This invention relates generally to optical imaging systems, and more particularly the invention relates to simulation and modeling of electromagnetic scattering of light in such imaging systems.  The invention has applicability to optical proximity correction in photo masks and mask image inspection, but the invention is not limited thereto. 
The general components of an optical lithography tool, shown schematically in the diagram of FIG.  1 are the illumination system, the projection system, the photomask (also called reticle), and the photoresist, spun on top of a semiconductor wafer.  The operation principle of the system is based on the ability of the resist to record an image of the pattern to be printed.  The mask, already carrying this pattern, is flooded with light and the projector forms an image of all mask patterns simultaneously onto (and into) the resist.  The inherent parallelism of this process is the main reason why optical photolithography is favored over any other lithography, since it facilitates a very high throughput of 30-120 wafers per hour.  The light intensity distribution on top of the resist surface is commonly referred to as aerial image.  The resist itself is a photosensitive material whose chemical composition changes during light exposure.  The pattern is thereby stored in form of a latent (bulk) image within the resist.  After exposure has occurred, the resist is developed by means of a chemical process that resembles the process of developing photographic film.  After development, the exposed parts of the resist remain or dissolve depending on its polarity (negative or positive, respectively).  The end-result of the lithography process is a more or less exact (scaled or not) replica of the mask pattern on the wafer surface that will play the role of a local protective layer (mask) for subsequent processing steps (etching, deposition, implantation). 
The role of the illumination system is to deliver a light beam that uniformly trans-illuminates the entire reticle.  See FIG.  2(a).  It typically consists of various optical elements, such as lenses, apertures, filters and mirrors.  The light source is responsible for generating very powerful and monochromatic radiation.  Power is necessary because it is directly related to throughput.  Monochromaticity is important because high quality refractive (or reflective, in the case of EUV lithography) optics can only be fabricated for a very narrow illumination bandwidth.  State-of-the-art optical lithography tools employ excimer lasers as their light source.  Deep Ultra Violet (DUV) lithography is the term used for lithography systems with illumination wavelengths λ=248 nm (excimer laser with KrF), λ=193 nm (excimer laser with ArF) and λ=157 nm (excimer laser with F2).  The successful development of current and future optical photolithography technologies is hinged upon research advances in both excimer laser technology and novel materials that possess the required properties (high optical transmission at DUV wavelengths, thermal properties, stability after heavy DUV radiation exposure) by which the optical elements of the system will be made. 
All illumination systems in optical projection printing tools are designed to provide what is known as Köhler illumination.  By placing the source or an image of the source in the front focal plane of the condenser column, the rays originating from each source point illuminate the mask as a parallel beam, as seen in FIG.  2.  Here, each source point emits a spherical wave that is converted by the illumination system into a plane wave incident on the object (photomask).  The angle of incidence of the plane wave depends on the location of the source point (α,β) with respect to the optical axis (0,0).  Each parallel beam is a plane wave whose direction of propagation depends on the relative position of the source point with respect to the optical axis.  Nonuniformity in the brightness of the source points is averaged out so that every location on the reticle receives the same amount of illumination energy.  As we will see in subsequent Sections Köhler illumination can be modeled in a concise mathematical way. 
In addition to dose uniformity, the lithography process should also maintain directional uniformity such that the same features are replicated identically regardless of their orientations.  The shape of the light source is therefore circular (or rotationally symmetric) in traditional optical lithography, although this is not true for certain advanced illumination schemes such as quadrupole illumination, where directional uniformity is sacrificed in order to maximize the resolution of features with certain orientations. 
The coherence of the light source is another important attribute.  Temporal coherence is usually not a big concern, since the narrow bandwidth of excimer lasers implies high temporal coherence.  Spatial coherence (or just plain coherence) on the other hand is always carefully engineered and, in most cases, adjustable.  Using special scrambling techniques, the light emitted from any point of the source is made completely uncorrelated (incoherent) to the light emitted from every other point.  However, light gathers coherence as it propagates away from its source.  The frequently quoted partial coherence factor a is a characteristic of the illumination system and is a measure of the physical extent and shape of the light source.  The larger the light source, the larger the partial coherence factor, and the light source has a lower degree of coherence.  In the limit of an infinite source, imaging is incoherent and σ=∞. On the other hand, the smaller the light source, the smaller the partial coherence factor, and the higher the degree of coherence.  Imaging with a point source is fully coherent and σ=0.  Note that a point source in a Köhler illumination will result in a single plane wave illuminating the mask and the angle of incidence of this wave depends on the relative position of the point source with respect to the optical axis.  For partial coherence factors between zero and infinity, imaging is partially coherent.  Typical partial coherence factors in optical lithography range from 0.3 to 0.9. 
