Abstract:
A method is provided for modeling lithographic processes in the design of photomasks for the manufacture of semiconductor integrated circuits, and more particularly for simulating intermediate range flare effects. For a region of influence (ROI) from first ROI1 of about 5λ/NA to distance ROI2 when the point spread function has a slope that is slowly varying according to a predetermined criterion, then mask shapes at least within the distance range from ROI1 to ROI2 are smoothed prior to computing the SOCS convolutions. The method provides a fast method for simulating intermediate range flare effects with sufficient accuracy.

Description:
TECHNICAL FIELD 
       [0001]    This invention relates generally to the field of optical lithography, and more particularly, to a method for incorporating intermediate-range flare effects for use in a model-based optical lithography simulation, to provide a fast and accurate correction of the shapes in a photo-mask or in mask verification during the design of a lithographic mask. 
       BACKGROUND OF THE INVENTION 
       [0002]    In the manufacture of integrated circuits, photolithography, or lithography, is typically used to transfer patterns relating to the layout of an integrated circuit onto a wafer substrate, including, but not limited to, materials such as silicon, silicon germanium (SiGe), silicon-on-insulator (SOI), or various combinations thereof. The drive to improve performance of very-large-scale integrated (VLSI) circuit-s results in increasing requirements to decrease the size of features and increase the density of layouts. This in turn increasingly requires the use of Resolution Enhancement Techniques (RET) to extend the capabilities of optical lithographic processes. RET includes but not limited to techniques such as the use of optical proximity correction (OPC), sub resolution assist feature (SRAF) enhanced lithography and phase-shifted-mask-enhanced lithography (PSM). 
         [0003]    In spite of the spectacular advancement of several forms of Resolution Enhancement Techniques (RET), the iterative Model-Based Optical Proximity Correction (MBOPC) methodology has established itself as a method of choice for compensation of the mask shapes for lithographic process effects during the process of designing such masks. Conventional MBOPC tools work include the following steps in a manner similar to the following. The shapes on the mask design (henceforth referred to as the mask) are typically defined as polygons. A pre-processing step is performed that divides the edges of each mask shape into smaller line segments. At the heart of the MBOPC tool is a simulator that simulates the image intensity at a particular point, which is typically at the center of each of the line segments. The segments are then moved back and forth, i.e., outward or inward from the feature interior, from their original position on the mask shape at each iteration-step of the MBOPC. The iteration stops when (as a result of the modification of the mask shapes) the image intensity at these pre-selected points matches a threshold intensity level, within a tolerance limit. 
         [0004]    The aforementioned methodology is illustrated in  FIG. 1 . In the current state of the art, an input mask layout  101  and a target image  106  are provided. The mask shapes are divided into segments  102 , where each segment is provided with a self-contained evaluation point ( 103 ). The optical and the resist image are then evaluated at evaluation points  104 . In step  105 , the images at each of the evaluation points are then checked against the tolerance of the target image  106 . If the image does not remain within tolerance the segment is iteratively moved forward or backward  107  until all segments reside within an accepted tolerance. Eventually, the final corrected mask layout is outputted  108 . 
         [0005]    Modeling aerial images is a crucial component of semiconductor manufacturing. Since present lithographic tools employ partially coherent illumination, such modeling is computationally intensive for all but the most elementary patterns. The aerial image generated by the mask, i.e., the light intensity of an optical projection system image plane, is a critically important parameter in micro-lithography for governing how well a developed photo-resist structure replicates a mask design and which, generally, needs to be computed to an accuracy of better than 1%. Such image models are used not only in mask correction (e.g. optical proximity correction methodologies) but also in other applications, such as mask verification methodologies. Mask verification is performed on a final mask design, after modification for example by optical proximity correction, to ensure that the final image will meet specified tolerances and not exhibit any catastrophic conditions such as opens, shorts, and the like. 
         [0006]    In an Aerial Image simulator, in addition to the diffraction of light in the presence of low order aberrations, the scattered light which affects the exposure over long distances on the wafer are recently being considered. Such long-range optical effects are generally referred to as “flare” in the literature. Flare affects the current extremely tight requirements on Across-Chip-Line-Width-Variation (ACLV). The flare effects are more pronounced in some novel RET methods requiring dual exposure such as alternating Phase Shifting Masks (Alt-PSM) or Double Di-Pole methodologies. The problem is even more pronounced in bright field masks that are used in printing critical levels which control the ultimate performance of the circuit, such as gate and diffusion levels. 
         [0007]    One significant difficulty when taking into consideration long range effects, such as flare, is the extent of the corrections flare effects required on the mask. The diffraction effects and corresponding optical lens aberrations that are modeled by the 37 lowest order Zernikes that dies off within a range of a few microns. The flare effect, on the other hand, extends up to a few mms, thus covering the entire chip area. 
         [0008]    Flare is generally considered to be the undesired image component generated by high frequency phase “ripples” in the wavefront corresponding to the optical process. Flare thus arises when light is forward scattered by appreciable angles due to phase irregularities in the lens. (An additional component of flare arises from a two-fold process of backscatter followed by re-scatter in the forward direction, as will be discussed hereinafter). High frequency wavefront irregularities are often neglected for three reasons. First, the wavefront data is sometimes taken with a low-resolution interferometer. Moreover, it may be reconstructed using an algorithm of an even lower resolution. Second, even when the power spectrum of the wavefront is known or inferred, it is not possible to include the effect of high frequency wavefront components on an image integral that is truncated at a short ROI distance, causing most of the scattered light to be neglected. Finally, it is not straightforward to include these terms in the calculated image. The present invention addresses these problems. 
