning device may be determined from the electromagnetic field of the radiation before the radiation reaches the patterning device and a function that characterizes the interaction. This function may be referred to as the mask transmission function (which can be used to describe the interaction by a transmissive patterning device and/or a reflective patterning device).

[0040] The mask transmission function may have a variety of different forms. One form is binary. A binary mask transmission function has either of two values (e.g., zero and a positive constant) at any given location on the patterning device. A mask transmission function in the binary form may be referred to as a binary mask. Another form is continuous. Namely, the modulus of the transmittance (or reflectance) of the patterning device is a continuous function of the location on the patterning device. The phase of the transmittance (or reflectance) may also be a continuous function of the location on the patterning device. A mask transmission function in the continuous form may be referred to as a continuous tone mask or a continuous transmission mask (CTM). For example, the CTM may be represented as a pixelated image, where each pixel may be assigned a value between 0 and 1 (e.g., 0.1, 0.2, 0.3, etc.) instead of binary value of either 0 or 1. In an embodiment, CTM may be a pixelated gray scale image, where each pixel has values (e.g., within a range [-255, 255], normalized values within a range [0, 1] or [-1, 1] or other appropriate ranges).

[0041] The thin-mask approximation, also called the Kirchhoff boundary condition, is widely used to simplify the determination of the interaction of the radiation and the patterning device. The thin-mask approximation assumes that the thickness of the structures on the patterning device is very small compared with the wavelength and that the widths of the structures on the mask are very large compared with the wavelength. Therefore, the thin-mask approximation assumes the electromagnetic field after the patterning device is the multiplication of the incident electromagnetic field with the mask transmission function. However, as lithographic processes use radiation of shorter and shorter wavelengths, and the structures on the patterning device become smaller and smaller, the assumption of the thin-mask approximation can break down. For example, interaction of the radiation with the structures (e.g., edges between the top surface and a sidewall) because of their finite thicknesses (“mask 3D effect” or “M3D”) may become significant. Encompassing this scattering in the mask transmission function may enable the mask transmission function to better capture the interaction of the radiation with the patterning device. A mask transmission function under the thin-mask approximation may be referred to as a thin-mask transmission function. A mask transmission function encompassing M3D may be referred to as a M3D mask transmission function.

[0042] Figure 2 schematically depicts an exemplary lithographic projection apparatus whose illumination source could be optimized utilizing the methods described herein. The apparatus comprises:

- an illumination system IL, to condition a beam B of radiation. In this particular case, the illumination system also comprises a radiation source SO;

- a first object table (e.g., mask table, patterning device table or reticle stage) MT provided with a patterning device holder to hold a patterning device MA (e.g., a reticle), and connected to a first positioner to accurately position the patterning device with respect to item PS;

- a second object table (substrate table or wafer stage) WT provided with a substrate holder to hold a substrate W (e.g., a resist-coated silicon wafer), and connected to a second positioner to accurately position the substrate with respect to item PS;

- a projection system (“lens”) PS (e.g., a refractive, catoptric or catadioptric optical system) to image an irradiated portion of the patterning device MA onto a target portion C (e.g., comprising one or more dies) of the substrate W.

[0043] As depicted herein, the apparatus is of a transmissive type (i.e., has a transmissive mask). However, in general, it may also be of a reflective type, for example (with a reflective mask). Alternatively, the apparatus may employ another kind of patterning device as an alternative to the use of a classic mask; examples include a programmable mirror array or LCD matrix.

[0044] The source SO (e.g., a mercury lamp or excimer laser) produces a beam of radiation. This beam is fed into an illumination system (illuminator) IL, either directly or after having traversed conditioning means, such as a beam expander Ex, for example. The illuminator IL may comprise adjusting means AD for setting the outer or inner radial extent (commonly referred to as o-outer and G-inncr, respectively) of the intensity distribution in the beam. In addition, it will generally comprise various other components, such as an integrator IN and a condenser CO. In this way, the beam B impinging on the patterning device MA has a desired uniformity and intensity distribution in its cross-section.

[0045] It should be noted with regard to Figure 2 that the source SO may be within the housing of the lithographic projection apparatus (as is often the case when the source SO is a mercury lamp, for example), but that it may also be remote from the lithographic projection apparatus, the radiation beam that it produces being led into the apparatus (e.g., with the aid of suitable directing mirrors); this latter scenario is often the case when the source SO is an excimer laser (e.g., based on KrF, ArF or Fz lasing).

[0046] The beam B subsequently intercepts the patterning device MA, which is held on a patterning device table MT. Having traversed the patterning device MA, the beam B passes through the lens PS, which focuses the beam B onto a target portion C of the substrate W. With the aid of the second positioning means (and interferometric measuring means IF), the substrate table WT can be moved accurately, e.g., so as to position different target portions C in the path of the beam B. Similarly, the first positioning means can be used to accurately position the patterning device MA with respect to the path of the beam B, e.g., after mechanical retrieval of the patterning device MA from a patterning device library, or during a scan. In general, movement of the object tables MT, WT will be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which are not explicitly depicted in Figure 2. However, in the case of a wafer stepper (as opposed to a step-and-scan tool) the patterning device table MT may just be connected to a short stroke actuator, or may be fixed.

[0047] The depicted tool can be used in two different modes:

- In step mode, the patterning device table MT is kept essentially stationary, and an entire patterning device image is projected in one go (i.e., a single “flash”) onto a target portion C. The substrate table WT is then shifted in the x or y directions so that a different target portion C can be irradiated by the beam B;

- In scan mode, essentially the same scenario applies, except that a given target portion C is not exposed in a single “flash”. Instead, the patterning device table MT is movable in a given direction (the so-called “scan direction”, e.g., the y direction) with a speed v, so that the projection beam B is caused to scan over a patterning device image; concurrently, the substrate table WT is simultaneously moved in the same or opposite direction at a speed V = Mv, in which M is the magnification of the lens PS (typically, M = 1/4 or 1/5). In this manner, a relatively large target portion C can be exposed, without having to compromise on resolution.

[0048] Figure 3 illustrates an exemplary flow chart for simulating lithography in a lithographic projection apparatus, according to an embodiment. As will be appreciated, the models may represent a different patterning process and need not comprise all the models described below. A source model 300 represents optical characteristics (including radiation intensity distribution, bandwidth and/or phase distribution) of the illumination of a patterning device. The source model 300 can represent the optical characteristics of the illumination that include, but not limited to, numerical aperture settings, illumination sigma (o) settings as well as any particular illumination shape (e.g., off-axis radiation shape such as annular, quadrupole, dipole, etc.), where o (or sigma) is outer radial extent of the illuminator.

[0049] A projection optics model 310 represents optical characteristics (including changes to the radiation intensity distribution and/or the phase distribution caused by the projection optics) of the projection optics. The projection optics model 310 can represent the optical characteristics of the projection optics, including aberration, distortion, one or more refractive indexes, one or more physical sizes, one or more physical dimensions, etc.

[0050] The patterning device / design layout model module 320 captures how the design features are laid out in the pattern of the patterning device and may include a representation of detailed physical properties of the patterning device, as described, for example, in U.S. Patent No. 7,587,704, which is incorporated by reference in its entirety. In an embodiment, the patterning device / design layout model module 320 represents optical characteristics (including changes to the radiation intensity distribution and/or the phase distribution caused by a given design layout) of a design layout (e.g., a device design layout corresponding to a feature of an integrated circuit, a memory, an electronic device, etc.), which is the representation of an arrangement of features on or formed by the patterning device. Since the patterning device used in the lithographic projection 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 projection apparatus including at least the illumination and the projection optics. The objective of the simulation is often to accurately predict, for example, edge placements and CDs, which can then be compared against the device design. The device design is generally defined as the pre-OPC patterning device layout, and will be provided in a standardized digital file format such as GDSII or OASIS. [0051] An aerial image 330 can be simulated from the source model 300, the projection optics model 310 and the patterning device / design layout model module 320. An aerial image (Al) is the radiation intensity distribution at substrate level. Optical properties of the lithographic projection apparatus (e.g., properties of the illumination, the patterning device, and the projection optics) dictate the aerial image.

