Patent Publication Number: US-10325757-B2

Title: Advanced dose-level quantization of multibeam-writers

Description:
RELATED APPLICATION 
     This application claims priority to U.S. provisional patent application Ser. No. 62/451,528 entitled “Advanced Dose-Level Quantization for Multibeam-Writers” filed on Jan. 27, 2017, which is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The invention generally relates to methods and apparatuses for generating a desired pattern on a specified target by irradiating a target with a beam of energetic radiation formed by electrically charged particles. 
     BACKGROUND OF THE INVENTION 
     The use of charged particle beams in the field of generating desired patterns such as for example those found on printed circuit boards is well known. One common technique for generating such patterns uses a Multibeam-Writer (MBW) which projects a charged particle beam through a series of apertures onto a desired surface. Such devices typically employ charged-particle optics systems, Pattern Definition (PD) devices, and a variety of methods for creating the ultimate pattern on the desired target surface. 
     In typical MBW systems the charged particle beam is moved along a predetermined path with respect to a target area, upon which a desired image is thereby created. 
     BRIEF SUMMARY OF THE INVENTION 
     Systems and methods in accordance with various embodiments of the invention provide a multibeam-writing system which projects a charged particle beam through a series of apertures onto a target area and methods for improving the clarity of the final desired pattern. In a number of embodiments the methods providing a pattern definition device having a plurality of apertures transparent to a source of radiation, and directing an illuminating wide beam through the apertures of the pattern definition device to form a patterned beam consisting of a corresponding plurality of beamlets, and illuminating a target with the patterned beam during a sequence of exposure intervals to form a pattern image on the target, wherein the pattern image further comprises a plurality of pattern pixels located on the target wherein the plurality of pattern pixels correspond to at least a portion of the plurality of apertures, and wherein during the sequence of exposure intervals, the at least a portion of the plurality of apertures are selectively controlled such that the plurality of pattern pixels are exposed to a respective dose amount in accordance with the desired pattern, generating a relative movement between the target and the pattern definition device to produce a stepwise movement of the pattern image on the target along a path over an exposure region, said path comprising a plurality sections which extend along a scanning direction, wherein the plurality of sections correspond to a plurality of exposure stripes that collectively cover the entirety of said exposure region over sequential exposures, wherein the exposure stripes are mutually overlapping and offset from each other in a direction transverse to the scanning direction, such that exposure region is exposed by at least two different exposure stripes at different transversal offsets, and calculating, for each pixel, a corrected dose amount by dividing the value of the nominal dose amount by a correction factor, wherein the same correction factor is used with pixels written by beamlets located at positions which differ only by said transversal offsets of overlapping stripes. 
     In other embodiments the method further provides that wherein during the step of calculating a corrected dose amount for each pixel, an available current density at the respective pixel is determined, wherein said maximum available current density is determined as the actual current density of the irradiating beam radiated through the aperture corresponding to the respective pixel, said correction factor of the respective pixel is calculated as the ratio of said available current density to the minimum current density across the overall beam array field, and correction factors are averaged among those pixels that are located at positions which differ only by said transversal offsets of overlapping stripes. 
     In yet other embodiments the further provides multiplicative renormalization of the correction factors, using a renormalizing factor chosen such that one of the largest value and the smallest value of the correction factors is renormalized to 1. 
     In still other embodiments the method provides that step of calculating a corrected dose amount for each pixel comprises calculating, for each pixel in a row of pixels parallel to the scanning direction within a respective exposure stripe, corrected dose amounts by dividing the values of the dose amounts by a row correction factor, wherein said row correction factor is uniformly applied to all pixels of a row of pixels. 
     In yet still other embodiments the method provides that said row correction factor is calculated for a respective row of pixels based on the values of current dose actually radiated through a series of apertures, said series of apertures containing all apertures within the pattern definition device which impart dose amounts to the respective row of pixels, wherein the row correction factor of a row of pixels is calculated as the ratio of actual current dose of an aperture, as averaged over the corresponding series of apertures, to a nominal current dose value assumed to be constant over the plurality of apertures of the pattern definition device. 
     In even other embodiments the method provides that said region where a beam exposure is to be performed is composed of a plurality of pattern pixels arranged in a regular arrangement, said region having a total width as measured across said scanning direction, said exposure stripes within said region running substantially parallel to each other along said scanning direction and having uniform widths as measured across said scanning direction. 
     In other embodiments the method provides that the exposure stripes are mutually overlapping, the position of the stripes differing by a transversal offset in a direction across the scanning direction, wherein the row correction factors of rows of pixels are averaged over those rows of pixels which are offset to each other by said transversal offset. 
     In still other embodiments the method provides that the correction factor varies between groups of pixels where said groups of pixels differ by an offset which does not correspond to a transversal offset of overlapping stripes. 
     In yet other embodiments the method further provides computing an exposure pattern suitable for exposing the desired pattern on a target using said pattern definition device for writing said desired pattern by exposing a multitude of pixels within said region on the target, wherein during exposing the desired pattern on a target: in said pattern definition device said plurality of blanking apertures is arranged in a predetermined arrangement defining mutual positions of the blanking apertures, each blanking aperture being selectively adjustable with regard to a dose value to be exposed through the respective blanking aperture onto a corresponding aperture image on the target during a respective exposure interval, said dose value taking a respective value in accordance with a discrete palette, said discrete palette including a number of gray values forming a scale ranging from a minimum value to a maximum value, during a writing process of said desired pattern, a sequence of exposure intervals is made, wherein in each exposure interval the blanking apertures are imaged onto the target, thus generating a corresponding plurality of aperture images, wherein the position of aperture images is kept fixed relative to the target at the position of a pixel during an exposure interval, but between exposure intervals the position of aperture images is shifted over the target, thus exposing the multitude of pixels on the target, and the aperture images are mutually overlapping on the target, and the aperture images have a nominal width which is greater than the distance between pixel positions of neighboring aperture images on the target, by an oversampling factor greater than one, wherein computing the exposure pattern comprises: determining the discrete palette, providing the desired pattern and calculating a nominal exposure pattern as a raster graphics defined on the multitude of pixel elements, said nominal exposure pattern being suitable to create a nominal dose distribution on the target realizing contour lines of the desired pattern and including for each pixel element a respective nominal dose value, and determining, for each pixel element, a discrete value which approximates the nominal dose value of the respective pixel element, said discrete value being selected from the discrete palette, wherein determining the discrete values includes employing ordered dithering using a dither matrix of a predefined size. 
     In yet still other embodiments the method provides that the dither matrix is a Bayer matrix. 
     In even other embodiments a pattern definition device is provided comprising: a plurality of apertures transparent to a source of radiation, a data processing unit having at least one input terminal and in communication with the plurality of apertures transparent to a source of radiation wherein the data processing unit is configured to receive a set of instructions defining a desired pattern image to be exposed on a target area, wherein at least a portion of the plurality of apertures correspond to a plurality of pattern image pixels on the target area and the at least a portion of the plurality of apertures are configured to expose the pattern image pixels to a respective dose amount in accordance with the desired pattern image, and wherein the data processing is further configured to calculate a correction needed for each of the pattern image pixels to correct for a pattern beam overlap by dividing a value of the nominal dose amount by a correction factor dose, and wherein the data processing unit is further configured to communicate the correction dose to the plurality of apertures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The description will be more fully understood with reference to the following figures, which are presented as exemplary embodiments of the invention and should not be construed as a complete recitation of the scope of the invention, wherein: 
         FIG. 1  illustrates an MBW system according to the state of the art in a longitudinal sectional view. 
         FIG. 2  illustrates a cross section view of an Aperture Array Plate in accordance with prior art for MBW systems. 
         FIG. 3  illustrates a writing strategy on a target using stripes arranged along a common scanning direction in accordance with prior art teachings. 
         FIG. 4  Illustrates an example of a pixel map of an exemplary pattern to be exposed in accordance with prior art teachings. 
         FIG. 5  Illustrates an arrangement of apertures as imaged onto a target in accordance with prior art teachings. 
         FIG. 6A  illustrates an arrangement of apertures in accordance with prior art teachings. 
         FIG. 6B  illustrates an example oversampling of the pixels in a “double grid” arrangement. 
         FIG. 7  illustrates an exemplary embodiment of an exposure scheme of pixels of one stripe in accordance with prior art teachings. 
         FIGS. 8A and 8B  illustrate the overlapping stripe (“multi-pass”) strategy for the example of two passes in accordance with many embodiments. 
         FIG. 9A  illustrates an intensity profile and a dose level in accordance with various embodiments. 
         FIG. 9B  illustrates another view of an intensity profile in accordance with various embodiments. 
         FIGS. 9C and 9D  illustrate MBW intensity profiles and related data as obtained for a simulation of a line in accordance with many embodiments. 