         [0009]    It is generally observed that the flare energy F({right arrow over (r)}) from a wavefront ripple follows approximately the inverse power law relationship of the form given by: F({right arrow over (r)})=K/({right arrow over (r)}−{right arrow over (r)}) γ , where {right arrow over (r)} is the location of the point of interest influenced by the flare energy, {right arrow over (r)}′ is the location of the source of flare, K is a constant to be fitted and the exponent γ is referred to as the flare kernel parameter and is determined experimentally. Flare energy is proportional to 1/dose. An example of a plot of flare observed experimental data points  201  is shown in  FIG. 2A .  FIG. 2A  is a dimensionless plot of flare energy as a function of distance from the source of light for a typical optical process of a numerical aperture (NA) of 0.75 and a pupil size (σ) of 0.3. The curve  202  fitted to the data points  201  yields a value for γ of 1.85. 
         [0010]    In order to compute the impact of the flare on the image intensity at a point the flare kernel is convolved or integrated with the mask shapes. The convolved contribution of the mask shapes are summed up to get the image intensity at a point. This step is shown in  FIG. 2B . 
         [0011]      FIG. 2B  is a simplified flowchart illustration of a method of simulating or recreating an output wafer image as is known in the art using the flare kernel described above. As is known in the art, method  200  provides a mask layout  201  and a set of flare parameters  202  as inputs. Parameters  202  may include for example the flare kernel parameter γ, the wavelength λ of the light used, source parameters such as inner and outer radius σ 1  and σ 2  of the source pupil, numerical aperture NA, and Zernike parameters Z 1 , Z 2 , . . . , Zn that define the lens aberrations of the optical system. 
         [0012]    In order to simulate optical image intensity at a point  251 , method  200  considers at step  203  an flare interaction region, or region of influence ROI,  252  surrounding the simulation point  251 . Interaction region  252  may typically be a square or circular area having dimensions of typically in the range 5-20 microns across that encloses all shapes that will have a significant optical influence on the image intensity at the simulation point  251 . As is known in the art, the size of the interaction region  252  is normally determined by the tradeoff between computational-speed versus desired accuracy. The image computation may typically proceed by computing the coherent kernels (Block  204 ), which are convolved with each of the mask functions (Block  205 ), and the convolutions are summed (Block  206 ) to obtain the simulated image on the wafer plane (Block  207 ) 
         [0013]    Since the effect of flare diminishes slowly but steadily across the chip, some prior arts make certain trade offs in computing the convolution process. The most accurate of the computation is the convolution with the actual geometry of the mask shapes. However, this methodology is very slow. On the other hand the impact of flare diminishes considerably beyond 10-15 microns. Beyond this range the geometric details of the mask shapes can be approximated by a density map or a pixelated image of the geometric shapes. In the closer range (less than 1-2 microns), however, it is important to use the exact polygonal shapes for the accuracy of the image computation. Since exact polygonal shapes are used in any case for computing the diffraction limited part of the aerial image for this range of 1-3 microns, short range computation of the flare does not add to any significant runtime penalty. 
         [0014]    Referring to  FIG. 3A , a layout  30  is shown for describing pixilated or density map, whereby the layout  30  has thereon a plurality of finite geometrical shapes  31 . The cell array layout  30  is divided or partitioned into a plurality of uniform patterns, illustrated as uniform rectangles  34 . Other pixel shapes may be used; for example, the cell array layout  30  may be partitioned into any type of polygon pattern that is capable of spanning and covering the whole layout including, but not limited to, regular or irregular, convex or concave, or any combination thereof. 
         [0015]    Once the layout  30  is divided into the plurality of uniform squares  34 , a density map  40  of the layout may be computed, as shown in  FIG. 3B . This is accomplished by initially dividing the layout  30  into each of the plurality of individual squares  34  followed by determining that portion of each square  34  that is covered by any finite geometrical shape(s)  31 . Once the amount of coverage of each of the uniform squares  34  has been computed, each square  34  is then assigned a number based upon how much of that square is covered by finite geometrical shape(s)  31 . For example, as shown in  FIG. 3B  the percentage of coverage of each square is illustrated, whereby this percentage represents a density number  45  for each square. 
         [0016]    In accordance with the invention, the overall density map may represent numerous different density effects including, but not limited to, geometries of the finite geometrical shapes, the coverage of such geometries (e.g., the percentage of the present model-based hierarchal prime cell level that is covered by finite geometrical shapes versus that portion not covered by such shapes, such as that shown in  FIG. 3B ), the amount of coverage of the cell array layout  30  portion, area coverage, coverage by the computed aerial, resist or any other form of wafer image, perimeter coverage or any other topological coverage, and even combinations thereof. 
         [0017]    After the overall density map  40  of the prime cell level is complete, i.e., once all density numbers  45  for the plurality of squares  34  have been computed, the density map  40  represents qausi-images of the shapes, rather then using exact geometries. Each density  45  operating at each of the plurality of squares  34  are convolved with the inverse power law kernel to obtain a plurality of convolved operating densities across the density map. 
         [0018]    The geometric convolution is described with the help of  FIGS. 4A ,  4 B,  4 C,  4 D.  FIG. 4A  shows a rectangular shape  400 .  FIG. 4B  shows the same rectangular shape partitioned into 4 sectors, viz.  401 ,  402 ,  403  and  404 . An example of a sector is shown as  450 . Each of these sectors are bounded by one horizontal and one vertical line. Rectangle  400  can be expressed as the follows: 
         [0019]    Sector  401 −Sector  402 −Sector  403 +Sector  404 . In a geometric convolution each of these sectors are convolved with the flare kernel. 