[0052] A resist layer on a substrate is exposed by the aerial image and the aerial image is transferred to the resist layer as a latent “resist image” (RI) therein. The resist image (RI) can be defined as a spatial distribution of solubility of the resist in the resist layer. A resist image 350 can be simulated from the aerial image 330 using a resist model 340. The resist model can be used to calculate the resist image from the aerial image, an example of which can be found in U.S. Patent Application No. 8,200,468, the disclosure of which is hereby incorporated by reference in its entirety. The resist model 340 typically describes the effects of chemical processes which occur during resist exposure, post exposure bake (PEB) and development, in order to predict, for example, contours of resist features formed on the substrate and so it typically related only to such properties of the resist layer (e.g., effects of chemical processes which occur during exposure, post-exposure bake and development). In an embodiment, the optical properties of the resist layer, e.g., refractive index, film thickness, propagation, and polarization effects — may be captured as part of the projection optics model 310.

[0053] So, in general, the connection between the optical and the resist model is a simulated aerial image intensity within the resist layer, which arises from the projection of radiation onto the substrate, refraction at the resist interface and multiple reflections in the resist film stack. The radiation intensity distribution (aerial image intensity) is turned into a latent “resist image” by absorption of incident energy, which is further modified by diffusion processes and various loading effects. Efficient simulation methods that are fast enough for full-chip applications approximate the realistic 3-dimensional intensity distribution in the resist stack by a 3-dimensional aerial (and resist) image.

[0054] In an embodiment, the resist image 350 can be used as an input to a post-pattern transfer process model module 360. The post-pattern transfer process model module 360 defines performance of one or more post-resist development processes (e.g., etch, development, etc.).

[0055] Simulation of the patterning process can, for example, predict contours, CDs, edge placement (e.g., edge placement error), etc. in the resist and/or etched image. Thus, the objective of the simulation is to accurately predict, for example, edge placement, and/or aerial image intensity slope, and/or CD, etc. of the printed pattern. These values can be compared against an intended design to, e.g., correct the patterning process, identify where a defect is predicted to occur, etc. The intended design is generally defined as a pre-OPC design layout which can be provided in a standardized digital file format such as GDSII or OASIS or other file format. [0056] Thus, the model formulation describes most, if not all, of the known physics and chemistry of the overall process, and each of the model parameters desirably corresponds to a distinct physical or chemical effect. The model formulation thus sets an upper bound on how well the model can be used to simulate the overall manufacturing process.

[0057] The following paragraphs describe a system and a method for selecting patterns for monitoring fading effect due to overlay correction in a lithography process. The patterns may be selected based on their fading sensitivity which is determined based on (a) a spatial derivative of a fading image, as described at least with reference to Figures 4-7B, or (b) a second order derivative of an aerial image of the patterns, as described at least with reference to Figures 8-9.

[0058] Figure 4 is a block diagram of an exemplary system for selecting patterns for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments. According to embodiments of the present disclosure, a fading source and a fading image are generated to determine fading sensitivity of a pattern. A set of patterns 430 is selected based on a spatial derivative 412 of a fading image 410 for monitoring a fading effect on the set of patterns 430. The selection process may include determining the fading image 410 using a fading source 402, determining the spatial derivative 412 of the fading image 410, determining a derivative 416 of (e.g., edge placement error (EPE)) with respect to a stage modulation of the wafer stage of a lithographic apparatus (e.g., moving standard deviation (MSD) of a moving trajectory profile of the wafer stage) based on the spatial derivative 412 of the fading image 410, and determining a derivative 418 of a fading sensitivity indicator 417 (e.g., EPE for a set of evaluations points) with respect to the MSD based on the derivative 416 of the EPE for each of the patterns 450 in a design layout to be printed on a substrate. The selection process may further include generating a feature vector 420 for each of the patterns 450 using the derivative 418 of the fading sensitivity indicator 417, grouping or ranking the feature vectors, and selecting the set of patterns 430 based on the grouping or ranking, all of which are described in greater detail in the following paragraphs.

[0059] The fading source 402 is constructed to represent a scanning location dependent illumination pupil profile of the lithographic apparatus. In some embodiments, it is determined based on a product of the illumination pupil profile and a location in a scanning direction of a slit in the lithographic apparatus. In some embodiments, a pupil shape changes along the scanning direction and this variation in pupil shape may also have an impact on the imaging of the patterns. Accordingly, the impact of the dynamic pupil shape is incorporated in selecting the patterns for monitoring fading effect. Figure 5 illustrates a dynamic pupil whose shape varies along the scanning direction, consistent with various embodiments. Figure 5 shows images 500 of pupil and illumination pupil profile 550 at various locations along the scanning direction 525. For example, if the scanning field is of a dimension “19x13” square units where “19” is in the scanning direction 525, a pupil shape may vary from a first pupil shape 505 to a nineteenth pupil shape 510 at each field point of the nineteen field points (e.g., a first scanning field point 515 Y;=l to a nineteenth scanning field point 520, Yi=19). The fading source 402 may be used to represent such a dynamic pupil.

[0060] The fading source 402 may be represented as:

_S(n _{= y}n_{S ki y})

... Eq. (l) where S, is an illumination pupil profile of a source (e.g., a source 12 A of Figure 1, or a source, SO of Figure 2) of the lithographic apparatus (e.g., the lithographic apparatus 10A of Figure 1, or the lithographic apparatus of Figure 2), y is a location in the scanning direction 525, k is indicative of a particular location in the pupil or an angle of incidence of illumination at the particular location, and n is an order of the fading source and can be any arbitrary number.

[0061] In some embodiments, a first order and second order of the fading source 402 are determined. For example, the first order fading source may be represented as: jO) = yS(k, y)

... Eq. (2)

[0062] Similarly, the second order fading source may be represented as:

St²) = y²S(k, y)

... Eq. (3)

[0063] A fading transmission cross-coefficient (TCC) 404 that is a TCC derived from the fading source 402 is determined. For example, a first order fading TCC, TCC^{( >}, and a second order fading TCC, TCC are determined based on the first order and second order fading source 402, respectively.

[0064] The fading image 410, FI, is obtained using the fading TCC 404 and a mask pattern 406 corresponding to the design layout (e.g., the mask pattern 406 is convolved with the fading TCC 404). In some embodiments, the fading image 410 is an image whose spatial derivative is indicative of the sensitivity of the corresponding aerial image to the MSD. A spatial derivative 412 of the fading image 410 is determined based on a sensitivity of the fading image 410 to each of various locations in the fading image 410. In some embodiments, the spatial derivative 412 includes a first order spatial derivative and a second order spatial derivative of the fading image 410. The first order spatial derivative of the fading image may be represented as:

... Ex. (4) where FI is the fading image 410, (x,y) is the spatial coordinate in the FI.

[0065] The second order spatial derivative of the fading image 410 may be represented as:

... Ex. (5)

[0066] The relation between the spatial derivative 412 of the fading image 410 and the sensitivity of the aerial image to MSD may be represented as:

... Eq. (6) where Al is the aerial image 414 corresponding to the mask pattern 406 (e.g., determined using the mask pattern 406 and a TCC 408 of a source of the lithographic apparatus), lx is the linear MSD in x-direction (e.g., wafer stage moving profile in the x-direction) and ly is the linear MSD in y-direction (e.g., wafer stage moving profile in the y-direction).

[0067] Accordingly, the sensitivity of the aerial image to the MSD is obtained using the spatial derivative 412 of the fading image 410.

[0068] In some embodiments, the fading sensitivity indicator 417 is a patterning parameter such as EPE. Accordingly, to determine a derivative 418 of the fading sensitivity indicator 417 with respect to the MSD, a derivative 416 of the EPE with respect to the MSD is determined. In some embodiments, the derivative 416 of the EPE includes the first and second order derivatives of the EPE to MSD. The first and second order derivatives of the EPE to MSD are determined using (a) the aerial image 414 and (b) the first order and second order derivatives of the aerial image (e.g., Eq. (6)), respectively, which are themselves determined based on the first order and second order spatial derivatives of the fading image 410 (e.g., expressions (4) and (5)). The first order derivative of the EPE with respect to MSD may be represented as: depe(x, l~) depe(x, l~) dl_x ’ dl y_v

... Ex. (7) where epe is the edge placement error at an evaluation point, x is spatial co-ordinate (e.g., in a vector form) of an evaluation point at which the EPE is determined, I is linear MSD (e.g., in vector form), lx is the linear MSD in x-direction (e.g., wafer stage moving profile in the x-direction) and ly is the linear MSD in y-direction (e.g., wafer stage moving profile in the y-direction).