The projection system typically consists of a multi-element lens column (up to 30-40 lenses) that may also have apertures, filters, or other optical elements, and it is a marvel of engineering precision in order to be able to reliably project images with minimum dimensions on the order of 100 nm for state-of-the-art systems.  One of the main reasons for the required high precision is control of the aberrations or deviations of the wavefront from its ideal shape.  Two relevant parameters of the projection system are the numerical aperture, NA, and the reduction factor, R. The numerical aperture is, by definition, the sine of the half-angle of the acceptance cone of light-rays as seen from the image side of the system.  The ratio of image-height to object-height is, by definition, the magnification factor M of the system.  The inverse of the magnification factor is the reduction factor, R. Since a typical system in photolithography projects at the image plane a scaled-down version of the object (mask), M is less than 1 and R is greater than 1.  State-of-the-art systems currently have reduction factors of R=4 or 5.  Note that two numerical apertures exist in the projection system, namely NAi (or simply NA) and NAo, which refer to the half-angle of the acceptance cone as seen from the image side and from the object (mask) side, respectively.  They are related through the reduction factor as follows:
                    R        =                                            NA              i                                      NA              o                                .                                     Equation        ⁢                                  ⁢        2        ⁢                  -                ⁢        1            
For a circularly shaped light source, the partial coherence factor σ mentioned above is related to the numerical apertures of both the projection system and the illumination system.  Specifically, σ is given by:
                    σ        =                                            NA              c                                      NA              p                                .                                     Equation        ⁢                                  ⁢        2        ⁢                  -                ⁢        2            where NAc is the numerical aperture of the condenser lens (illumination system) and NAp is the numerical aperture of the projector lens.  Some confusion arises from the fact that, in the above equation, the reduction factor of the imaging system is implicitly taken into account.  FIG.  2(b) clarifies the situation by showing simplified diagrams of two optical systems with parameters NA=0.5, σ=0.5 and R=5 or R=1. 
The photomask, also called reticle, carries the pattern to be printed at a given lithography processing step.  The masks of integrated circuits having large die-sizes or footprints (that is, occupying large areas on the semiconductor wafer), typically carry just one copy of the chip pattern.  A matrix of several chip patterns is contained in one mask whenever the chip size permits.  Note that the mask is drawn R times the actual size on the semiconductor wafer, since the dimensions of the circuit will be scaled down by the reduction factor, R. For this reason it is not sufficient to just provide feature sizes, since it may not be immediately obvious from the context whether these are photomask (object) or resist (image) sizes.  A typical convention for distinguishing photomask feature sizes from resist feature sizes is to include in parenthesis the reduction factor R. For example, a 600 nm (4×) line has a size of 600 nm on the mask, and would produce a 600 nm/4=150 nm line if used in a 4×imaging system.  Similarly, a 130 nm (1×) line refers to the size of a line at the image (wafer) plane and would result from the printing of a 130 nm line on the mask for a system with R=1, or a 520 nm (4×130 nm) line on the mask for a system with R=4, or a 1.3 μm (10×130 nm) line on the mask for a system with R=10. 
Depending on their operation principle, photomasks can be divided into two broad categories: conventional binary or chrome-on-glass (COG) masks and advanced phase-shifting masks (PSM). 
A binary or COG mask consists of a transparent substrate (mask blank), covered with a thin opaque film that bears the desired pattern.  Light can either pass unobstructed through an area not covered by the opaque film or be completely blocked if it is incident on an area that is protected by the film.  This binary behavior of the transmission characteristic of the mask is responsible for its name.  The mask blank for DUV lithography typically consists of fused silica glass that has excellent transmission at λ=248 nm and somewhat poorer but acceptable transmission at λ=193 nm and λ=157 nm.  The opaque film is typically on the order of 100 nm thick and has a chromium (Cr) composition. 