         [0020]    The number of sectors for a shape is linearly proportional to the number of vertices of that shape. This is explained in  FIGS. 4C and 4D .  FIG. 4C  shows a polygon  410  with 10 vertices.  FIG. 4D  shows the corresponding shape in terms of sectors. Shape  410  can be parsed in terms of sectors as: (Sector  411 −Sector  412 −Sector  413 +Sector  414 )+(Sector  415 −Sector  416 −Sector  417 +Sector  418 )+(Sector  419 −Sector  420 −Sector  421 +Sector  422 ). 
         [0021]    Therefore, as the number of vertices increases in a shape the geometric convolution becomes more and more computationally expensive since the number of sectors increases. 
         [0022]    The differences between the pixel based and geometry based approach are further elaborated using  FIGS. 6A and 6B .  FIG. 6A  shows an exemplary mask  600  with 4 micron×4 micron area with vertices A, B, C and D. The distance between vertices A and B, B and C, C and D, and, D and A are 4 microns. Mask  600  consists of 4 copies of the shape  601  at the shown locations, so that the distance between them is  605 . Shape  601  is an ortho-normal shape with edges parallel to the x and the y axes only. In this example  605  has a value of 0.700 micron. The left most shape  601  is placed at a distance of  606  from the edge AD and at a distance  607  from the edge AB, respectively, of mask  600 . In this example  606  has a value of 0.350 micron and  607  has a value of 0.500 microns respectively. Shape  601  have 12 vertices, viz., a, b, c, d, e, f, g, h, i, j, k, l. In this particular example, the distances between respective vertices are given as follows: a and b: 0.295 micron, b and c: 0.300 micron, c and d: 0.005 micron, d and e: 0.300 micron, e and f: 0.005 micron, f and g: 2 micron, g and h: 0.005 micron, h and i: 0.300 micron, i and j: 0.005 micron, j and k: 0.300 micron and k and l: 0.295 micron. 
         [0023]    The shapes in mask  600  are used to compute the flare intensity at a point  606  at the center of the edge AD of the mask  600 . The flare kernel is shown as the curve  610  which is an inverse power-law kernel with the value of γ=1.5. 
         [0024]    Using geometric convolutions as explained above requires that each shape  601  of the mask is represented by 12 sectors. Therefore after 48 convolution computations (for 4 shapes) the value of the flare intensity as computed at point  606  is 0.032692. The above computation is the most accurate computation barring numerical errors. 
         [0025]      FIG. 6B  shows the effect of pixelization of the mask  600 . Mask  600  is pixelized into 16 pixels, viz.,  621  through  636 . Each pixel size is 1 micron×1 micron. The density values of the pixels are as follows: For pixels  621 , 622 ,  623 ,  624 : 0.2153; For pixels  625 ,  626 , 627 ,  628 : 0.4456; For pixels  629 ,  630 ,  631 ,  632 : 0.4785; and For pixels  633 ,  634 , 635 ,  636 : 0.1675. Using the above pixelized representation, the flare intensity at point  606  is computed as 0.029103. This result has a 10% error and uses 16 convolutions. If the pixels are made smaller the error reduces, but the number of convolution computation increases. 
         [0026]    There is a region that is in between the short and the long range, for example, between from 2-10 microns. This region is referred to as the “Intermediate Range.” It is important to make a very careful speed accuracy trade off for the computation in this region. There are several reasons, why intermediate-range computation of flare is very important. With better optics (high gamma) intermediate range flare dominates the longer range. Accuracy is more important for the intermediate range than the longer range. 
         [0027]    There are known methods using the density based approach or a pixelated polygons in the intermediate region for obtaining efficient computation, but that have the disadvantage of diminished accuracy. Yet other methods use exact mask geometries for accuracy at the significant cost of computational efficiency. Intermediate range flare dominates the flare computation both memory and runtime wise more than the short or the long range flares. 
         [0028]    Accordingly, it would be desirable to provide a method for computing the convolution of intermediate range flares with mask shapes in a manner that improves the efficiency of lithographic process models for use in MBOPC or mask verification, while not reducing the quality or accuracy of the simulations. 
       SUMMARY OF THE INVENTION 
       [0029]    According to a first aspect, a method is provided for designing a lithographic mask, including the use of a lithographic process model for simulating an image formed by illumination of the lithographic mask in a lithographic system, the method comprising: determining a first region of influence (ROI 1 ) around a point of interest on the mask design, such that mask features within said first ROI 1  will contribute a relatively large amount of flare energy at said point of interest; determining a second region of influence (ROI 2 ) around said point of interest, such that mask features outside of said ROI 2  will contributed a relatively small amount of flare energy at said point of interest in accordance with a predetermined criterion, such that the region between ROI 1  and ROI 2  comprises an intermediate region of influence (intermediate ROI); identifying an initial mask polygon shape having a first plurality of vertices located within the intermediate ROI; and smoothing said initial mask polygon shape to form a smoothed mask polygon shape that has fewer vertices within the intermediate ROI than said first plurality of vertices; determining a smoothed flare contribution at said point of interest from said vertices of said smoothed mask polygon within the intermediate ROI; and determining an image at the point of interest comprising using said smoothed flare contribution in the lithographic process model rather than a flare contribution from said initial mask polygon shape. 
         [0030]    According to another aspect of the invention, the ROI 1  has an outer boundary at a distance of about 5λ/NA around the point of interest, where 2 is the wavelength of the illumination energy and NA is the numerical aperture of the lithographic system. 
         [0031]    According to yet another aspect of the invention, the predetermined criterion comprises a slope cutoff for determining when a slope of the point spread function of the lithographic system is close to zero. 
         [0032]    According to a further aspect of the invention, the point spread function h/({right arrow over (r)}−{right arrow over (r)} Avg ) is a function of distance {right arrow over (r)}−{right arrow over (r)} Avg  from the point of interest {right arrow over (r)}, and the point spread function has the form h∝K/({right arrow over (r)}−{right arrow over (r)}−) γ , where K and γ are experimentally determined, and wherein the slope of the point spread function is given by 
         [0000]    
       