[0069] The second order derivative of the EPE with respect to MSD may be represented as: d² epe(x, l~) d²epe(x, l~) d²epe(x, l~) dl_x ' dl y² ' dl_x dl y_v

... Ex. (8)

[0070] As mentioned above, the fading sensitivity indicator 417, FSl, is determined based on EPE for each evaluation point of a number of evaluation points on a pattern. For example, the fading sensitivity indicator may be represented as:

FSl = J ^eP^ek k

... Eq. (9) where epe is the edge placement error at an evaluation point, k is the number of evaluation points on a pattern, and n is the order of the equation.

[0071] The derivative 418 of the fading sensitivity indicator 417 with respect to the MSD, which is indicative of the fading impact on a pattern, that is, the sensitivity of the pattern to image fading, is determined using the derivative 416 of the EPE. In some embodiments, the derivative 418 of the fading sensitivity indicator 417 includes a first order and second order derivative of the fading sensitivity indicator 417 with respect to the MSD. For example, the first order and second order derivatives of the fading sensitivity indicator 417 are determined using the first order and second order derivatives of the EPE (e.g., expressions (7) and (8)). The first order derivative of the fading sensitivity indicator 417 with respect to MSD may be represented as: d FS1 d FSl dl_x ’ dl y_v

... Ex. (10) where FSl is the fading sensitivity indicator 417 for all evaluation points, lx is the linear MSD in x-direction (e.g., wafer stage moving profile in the x-direction) and ly is the linear MSD in y-direction (e.g., wafer stage moving profile in the y-direction).

[0072] The second order derivative of the fading sensitivity indicator 417 with respect to MSD may be represented as: d² FSI d²FSI d²FSI dl² ' dl y² ' dl_x dl y_v

... Ex. (11)

[0073] In some embodiments, each of the patterns 450 is represented as a feature vector of derivatives 418 of the fading sensitivity indicator 417. For example, the feature vector is determined based on a magnitude of the MSD of the lithographic apparatus, a, and the first order and second order derivatives of the fading sensitivity indicator 417 with respect to the MSD. An example feature vector 420 of the pattern determined using the derivatives 418 of the fading sensitivity indicator 417 may be represented as follows:

... Eq. (12)

[0074] The feature vectors may be ranked or grouped based on a specified criterion (e.g., any of a known number of grouping or ranking methods) and a set of patterns 430 may be selected based on the ranking or grouping. For example, the feature vectors may be grouped using K-means to generate a number of groups 425i-425_n, and the set of patterns 430 may be selected from one or more of the groups. In some embodiments, if the feature vectors are ranked, the set of patterns 430 may be selected based on the ranking (e.g., patterns corresponding to top n ranked feature vectors).

[0075] Regarding the selection of the cutlines 480, a fading sensitivity indicator 419 is determined for each cutline of a number of cutlines that can be placed on a pattern. Figure 6 illustrates placement of cutlines on a pattern, consistent with various embodiments. As illustrated in Figure 6, a number of candidate cutlines 480 such as a first cutline 605, a second cutline 607 and a third cutline 609 are considered for placement on a pattern such as a contact hole 650. Each of the cutlines is associated with two endpoints. For example, the first cutline 605 is associated with a pair of endpoints 606 and 616, the second cutline with a pair of endpoints 604 and 614, and the third cutline 609 with a pair of endpoints 602 and 612. The fading sensitivity indicator 419 is determined for each cutline. In some embodiments, the fading sensitivity indicator 419, FSI, for cutline selection is determined using EPE similar to the fading sensitivity indicator 417 for pattern selection. However, the difference is in how the EPE is computed for determining the fading sensitivity indicator 419 for cutline selection. For example, the FSI is determined as an EPE for a pair of evaluation points corresponding to the two endpoints of a cutline. That is, in the Eq. 9, k is assigned a value of “2” (e.g., indicative of the two endpoints of a cutline) instead of the total number of evaluation points on a pattern that is used for determining the fading sensitivity indicator 417 for pattern selection process. Such a fading sensitivity indicator 419 is determined for each of the cutlines 480. The derivatives 418 of the fading sensitivity indicator 419 with respect to the MSD, the feature vector of the derivatives 418 of the fading sensitivity indicator 419, are determined for each cutline in a way similar to the derivatives 418 of the fading sensitivity indicator 417 and the feature vector 420, respectively, as described above for the pattern selection. The feature vectors of the derivatives 418 of the fading sensitivity indicator 419 of all cutlines 480 are grouped or ranked (e.g., similar to that described for the pattern selection above) and a set of cutlines 490 may be selected based on the grouping or ranking. For example, the feature vectors may be grouped using K-means to generate a number of groups 485i-485_x, and the set of cutlines 490 may be selected from one or more of the groups. In some embodiments, if the feature vectors are ranked, the set of cutlines 490 may be selected based on the ranking (e.g., cutlines corresponding to top n ranked feature vectors).

[0076] Figures 7A and 7B are flow diagrams of an exemplary method for selecting patterns for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments. The method is described at least with reference to Figures 4-6 above. At process P702, an illumination pupil profile of a source of a lithographic apparatus is obtained. For example, the illumination pupil profile 550 that is descriptive of the illumination pupil profile at various locations along the scanning direction of the slit in the lithographic apparatus is obtained.

[0077] At process P704, a fading source representing a scanning location dependent illumination pupil profile of the lithographic apparatus is obtained. In some embodiments, the fading source 402 is determined based on a product of the illumination pupil profile (e.g., obtained in process P702 above) and a location in a scanning direction of the slit. For example, a first order fading source and a second order fading source are determined (e.g., using Eqs. (2) and (3)).

[0078] At process P706, a fading image of a pattern is determined using the fading source. In some embodiments, the fading image 410 is an image whose spatial derivative is indicative of the sensitivity of the corresponding aerial image with respect to the stage modulation of a wafer stage of the lithographic apparatus (e.g., MSD of a moving profile of the wafer stage). In some embodiments, obtaining the fading image 410 includes obtaining a first order fading image and second order fading image (e.g., as indicated using expressions (4) and (5) above). A fading TCC 404, which is a TCC of the fading source 402 is determined, and then used along with the mask pattern 406 to determine the fading image 410. For example, a first order fading TCC, TCC¹, and a second order fading TCC, TCC², are determined based on the first order and second order fading source 402, respectively. The first order and second order fading image are obtained using the mask pattern 406 and the first order and second order fading TCCs, respectively (e.g., by convolving the mask pattern 406 with the fading TCCs).

[0079] At process P708, a fading sensitivity of the patterns is determined based on a spatial derivative of the fading image. For example, the fading sensitivity of the pattern is determined using derivatives of a fading sensitivity indicator (e.g., EPE) with respect to MSD. The derivatives 418 of the fading sensitivity indicator 417 are determined using the spatial derivatives 412 of the fading image 410, which is described in more detail at least with reference to Figure 4 above and Figure 7B below.

[0080] After determining the derivatives 418 of the fading sensitivity indicator 417, at process P710, each of the patterns is represented as a feature vector of derivatives 418 of the fading sensitivity indicator 417. For example, the feature vector 420 is determined based on a magnitude of the MSD of the lithographic apparatus, a, and the first and second order derivatives of the fading sensitivity indicator 417 with respect to MSD. An example feature vector 420 of the pattern is shown at least with reference to Eq. (12) above.

[0081] At process P712, a set of patterns 430 from the patterns 450 is selected based on the fading sensitivity indicator 417 for monitoring fading effect. In some embodiments, the selection process may include ranking or grouping the feature vectors based on a specified criterion (e.g., using any of a known number of grouping or ranking methods) and selecting the set of patterns 430 based on the ranking or grouping. For example, the feature vectors may be grouped using K-means to generate a number of groups 425i-425_n, and the set of patterns 430 may be selected from one or more of the groups. In some embodiments, if the feature vectors are ranked, the set of patterns 430 may be selected based on the ranking (e.g., patterns corresponding to top n ranked feature vectors).