         FIG. 10  illustrates an intensity profile generated by exposure of a single exposure spot in accordance with many embodiments 
         FIG. 11A  illustrates an intensity profile generated from the exposure of a line of a determined width. 
         FIGS. 11B and 11C  illustrate the fine adjustment of the position of one edge ( FIG. 11B ) or both edges ( FIG. 11C ) of the line of  FIG. 11A  via suitable modifications of the dose levels corresponding the exposure spots. 
         FIG. 12  illustrates an example of a measured current density map in accordance with many embodiments. 
         FIG. 13  is illustrative of an exemplary embodiment of a data path of an MBW. 
         FIG. 14  illustrates an exemplary embodiment of double-grid oversampling. 
         FIG. 15  illustrates possible configurations of dose in accordance with various embodiments. 
         FIG. 16  illustrates an embodiment of dithering with o=2. 
         FIG. 17A  illustrates the index matrix used in the dithering matrix in accordance with  FIG. 16 . 
         FIG. 17B  illustrates the threshold matrix used in the dithering matrix in accordance with  FIG. 16 . 
         FIG. 18  illustrates an embodiment of dithering with o=4. 
         FIG. 19A  illustrates the index matrix used in the dithering matrix in accordance with  FIG. 18 . 
         FIG. 19B  illustrates the threshold matrix used in the dithering matrix in accordance with  FIG. 18   
         FIG. 20  illustrates an exemplary embodiment of the dithering process. 
         FIG. 21  illustrates another exemplary embodiment of the dithering process. 
         FIG. 22  illustrates an exemplary embodiment of the dithering process further illustrating a correction of beam current inhomogeneity. 
         FIG. 23  illustrates an exemplary embodiment further illustrating a correction of beam current inhomogeneity which is uniform along the scanning direction. 
         FIGS. 24 and 25  illustrate other exemplary embodiments of dithering in accordance with the invention. 
         FIG. 26  illustrates a plot of the rounding error as a function of the nominal target dose. 
         FIG. 27A  shows an exemplary embodiment of a typical current profile across an image field. 
         FIG. 27B  shows the current profile of  FIG. 27A  averaged along the X-direction 
         FIG. 28A  shows correction dose factors for the current profile of  FIG. 27A   
         FIG. 28B  shows correction dose factors for a uniform correction, i.e. uniform along the X-direction. 
         FIG. 29A  shows the corrected dose profile as obtained from  FIGS. 27A and 28A . 
         FIG. 29B  shows the corrected dose profile as obtained from  FIGS. 27B and 28B . 
         FIGS. 30A to 32B  show current profiles, correction dose factors and corrected dose profiles in a manner analogous to  FIGS. 27A to 29B , respectively, in accordance with various embodiments of the invention. 
         FIGS. 33A to 35B  show current profiles, correction dose factors and corrected dose profiles in a manner analogous to  FIGS. 27A to 29B  respectively in accordance with various embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Turning now to the description and drawings, methods and apparatus for improving a multibeam-writing systems are provided. In many embodiments of a multibeam-writer, a charged-particle beam is directed onto a desired target area as the beam device moves in relation to the target area. In such embodiments, a desired pattern is translated onto the target as the charged particle beam passes through a series of imaging apertures and/or subsequent lenses and deflectors that direct the beam onto the target in the desired positions (i.e. pixel locations) as the device is moved over the target area. In various embodiments exposure stripes are formed over the region to be exposed as the device moves; creating mutually overlapping exposure areas from each subsequent pass. In embodiments, at each desired pixel location correction factors are applied to account for image blurring from the exposure stripe overlap. 
     A typical implementation of a Multibeam-Writer (MBW) utilizes a 50 keV electron writer tool resulting in a total beam size of 20 nm which comprises a 512×512 grid of 262,144 total programmable beamlets within a beam array field of 81.92 μm×81.92 μm at the target area. An MBW of this type typically utilizes a 152.4 mm×152.4 mm substrate that is approximately 6.35 mm thick. The substrate is typically covered in an electron beam sensitive resist. 
     Beam size can be reduced from the typical 20 nm to 10 nm through a variety of changes. Reducing the beam size from 20 to 10 nm can typically be achieved by using a different aperture array plate (AAP), with 2 μm×2 μm opening size of the apertures instead of 4 μm×4 μm opening size. As outlined in U.S. Pat. No. 8,546,767, whose disclosure is incorporated herein by reference, a change of the beam size may also be realized in-situ by spatial adjustment of the AAP having multiple aperture arrays of different geometric parameters, such a total size, aperture spacing, aperture shapes etc. 
     Even with the reduction of beam size typical overlapping and column blur can occur. For example, a 10 nm beam size and a substrate with a current density of 4 A/cm 2 , would produce a maximum individual beamlet current of 1.05 μA for each of the 262,144 programmable beamlets; if all beamlets were activated. This example would still produce a 1 sigma blur of the column. 
     First generation MBW production machines use 20 nm and 10 nm beams providing up to approximately 1 μA current for all 262,144 programmable beamlets. New MBW production machines use even smaller beam size. In some cases, an 8 nm beam size would provide a 640×640 array with 409,600 beamlets within the 81.92 μm×81.92 μm beam array field at the substrate. Keeping the maximum current density at 4 A/cm 2  will ensure that the maximum current (with all beamlets “on”) is 1.05 μA. In other cases, a 5 nm beam size would provide a 1024×1024 array equaling 1,048,576 programmable beamlets at the substrate; again, at a maximum current density of 4 A/cm 2  the maximum current (with all beamlets “on”) is 1.05 μA. 
     In many applications the MBW performance becomes increasingly more demanding as the Critical Dimension requirements become increasingly smaller for example at the nanometer level. In some applications Local Critical Dimension Uniformity (LCDU) and Global Critical Dimension Uniformity (GCDU) are required to be within a 3 sigma or 6 sigma variation at the nanometer level over the entire MBW writing field. 
     Therefore, it is desirable to finely-adjust the line edge position by means of a specifically adapted exposure dose profile. Such a fine-adjustment should not only be adaptable within the MBW beam array field (local) but also over the whole MBMW writing field on a substrate (global). However, in order to fulfill the very demanding MBW specifications of very low LCDU and GCDU values, there is the need for additional fine corrections. Here, the terms “local” and “global” refer again to small fields (e.g. the area of the MBW beam array field) and the whole MBW writing field on a substrate, respectively. 
     In addition to the blurring and clarity issues the use of MBWs often involves the use of exposure stripes for which an effect called “substripes” typically occurs which can affect the overall clarity of the projected image. Essentially the beamlets may be affected by imperfections arising from spatial variations in the current density within the beam illuminating the Aperture Array Plate (AAP). Additionally, imperfections in the AAP may contribute to the substripes. Consequently, there is a need to allow for crisper line creation and the reduction of substripes. 
     Such clarity issues as previously discussed may be corrected in accordance with the various embodiments described herein. 
     According to an exemplary embodiment of the invention, it is possible to correct the “substripe” effect by applying a correction factor to the nominal dose amounts at the pixels, where the presence of overlapping stripes is taken into account by averaging of the correction factors. More specifically where the exposure stripes mutually overlap in the transverse direction to the scanning direction the corrected dose factor can be calculated. The desired image pattern will produce a number of image pixels that correspond to a variety of apertures. The pixels required the corrected dose amount calculated based on the substripe overlap wherein the corrected dose for each pixel is calculated by dividing the value of the nominal dose by a correction factor for each pixel. The same correction factor may be used for pixels that have equivalent positions with respect to the mutually overlapping substripes. Otherwise the correction factor may, in general, vary, in particular between pixels (or groups of pixels) which do not have equivalent positions. 
     In accordance with one embodiment of the invention, the correction factors are calculated to correct variations of the current density within the irradiating beam. Thus, during the step of calculating a corrected dose amount for each pixel, the following steps may be performed:
         an available current density at the respective pixel is determined, wherein said available current density is determined as the actual current density of the irradiating beam radiated through the aperture corresponding to the respective pixel;   the correction factor of the respective pixel is calculated as the ratio of said available current density to the minimum current density across the overall beam array field; and   an averaging of correction factors is made within respective sets of pixels, namely by averaging the correction factors among those pixels that are located at positions which differ with respect to the transversal offsets of overlapping stripes.       

     Optionally, a multiplicative renormalization of the correction factors, in particular the averaged correction factors, may be added, for instance such that the largest or, preferably, the smallest (averaged) correction factor is set to 1. 
     A further development of this method extends the range of averaging to an entire row of pixels. Thus, in this case the step of calculating a corrected dose amount for each pixel would comprise calculating, for each pixel in a row of pixels parallel to the scanning direction within a respective exposure stripe, corrected dose amounts by dividing the values of the dose amounts by a row correction factor, wherein said row correction factor is uniformly applied to all pixels of a row of pixels. Additionally, the row correction factor may be calculated for a respective row of pixels based on the values of current dose actually radiated through a series of apertures. Such a series of apertures may contain all apertures within the pattern definition device which impart dose amounts to the respective row of pixels. The row correction factor of a row of pixels is calculated as the ratio of actual current dose of an aperture averaged over the corresponding series of apertures, where a nominal current dose value assumed to be constant over the plurality of apertures of the pattern definition device. 