         
           
             
               
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         [0033]    According to yet another aspect of the invention, the intermediate ROI is further divided into a plurality of sub-intermediate ROIs, and wherein a different amount of smoothing is performed in at least one of said plurality of sub-intermediate ROIs than in another of said plurality of sub-intermediate ROIs. The amount of smoothing in a sub-intermediate ROI preferably depends on the proximity of said sub-intermediate ROI to said point of interest. 
         [0034]    According to another aspect of the invention, smoothing is performed by a sequential grow and shrink operation or a low-pass filtering in the spatial frequency domain. 
         [0035]    According to another aspect of the invention, the step of determining an image at the point of interest comprises determining a flare contribution from within said first ROI 1  using mask features within said first ROI 1  that are not smoothed. 
         [0036]    According to another aspect of the invention, the step of determining an image at the point of interest comprises determining a flare contribution from mask features located beyond said second ROI 2  using a density mapping approach. 
         [0037]    According to yet another aspect of the invention, the resulting image may be provided for use in an optical proximity correction methodology or in a mask verification methodology. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0038]    These and other features, aspects, and advantages will be more readily apparent and better understood from the following detailed description of the invention, with reference to the following figures wherein like designations denote like elements, which are not necessarily drawn to scale. 
           [0039]      FIG. 1  illustrates an exemplary method for performing optical proximity correction. 
           [0040]      FIG. 2A  illustrates a plot of observed flare as a function of distance from the source of light. 
           [0041]      FIG. 2B  illustrates an exemplary method for simulating an image including flare. 
           [0042]      FIG. 3A  illustrates a mask layout partitioned for simulating an image using a pixilated or density mapping approach. 
           [0043]      FIG. 3B  illustrates an assignment of density mapping values for the mask layout illustrated in  FIG. 3A . 
           [0044]      FIGS. 4A-4D  illustrate partitioning of mask shapes for use in a geometric convolution approach for simulating an image. 
           [0045]      FIG. 5  illustrates a plot of the slope of the point spread function used for determining ROI 2  in accordance with the invention. 
           [0046]      FIG. 6A  illustrates mask shapes having segments and vertices used in a geometric convolution approach for simulating an image. 
           [0047]      FIG. 6B  illustrates density values assigned to pixels representing mask shapes in a pixelization approach for simulating an image. 
           [0048]      FIGS. 7A-7B  illustrate smoothed shapes used in flare computations in accordance with the invention. 
           [0049]      FIGS. 8A-8E  illustrate embodiments of regions of influence (ROIs) around an evaluation point or point of interest in a mask layout, in accordance with the invention. 
           [0050]      FIG. 9  illustrates an embodiment of a method for computing flare energy at a point of interest, in accordance with the invention. 
           [0051]      FIG. 10  illustrates an embodiment of a computer system and computer program product configured to perform a method for computing flare energy, in accordance with the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0052]    An objective of the embodiments of the invention described herein is to provide a method and system by which evaluation of flare, particularly in the intermediate distance range, is done efficiently and accurately in a lithographic process model or simulator used in a designing a mask, for example, to perform optical proximity correction (OPC) or mask verification. Thus, according to embodiments of the invention, unnecessary variations are removed from the neighboring mask shapes that influence the computation of convolution of the flare in an intermediate distance ranging from a first ROI (ROI 1 ), preferably having a radius of about 5λ/NA around the point of interest where the image is to be simulated, to a second region of influence ROI 2  around the point of interest, determined according to a predetermined small flare influence criteria, beyond which the effect of flare would be sufficiently small so that a density mapping approach provides sufficient accuracy. As described with reference to  FIG. 2D , a density mapping approach represents the influence on the point of interest expressed as an average weighting of shapes within a pixel. In a preferred embodiment, the predetermined small flare criterion is where the ROI 2  has an outer boundary whose distance from the point of interest is such that the slope of the point spread function of the optical system is equal to or less than a slope cutoff close to zero. The slope cutoff is the predetermined small flare criterion in this embodiment. 
         [0053]    In accordance with the invention, smoothed versions of shapes are used corresponding to mask shapes that are in the range of intermediate flare influence between the region of high flare influence ROI 1  and a second region of small flare influence ROI 2 , as determined by a small flare influence criterion. The overall efficiency of flare calculations overall is preferably obtained by combining a rigorous flare calculation for shapes in the distance range less than a first radius of high flare influence for the ROI 1  (i.e. less than about 5λ/NA), the intermediate flare calculations using smoothed shapes in the intermediate range from ROI 1  to ROI 2  in accordance with the invention, and a density mapped computation of flare influence for shapes at distances greater than the outer boundary of ROI 2 . The ROI, according to the invention, is not limited to having a radial distance from the point of interest, but is also intended to encompass any distance from the point of interest that may be used to indicate the range of influence, such as a horizontal or vertical distance from the point of interest in a cartesian coordinate system. 
         [0054]    The advantage of the present invention is that it will reduce the number of unnecessary sectors for mask shapes in the intermediate flare range, which would improve the efficiency of the MBOPC iterations over the prior art. The reduced number of sectors in the intermediate range will also improve memory utilization of the MBOPC and also result in improved hierarchical handling for the OPC. Accuracy in the short range of high influence less than ROI 1  of about 5λ/NA may be maintained with a rigorous calculation using the original unsmoothed shapes. Additional efficiency may be obtained by using a density mapped representation of the shapes for distances greater than ROI 2 . 
         [0055]    The image intensity at a point on the wafer is modeled by the Hopkin&#39;s Equation described below in Equation (1). 
         [0000]        I   0 ( {right arrow over (r)} )=∫∫∫∫ d{right arrow over (r)}′d{right arrow over (r)}″h ( {right arrow over (r)}−{right arrow over (r)} ′) h *( {right arrow over (r)}−{right arrow over (r)} ″) j ( {right arrow over (r)}″−{right arrow over (r)} ″) m ( {right arrow over (r)} ′) m *( {right arrow over (r)} ″),   (1) 
         [0056]    where
       h is the lens point spread function of the optical system or kernel;   j is the partial coherence of the optical wave;   m is the mask amplitude function; and   {right arrow over (r)}′ and {right arrow over (r)}″ are dummy distance variables in the optical coordinate system from the point of interest {right arrow over (r)}.       
 