[0082] Figure 7B is a flow diagram of an exemplary method for determining derivatives of a fading sensitivity indicator with respect to an MSD for a pattern, consistent with various embodiments. In some embodiments, the method of Figure 7B may be executed as part of process P708 of the method of Figure 7 A. The embodiments determine the derivatives 418 of the fading sensitivity indicator 417 with respect to the MSD using the fading image 410. At process P752, spatial derivatives 412 of the fading image 410 are determined. A spatial derivative 412 of the fading image 410 is determined based on a sensitivity of the fading image 410 to each of various locations in the fading image 410. For example, a first order spatial derivative and a second order spatial derivative of the fading image 410 are obtained as indicated by the expressions (4) and (5) above. In some embodiments, the spatial derivative 412 of the fading image 410 is indicative of the sensitivity of the corresponding aerial image with respect to the MSD. Accordingly, the first order and second order derivatives of the aerial image with respect to the MSD are obtained using the first order and second order spatial derivatives of the fading image 410 (e.g., using Eq. (6)).

[0083] At process P754, a derivative 416 of an EPE with respect to the MSD is determined using the derivative of the aerial image to the MSD obtained in process P752 above. For example, the first order derivative of the EPE and the second order derivative of the EPE (e.g., indicated using expressions (7) and (8) above) may be obtained using the first order and second order derivatives of the aerial image with respect to the MSD, respectively.

[0084] At process P756, a derivative 418 of the fading sensitivity indicator (e.g., EPE) 417 to with respect to the MSD is determined using the derivative 416 of EPE obtained in process P754 above. For example, the first order and second order derivatives of the fading sensitivity indicator 417 (e.g., expressions (10) and (11)) are determined using the first order and second order derivatives of the EPE. After determining the derivatives 418 of the fading sensitivity indicator 417, the method returns to process P710 of Figure 7A for pattern selection. In some embodiments, in determining the derivative 418 of the fading sensitivity indicator 417, the fading sensitivity indicator 417 is computed based on EPE at each of a number of evaluation points on the pattern (e.g., using Eq. (9)).

[0085] While the foregoing paragraphs of Figures 7A and 7B describe a method for selecting patterns for monitoring the fading effect, the method may also be implemented for selecting cutline placements for monitoring the fading effect. For example, in process P756 of Figure 7B, a fading sensitivity indicator 419 is computed based on EPE for a pair of evaluation points corresponding to the two endpoints of a cutline instead of a number of evaluation points on the pattern as used in determination of the fading sensitivity indicator 417. That is, in the Eq. (9), for cutline selection process, k is assigned a value of “2” (e.g., indicative of the two endpoints of a cutline) instead of the total number of evaluation points on a pattern assigned in the determination of the fading sensitivity indicator 417 in the pattern selection process. Such a fading sensitivity indicator 419 is determined for each of a number of candidate cutlines. After determining the fading sensitivity indicator 419, the process of determining the derivative 418 of the fading sensitivity indicator 419 with respect to the MSD, generating the feature vectors including the derivatives 418 of the fading sensitivity indicator 419, grouping or ranking the feature vectors, and selecting the cutlines based on the grouping or ranking is similar to the process described at least with reference to P708-P712 of Figure 7A above. [0086] Figure 8 is a block diagram of an exemplary system for selecting patterns based on second order spatial derivatives of an aerial image for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments. Figure 9 is a flow diagram of an exemplary method for selecting patterns based on second order derivatives of an aerial image for monitoring fading effect due to overlay correction in a lithography process, consistent with various embodiments. [0087] At process P902, an aerial image corresponding to the mask pattern 406 is obtained. The aerial image may be obtained in a number of ways. For example, an aerial image 414 may be obtained based on the mask pattern 406 and TCC 408 of a source of the lithographic apparatus (e.g., by convolving the mask pattern 406 with the TCC 408). In another example, the aerial image 414 may be obtained using one or more models, machine learning (ML) models or non-ML models. In another example, the aerial image 414 may be obtained using a simulation of the lithography process, as described at least with reference to Figure 3.

[0088] At process P904, a fading sensitivity of the patterns is determined based on second order spatial derivatives of the aerial image. The second order spatial derivatives of the aerial image 414 may be indicative of a sensitivity of a pattern to image fading. In some embodiments, a spatial derivative of the aerial image 414 is determined based on a sensitivity of the aerial image 414 to each of various locations in the aerial image 414. The second order spatial derivatives of the aerial image may be denoted as: d²AI d²AI d²AI d²AI dx² ' dy² ' dx dy ’ dy dx

... Ex. (13) where Al is the aerial image 414, (x,y is the spatial coordinate in the Al.

[0089] In some embodiments, the second order spatial derivatives of the aerial image 414 may be determined as a Hessian matrix 804, which may be represented as below:

... Eq. (14)

[0090] The Hessian matrix 804 of the second order spatial derivatives of the aerial image may be used to indicate of sensitivity of the pattern to image fading. For example, a magnitude 806 of the Hessian matrix represents the sensitivity of the patterns to image fading. The magnitude 806 may be determined in a number of ways, e.g., as the largest singular value of the Hessian matrix - eigenvalue, 2₀, or as an average of eigenvalues, 2₀ and

or in other ways. The eigenvalues may be determined in a number of ways (e.g., singular value decomposition, eigen decomposition). In some embodiments, using the largest singular value of the Hessian matrix, 2₀, aids in identifying patterns that are most sensitive to image fading. A fading sensitivity indicator 812 may be determined based on the second order spatial derivatives of the aerial image. In embodiments where the second order spatial derivatives of the aerial image are represented the Hessian matric, the fading sensitivity indicator 812 may be determined based on the magnitude 806 of the Hessian matrix (e.g., 2₀). The fading sensitivity indicator 812 may be further determined based on a metric 808 associated with the aerial image 414 (e.g., image log slope (ILS)). As an example, the fading sensitivity indicator 812 may be represented as:

FSI = .₀/ILS

... Eq. (15)

[0091] The fading sensitivity indicator 812 is computed for each evaluation point of a number of evaluation points on a pattern and for some or all patterns in the design layout.

[0092] At process P906, a set of patterns 830 is selected based on the fading sensitivity indicator. The set of patterns 830 are selected based on the fading sensitivity indicator values satisfying a specified criterion. For example, those patterns that contain evaluation points with the largest values of the fading sensitivity indicator 812 (e.g., ( ₀/ILS~) value) are selected for monitoring the fading effect. Similarly, a set of cutlines 890 are chosen based on the fading sensitivity indicator values associated with a pair of evaluation points corresponding to its endpoints satisfying a specified criterion. For example, those cutlines with the largest sum of fading sensitivity indicator values for the pair of evaluation points corresponding to its endpoints may be selected for placement on the selected patterns. Referring to Figure 6, the fading sensitivity indicator values may be calculated for all cutlines, and the cutline 607 may be selected for placement on the pattern 650 based on the sum of fading sensitivity indicator values of its endpoints 604 and 614 being the largest. Similarly, the cutline 609 may be selected based on the sum of fading sensitivity indicator values of its endpoints 602 and 612 being the largest.

[0093] The foregoing paragraphs describe selecting patterns for monitoring the fading effect based on their fading sensitivity indicator values. The following paragraphs describe reducing the fading sensitivity of the patterns. In some embodiments, a source mask optimization (SMO) process, which is a process that optimizes at least one of source parameters (e.g., illumination pupil profile), mask parameters (e.g., mask pattern), or wavefront in order to increase imaging quality (e.g., one or more parameters such as image contrast, edge placement error, CD uniformity, resist contours, depth of focus, throughput, etc.) of the patterning process, may be enhanced or adjusted to reduce the fading sensitivity of the patterns as well. A cost function of the SMO may be modified to include parameters that have an impact on the fading sensitivity of the patterns. For example, second order spatial derivatives of an aerial image of a pattern (e.g., Hessian matrix), which have an impact on the fading effect of a pattern (e.g., as described at least with reference to Figures 8-9), may be added as a cost term to the cost function of the SMO process to generate a “fading-aware” cost function. In another example, a stage modulation of a wafer stage of the lithographic apparatus (e.g., MSD), which has an impact on the fading effect of a pattern (e.g., as described at least with reference to Figures 4-7B), may be incorporated in the cost function of the SMO process to generate a fading-aware cost function. After performing the SMO with the fading-aware cost function, the resulting source parameters, mask parameters, or wavefront are optimized to reduce the fading sensitivity of the patterns.