Adding phase modulation to the photomask can profoundly increase the attainable resolution.  This is the principle followed by phase-shifting masks (PSM), which employ discrete transmission and discrete phase modulation.  There are many different flavors of PSMs depending on the way that the phase modulation is achieved.  One of the most promising PSM technologies is what is known as alternating phase-shifting mask (alt.  PSM, or APSM).  Here are cut-planes of geometry of a binary (COG) mask (a) and an alternating phase-shift mask (b).  The ideal electric field distribution for the binary mask (c) leads to a poor image intensity distribution (e) at the image plane, whereas the ideal electric field distribution for the alt.  PSM (d), because of destructive interference, leads to a robust image.  The principle of an alt.  PSM is compared with that of a binary mask in FIG.  3.  The center line is bordered by transmitting regions with 180° phase difference on an alt.  PSM and by clear areas of the same phase on a binary mask.  The phase difference on the alt.  PSM leads to destructive interference, resulting in a sharp dark image.  The binary mask image is not as sharp because of the lack of phase interaction.  The 180° phase difference is created by etching trenches, also called phase-wells, into the fused silica substrate during the alt.  PSM fabrication process, which is now more complex than the COG fabrication process.  The difference in the amount of material removed detch is such that the path length difference between light passing through the different phase regions is half of the wavelength in air. 
Sub-wavelength lithography, where the size of printed features is smaller than the exposure wavelength, places a tremendous burden on the lithographic process.  Distortions of the intended images inevitably arise, primarily because of the nonlinearities of the imaging process and the nonlinear response of the photoresist.  Two of the most prominent types of distortions are the wide variation in the linewidths of identically drawn features in dense and isolated environments (dense-iso bias) and the line-end pullback or line-end shortening (LES) from drawn positions.  The former type of distortion can cause variations in circuit timing and yield, whereas the latter can lead to poor current tolerances and higher probabilities of electrical failure. 
Optical proximity correction or optical proximity compensation (OPC) is the technology used to compensate for these types of distortions.  OPC is loosely defined as the procedure of compensating (pre-distorting) the mask layout of the critical IC layers for the lithographic process distortions to follow.  This is done with specialized OPC software.  In the heart of the OPC software is a mathematical description of the process distortions.  This description can either be in the form of simple shape manipulation rules, in which case the OPC is referred to as “rule-based OPC,” or a more detailed and intricate process model for a “model-based OPC.”  The OPC software automatically changes the mask layout by moving segments of line edges and adding extra features that (pre-) compensate the layout for the distortions to come.  Although after OPC has been performed the mask layout may be quite different than the original (before OPC) mask, the net result of this procedure is a printed pattern on the wafer that is closest to the IC designer's original intent. 
In the early 1990's the problem of OPC for large mask layouts was addressed formally as an optimization problem.  The size of such a problem is formidable, but through introduction of appropriate constraints—induced primarily from mask fabrication constraints, local optimal mask “points” were successfully demonstrated.  This research has led to commercially available software tools that perform OPC on a full-chip scale.  All three approaches rely heavily on speedy calculations of the image intensity at selected points of the image field.  Although the methods by which they achieve this appear seemingly different, they are nevertheless the same, in the sense that they all rely on a decomposition of the kernel of the imaging equation for partially coherent light. 
Domain decomposition techniques, where a large electromagnetic problem is broken up into smaller pieces and the final solution is arrived at by synthesizing (field-stitching) the elemental solutions, have been proposed for the study of one-dimensional binary (phase only) diffractive optical elements.  Others working on the same problem have demonstrated how to create and use a perturbation model for binary edge-transitions based on the product of the ideal, sharp transition and the continuous field variations in the vicinity of the edge.  Independently, a similar technique to field-stitching has been proposed for the simulation of two-dimensional layouts of advanced photomasks (alternating PSM, masks with OPC) that properly models interactions from neighboring apertures and furthermore takes advantage of the spectral properties of the diffracted fields to come up with a compact model for the edge-transitions. 
The present invention is directed to an improved methodology for OPC on optical masks.