         [0061]    The above Equation (1) expression for intensity I({right arrow over (r)}) at the point of interest {right arrow over (r)} can be approximated by the Sum of Coherent Systems (SOCS) as: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0062]    where 
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         [0000]    and ROI is the region of interaction or influence. 
         [0063]    Defining {right arrow over (Δ)}={right arrow over (r)}″−{right arrow over (r)}′ and {tilde over ({right arrow over (r)}−={right arrow over (r)}−{right arrow over (r)} Avg , at large values of {tilde over ({right arrow over (r)}, use the following approximation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                   ) 
                 
               
             
           
         
       
     
         [0064]    Equation (2) can be further approximated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                      
                     
                       ( 
                       
                         r 
                         → 
                       
                       ) 
                     
                   
                   ≅ 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                            
                           
                             
                               ∫ 
                               ROI 
                               
                                   
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   
                                     r 
                                     → 
                                   
                                   Avg 
                                 
                               
                                
                               
                                 
                                   
                                     h 
                                     ~ 
                                   
                                   i 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       → 
                                     
                                     - 
                                     
                                       
                                         r 
                                         → 
                                       
                                       Avg 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 m 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       → 
                                     
                                     Avg 
                                   
                                   ) 
                                 
                               
                             
                           
                            
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ∫ 
                           ∫ 
                         
                         
                           outside 
                           ROI 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           2 
                         
                          
                         
                           
                             r 
                             → 
                           
                           Avg 
                         
                       
                        
                       
                         
                           
                              
                             
                               h 
                                
                               
                                 ( 
                                 
                                   
                                     r 
                                     → 
                                   
                                   - 
                                   
                                     
                                       r 
                                       → 
                                     
                                     Avg 
                                   
                                 
                                 ) 
                               
                             
                              
                           
                           2 
                         
                         [ 
                         
                           
                             
                               ∫ 
                               ∫ 
                             
                             
                               outside 
                               ROI 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                                
                               2 
                             
                              
                             
                               Δ 
                               → 
                             
                           
                            
                           
                             j 
                              
                             
                               ( 
                               
                                 Δ 
                                 → 
                               
                               ) 
                             
                           
                            
                           
                             m 
                             ( 
                             
                               
                                 
                                   r 
                                   → 
                                 
                                 Avg 
                               
                               - 
                               
                                 
                                   Δ 
                                   → 
                                 
                                 2 
                               
                             
                             ) 
                           
                            
                           
                             
                               m 
                               * 
                             
                             ( 
                             
                               
                                 
                                   r 
                                   → 
                                 
                                 Avg 
                               
                               + 
                               
                                 
                                   Δ 
                                   → 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where {tilde over (h)} i  is the conventional SOCS approximation to the point spread function or kernel, within the diffraction limited ROI 1 . 
         [0065]    We will describe the first term of the equation (4) as the diffraction limited part of the image or the SOCS image I SOCS ({right arrow over (r)}): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       SOCS 
                     
                      
                     
                       ( 
                       
                         r 
                         → 
                       
                       ) 
                     
                   
                   ≡ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                       
                         
                            
                           
                             
                               ∫ 
                               ROI 
                               
                                   
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                    
                                   2 
                                 
                                  
                                 
                                   
                                     r 
                                     → 
                                   
                                   Avg 
                                 
                               
                                
                               
                                 
                                   
                                     h 
                                     ~ 
                                   
                                   i 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       → 
                                     
                                     - 
                                     
                                       
                                         r 
                                         → 
                                       
                                       Avg 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 m 
                                  
                                 
                                   ( 
                                   
                                     
                                       r 
                                       → 
                                     
                                     Avg 
                                   
                                   ) 
                                 
                               
                             
                           
                            
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     4 
                      
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
         [0066]    At a long distance, j({right arrow over (Δ)})→δ({right arrow over (Δ)}), where δ({right arrow over (Δ)}) is an impulse response function. 
         [0067]    Substituting δ({right arrow over (Δ)}) for j({right arrow over (Δ)}) in Equation (4), we get: 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                      
                     
                       ( 
                       
                         r 
                         → 
                       
                       ) 
                     
                   
                   ≅ 
                   
                     
                       
                         I 
                         SOCS 
                       
                        
                       
                         ( 
                         
                           r 
                           → 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           ∫ 
                           ∫ 
                         
                         
                           outside 
                           
                             ROI 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           2 
                         
                          
                         
                           
                             r 
                             → 
                           
                           Avg 
                         
                       
                        
                       
                         
                            
                           
                             h 
                              
                             
                               ( 
                               
                                 
                                   r 
                                   → 
                                 
                                 - 
                                 
                                   
                                     r 
                                     → 
                                   
                                   Avg 
                                 
                               
                               ) 
                             
                           
                            
                         
                         2 
                       
                        
                       
                         m 
                          
                         
                           ( 
                           
                             
                               r 
                               → 
                             
                             Avg 
                           
                           ) 
                         