     Another embodiment of the invention may be directed to the region where a beam exposure is to be performed. Such embodiment may comprise of a plurality of pattern pixels arranged in a regular arrangement, said region having a total width as measured across said scanning direction. Additionally, the exposure stripes within said region will run substantially parallel to each other along said scanning direction and have uniform widths as measured across the scanning direction. A typical implementation of this embodiment may provide that exposure stripes are mutually overlapping, the position of the stripes differing by a transversal offset in a direction across the scanning direction; in this case it may be suitable to average row correction factors of rows of pixels over those rows of pixels which are offset to each other by said transversal offset. 
     Another exemplary embodiment, which further improves avoiding the “substripe” effect and similar rasterizing effects, relates to computing an exposure pattern which is suitable for exposing a desired pattern on a target in a charged-particle lithography apparatus as mentioned above. During the exposure of the desired pattern on the target, a particle beam is directed to and illuminates a pattern definition device comprising an aperture array composed of a plurality of blanking apertures through which said particle beam penetrates for writing said desired pattern by exposing a multitude of pixels within a region (region of exposure) on the target. In the pattern definition device, the plurality of blanking apertures are arranged in a predetermined arrangement defining mutual positions of the blanking apertures. Each of the various apertures selectively adjustable with regard to a dose value to be exposed through the respective blanking aperture onto a corresponding aperture image on the target during a respective exposure interval. The dose value is a respective value in accordance with a discrete palette, where the palette includes a number of gray values forming a scale ranging from a minimum value to a maximum value. During a writing process of the desired pattern, a sequence of exposure intervals is made, wherein in each exposure interval the blanking apertures are imaged onto the target, thus generating a corresponding plurality of aperture images. The position of aperture images is thereby kept fixed relative to the target at the position of a pixel during an exposure interval. However, between exposure intervals the position of aperture images is shifted over the target, thus exposing the multitude of pixels on the target; and the aperture images are mutually overlapping on the target. The nominal width of the aperture image is typically greater than the distance between pixel positions of neighboring aperture images on the target, by an oversampling factor greater than one. In this context, according to the various embodiments, computing the exposure pattern comprises:
     (i) determining the discrete palette,   (ii) providing the desired pattern and calculating a nominal exposure pattern as a raster graphics defined on the multitude of pixel elements, said nominal exposure pattern being suitable to create a nominal dose distribution on the target realizing contour lines of the desired pattern and including for each pixel element a respective nominal dose value, and   (iii) determining, for each pixel element, a discrete value (h) which approximates the nominal dose value of the respective pixel element, said discrete value being selected from the discrete palette,   wherein in step (iii), determining the discrete values includes employing ordered dithering using a dither matrix, for instance a Bayer matrix, of a predefined size.   

     Compared to other dithering methods, ordered dithering has several advantages in context of a charged particle multibeam writer. Firstly, it is computationally inexpensive which is highly important for a fast (i.e. real-time) data processing in the datapath. Secondly, it is a deterministic procedure, which means that its results are uniquely reproducible. Thirdly, the ordered dithering matrix can be chosen (i.e. optimized) in a way such that line edge placement and line edge roughness in specific directions becomes optimal. This is particularly useful since the layout of semi-conductor devices mainly have two preferred (orthogonal) axes, i.e. usually horizontal and vertical lines are more important and dominant as compared to lines in arbitrary direction. 
     Further details of the aforementioned embodiments of a novel dose-level quantization method for pixel data, as part of the on-line data path of a lithographic charged particle multibeam exposure tool are discussed in the subsequent sections. 
     Various aspects of the multibeam exposure tool are further discussed in U.S. Pat. Nos. 6,768,125 and 7,781,748, whose disclosures are incorporated herein by reference. 
     Lithographic Apparatus 
     An overview of a lithographic apparatus suitable to employ exemplary embodiments of the invention is illustrated in  FIGS. 1 and 2 . In accordance with many embodiments the lithographic apparatus as illustrated in  FIG. 1  may comprise an illumination system  3  coupled to a Pattern Definition (PD) system  4  which focuses the beam through a projecting system  5  onto a target station  6 . The target station generally contains a substrate for which the projected image is to be directed to. 
     In many embodiments the illumination system  3  comprises, for instance, an electron gun  7 , an extraction system  8  as well as a condenser lens system  9 . It should, however, be noted that in place of electrons, in general, other electrically charged particles can be used as well. Apart from electrons these can be, for instance, hydrogen ions or heavier ions, charged atom clusters, or charged molecules. 
     The extraction system  8  accelerates the particles to a defined energy of typically several keV, e.g. 5 keV. By means of a condenser lens system  9 , the particles emitted from the illumination source  7  are formed into a broad, substantially telecentric particle beam  50  serving as Lithography Beam  19   a . The Lithography Beam  19   a  then irradiates a PD system  4  which comprises a number of plates with a plurality of openings (also referred to as apertures). The PD system  4  is held at a specific position in the path of LB, which thus irradiates the plurality of apertures and/or openings and is split into a number of beamlets  51  and  52 . 
     Some of the apertures/openings are “switched on” or “open” so as to be transparent to the incident beam in the sense that they allow the portion of the beam that is transmitted through it, i.e. the beamlets  51 , to reach the target; the other apertures/openings are “switched off” or “closed”, i.e. the corresponding beamlets  52  cannot reach the target. Thus, effectively these apertures/openings are non-transparent (opaque) to the beam. Thus, the Lithography Beam  19   a  is structured into a Patterned Beam  19   b , emerging from the PD system  4 . The pattern of switched on apertures is chosen according to the pattern to be exposed on the substrate  16  covered with charged-particle sensitive resist  17 . It has to be noted that the “switching on/off” of the apertures/openings is usually realized by a suitable type of deflection means provided in one of the plates of the PD system  4 : “Switched off” beamlets  52  are deflected off their path (by sufficient albeit very small angles) so they cannot reach the target but are merely absorbed somewhere in the lithography apparatus, e.g. at an absorbing plate  11 . 
     The pattern as represented by the patterned beam  19   b  is then projected by means of an electro-magneto-optical projection system  5  onto the substrate  16  where the beam forms an image of the “switched-on” apertures and/or openings. In accordance with various embodiments, the projection system  5  may implement a demagnification of, for instance, 200:1 with two crossovers c 1  and c 2 . In many embodiments the substrate  16  may be a 6″ mask blank or a silicon wafer covered with a particle sensitive resist layer  17 . The substrate is held by a chuck  15  and positioned by a substrate stage  14  of the target station  6 . 
     The information regarding the pattern to be exposed is supplied to the PD system  4  by the data path realized by means of an electronic pattern information processing system  18 . The data path is explained further below in section “Datapath.” 
     In accordance with many embodiments the projection system  5  may include a number of consecutive electro-magneto-optical projector stages  10   a ,  10   b ,  10   c , which preferably include electrostatic and/or magnetic lenses, and possibly other deflection means. The projection system  5  employs a demagnifying imaging through crossovers c 1 , c 2 . The demagnification factor for both stages is chosen such that an overall demagnification of several hundred results, e.g. 200:1 reduction. A demagnification of this order is in particular suitable with a lithography setup, in order to alleviate problems of miniaturization in the PD device. 
     In the whole projection system  5 , provisions may be made to extensively compensate the lenses and or deflection means with respect to chromatic and geometric aberrations. As a means to shift the image laterally as a whole, i.e. along a direction perpendicular to the optical axis  19 , deflection means  12   a ,  12   b , and  12   c  may be provided in the condenser  3  and projection system  5 . The deflection means may be realized as a multi-pole electrode system which is either positioned near the source extraction system  8  or one of the crossovers c 1  and c 2 , as shown in  FIG. 1  with the deflection means  12   b , or after the final lens  10   c  of the respective projector, as in the case with the stage deflection means  12   c  in  FIG. 1 . In many embodiments, a multipole electrode arrangement is used as deflection means both for shifting the image in relation to the stage motion and for correction of the imaging system in conjunction with the charge-particle optics alignment system. These deflection means  10   a ,  10   b ,  10   c  are not to be confused with the deflection array means of the PD system  4  in conjunction with the stopping plate  11 , as the latter are used to switch selected beamlets of the patterned beam  19   b  “on” or “off”, whereas the former only deal with the particle beam as a whole. In some embodiments a solenoid  13  may be used to rotate the ensemble of programmable beams providing an axial magnetic field. 