[0094] Figure 10 is a flow diagram of a process for reducing fading sensitivity of the patterns by performing an SMO process, consistent with various embodiments. At process P1002, a set of patterns 1002 for which fading sensitivity is to be reduced is obtained from a design layout to be printed on a substrate. The set of patterns 1002 may include all or some of the patterns from the design layout. In some embodiments, the set of patterns 1002 may be user selected patterns.

[0095] In some embodiments, the set of patterns 1002 may be similar to the set of patterns 430 of Figure 4, which is selected based on their fading sensitivity indicator values. As a first example, as described at least with reference to Figure 7A, the selection process may include ranking or grouping feature vectors of the patterns based on a specified criterion (e.g., using any of a known number of grouping or ranking methods such as K-means) to generate a number of groups 4251 -425 _n, and selecting the set of patterns 430 based on the ranking or grouping (e.g., patterns corresponding to top n ranked feature vectors).

[0096] In some embodiments, the set of patterns 1002 may be similar to the set of patterns 830 of Figure 8, which is selected based on the fading sensitivity indicator satisfying a specified criterion. For example, those patterns that contain evaluation points with the largest values of the fading sensitivity indicator 812 (e.g., ( ₀/ILS~) value) are selected.

[0097] At process P1004, a set of mask patterns 1004 corresponding to the set of patterns 1002 is obtained.

[0098] At process P1006, the fading sensitivity of the set of pattern 1002 is reduced by adjusting at least one of (a) source parameters (e.g., an illumination pupil profile of a source of the lithographic apparatus), (b) mask parameters (e.g., the mask pattern), or (c) wavefront based on a cost function. In some embodiments, the source or mask parameters or wavefront are adjusted by performing an SMO process with an adjusted cost function. The cost function of the SMO process may be adjusted to generate a “fading-aware” cost function by including parameters (e.g., second order derivatives of an aerial image of the set of patterns 1002 or stage modulation (MSD) of a wafer stage of the lithographic apparatus) that have an impact on the fading sensitivity of the patterns. The SMO process with the fading-aware cost function is executed, which generates output parameters 1006 including at least one of the source parameters (e.g., illumination pupil profile shape), mask parameters (e.g., mask pattern), or wavefront that are optimized to reduce the fading sensitivity of the set of patterns 1002.

[0099] In some embodiments, the cost function of an SMO process is an EPE based cost function, which may be represented as:

... Eq. (16) wherein the cost function s is specified in terms of variables of the illumination mode (v_src), variables of creating the mask pattern (v_mask), variables of the wavefront (e.g., the projection system) (^Vwavefront)- Further, pw corresponds to the process window conditions simulated for a set of parameters (e.g., focus, dose, etc.), eval corresponds to the evaluation features placed within the design pattern, w is a weighting factor for the particular pw and/or evaluation feature eval, EPE is edge placement error being evaluated for the particular combination of pw, and evaluation feature eval, index p is a natural number for the approximation of the cost function, p_Sideiobei^{s a} penalty corresponding to undesired side edge printing of the pattern, p_siOpei^{s a} penalty corresponding to the image slope (e.g., image log slope) of the pattern image, PMRC i^{s a} penalty corresponding to one or more patterning device manufacturing rule checks, and p_src penalty corresponding to the design of the illumination mode. As will be appreciated, less (including none), more or different penalties can be applied. Additionally, the cost function may also be defined in a number of ways and EPE is just one example implementation of the cost function of the SMO process.

[00100] The cost function, s, may be modified to a fading-aware cost function in any of a number of ways to include parameters that have an impact on the fading sensitivity of the patterns. Two examples of modifying the cost function to generate a fading-aware cost function are described at least with reference to Figures 11 A and 1 IB below, respectively.

[00101] Figure 11 A is a flow diagram of a method for modifying a cost function of an SMO process to a fading-aware cost function based on second order spatial derivates of an aerial image of a pattern, consistent with various embodiments. In some embodiments, the method of Figure 11 A may be implemented as part of process P1006 of Figure 10.

[00102] At process Pl 122, second order spatial derivatives of an aerial image of the set of patterns 1002 are obtained. For example, the second order spatial derivates such as the ones indicated in Eq. (13) are obtained based on an aerial image of the first pattern of the set of patterns 1002. In some embodiments, the second order spatial derivatives of the aerial image of the pattern is computed as a Hessian matrix. For example, the Hessian matrix such as the one in Eq. (14) may be computed. In some embodiments, a number of evaluation points may be identified on the pattern and the Hessian matrix may be computed for each evaluation point.

[00103] At process Pl 124, a cost term 1124 may be defined based on the second order spatial derivatives of the aerial image of the pattern. For example, the cost term, Pf_adi_ng, may be represented as: (norm CHi))^P

... Eq. (17) where i is the i-th evaluation point, w is a weight factor, p is a positive power, H, is Hessian matrix, and norm(H) is any type of matrix norm function. An example norm function may be a Frobenius norm function. The Frobenius norm function of the Hessian matrix may be represented as:

... Eq. (18)

[00104] At process Pl 126, the cost term 1124, Pf_adi_ng, may be added to the cost function, s, of the SMO process indicated in Eq. (16) to generate a fading-aware cost function 1126, Sf_ading. The fading-aware cost function 1126 may be represented as:

Sfading (^vsrc> ^vmask> ^Vwavefront) ^— ’ ■■■ + Psidelobe + Pslope + PMRC + p_src + Pfading ■ ■■ pw,eval

... Eq. (19)

[00105] Figure 1 IB is a flow diagram of another method for modifying a cost function of an SMO process to a fading-aware cost function based on a stage modulation of a wafer stage of a lithographic apparatus, consistent with various embodiments. In some embodiments, the method of Figure 1 IB may be implemented as part of process P1006 of Figure 10. At process Pl 142, a stage modulation 1142 of the wafer stage (e.g., MSD of a moving trajectory profile of the wafer stage) is obtained.

[00106] At process Pl 144, a set of parameters 1144 based on which the process window, pw, of cost function is computed is obtained. For example, the set of parameters 1144 may include dose and focus. The process window, pw, may be represented as pw (dose, focus, ... ).

[00107] At process Pl 146, the MSD 1142 is added to the set of parameters 1144 to generate a fading-aware cost function 1146. The process window is computed based on the modified set of parameters, which includes the MSD 1142. The modified process window, pWf_ading, may be represented as pWf_ading (dose, focus, ... , MSD), and the fading-aware cost function 1146, S _ading, may be represented as:

4” Psrc ■■■

... Eq. (20)

[00108] Figure 12 is a block diagram illustrating a comparison of results between an SMO process and an enhanced SMO process with fading-aware cost function, consistent with various embodiments. Figure 12 shows a first illumination pupil profile 1202 and a first mask pattern 1212 generated by a non-fading-aware SMO process for a target pattern 1204. The Figure 12 also shows a second illumination pupil profile 1206 and a second mask pattern 1216 generated for the target pattern 1204 by an enhanced SMO process using the fading-aware cost function (e.g., fading-aware cost function 1126). Note that the second illumination pupil profile 1206 is different from the first illumination pupil profile 1202, and the second mask pattern 1216, which is larger than the first mask pattern 1212, are both optimized for reducing the fading sensitivity of the target pattern 1204.

[00109] Figure 13 is another block diagram illustrating a comparison of results between an SMO process and an enhanced SMO process with fading-aware cost function, consistent with various embodiments. Figure 13 shows a first mask pattern 1302 generated for a target pattern (not illustrated) at a nominal condition and a second mask pattern 1304 generated by a non-fading-aware SMO process. Note that the CD 1322 is sensitive (e.g., varies) with respect to MSD linear in x direction. The Figure 12 also shows a third mask pattern 1314 generated for the target pattern by an enhanced SMO process using the fading-aware cost function (e.g., fading-aware cost function 1146). Note that the sensitivity of the CD (e.g., variation of the CD) to MSD linear x is reduced significantly for the third mask pattern 1314 compared to the second mask pattern 1304. The variation of CD with respect to MSD linear x for the second mask pattern 1304 generated by the non-fading-aware process and the third mask pattern 1314 generated by the enhanced SMO process using the fading-aware cost function are illustrated using the first graph 1306 and the second graph 1316, respectively. As can be appreciated, the fading-aware cost function based SMO process reduces the variation of CD with respect to MSD linear x significantly compared to variation of the CD of the second mask pattern 1304 generated by the non-fading-aware process.