                       
                        
                       
                         
                           m 
                           * 
                         
                          
                         
                           ( 
                           
                             
                               r 
                               → 
                             
                             Avg 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                            
                           
                             h 
                              
                             
                               ( 
                               
                                 
                                   r 
                                   → 
                                 
                                 - 
                                 
                                   
                                     r 
                                     → 
                                   
                                   Avg 
                                 
                               
                               ) 
                             
                           
                            
                         
                         2 
                       
                        
                       
                         
                           ∫ 
                           ∫ 
                         
                         
                           outside 
                           
                             ROI 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           2 
                         
                          
                         
                           
                             r 
                             → 
                           
                           Avg 
                         
                       
                        
                       
                         m 
                          
                         
                           ( 
                           
                             
                               r 
                               → 
                             
                             Avg 
                           
                           ) 
                         
                       
                        
                       
                         
                           m 
                           * 
                         
                          
                         
                           ( 
                           
                             
                               r 
                               → 
                             
                             Avg 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0068]    Where ROI 1  is the range of the diffraction limited optics determined by a rule of thumb given as ROI 1 ˜5λ/NA, where NA is the numerical aperture of the optical system and λ is the wavelength of light. 
         [0069]    I SOCS ({right arrow over (r)}) is the SOCS approximation from the first term of Equation 4 as described by Equation (4A) above. 
         [0070]    The other two terms of Equation (5) are due to the flare energy of the optical light. 
         [0071]    The second term is referred to as the Intermediate Range Flare and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       ∫ 
                     
                     
                       outside 
                       
                         ROI 
                          
                         
                             
                         
                          
                         1 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       2 
                     
                      
                     
                       
                         r 
                         → 
                       
                       Avg 
                     
                   
                    
                   
                     
                        
                       
                         h 
                          
                         
                           ( 
                           
                             
                               r 
                               → 
                             
                             - 
                             
                               
                                 r 
                                 → 
                               
                               Avg 
                             
                           
                           ) 
                         
                       
                        
                     
                     2 
                   
                    
                   
                     m 
                      
                     
                       ( 
                       
                         
                           r 
                           → 
                         
                         Avg 
                       
                       ) 
                     
                   
                    
                   
                     
                       
                         m 
                         * 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             → 
                           
                           Avg 
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     B 
                   
                   ) 
                 
               
             
           
         
       
     
         [0072]    The third term of Equation 5 is referred to as the Long Range Flare, and is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         h 
                          
                         
                           ( 
                           
                             
                               r 
                               → 
                             
                             - 
                             
                               
                                 r 
                                 → 
                               
                               Avg 
                             
                           
                           ) 
                         
                       
                        
                     
                     2 
                   
                    
                   
                     
                       ∫ 
                       ∫ 
                     
                     
                       outside 
                       
                         ROI 
                          
                         
                             
                         
                          
                         2 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                     
                        
                       2 
                     
                      
                     
                       
                         r 
                         → 
                       
                       Avg 
                     
                   
                    
                   
                     m 
                      
                     
                       ( 
                       
                         
                           r 
                           → 
                         
                         Avg 
                       
                       ) 
                     
                   
                    
                   
                     
                       
                         m 
                         * 
                       
                        
                       
                         ( 
                         
                           
                             r 
                             → 
                           
                           Avg 
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     5 
                      
                     C 
                   
                   ) 
                 
               
             
           
         
       
     
         [0073]    The boundary between the SOCS approximation and the Intermediate Range Flare is determined by ROI 1  which is given as ˜5λ/NA. 
         [0074]    Referring to  FIG. 5 , a preferred embodiment in accordance with the invention is described for determining the boundary ROI 2  between the Intermediate Range Flare and the Long Range Flare.  FIG. 5  illustrates plots of the slope (plotted in normalized units along the vertical axis, where a slope of zero is indicated by reference numeral  500 ) of the point spread function h({right arrow over (r)}−{right arrow over (r)} Avg ) versus distance {right arrow over (r)}−{right arrow over (r)} Avg  from the point of interest {right arrow over (r)}. The point spread function or kernel h({right arrow over (r)}−{right arrow over (r)} Avg ) has the form h∝K/({right arrow over (r)}−{right arrow over (r)}′) γ , where K and γ are experimentally determined by measuring flare and fitting the power law function to the measured flare data. The slope of the point spread function is given by 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   h 
                    
                   
                     ( 
                     
                       
                         r 
                         → 
                       
                       - 
                       
                         
                           r 
                           → 
                         
                         Avg 
                       
                     
                     ) 
                   
                 
               
               
                 ∂ 
                 
                   r 
                   → 
                 
               
             
             , 
           
         
       