     Turning now to  FIG. 2  which illustrates an exemplary embodiment of a PD system  4 . The embodiment of a PD system  4 , may comprise three plates stacked in a consecutive configuration: An “Aperture Array Plate” (AAP)  20 , a “Deflection Array Plate” (DAP)  30  and a “Field-boundary Array Plate” (FAP)  40 . It is worthwhile to note that the term ‘plate’ refers to an overall shape of the respective device, but does not necessarily indicate that a plate is realized as a single plate component even though the latter is usually the preferred way of implementation; still, in certain embodiments, a ‘plate’, such as the aperture array plate, may be composed of a number of sub-plates. The plates are preferably arranged parallel to each other, at mutual distances along the Z direction which is represented by the vertical axis in  FIG. 2 . 
     In various embodiments the flat upper surface of AAP  20  forms a defined potential interface to the charged-particle condenser optics/illumination system  3 . The AAP may, for example be made from a square or rectangular piece of a silicon wafer (approx. 1 mm thickness)  21  with a thinned center part  22 . The plate may be covered by an electrically conductive protective layer  23  which will be particularly advantageous when using hydrogen or helium ions further illustrated in U.S. Pat. No. 6,858,118 which disclosure is incorporated herein by reference. When using electrons or heavy ions (e.g. argon or xenon), the layer  23  may also be of silicon provided by the surface section of  21  and  22 , respectively, so that there is no interface between layer  23  and the bulk parts  21 ,  22 . 
     In accordance with many embodiments the AAP  20  may comprise a plurality of apertures  24  formed by openings traversing the thinned part  22 . The apertures  24  are arranged in a predetermined arrangement within an aperture area provided in the thinned part  22 , thus forming an aperture array  26 . The arrangement of the apertures in the aperture array  26  may be, for instance, a staggered arrangement or a regular rectangular or square array. In various embodiments, the apertures  24  are realized having a straight profile fabricated into the layer  23  and a “retrograde” profile in the bulk layer of the AAP  20  such that the downward outlets  25  of the openings are wider than in the main part of the apertures  24 . Both the straight and retrograde profiles can be fabricated with state-of-the-art structuring techniques such as reactive ion etching. The retrograde profile strongly reduces mirror charging effects of the beam passing through the opening. 
     As illustrated in  FIG. 2  according to various embodiments the DAP  30  may be a plate provided with a plurality of openings  33 , whose positions correspond to those of the apertures  24  in the AAP  20 . Additionally, the openings  33  may be configured with electrodes  35 ,  38  configured for deflecting the individual beamlets passing through the openings  33  selectively from their respective paths. The DAP  30  can, for instance, be fabricated by post-processing a CMOS wafer with an ASIC circuitry. In accordance with some embodiments the DAP  30  may be of a square or rectangular shape and comprise a thicker part  31  forming a frame holding a center part  32  which has been thinned (but may be suitably thicker as compared to the thickness of the tinned part  22 ). The aperture openings  33  in the center part  32  are wider compared to  24  (by approx. 2 μm at each side for instance). In some embodiments CMOS electronics  34  may be utilized to control the electrodes  35 ,  38 , which are provided by means of MEMS techniques. In accordance with various embodiments each opening  33 , may contain a “ground” electrode  35  and a deflection electrode  38  adjacent thereto. The ground electrodes  35  may be electrically interconnected, connected to a common ground potential, and comprise a retrograde part  36  to prevent charging and an isolation section  37  in order to prevent unwanted shortcuts to the CMOS circuitry. The ground electrodes  35  may also be connected to those parts of the CMOS circuitry  34  which are at the same potential as the silicon bulk portions  31  and  32 . 
     In accordance with some embodiments the deflection electrodes  38  may be configured to be selectively applied with an electrostatic potential; when such electrostatic potential is applied to an electrode  38 , this will generate an electric field causing a deflection upon the corresponding beamlet, deflecting it off its nominal path. Additionally, the electrodes  38  may have a retrograde section  39  in order to avoid charging. Each of the electrodes  38  is connected at its lower part to a respective contact site within the CMOS circuitry  34 . 
     In many embodiments the ground electrodes  35  may be higher than the height of the deflection electrodes  38  in order to suppress cross-talk effects between channels. Although the electrodes  35  and  38  are illustrated in a specific configuration, it should be understood that they may take on any suitable configuration for example the electrodes may face upstream rather than downstream as illustrated in  FIG. 2 . 
     The arrangement of a PD system  4  with a DAP  30  shown in  FIG. 2  is only one of several possibilities. In a variant (not shown) the ground and deflection electrodes  35 ,  38  of the DAP may be oriented upstream (facing upward), rather than downstream. Further DAP configurations, may be utilized, such as those described in U.S. Pat. No. 8,198,601 whose disclosure is incorporated herein by reference. 
     In accordance with various embodiments the third plate  40  serving as FAP may have a flat surface facing the first lens part of the down-stream demagnifying charged-particle projection optics  5  and thus provides a defined potential interface to the first lens  10   a  of the projection optics. The thicker part  41  of FAP  40  may be a square or rectangular frame made from a part of a silicon wafer. The FAP  40  may further comprise a thinned center section  42 . In accordance with many embodiments the FAP  40  may be provided with a plurality of openings  43  which correspond to the openings  24 ,  33  of the AAP  20  and DAP  30  but are wider as compared to the latter. 
     In accordance with many embodiments the PD system  4 , and in particular the first plate of it, the AAP  20 , may be illuminated by a broad charged particle beam  50  (herein, “broad” beam means that the beam is sufficiently wide to cover the entire area of the aperture array formed in the AAP), which is thus divided into many thousands of micrometer-sized beamlets  51  when transmitted through the apertures  24 . The beamlets  51  will traverse the DAP and FAP unhindered. 
     As already mentioned, whenever a deflection electrode  38  is powered through the CMOS electronics, an electric field will be generated between the deflection electrode and the corresponding ground electrode, leading to a small but sufficient deflection of the respective beamlet  52  passing therethrough ( FIG. 2 ). The deflected beamlet can traverse the DAP and FAP unhindered as the openings  33  and  43 , respectively, are made sufficiently wide. However, the deflected beamlet  52  is filtered out at the stopping plate  11  of the sub-column ( FIG. 1 ). Thus, only those beamlets which are unaffected by the DAP will reach the substrate. 
     The ensemble of (unaffected) beamlets  51  as formed by AAP may be projected to the substrate with a predefined reduction factor R of the projection charged-particle optics. Thus, at the substrate a “beam array field” (BAF) is projected having widths BX=AX/R and BY=AY/R, respectively, where AX and AY denote the sizes of the aperture array field along the X and Y directions, respectively. The nominal width of a beamlet at the substrate (i.e. aperture image) is given by bX=aX/R and bY=aY/R, respectively, where aX and aY denote the sizes of the beamlet  51  as measured along the X and Y directions, respectively, at the level of the DAP  30 . Thus, the size of a single aperture image formed on the target is bX×bY 
     The reduction factor of the demagnifying charged-particle optics  5 , as illustrated in  FIG. 1 , may be chosen in accordance with many embodiments in view of the dimensions of the beamlets and their mutual distance in the PD device  4  and the desired dimensions of the structures at the target. This will allow for micrometer-sized beamlets at the PD system whereas nanometer-sized beamlets are projected onto the substrate. 
     In many embodiments the beamlets  51  and  52  as represented in  FIGS. 1 and 2 , equate to a plurality of programmable beamlets. It is worthwhile to note that the individual beamlets  51 ,  52  depicted in  FIGS. 1 and 2  represent a much larger number of beamlets, typically many thousands, arranged in a two-dimensional X-Y array. In accordance with some embodiments there may be multi-beam charged-particle optics with a reduction factor of R=200 for ion as well as electron multi-beam columns with many thousands (e.g., 262,144) programmable beamlets. In some embodiments columns with a BAF of approx. 82 μm×82 μm may exist at the substrate. 
     Pattern Exposure 
     Turning now to  FIG. 3  regarding defining the pattern to be exposed on the target. In many embodiments a pattern image  27 A to be produced on the target  16  is defined by the PD system  4 . The target surface covered with the charged-particle sensitive resist layer  17  may comprise one or more areas  27 C to be exposed. Generally, the pattern image  27 A exposed on the target has a finite size y 0  which is usually smaller than the width of the area  27 C to be exposed. Therefore, a scanning stripe exposure strategy may be utilized, where the target is moved under the incident beam, thus changing perpetually changing the position of the beam on the target. It is emphasized that for the purpose of the invention only the relative motion of the pattern image  27 A on the target is relevant. In many embodiments the relative movement of the pattern image  27 A over the target area  27 C may form a sequence of exposure stripes s 1 , s 2 , s 3 , . . . sn having the same width as the pattern image  27 A. The complete set of stripes covers the total area of the substrate surface. In various embodiments the scanning direction  27 B may be uniform or may alternate from one stripe to the next. 