[00110] Figure 14 is a block diagram that illustrates a computer system 100 which can assist in implementing various methods and systems disclosed herein. The computer system 100 may be used to implement any of the entities, components, modules, or services depicted in the examples of the figures (and any other entities, components, modules, or services described in this specification). The computer system 100 may be programmed to execute computer program instructions to perform functions, methods, flows, or services (e.g., of any of the entities, components, or modules) described herein. The computer system 100 may be programmed to execute computer program instructions by at least one of software, hardware, or firmware.

[00111] Computer system 100 includes a bus 102 or other communication mechanism for communicating information, and a processor 104 (or multiple processors 104 and 105) coupled with bus 102 for processing information. Computer system 100 also includes a main memory 106, such as a random access memory (RAM) or other dynamic storage device, coupled to bus 102 for storing information and instructions to be executed by processor 104. Main memory 106 also may be used for storing temporary variables or other intermediate information during execution of instructions to be executed by processor 104. Computer system 100 further includes a read only memory (ROM) 108 or other static storage device coupled to bus 102 for storing static information and instructions for processor 104. A storage device 110, such as a magnetic disk or optical disk, is provided and coupled to bus 102 for storing information and instructions.

[00112] Computer system 100 may be coupled via bus 102 to a display 112, such as a cathode ray tube (CRT) or flat panel or touch panel display for displaying information to a computer user. An input device 114, including alphanumeric and other keys, is coupled to bus 102 for communicating information and command selections to processor 104. Another type of user input device is cursor control 116, such as a mouse, a trackball, or cursor direction keys for communicating direction information and command selections to processor 104 and for controlling cursor movement on display 112. This input device typically has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allows the device to specify positions in a plane. A touch panel (screen) display may also be used as an input device.

[00113] According to one embodiment, portions of one or more methods described herein may be performed by computer system 100 in response to processor 104 executing one or more sequences of one or more instructions contained in main memory 106. Such instructions may be read into main memory 106 from another computer-readable medium, such as storage device 110. Execution of the sequences of instructions contained in main memory 106 causes processor 104 to perform the process steps described herein. One or more processors in a multi-processing arrangement may also be employed to execute the sequences of instructions contained in main memory 106. In an alternative embodiment, hard-wired circuitry may be used in place of or in combination with software instructions. Thus, the description herein is not limited to any specific combination of hardware circuitry and software.

[00114] The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to processor 104 for execution. Such a medium may take many forms, including but not limited to, non-volatile media, volatile media, and transmission media. Nonvolatile media include, for example, optical or magnetic disks, such as storage device 110. Volatile media include dynamic memory, such as main memory 106. Transmission media include coaxial cables, copper wire and fiber optics, including the wires that comprise bus 102. Transmission media can also take the form of acoustic or light waves, such as those generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD- ROM, DVD, any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave as described hereinafter, or any other medium from which a computer can read.

[00115] Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor 104 for execution. For example, the instructions may initially be borne on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to computer system 100 can receive the data on the telephone line and use an infrared transmitter to convert the data to an infrared signal. An infrared detector coupled to bus 102 can receive the data carried in the infrared signal and place the data on bus 102. Bus 102 carries the data to main memory 106, from which processor 104 retrieves and executes the instructions. The instructions received by main memory 106 may optionally be stored on storage device 110 either before or after execution by processor 104.

[00116] Computer system 100 also preferably includes a communication interface 118 coupled to bus 102. Communication interface 118 provides a two-way data communication coupling to a network link 120 that is connected to a local network 122. For example, communication interface 118 may be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding type of telephone line. As another example, communication interface 118 may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also be implemented. In any such implementation, communication interface 118 sends and receives electrical, electromagnetic, or optical signals that carry digital data streams representing various types of information.

[00117] Network link 120 typically provides data communication through one or more networks to other data devices. For example, network link 120 may provide a connection through local network 122 to a host computer 124 or to data equipment operated by an Internet Service Provider (ISP) 126. ISP 126 in turn provides data communication services through the worldwide packet data communication network, now commonly referred to as the “Internet” 128. Local network 122 and Internet 128 both use electrical, electromagnetic, or optical signals that carry digital data streams. The signals through the various networks and the signals on network link 120 and through communication interface 118, which carry the digital data to and from computer system 100, are exemplary forms of carrier waves transporting the information.

[00118] Computer system 100 can send messages and receive data, including program code, through the network(s), network link 120, and communication interface 118. In the Internet example, a server 130 might transmit a requested code for an application program through Internet 128, ISP 126, local network 122 and communication interface 118. One such downloaded application may provide for the illumination optimization of the embodiment, for example. The received code may be executed by processor 104 as it is received, or stored in storage device 110, or other non-volatile storage for later execution. In this manner, computer system 100 may obtain application code in the form of a carrier wave.

[00119] While the concepts disclosed herein may be used for imaging on a substrate such as a silicon wafer, it shall be understood that the disclosed concepts may be used with any type of lithographic imaging systems, e.g., those used for imaging on substrates other than silicon wafers. [00120] The terms “optimizing” and “optimization” as used herein refers to or means adjusting a patterning apparatus (e.g., a lithography apparatus), a patterning process, etc. such that results and/or processes have more desirable characteristics, such as higher accuracy of projection of a design pattern on a substrate, a larger process window, etc. Thus, the term “optimizing” and “optimization” as used herein refers to or means a process that identifies one or more values for one or more parameters that provide an improvement, e.g., a local optimum, in at least one relevant metric, compared to an initial set of one or more values for those one or more parameters. "Optimum" and other related terms should be construed accordingly. In an embodiment, optimization steps can be applied iteratively to provide further improvements in one or more metrics.

[00121] Aspects of the invention can be implemented in any convenient form. For example, an embodiment may be implemented by one or more appropriate computer programs which may be carried on an appropriate carrier medium which may be a tangible carrier medium (e.g., a disk) or an intangible carrier medium (e.g., a communications signal). Embodiments of the invention may be implemented using suitable apparatus which may specifically take the form of a programmable computer running a computer program arranged to implement a method as described herein. Thus, embodiments of the disclosure may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the disclosure may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine -readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine -readable medium may include read only memory (ROM); random access memory (RAM); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical, or other forms of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.), and others. Further, firmware, software, routines, instructions may be described herein as performing certain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc.

[00122] Embodiments of the present disclosure can be further described by the following clauses. 1. A method for monitoring fading effect in lithography, the method comprising: obtaining an illumination pupil profile of a source of a lithographic apparatus; determining a fading source representing a scanning location dependent illumination pupil profile of the lithographic apparatus, wherein the fading source is determined based on a product of the illumination pupil profile and a location in a scanning direction of a slit in the lithographic apparatus; determining a fading image of a pattern using the fading source; and determining a fading sensitivity of the patterns based on a spatial derivative of the fading image.

2. The method of clause 1, wherein the fading image is an image whose spatial derivative indicates a sensitivity of the corresponding aerial image with respect to a stage modulation of a wafer stage of the lithographic apparatus.

3. The method of clause 2, wherein the spatial derivative of the fading image includes at least one of a first order spatial derivative or a second order spatial derivative of the fading image.

4. The method of clause 2, wherein the stage modulation is programmed for overlay correction of an overlay between patterns of different layers of a design layout.

5. The method of clause 2, wherein the stage modulation is characterized by moving standard deviation (MSD) of a moving trajectory profile of the wafer stage.

6. The method of clause 1, wherein the fading source includes: a first order fading source corresponding to a product of the illumination pupil profile and the location in the scanning direction of the slit, and a second order fading source corresponding to a product of the illumination pupil profile and a power of two of the location in the scanning direction of the slit.

7. The method of clause 1, wherein determining the fading image includes: determining a fading transmission cross-coefficient (TCC) that is a TCC derived from the fading source; and determining the fading image based on the fading TCC.

8. The method of clause 7, wherein the fading TCC includes a first order fading TCC and a second order fading TCC, which are determined based on a first order and a second order of the fading source, respectively, wherein the first order fading source corresponds to a product of the illumination pupil profile and the location in the scanning direction of the slit, and the second order fading source corresponds to a product of the illumination pupil profile and a power of two of the location in the scanning direction of the slit.