     
         [0000]    and is plotted in  FIG. 5  for various values of the flare kernel parameter γ. Typical values of γ current optical systems range between about 2 to 3. The curve  501  is the slope of h({right arrow over (r)}−{right arrow over (r)} Avg ) for γ=2, the curve  502  is the slope of h({right arrow over (r)}−{right arrow over (r)} Avg ) for γ=2.5, and the curve  503  is the slope h({right arrow over (r)}−{right arrow over (r)} Avg ) for γ=3. Note that the slopes  501 ,  502 ,  503  all converge to zero slope  500  for large values of {right arrow over (r)}−{right arrow over (r)} Avg , since the influence of flare diminishes with increasing distance from the point of interest. A slope cutoff criterion  510  may be chosen for a value of slope that sufficiently close to the value zero ( 500 ) such that the influence of flare beyond that distance is sufficiently small for the given mask layout and lithographic process. If the point spread function h({right arrow over (r)}−{right arrow over (r)} Avg ) is determined experimentally to have γ=2.5, which a reasonable value for current lenses, then for given a slope cutoff value  510 , the small flare radius of influence ROI 2  ( 525 ) is the distance given by the intersection of the slope of the point spread function h({right arrow over (r)}−{right arrow over (r)} Avg ) with the slope cutoff value  510 . 
         [0075]    In one embodiment in accordance with the present invention, multiple ROIs are defined around the evaluation point on an edge whose image intensity are to be evaluated, and the influencing neighboring shapes are smoothed to progressively remove details as the neighboring shape is located outside of a given ROI. 
         [0076]    Thus, in accordance with one embodiment of the invention, the amount by which a neighboring shape is smoothed depends on its proximity to the point of interest on the main shape. 
         [0077]      FIGS. 7A and 7B  illustrate modifications of mask shapes used in flare computations in accordance with the invention.  FIG. 7A  illustrates one embodiment of smoothed shapes  701  of the mask  600  as in  FIG. 6A  with the original shapes  601  now shown in dashed lines. The point of interest  606  is the point at which the flare intensity is to be computed. The radius of small flare influence ROI 2  is indicated by reference numeral  525 . 
         [0078]    Referring to  FIG. 7A , according to one embodiment, shapes  701  are created by removing unnecessary variations of the shape  601  for all shapes within the radius of small flare influence ROI 2   525  from the point of interest  606 . This operation of removing unnecessary variations is referred to as smoothing. 
         [0079]    For example, if the flare intensity is computed at point  606  using the geometric convolution method and using smoothed shapes  701  (instead of original shapes  601 ) it would result in a flare intensity value of 0.031513. This value has an error of 3.6% relative to the geometric convolution using the original unsmoothed shapes (see  FIG. 6A  and the related discussion above). However, the method using smoothed shapes in accordance with the invention only requires 16 convolution computations as compared to 48 convolutions using unsmoothed shapes. Note that this method achieves much better accuracy than the pixelized method with the same number of convolution computations. 
         [0080]    The accuracy of this computation can be further increased with some computation cost, as in another embodiment illustrated in  FIG. 7B . In the embodiment illustrated in  FIG. 7B , a distance  521  of high flare influence ROI 1  is identified, preferably about 5λ/NA, and all shapes  601  closest to the point  606  within ROI 1   521  are not smoothed at all. However, the shapes within the intermediate distance range  535  between ROI 1   521  and ROI 2   525  (the distance of small flare influence), are smoothed to result in smoothed shapes  701  prior to computing the influence of flare effects. Using the shapes shown in  FIG. 7B  the flare intensity is computed at point  606  using the geometric method would result in a value of 0.032508. This value has an error of 0.56% relative to geometric convolution using the original unsmoothed shapes, while requiring 24 convolution computations (as compared to the 48 convolutions using the unsmoothed shapes). 
         [0081]    In another embodiment in accordance with the present invention, the region of intermediate flare interaction ROI 2  may be divided into several sub-ROIs, each having progressively decreasing flare influence as distance increases from the point of interest  606 . The sub-ROIs may be defined as being contiguous or non-contiguous, and the embodiment is not intended to be a limiting example. The shapes within further sub-ROI&#39;s would have an increased amount of smoothing within the sub-ROI region relative to a sub-ROI region that is closer to the point of interest  606 . 
         [0082]    This is explained with the help of  FIGS. 8A ,  8 B and  8 C.  FIG. 8A  shows a mask shape  801  having a mask edge mask edge  810  and the corresponding evaluation point  815 . The radius of low flare influence ROI 2  corresponding to  815  is defined as the rectangular region  820 . Note that those skilled in the art may also use other appropriate shapes of ROI 2  such as a sphere or an ellipse or a other form of polygonal shapes or any Boolean combinations thereof. There are four shapes, viz.,  802 ,  803 ,  804  and  805  within ROI 2   820  that would influence the computation of the flare intensity at the point  815 . 
         [0083]    Now referring to  FIG. 8B , the ROI 2   820  is further divided into four regions  830 ,  840 ,  850  and  860 . In this embodiment ROI 2   820  is divided uniformly by distance. However, in another embodiment, region ROI 2   820  can be divided uniformly based on the flare contribution energy. Further in another embodiment the ROI 2  can be divided uniformly based on pattern density. The region  830  may be defined as a region of high flare influence ROI 1 . Shape  802  is within the high flare influence region ROI 1   830 . Shape  803  is within sub-ROI region  840 , shape  804  is within sub-ROI region  850  and shape  805  is within sub-ROI region  860 . 
         [0084]    Now referring to  FIG. 8C , shape  802  which is within ROI 1  region  830  and closest to the point of interest  815  is the unsmoothed original shape  802  used in computing the flare intensity. Shape  803  which is within sub-ROI region  840  is smoothed to result in shape  813  before it is used computing the flare intensity. Shape  804  which is within sub-ROI region  850  which is further away from point of interest  815  is smoothed even more to result in smoothed shape  814  before it is used computing the flare intensity at point  815 . Similarly, shape  805  which is within sub-ROI region  860 , which is furthest away from point  815 , is smoothed even more to result in smoothed shape  816  before it is used computing the flare intensity at point  815 . 
         [0085]    It is assumed in the current embodiment that all the variations are significant for the shapes that are closest to the main shape. However, in some designs, mask shapes may include sub-resolution features that are lithographically insignificant at any distance. These sub-resolution features may be pre-smoothed in the design before applying the model based OPC. 