     Turning now to  FIG. 4 , many embodiments of the invention may have various image patterns  28 A as illustrated by the pixel  28 B pattern of  FIG. 4 . In accordance with many embodiments the pixel  28 B may be exposed with varying levels of intensity thus producing an image level different in a variety of pixels, which is best illustrated by the table in  FIG. 4 .  401 ,  402 , and  403  illustrate the various levels of intensity in accordance with some embodiments. The full gray level  401  can be seen in various pixels  28 C while a reduced gray level can be seen in other pixels  28 D. Thus the “switched on” beamlets as determined by the PD system  4  and the Pattern Information Processing System  18 , may be capable of exposing the substrate to a variety of intensity levels to produce the desired image. The number and intensity of pixels exposed at any given time during the process may vary as the number of apertures capable of being “switched on” at a given time is predetermined by the physical parameters of the PD system. 
     In accordance with many embodiments the intensity or varying level of which the pixel is exposed may be determined by the sequence of apertures activated to produce the pattern on the desired pixel location. For example, while the substrate  16  moves, the same image element corresponding to a pattern pixel  28 B on the target may be exposed many times by the images of a sequence of apertures. 
     The image pattern may be shifted through the apertures of the PD system. In some embodiments, for example, all apertures may be switched on when such apertures are directed to a specific pixel location. The result would be the maximum exposure level to that pixel thus producing a “white” shade. Yet in many embodiments the number of “switched on” apertures may vary thus producing a variety of exposure dose levels; accordingly producing a variety of gray levels on the substrate. In some embodiments the dose level may be minimum equating to a “black” shade. Thus in an actual pattern not all pixels are exposed to the full dose of the aperture array plate due to various apertures being “switched off.” 
     In some embodiments the dose level is regulated by reducing the duration of unblanked exposure for the apertures involved. Thus, the exposure duration of one aperture image may be controlled by a discrete number of gray levels; each of which represents a particular dose to be applied on the substrate/target, e.g. 0, 1/(n Y −1) . . . , i/(n Y −1), . . . , 1 with n Y  being the number of available “pixel gray levels” and i an integer (“gray index”, 0≤i≤n Y ). Generally, however, the dose increments need not be equidistant and form a non-decreasing sequence between 0 and 1. The exposed aperture image may be the manifestation of one of a given numbers of gray shades that correspond to zero and the maximum exposure duration and dose level. 
       FIG. 5  illustrates an arrangement of apertures in the aperture field of the PD device, in accordance with various embodiments of the invention. Aperture images  53 A as may be projected to the target are illustrated in dark. The main axes X and Y correspond to the direction of advance of the target motion (scanning direction  27 B) and the perpendicular direction, respectively. Each aperture image has widths bX and bY along the directions X and Y respectively. The apertures are arranged along lines and rows having MX and MY apertures, respectively, with the offset between neighboring apertures in a line and row being NX·bX and NY·bY respectively. In accordance with many embodiments based on the layout of the apertures each aperature image may have a resulting conceptual cell  53 B having an area of NX·bX·NY·bY, and the aperture arrangement contains MX·MY cells arranged in a rectangular way. For simplicity the conceptual cell may be referred to as an “exposure cells”. In accordance with many embodiments the complete aperture arrangement, as projected onto the target, has dimensions of BX=MX·NX·bX by BY=MY·NY·bY. In some embodiments the grid may assume a square shape as a special case of a rectangular grid, and set b=bX=bY, M=MX=MY, and N=NX=NY with M being an integer. In such configurations an “exposure cell”  53 B has a size of N·b×N·b on the target substrate. 
     Turning now to  FIG. 6A , in accordance with many embodiments the pitch between two neighboring exposure positions may be denoted as e. In general, the distance e can be different from the nominal width b of an aperture image. In some embodiments b=e, which is illustrated in  FIG. 6A .  FIG. 6 a    in accordance with various embodiments illustrates an arrangement of exposure cells  53 B in a 2×2 pattern where one aperture image  54 A covers (the nominal position of) one pixel. In other embodiments, as illustrated in  FIG. 6B , e may be a fraction b/o of the width b of the aperture image, with o&gt;1 being preferably (but not necessarily) an integer which we also refer to as the oversampling factor. Such examples are further illustrated by U.S. Pat. Nos. 8,222,621 and 7,276,714 which disclosures are incorporated herein by reference. In this case the aperture images, in the course of the various exposures, will spatially overlap, allowing a higher resolution of the placement of the pattern to be developed. It follows that each image of an aperture will, at one time, cover multiple pixels, namely o 2  pixels. The entire area of the aperture field as imaged to the target will comprise (NMo) 2  pixels. From the point of view of placement of aperture image, this oversampling corresponds to a so-called placement grid which is different (since it is finer in spacing) than what would be necessary to simply cover the target area. 
       FIG. 6B  further illustrates an exemplary embodiment of an oversampling of o=2 combined with placement grids, namely, the image of an aperture array with an exposure cell C 4  having parameters o=2, N=2 (“double-grid”). Thus, on each nominal location (small square fields in  FIG. 6B ) four aperture images  55 A (dashed lines) are printed, which are offset on a regular grid by pitch e in both X and Y directions. While the size of the aperture image still is of the same value b, the pitch e of the placement grid is now b/o=b/2. The offset to the previous nominal location (offset of the placement grid) is also of size b/2. At the same time, the dose and/or the gray shade of each pixel may be adapted (reduced), by choosing a suitable gray value for the aperture image that cover the respective pixel. As a result, an area of size b×b is printed but with an enhanced placement accuracy due to the finer placement grid. Direct comparison of  FIG. 6B  with  FIG. 6A  shows that locations of aperture images are just arranged on a placement grid twice (generally, o times) as fine as before, while the aperture images themselves overlap. The exposure cell C 4  now contains (No) 2  locations (i.e., “pixels”) to be addressed during the write process and thus, by a factor of o 2 , more pixels than before. Correspondingly, the area bi 1  with the size of an aperture image b×b is associated with o 2 =4 pixels in the case of oversampling with o=2 in  FIG. 6B . Of course, o may take any other integer value as well, in particular 4 (“quad-grid”, not shown) or 8, or also a non-integer value greater one, such as √2=1.414. 
     In accordance with many embodiments an exposure scheme of the pixels, which is suitable for the invention is illustrated in  FIG. 7 . As illustrated in  FIG. 7  is a sequence of frames, with increasing time from top (earlier) to bottom (later). The parameter values in this figure are o=1, N=2; also, a rectangular beam array is assumed with MX=8 and MY=6. In some embodiments, the target may move continuously to the left, whereas the beam deflection is controlled with a seesaw function as shown on the left side of  FIG. 7 . During each time interval of length T 1 , the beam image stays fixed on a position on the target (corresponding to a position of a “placement grid”). Thus, the beam image is shown to go through a placement grid sequence p 11 , p 21 , p 31 . One cycle of placement grids is exposed within a time interval L/v=NMb/v, by virtue of the target motion v. The time T 1  for exposure at each placement grid corresponds to a length L G =vT 1 =L/(No) 2 =bM/No 2 , which we call “exposure length.” 
     In many embodiments the beamlets are moved over the distance of L G  during the exposure of one set of image elements together with the target. In other words, all beamlets maintain a fixed position with regard to the surface of the substrate during the time interval T 1 . After moving the beamlets with the target along distance L G , the beamlets are relocated to start the exposure of the image elements of the next placement grid. After a full cycle through the positions p 11  . . . p 31  of a placement grid cycle, the sequence starts anew, with an additional longitudinal offset L=bNM parallel to the X direction (scanning direction). At the beginning and at the end of the stripe the exposure method may not produce a contiguous covering, so there may be a margin of length L that is not completely filled. 
     In various embodiments an overlapping stripe (“multi-pass”) strategy for error reduction may be used. Similar strategies are described in U.S. Pat. No. 9,053,906 B2 which disclosure is incorporated herein by reference. An exemplary embodiment (“double-pass”) is illustrated in  FIGS. 8A and 8B , which show an exemplary sub-area of the target to be exposed in two passes ps 1 , ps 2 . In the first pass ps 1  the stripes s 11 , s 12 , s 13  are exposed in consecutive order, thus exposing the pixels belonging to a partial grid G 1  (the number of pixels within each of the stripes is reduced in the depiction of  FIGS. 8A and 8B  for the sake of clarity and may be higher in typical embodiment). In  FIG. 8A , letters A, C, E denote the pixels which are exposable through stripes s 11 , s 12 , and s 13 , respectively. The stripes s 11 -s 13  of one pass are preferably located side-by-side, so as to produce a continuous grid over the area on the target. In this way, the stripes, each having individual width y 0 , may cover the total width Ry of the area Rr to be exposed along the Y direction (i.e., across the scanning direction sd). In many embodiments, the stripes s 11 -s 13  may extend to either side of the area shown, and the first pass ps 1  may continue with further stripes (not shown) after the stripe s 13  has been imaged. After completion of all stripes of the first pass ps 1 , the stripes of another pass ps 2  are performed, as illustrated in  FIG. 8B . 