9. The method of clause 1, wherein determining the fading sensitivity of the patterns includes: determining a derivative of a first fading sensitivity indicator to a stage modulation of a wafer stage of the lithographic apparatus based on the spatial derivative of the fading image. 10. The method of clause 9, wherein determining the derivative of the first fading sensitivity indicator includes: determining a derivative of an edge placement error to the stage modulation, wherein the first fading sensitivity indicator is determined based on the edge placement error associated with a plurality of evaluation points on each of the patterns.

11. The method of clause 10, wherein determining the derivative of the edge placement error to the stage modulation includes: determining a derivative of an aerial image of the patterns to the stage modulation.

12. The method of clause 11, wherein the derivative of the aerial image to the stage modulation is determined based on the spatial derivative of the fading image.

13. The method of clause 1, wherein each of the patterns is represented as a feature vector of a derivative of a first fading sensitivity indicator to a stage modulation of a wafer stage of the lithographic apparatus.

14. The method of clause 13, wherein the feature vector is determined based on a first order derivative of the first fading sensitivity indicator, a second order derivative of the first fading sensitivity indicator, and a constant that is representative of the stage modulation.

15. The method of clause 1 further comprising: selecting a set of patterns from the patterns based on a first fading sensitivity indicator for monitoring fading effect.

16. The method of clause 15, wherein selecting the set of patterns includes: grouping feature vectors of the patterns based on a specified criterion; and selecting the set of patterns based on the grouping.

17. The method of clause 1, wherein determining the fading sensitivity of the patterns further includes: determining the fading sensitivity of cutlines of the patterns based on a derivative of a second fading sensitivity indicator to a stage modulation of a wafer stage of the lithographic apparatus.

18. The method of clause 17, wherein the second fading sensitivity indicator is determined based on an edge placement error associated with pairs of evaluation points on the patterns corresponding to the cutlines, wherein each pair of evaluation points on a pattern corresponds to two endpoints of a cutline to be placed on the pattern.

19. The method of clause 17, wherein each of the cutlines is represented as a feature vector of a first order derivative and a second order derivative of the second fading sensitivity indicator to the stage modulation, and a constant that is representative of the stage modulation.

20. The method of clause 17 further comprising: selecting a set of cutlines from the cutlines based on the second fading sensitivity indicator for monitoring fading effect.

21. The method of clause 20, wherein selecting the set of cutlines includes: grouping feature vectors of the cutlines based on a specified criterion; and selecting the set of cutlines based on the grouping.

22. The method of clause 1 further comprising: selecting a set of patterns from the patterns based on a first fading sensitivity indicator; and adjusting (a) the illumination pupil profile or (b) a mask pattern corresponding to a first pattern of the set of patterns based on a cost function to reduce the fading sensitivity of the first pattern.

23. The method of clause 22, wherein the cost function is an edge placement error (EPE) based cost function.

24. The method of clause 22, wherein adjusting the illumination pupil profile or the mask pattern includes: defining a cost term based on second order spatial derivatives of an aerial image of the first pattern, and adding the cost term to the cost function.

25. The method of clause 24, wherein the second order spatial derivatives of the aerial image are determined as a Hessian matrix.

26. The method of clause 24, wherein the cost term is computed as a product of a weighting factor and a norm of Hessian matrix of the second order spatial derivatives of the aerial image.

27. The method of clause 24, wherein the cost term is computed for each evaluation point of a plurality of evaluation points associated with the first pattern.

28. The method of clause 22, wherein the cost function is computed for a process window condition, wherein the process window condition is computed for a set of parameters.

29. The method of clause 28, wherein adjusting the illumination pupil profile or the mask pattern includes: adding a stage modulation of a wafer stage of the lithographic apparatus to the set of parameters to generate an updated set of parameters, and computing the process window condition based on the updated set of parameters.

30. The method of clause 29, wherein the stage modulation is characterized by moving standard deviation (MSD) of a moving trajectory profile of the wafer stage.

31. The method of clause 22 further comprising: adjusting a wavefront based on the cost function to reduce the fading sensitivity of the first pattern.

32. A method for simulating fading effect in lithography, the method comprising: obtaining an aerial image of patterns to be printed on a substrate; determining fading sensitivity of the patterns based on second order spatial derivatives of the aerial image; and selecting a set of patterns from the patterns based on the fading sensitivity for monitoring fading effect.

33. The method of clause 32, wherein selecting the set of patterns based on the fading sensitivity includes: determining a value of a fading sensitivity indicator for each of the patterns; and selecting those patterns with values of the fading sensitivity indicator satisfying a first specified criterion as the set of patterns for monitoring fading effect.

34. The method of clause 32, wherein determining the fading sensitivity includes: determining a value of a fading sensitivity indicator for each of the patterns based on the second order spatial derivatives of the aerial image.

35. The method of clause 34, wherein the second order spatial derivatives are determined as a Hessian matrix.

36. The method of clause 35, wherein the fading sensitivity indicator is determined based on a magnitude of the Hessian matrix.

37. The method of clause 36, wherein the magnitude is representative of on an eigen value associated with the Hessian matrix.

38. The method of clause 34, wherein the fading sensitivity indicator is determined based on a metric associated with the aerial image.

39. The method of clause 38, wherein the metric is indicative of an image log slope of the aerial image.

40. The method of clause 34, wherein the fading sensitivity indicator is determined based on (a) an eigen value of a Hessian matrix of the second order spatial derivatives of the aerial image, and (b) an image log slope of the aerial image.

41. The method of clause 33, wherein determining the value of the fading sensitivity indicator includes: determining the value of the fading sensitivity indicator for each evaluation point of a plurality of evaluation points associated with a pattern of the patterns.

42. The method of clause 33 further comprising: determining the value of the fading sensitivity indicator for each cutline of a plurality of cutlines of a pattern of the patterns; and selecting those cutlines with values of the fading sensitivity indicator satisfying a second specified criterion as a set of cutlines for the pattern for monitoring fading effect.

43. The method of clause 42, wherein determining the value of the fading sensitivity indicator for each cutline includes: determining a pair of evaluation points on the pattern that corresponds to two endpoints of the corresponding cutline; and determining the value of the fading sensitivity indicator for each evaluation point of the pair of evaluation points.

44. A method for reducing fading sensitivity of a pattern by performing source mask optimization, the method comprising: obtaining a set of patterns from a design layout to be printed on a substrate using a lithographic apparatus; obtaining a mask pattern corresponding to a first pattern of the set of patterns; and reducing fading sensitivity of the first pattern by adjusting at least one of (a) an illumination pupil profile of a source of the lithographic apparatus or (b) the mask pattern based on a cost function.

45. The method of clause 44, wherein the cost function is an edge placement error (EPE) based cost function.

46. The method of clause 44, wherein adjusting the illumination pupil profile or the mask pattern includes: defining a cost term based on second order spatial derivatives of an aerial image of the first pattern, and adding the cost term to the cost function.

47. The method of clause 46, wherein the second order spatial derivatives of the aerial image are determined as a Hessian matrix.

48. The method of clause 46, wherein the cost term is computed as a product of a weighting factor and a norm of Hessian matrix of the second order spatial derivatives of the aerial image.

49. The method of clause 46, wherein the cost term is computed for each evaluation point of a plurality of evaluation points associated with the first pattern.

50. The method of clause 44, wherein the cost function is computed for a process window condition, wherein the process window condition is computed for a set of parameters.

51. The method of clause 50, wherein adjusting the illumination pupil profile or the mask pattern includes: adding a stage modulation of a wafer stage of the lithographic apparatus to the set of parameters to generate an updated set of parameters, and computing the process window condition based on the updated set of parameters.

52. The method of clause 51, wherein the stage modulation is characterized by moving standard deviation (MSD) of a moving trajectory profile of the wafer stage.

53. The method of clause 44 further comprising: adjusting a wavefront based on the cost function to reduce the fading sensitivity of the first pattern.

54. A method for reducing fading sensitivity of a pattern by performing source mask optimization, the method comprising: obtaining a set of patterns from a design layout to be printed on a substrate using a lithographic apparatus; obtaining a mask pattern corresponding to a first pattern of the set of patterns; and adjusting at least one of (a) an illumination pupil profile of a source of the lithographic apparatus or (b) the mask pattern based on a cost function to reduce fading sensitivity of the first pattern, wherein the cost function includes a cost term that is defined based on second order spatial derivatives of an aerial image of the first pattern.