         [0086]    Smoothing may be performed by any suitable method, such as by sequential grow and shrink operations, for example, in a manner similar to Minkowski&#39;s Sum and Difference, described further below and discussed in co-assigned U.S. Pat. No. 7,261,981, the contents of which are incorporated herein by reference. Other suitable smoothing methods may be used, such as low-pass filtering in the spatial frequency domain, and may include any smoothing methods known presently to those skilled in the art or developed in the future. 
         [0087]    A Minkowski&#39;s Sum of an object in the two-dimensional Euclidean domain is defined by rolling a ball of a given radius along the exterior boundary of the object and taking the point-set union of the original object and the area swept by the rolling ball. A Minkowski&#39;s Difference on an object in the two dimensional Euclidean domain is defined by rolling a ball of a given radius along the interior boundary of the object and taking the point-set difference of the area swept by the rolling ball from the original object. In this embodiment, since for manufacturing purposes, the mask shapes have edges that are in general substantially orthogonal in nature, smoothing is preferably performed using a sequential shrink and grow operations similar to Minkowski&#39;s Sum and Difference smoothing, where the shrink and grow smoothing operation is performed using an ortho smoothing object having edges parallel to the substantially orthogonal edges of the object. 
         [0088]    Though the above embodiment of the invention has been demonstrated for small neighboring shapes, the inventive methodology can be applied to neighboring shapes that span several sub-regions or sub-ROIs of the intermediate range within the small flare influence distance ROI 2 . Referring to  FIG. 8D , there is a mask shape  821  having a mask edge  822  with an evaluation point  825 . The corresponding ROI 2  is shown as  824 . Another shape  823  is shown within the ROI 2   824  that influences the computation of the flare intensity at point  825 . 
         [0089]    Now referring to  FIG. 8E , the ROI 2   824  is further divided into 4 sub-ROIs, viz.,  831 ,  832 ,  833  and  834 . Shape  823  spans two of these sub ROI-s, viz., sub-ROI&#39;s  832  and  833 . 
         [0090]    In an embodiment in accordance with the current invention, a first portion  826  of shape  823  that is within the sub-ROI region  832  is smoothed differently than the portion  827  of shape  823  that is within the sub-ROI region  833 . 
         [0091]      FIG. 9  illustrates a flow diagram of a preferred embodiment of the present invention for computation of the flare intensity at a given point. First, a mask layout is provided, having a List  1  of M shapes (Block  901 ). Then an amount or degree of smoothing for each of the sub-ROI&#39;s to be considered, for example, a smoothing of degree n=1, . . . , N, is applied to each of these M shapes, where N is the total number of sub-ROI&#39;s around each shape. The nth degree of smoothing refers to any amount of smoothing that increases as n, or as the effective influence of features within a sub-ROI on optical processing of the shape of interest m decreases, for example as the distance of a neighboring shape from the evaluation point increases. The amount of increased smoothing of a neighboring shape as a function of distance from the evaluation point can be any appropriate amount, and need not be limited to a fixed factor or monotonic increase of smoothing. A preferred value for N is 4, but other values may be appropriate depending on the trade-offs between shape influence and computation time. 
         [0092]    In Step  902 , for all of shapes, m=1, . . . , M in the List  1  they are smoothed by the given amount and put in another list List  2 . 
         [0093]    In Step  903 , for all of shapes, m=1, . . . , M in the List  2  they are smoothed again by the given amount and put in another list List  3 . 
         [0094]    In Step  904 , for all of shapes, m=1, . . . , M in the List  3  they are smoothed by the given amount and put in another list List  4 . 
         [0095]    For all of shapes, m=1, . . . , M (Block  905  and  906 ), a main ROI around the evaluation point is obtained. The shapes are then (Block  907 ) divided into N sub-ROI&#39;s (Block  904 ), viz., R 1 , R 2 , R 3  and R 4 . 
         [0096]    In Step  908 , all the shapes of List  1  that are partially or completely within R 1 , are convolved with the flare kernel to compute the flare energy within region R 1 . 
         [0097]    In Step  909 , all the shapes of List  2  that are partially or completely within R 2 , are convolved with the flare kernel to compute the flare energy within region R 2 . 
         [0098]    In Step  910 , all the shapes of List  3  that are partially or completely within R 3 , are convolved with the flare kernel to compute the flare energy within region R 3 . 
         [0099]    In Step  911 , all the shapes of List  4  that are partially or completely within R 4 , are convolved with the flare kernel to compute the flare energy within region R 4 . 
         [0100]    In the final step  912 , the flare energy as computed in steps  908 ,  909 ,  910  and  911  are summed up to output the total flare energy at the given point. 
         [0101]    Methods of obtaining effective bounds on process parameters as described above may be implemented in a machine, a computer, and/or a computing system or equipment.  FIG. 10  is a simplified diagram illustration of a computing system  1000  according to one embodiment of the present invention. Computing system or computer system  1000  may include, inter alia, a central processing unit (CPU)  1001  for data processing, at least one input/output (I/O) device  1002  (such as a keyboard, a mouse, a compact disk (CD) drive, a display device, or a combination thereof or the like) for accepting instructions and/or input from an operator or user and outputting results from CPU processing data during simulation or computation, a controller  1003  capable of controlling the operation of computing system  1000 , a storage device or medium  1004  capable of reading and/or writing computer readable code, and a memory device or medium  1005 —all of which are operationally connected, e.g., by a bus or a wired or wireless communications network ( 1006 ). Embodiments of the present invention may be implemented as a computer program product stored on a computer readable medium such as storage device  1004 , or memory device  1005 , a tape or a compact disk (CD). The computer program product may contain instructions which may implement the method according to embodiments of the present invention on the computer system  1000 . Finally, the present invention can also be implemented in a plurality of distributed computers where the present items may reside in close physical proximity or distributed over a large geographic region and connected by a communications network. 
         [0102]    While certain features of the invention have been illustrated and described herein, many modifications, substitutions, changes, and equivalents will now occur to those of ordinary skill in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the spirit of the invention.