     As illustrated in  FIG. 8B , the stripes s 21 , s 22  may expose pixels formed within the second partial grid G 2 .  FIG. 8B  shows two stripes s 21 , s 22 , which expose the pixels denoted by letters B and D, respectively. Thus, each pass ps 1 , ps 2  is associated with one of the partial grids G 1 , G 2  of pattern pixels which are exposable during the respective pass. Taken together, the pixels of the grids G 1 , G 2  combine to the complete plurality of pattern pixels in the region which is to be exposed. In other words, the second pass ps 2  exposes those pixels which are left out in the first pass ps 1 , and vice versa. With regard to the Y axis the exposure stripes of different passes are mutually overlapping, preferably in a regular manner wherein the overlapping stripes, for instance of stripes s 11  and s 21 , differ by a transversal offset Y 1  along the Y direction (which is the direction across the orientation of the stripes, identical to the scanning direction). For exposing the first half of the stripe s 11 , and to also cover this part of the total width Ry, an additional ‘edge stripe’ s 20  (not indicated in the pixel pattern) may be performed, in which only the upper half of the pixels are exposed, while the lower half of the pixels are kept switched-off along the entire length of the stripe s 20 . In accordance with many embodiments of the invention there may be more than two passes; for instance, in a “quad-pass” writing strategy, four partial grids written in four passes may be combined to form the complete plurality of pattern pixels. Further details concerning the exposure of the pixels through exposure stripes and partial grids are described in U.S. Pat. Pub. No. 2016/0276131 A1 which disclosure is incorporated herein by reference. 
     Turning now to  FIG. 9A , a graphical illustration of the ideal intensity profile  71  for a line of a width 30 nm, in the idealized case of zero blur is presented. When using “quad-grid” (o=4) multi-beam exposure the overlap is a quarter of the beam size. Thus, for the case of 20 nm beam size the physical grid size is 5 nm. A discrete dose level can be assigned to each area of the physical grid, which may be 5 nm×5 nm in some embodiments. The line  72  in  FIG. 9A  indicates the superposition of the intensity (or total dose) as it is composed by the overlapping exposure spots with discrete dose levels assigned to the pixel positions for generating the 30 nm line, whereas for better visibility the blur has been set to zero (so that the dose distribution of a single exposure spot becomes a rectangle). If the blur has a realistic value, the step function at the edge of the rectangle is convoluted with a Gaussian function, which eventually transforms to a Gaussian shape. In that sense the line  72  can be seen as superposition of Gaussian functions at blur zero. In many embodiments the dose level histogram will not be symmetrical in order to position the left and right edge at pre-defined positions. 
     In accordance with many embodiments,  FIG. 9B  represents a graphical illustration of a simulation for a line of 30.0 nm width, with the left edge to be positioned at 0.0 nm and the right edge at 30.0 nm. For the simulation, it was assumed that beam spots of 20 nm are exposed with 5.1 nm 1sigma blur (i.e., 12.0 nm FWHM blur). The intensity profile  76  is formed by overlapping the profiles of the exposure spots  73 ,  74 , and  75 . The dose level of the leftmost exposure spot  74  is adjusted such that the 30 nm line starts at the desired start position  77 , i.e. at 0 nm. The dose level of the rightmost exposure spot  75  is adjusted such that exposed line ends at position  78  at 30.0 nm. As can be seen in  FIG. 9B , in accordance with “quad-grid” exposure, the overlap of the exposure spots  73 ,  74 ,  75  is a quarter of the beam size, i.e. 5 nm. 
       FIGS. 9C and 9D  illustrate how the invention enables the MBW device to write lines with precise edge definitions in accordance with many embodiment of the invention. In each figure, the top frame shows the edge position error vs. line width, the middle frame illustrates the intensity profile, and the bottom frame shows the edge position deviation when enhancing the exposure dose by 10% vs. line width.  FIG. 9C  shows the intensity profile obtained for a 31.4 nm line width, and  FIG. 9D  for a 40.0 nm line width. Using the MBW with 20 nm beam size and quad-grid exposure (5 nm physical grid size), the line width of the structure generated by the exposure can be changed in steps of 0.1 nm. Because of the integer dose levels, there may be slight deviations from the 0.1 nm address grid. These deviations are indicated as “edge position error” (top frames), as functions of the desired line width, in 0.1 nm steps between 30.0 nm and 40.0 nm. As can be seen the deviations are within ±0.05 nm. Furthermore, the change of edge position with 10% change of dose is only approx. 1 nm, varying only slightly with change of line width as shown in the bottom frames. In other words, since the dose is controlled in a MBW to better than 1%, the change of edge position with 1% change of dose is within approx. one atomic layer. 
     In accordance with many embodiments of the invention dose variations may be utilized in the MBMW to achieve edge placement with sub-pixel precision.  FIG. 10  illustrates the exposure of one exposure spot with a maximum dose level. In the exemplary case of a 4-bit coding, there are 16 dose levels (0, 1, 2, . . . 15), i.e. the maximum dose level is the sum of 15 dose level increments  64 . 
     Turning now to  FIGS. 11A, 11B, and 11C , intensity profile diagrams illustrating how the multi-beam exposure methods can achieve a fine positioning of structure feature with resolution smaller than the grid size. In the intensity profile diagrams, like those of  FIGS. 11A-C , the discrete dose levels are visualized as rectangles  64  of uniform height, piled up in a “brick-layer” arrangement; of course, this “brick-layer” depiction is only symbolical and intended to facilitate interpretation of the drawings. 
       FIG. 11A  shows a dose level histogram, for an exemplary embodiment of a line of 30 nm width exposed by means of a 4 bit (i.e., 15 dose levels per spot) exposure in a quad-grid with a beam spot size of 20 nm width. The grid size  62  is ¼ of the linear size of the exposure spots, which are symbolized as rectangles piled up in a “brick-layer” arrangement, and the resulting dose level distribution  65  is outlined as a bold line. 
     The line width can be made smaller or larger in very fine steps, which are smaller than the grid size, in this case the quad-grid size  62 . Reducing the line width can be achieved by lowering the dose level of the outermost exposure spots and/or omitting exposure spots (the latter when the reduction is at least about one half of an exposure spot size). Increasing the line width can be achieved by enhancing the dose level of the outermost exposure spots and/or, in particular when the maximum dose level has been reached, by adding an additional, preferably overlapping, exposure spot. The latter aspect is illustrated in  FIG. 11B : an exposure spot  66  having a defined dose level is added, resulting in a dose level histogram  67  for the line with larger width compared to  65 . By combining these effects of decreasing and increasing on either side, there is also the possibility to shift the line position in very fine steps.  FIG. 11C  illustrates a shift of the line without changing the width, which is achieved by removing dose levels from spot  68  and adding dose levels from spot  69 , resulting in the dose level histogram  70  which corresponds to a line shifted to the right as compared to the line of  FIG. 11A . 
     The intensity profiles illustrated in  FIGS. 11A-C  are shown along the X direction of the target plane. It should be understood that the multi-beam exposure methods illustrated here may be applied to lines along other directions as well, and fine positioning can be achieved for lines at any angle on the target plane. 
     Dose Inhomogeneity Correction 
     U.S. Pat. Pub. No. 2015/0347660 A1, which disclosure is incorporated herein by reference, illustrates the current transmitted by each beamlet (or aperture) may not be uniform but may vary, mainly as a function of the distance to the optical axis  19  ( FIG. 1 ). This effect is due to imperfections in the charged particle source. Without further corrections, the dose a pixel may receive will thus vary depending on the beamlet writing said pixel, leading to systematic edge placement errors. 
       FIG. 12  illustrates an exemplary embodiment of a current density map Mp. More precisely, it shows a 8×8 coarse grained map of the deviation α(X,Y) (quantified in percentage [%]) of the current of a single beamlet located at a position/areal (X,Y) relative to the mean current across the image field. Typically, in the map the current dose values near the corners of the beam array are either reduced or enhanced with regard to the average over the map. In the example of  FIG. 12  the beam array field of 82 μm×82 μm at the target consisted of 512×512=262,144 programmable beamlets. As shown, an 8×8 matrix of the current dose distribution was measured, wherein each measured value comprises 262,144/64=16,384 beamlets used to generate the respective value. The electron source underlying  FIG. 12  was of the type of a thermal field emission cathode with a flat emitter surface (single crystal, e.g. Tungsten or LaB 6 ). Since the electrons are emitted from a larger surface (typically 20 μm), it is unavoidable due to mechanical imperfection (e.g. alignment of emitter surface with respect to anode) or local differences in the extraction field strength, that the angular current density varies across the emitter. 