55. The method of clause 54, wherein the cost function is an edge placement error (EPE) based cost function.

56. The method of clause 54, wherein the second order spatial derivatives of the aerial image are determined as a Hessian matrix.

57. The method of clause 54, wherein the cost term is computed as a product of a weighting factor and a norm of Hessian matrix of the second order spatial derivatives of the aerial image.

58. The method of clause 54, wherein the cost term is computed for each evaluation point of a plurality of evaluation points associated with the first pattern.

59. A method for reducing fading sensitivity of a pattern by performing source mask optimization, the method comprising: obtaining a set of patterns from a design layout to be printed on a substrate using a lithographic apparatus; obtaining a mask pattern corresponding to a first pattern of the set of patterns; and adjusting at least one of (a) an illumination pupil profile of a source of the lithographic apparatus or (b) the mask pattern based on a cost function to reduce fading sensitivity of the first pattern, wherein the cost function is computed for a process window condition, wherein the process window condition is computed based on a stage modulation of a wafer stage of the lithographic apparatus.

60. The method of clause 59, wherein the stage modulation is characterized by moving standard deviation (MSD) of a moving trajectory profile of the wafer stage.

61. The method of clause 59 further comprising: adjusting a wavefront based on the cost function to reduce the fading sensitivity of the first pattern.

62. An apparatus, the apparatus comprising: a memory storing a set of instructions; and a processor configured to execute the set of instructions to cause the apparatus to perform a method of any of the above clauses.

63. A non-transitory computer-readable medium having instructions recorded thereon, the instructions when executed by a computer implementing the method of any of the above clauses. [00123] In block diagrams, illustrated components are depicted as discrete functional blocks, but embodiments are not limited to systems in which the functionality described herein is organized as illustrated. The functionality provided by each of the components may be provided by software or hardware modules that are differently organized than is presently depicted, for example such software or hardware may be intermingled, conjoined, replicated, broken up, distributed (e.g., within a data center or geographically), or otherwise differently organized. The functionality described herein may be provided by one or more processors of one or more computers executing code stored on a tangible, non-transitory, machine -readable medium. In some cases, third party content delivery networks may host some or all of the information conveyed over networks, in which case, to the extent information (e.g., content) is said to be supplied or otherwise provided, the information may be provided by sending instructions to retrieve that information from a content delivery network.

[00124] Unless specifically stated otherwise, as apparent from the discussion, it is appreciated that throughout this specification discussions utilizing terms such as “processing,” “computing,” “calculating,” “determining” or the like refer to actions or processes of a specific apparatus, such as a special purpose computer or a similar special purpose electronic processing/computing device. [00125] The reader should appreciate that the present application describes several inventions. Rather than separating those inventions into multiple isolated patent applications, these inventions have been grouped into a single document because their related subject matter lends itself to economies in the application process. But the distinct advantages and aspects of such inventions should not be conflated. In some cases, embodiments address all of the deficiencies noted herein, but it should be understood that the inventions are independently useful, and some embodiments address only a subset of such problems or offer other, unmentioned benefits that will be apparent to those of skill in the art reviewing the present disclosure. Due to cost constraints, some inventions disclosed herein may not be presently claimed and may be claimed in later filings, such as continuation applications or by amending the present claims. Similarly, due to space constraints, neither the Abstract nor the Summary sections of the present document should be taken as containing a comprehensive listing of all such inventions or all aspects of such inventions.

[00126] It should be understood that the description and the drawings are not intended to limit the present disclosure to the particular form disclosed, but to the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the inventions as defined by the appended claims.

[00127] Modifications and alternative embodiments of various aspects of the inventions will be apparent to those skilled in the art in view of this description. Accordingly, this description and the drawings are to be construed as illustrative only and are for the purpose of teaching those skilled in the art the general manner of carrying out the inventions. It is to be understood that the forms of the inventions shown and described herein are to be taken as examples of embodiments. Elements and materials may be substituted for those illustrated and described herein, parts and processes may be reversed or omitted, certain features may be utilized independently, and embodiments or features of embodiments may be combined, all as would be apparent to one skilled in the art after having the benefit of this description. Changes may be made in the elements described herein without departing from the spirit and scope of the invention as described in the following claims. Headings used herein are for organizational purposes only and are not meant to be used to limit the scope of the description. [00128] 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 includes A or B, then, unless specifically stated otherwise or infeasible, the component may include A, or B, or A and B. As a second example, if it is stated that a component includes A, B, or C, then, unless specifically stated otherwise or infeasible, the component may include A, or B, or C, or A and B, or A and C, or B and C, or A and B and C. Expressions such as “at least one of’ do not necessarily modify an entirety of a following list and do not necessarily modify each member of the list, such that “at least one of A, B, and C” should be understood as including only one of A, only one of B, only one of C, or any combination of A, B, and C. The phrase “one of A and B” or “any one of A and B” shall be interpreted in the broadest sense to include one of A, or one of B.

[00129] The descriptions herein are intended to be illustrative, not limiting. Thus, it will be apparent to one skilled in the art that modifications may be made as described without departing from the scope of the claims set out below.

Claims

1. A method for simulating fading effect in lithography, the method comprising: obtaining an aerial image of patterns to be printed on a substrate; determining fading sensitivity of the patterns based on second order spatial derivatives of the aerial image; and selecting a set of patterns from the patterns based on the fading sensitivity for monitoring fading effect.

2. The method of claim 1, wherein selecting the set of patterns based on the fading sensitivity includes: determining a value of a fading sensitivity indicator for each of the patterns; and selecting those patterns with values of the fading sensitivity indicator satisfying a first specified criterion as the set of patterns for monitoring fading effect.

3. The method of claim 1, wherein determining the fading sensitivity includes: determining a value of a fading sensitivity indicator for each of the patterns based on the second order spatial derivatives of the aerial image, wherein the second order spatial derivatives are determined as a Hessian matrix.

4. The method of claim 3, wherein the fading sensitivity indicator is determined based on a magnitude of the Hessian matrix.

5. The method of claim 4, wherein the magnitude is representative of on an eigen value associated with the Hessian matrix.

6. The method of claim 2, wherein the fading sensitivity indicator is determined based on at least one of: (a) an eigen value of a Hessian matrix of the second order spatial derivatives of the aerial image, and(b) an image log slope of the aerial image.

7. The method of claim 2, wherein determining the value of the fading sensitivity indicator includes: determining the value of the fading sensitivity indicator for each evaluation point of a plurality of evaluation points associated with a pattern of the patterns.

8. The method of claim 2 further comprising: determining the value of the fading sensitivity indicator for each cutline of a plurality of cutlines of a pattern of the patterns; and selecting those cutlines with values of the fading sensitivity indicator satisfying a second specified criterion as a set of cutlines for the pattern for monitoring fading effect.

9. A method for reducing fading sensitivity of a pattern by performing source mask optimization, the method comprising: obtaining a set of patterns from a design layout to be printed on a substrate using a lithographic apparatus; obtaining a mask pattern corresponding to a first pattern of the set of patterns; and reducing fading sensitivity of the first pattern by adjusting at least one of (a) an illumination pupil profile of a source of the lithographic apparatus or (b) the mask pattern based on a cost function.

10. The method of claim 9, wherein the cost function is an edge placement error (EPE) based cost function, and wherein the cost function is computed for a process window condition, wherein the process window condition is computed for a set of parameters.

11. The method of claim 9, wherein adjusting the illumination pupil profile or the mask pattern includes: calculating a cost function comprising a cost term indicative of second order spatial derivatives of an aerial image of the first pattern.

12. The method of claim 11, wherein the second order spatial derivatives of the aerial image are determined as a Hessian matrix.

13. The method of claim 11, wherein the cost term is computed for each evaluation point of a plurality of evaluation points associated with the first pattern, wherein the cost term is computed as a product of a weighting factor and a norm of Hessian matrix of the second order spatial derivatives of the aerial image.

14. The method of claim 10, wherein adjusting the illumination pupil profile or the mask pattern includes: adding a stage modulation of a wafer stage of the lithographic apparatus to the set of parameters to generate an updated set of parameters, and computing the process window condition based on the updated set of parameters.

15. The method of claim 1 further comprising: adjusting a wavefront based on the cost function to reduce the fading sensitivity of the first pattern.