     In accordance with many embodiments dose variations may be corrected by updating the dose corresponding to a pixel by dividing by a homogenizing “dose correction factor” q depending on the beamlet writing said pixel, which is given by q=C(1+α), where C=1/[min [X,Y]  (1+α(X,Y))] is a constant that fixes the minimum dose inhomogeneity correction factor at 1. (Here the symbol min [X,Y]  is the minimum value among the values within the entire range of interest of X and Y coordinates.) This correction typically happens on-line as part of the data path. 
     Data Path 
     In accordance with many embodiments the MBW integrates a processing system  18 , as illustrated in  FIG. 1 , that converts the patterns to be written to beamlet dose assignments (as described above), which can be used in the writing process, is referred to as “data path” system. In accordance with many embodiments,  FIG. 13  shows a flowchart of a data path  170  in the context of the invention. The data path is preferably performed in real time; in a variant, part or all of the calculations of the data path may be performed in advance, for instance in a suitable computer. 
     The complete pattern image may comprise a vast amount of image data, which is why for efficient computation of those data a high-speed data path that generates the pixel data to be exposed, preferably in real-time, will be suitable. The pattern to be exposed is typically described in a vector format, for example as a collection of geometries like rectangles, trapezoids or general polygons, which typically offers better data compaction and therefore reduces the requirements on data storage. Therefore, in accordance with many embodiments, the data path may consist of three major parts:
         a vector-based physical correction process (step  160 ),   rasterization processes to translate the vector to pixel data (steps  161  to  164 ), and   buffering of pixel data for temporary storage for the writing process (steps  165  and  166 ).       

     As illustrated in  FIG. 13 , the data path starts upon being supplied a pattern PDATA to be exposed at step  160 . In step  160 , generally, the pattern PDATA to be exposed may be split into a large number of small data chunks, possibly with geometric overlaps. Corrections that can be applied in the vector domain (e.g. proximity effect correction) may be carried out to all chunks independently, possibly in parallel, and the resulting data is sorted and coded so as to improve computation speed of subsequent steps. The output is a collection of chunks where all chunks contain a collection of geometries. 
     Stage  161  as illustrated in  FIG. 13  may be referred to as the Rasterization stage: RAST. The geometries of every chunk are converted into rasterized pixel graphics. In this step, each pixel may be assigned a floating-point gray scale intensity depending on the geometric overlap of the corresponding surface of the raster-grid cell with the pattern to be exposed, i.e. the entity of all associated chunks. This floating-point intensity represents the ideal physical exposure dose to be delivered onto the target at the respective pixel location. Furthermore, every pixel that is completely inside a geometry may be assigned the maximal intensity, whereas the intensity of pixels that crosses an edge of a geometry is weighted by the fraction of the area of the pixel that is covered by the geometry. This method implies a linear relation between the area of the geometry and the total dose after the rasterization. 
     Stage  162  as illustrated in  FIG. 13  may be referred to as the Pixel-to-beamlet assignment stages: ASSIGN. In this step, given a particular write sequence, it is determined which pixel will be written by which beamlet. 
     Stage  163  as illustrated in  FIG. 13  may be referred to as the Pixel based corrections stage: CORR 1 . In this step, all corrections that can be applied in the pixel domain may be performed. These corrections comprise compensation of deviations from a uniform current density of the beam  50  over the aperture field, as described earlier and further illustrated in U.S. Pat. No. 9,495,499; whose disclosure is incorporated herein by reference. Additionally, the corrections may be for individual defective beam deflectors in the DAP  30 . Further illustration is provided in U.S. Pat. No. 9,269,543; which disclosure is incorporated herein by reference. Pixel based corrections are realized by modifying the floating-point intensity of each individual pixel. This is being done with respect to the Pixel-to-beamlet assignment of Stage  162 , which makes it possible to define and apply a compensation dose-factor q (or, equivalently a dose-shift s) for each pixel depending on by which beamlet it is written, and/or by which beamlets the neighboring pixels are written. 
     Stage  164  as illustrated in  FIG. 13  may be referred to as the Quantization stage: QUANT. The quantization process converts the possibly corrected, floating-point intensity of each pixel into a quantized (or equivalently ‘discrete’) gray level, given a predetermined gray value scale. 
     Stage  165  as illustrated in  FIG. 13  may be referred to as a second pixel based correction stage or CORR 2  may be for further optional pixel based corrections in the gray-level pixel data domain (not part of the present invention). 
     Stage  166  as illustrated in  FIG. 13  may be referred to as Pixel packaging, PPACK. The pixel image obtained from stage  164  is sorted according to the placement grid sequence and sent to a pixel buffer PBUF which is provided in the processing system  18  of the writer tool ( FIG. 1 ). The pixel data is buffered until a sufficient amount of data, typically at least the length of a stripe, is present, which triggers the exposure of the stripe (see  FIG. 7 ). The data is taken out of the buffer during the writing process. After the stripe has been written, the process described above starts anew for the pattern data of the next region, such as the next stripe. 
     Dose-Level Quantization 
     The present invention pertains to the QUANT stage  164  of the data path, which converts the floating-point (or equivalently high-resolution) intensity data into a quantized (i.e. discrete) gray level scale. In a typical realization of the invention the gray-level data is finally represented by a low-bit code, i.e., a code expressed through a small number of data bits. For instance, in a scenario where every pixel is described by 4 bits, pixels that are switched on have 2 4 =16 possible configurations, i.e. n Y =16 dose levels (0, 1, 2, . . . , 15). In a realization where the minimum dose 0% and the maximum dose 100% is equidistantly divided into 16 discrete dose levels, the step between two dose levels is 100%/15=6.67%. 
     In accordance with various embodiments finer dose-steps may be achieved via a suitable approach that exploits for oversampling o&gt;1 to improve the discretization by means of a dithering process. The main principle of altering the dose in steps finer than 6.67% is illustrated in  FIGS. 14 and 15 . In  FIG. 14 , four neighboring pixels p 1 , p 2 , p 3  and p 4  for the example of double-grid oversampling (o=2) are shown. Since the pitch between neighboring pixels is e=b/2, each overlapping area of size e 2 =b 2 /4 is simultaneously covered by four beamlets. As the individual pixels each carry a 4-bit wide information, there are now 4×15+1=61 possible dose levels. Thus, the dose on the substrate can effectively be altered in steps of 6.67%/4=1.67% in contrast to the 6.67% for the individual pixels. In  FIG. 15 , the dose of the overlapping area o 1  in the middle (dashed area) is considered. Starting from dose zero on all four neighboring pixels cfg 1 , the dose-level of o 1  is increased by one, resulting in configuration cfg 2  where the dose in the overlapping area is now 1 out of 4×15=60 possible non-zero dose-level configurations. Thus, the overall dose has increased by only 1/60=1.67%.  FIG. 15  shows all possible 4+1=5 dose configurations cfg 1 , cfg 2 , cfg 3 , cfg 4 , cfg 5  of the overlapping area for the case of two-dose levels, e.g. the pair  0  and  1 , for the individual pixels p 1 , p 2 , p 3  and p 4 . The skilled person easily realizes that any of the 60 possible non-zero dose-levels can be obtained by a combination of 4 pixels with 15 non-zero dose-levels. The finer gray level scale achievable in the overlapping area will be referred to as “effective gray levels” in the following. 
     As the number of overlapping pixels only depends on the oversampling factor o, it is straight-forward to compute the number of dose levels for any combination of oversampling o and gray-level n-bit resolution. In detail, the number of overlapping pixels is o 2 , resulting in o 2 ×(2 n −1)+1 effective gray-levels in steps of 1/(o 2 ×(2n−1)+1). Besides the mentioned cases of oversampling o=2 and bit-resolution n=4, another interesting scenario with respect to the implementation of the applicant is o=4 (so-called quad-grid-mode) and n=4, where n Y =4 2 ×(2 4 −1)+1=241 effective dose-levels are available, which can be varied in steps of 0.4167%. It will be evident to the skilled person that other combinations of o and n may be suitable depending on the individual implementation. 
     A computationally inexpensive algorithm is required which, starting from a desired floating-point pixel intensity, determines a proper discrete gray-level assignment for neighboring pixels. Besides the requirement of being computationally inexpensive, this algorithm will have to ensure that the entire range of o 2 ×(2 n -1)+1 effective dose-levels can be exploited. 
     Quantization Using Ordered Dithering 
     Due to its ease of calculation and deterministic behavior, ordered dithering is a method particularly suited for dose level discretization. During ordered dithering quantization, for every pixel (aperture image position) the fractional gray level value of the nominal dose is compared against a threshold value in a regular pattern obtained from a Bayer index matrix which is used as dithering matrix. The dose is then rounded up to the next dose step if it surpasses the threshold and rounded down otherwise. In general, the relationship between threshold matrix T and Bayer index matrix B (which describes the order in which beamlets are rounded up for increasing target dose) is given by 
               T   ij     =           2   ⁢     B   ij       -   1       2   ⁢     d   2         ⁢     (     i   ,     j   =   1     ,   …   ⁢           ,   d     )             
where d is the dithering order, i.e., the size of the dithering matrix B, which is usually quadratic. The dithering order d may be conveniently chosen equal to the oversampling factor o. Given the fundamental Bayer matrix B 2  of  FIG. 17A , Bayer matrices of arbitrary dimensions which are powers of two can be calculated recursively, namely, starting from the 2×2 matrix B 2 =B 2  and using the recursion rule
 
               B     2   ⁢   n       =     (             4   ×     (       B   n     -   1     )       +   1             4   ×     (       B   n     -   1     )       +   3                 4   ×     (       B   n     -   1     )       +   4             4   ×     (       B   n     -   1     )       +   2           )           
where B n  is a Bayer matrix of dimension n×n
 
     A simplified example for the application of a dithering matrix for rounding is presented in  FIG. 16 , relating to the case of two pixel gray levels (that is, every pixel has 1-bit data; thus, it can only be switched on or off) and double-dithering (i.e. d=2). The array NP 2  represents an example of a desired pattern, in which each entry contains a value of the dose to be exposed at the respective area element on the target (target dose values, also referred to as nominal dose values). In the example shown, the array contains several instances of values 0, 0.1, 0.5, and 1.  FIG. 17A  shows a dithering matrix B 2  of size 2×2 for d=2, and  FIG. 17B  shows the threshold matrix T 2  resulting from the dithering matrix B 2 . The threshold matrix T 2  is tiled in the plane in a regular pattern to obtain the threshold pattern TP 2  (left-hand part of  FIG. 16 ). The dimensions of the threshold pattern TP 2  will suitably conform to the dimensions of the desired pattern array NP 2  (top of  FIG. 17 ). According to the dithering procedure, rounding is achieved comparing the nominal dose value in each entry of the pattern NP 2  with the corresponding entry of the threshold pattern TP 2 , resulting in a quantized array QP 2 , which contains an array of dose values quantized to conform to the available gray values. In this example, the nominal dose values of 0.5 (half a pixel gray level) are rounded up and down in an alternating pattern, whereas doses of 0.1 are rounded down everywhere. The other dose values which occur in the target dose, i.e. 0 and 1, remain unchanged since they exactly match a pixel gray level 
     Another example is given in  FIGS. 18, 19A, and 19B  for the same target pixel doses y in a pattern array NP 4 , but using quad-dithering (d=4) with a 4×4 dithering matrix B 4  ( FIG. 19A ), and 4 (2-bit) pixel gray levels. The 2-bit information of one gray value means that a the available discrete gray values h form a palette having 2 2 =4 values, for instance, 0 (zero), ⅓, ⅔, and 1. Each element will be assigned a quantized value h chosen from the palette based on the original pattern.  FIG. 19B  shows the threshold matric T 4  which is calculated from the dithering matrix B 4 . In  FIG. 18  the resulting threshold pattern TP 4  is illustrated. During the process of dithering from the desired pattern array NP 4  to the quantized array QP 4 , the nominal dose values of 0.5 are rounded in an alternating pattern to the next pixel gray levels of ⅓ and ⅔, since the remainder of the subtraction 0.5−⅓=⅙, when divided by the step-size of the gray scale, i.e. ⅓, gives 3/6, which is larger than 5/32 and 15/32, but smaller than 20/32 and 17/32. Similarly, half of the pixel dose values of 0.1 are rounded up to ⅓ (since 0.1/(⅓)=0.3 is greater than 9/32 and 1/32) and the other half are rounded down to 0 (since 0.3 is smaller than 19/32 and 25/32). It should be noted that the values of the entries shown in the matrices T 4  and TP 4  are scaled by an overall factor of 1/32 (indicated outside the respective matrix). 
     The general procedure for arbitrary dithering order d and bits per pixel n is as follows:
         1. Decompose every nominal dose value into the form y=c*k+r, where the dividend k is the step-size, i.e. the step width of the gray scale, as determined by the number of bits n, i.e. k=1/(2 n −1), and where the integer quotient c and (positive real-valued) remainder r are determined uniquely according to the Euclidean division theorem. In particular, the quotient c is a non-negative integer, and the remainder is a non-negative number r&lt;k.   2. Compare the value v=r/k with the associated entry of the dithering threshold matrix. If this value v is larger than the entry from the dithering threshold matrix, the pixel is a assigned the discrete grey-level h=(c+1)*k; whereas if v is less than or equal to said threshold value, the pixel is assigned the discrete grey-level h=c*k.       

     Thus, in the second step, a floating point nominal dose value y (unless it already coincides with one of the values in the gray scale palette) is either rounded up or down, relative to the discrete grey-level scale with step-size k. 
     Ordered Dithering and Dose Corrections 
     The combination of oversampling, dose corrections, and ordered dithering can lead to complex stochastical effects. Various exemplary embodiments consider the case of double-grid exposure (o=2) and n Y =16, i.e. 16 gray levels (4-bit) for every pixel/beamlet and double-dithering (using the dithering matrices of  FIGS. 17A +B). As in  FIG. 14 , an overlapping area o 1  is exposed by four overlapping aperture image positions p 1 , p 2 , p 3 , p 4 , giving a total of 61 gray levels. Since this corresponds to dose increments of 1/60=1.67%, the maximal dose error due to quantization is 0.84% (½ effective gray level or ⅛ pixel gray level) when using optimal rounding. When using ordered dithering for quantization, this error may become larger, particularly in the presence of inhomogeneous beam current and corresponding corrections as described earlier. 
       FIG. 20  illustrates an exemplary embodiment for the case of homogeneous beam current. Consider a typical use-case, where a line through the upper half of the overlapping area o 1  is to be exposed, as illustrated in the target dose array NP 20 . The beamlets at pixels p 3 , p 4  are to deliver a maximal dose of 1, and the beamlets at p 1 , p 2  a dose of 7.125/15=0.475 to produce a total o 1  dose of 44.25/60=0.7375 of the maximal overlap dose  4 . Applying the dithering matrix to obtain the quantized array QP 20 , the pixel p 1  is rounded up to the next discrete gray level  8 , whereas the pixel p 2  is rounded down to 7; the pixels b 3 , b 4  remain unchanged. The two roundings combine to an overlap dose error of 0.75/60=1.25% (which experimentally corresponds to a line edge placement error in the order of approx. 0.4 nm). 
     While in this example, the rounding error is larger than the ideal quantization error, it is the worst case scenario when writing a line with this dithering matrix and homogeneous beamlet doses. Also note that this is not a generic error that occurs when using ordered dithering, which means that there are also cases where there is no rounding error at all. Consider for example the case in  FIG. 21 . Here, the target dose value (array NP 21 ) for the overlapping area o 1 , namely 45, is exactly matched when using the ordered dithering process as visible from the resulting quantized array QP 21 , even though the target dose of pixels p 1 , p 2  lies between two discrete values; namely 7 and 8. 
     In the case of inhomogeneous beamlet current the rounding behavior can be much worse. This is due to the fact that the corrected doses may align with the dithering thresholds in an unfavorable way. An example is given in  FIG. 22 , comprising the same target dose values NP 20  as in the case of  FIG. 20 , but with additional dose inhomogeneity correction factors. While the target dose is the same as in  FIG. 20 , due to the fact that the beamlets carry different amounts of current, the doses assigned to each pixel have to be adjusted with beamlet-dependent dose factors DF 22  to effectively deliver the target doses DP 22 . Each pixel is now assigned an individual corrected dose, which, in some cases, can align unfavorably with the dithering thresholds. In  FIG. 22 , the worst possible rounding behavior is shown in the array NQ 22  (all pixels are rounded up), leading to a total dose error of approximately 3.3% (45−43=2 out of 60 non-zero effective gray levels, or, equivalently ½ single pixel gray level), which, for the MBMW of the applicant, may experimentally correspond to an edge placement error of about 1.32 nm. This dose error also approximately remains in the effective dose (i.e., accounting for the different beamlet currents). 
     In accordance with many embodiments, the above scenario can be resolved by using the same or very similar dose correction values for overlapping pixels which undergo the dithering process. Consider, for example,  FIG. 23 ; where, a common dose-correction-factor 0.972 (array DF 23  has a uniform value) among the neighboring beamlets is applied which preserves the exact target dose, as can be seen in arrays DP 23  and NQ 23 . That is, in the quantized dose distribution NQ 23  the target dose of overlap o 1  is still 43 as before, but with the advantage that when the ordered dithering is applied, no rounding error occurs at all. 
       FIG. 26  illustrates a plot of the rounding errors (in terms of effective gray levels) for the target dose values from  FIGS. 20 to 23  for the range 0.84 to 1.0 of common dose-correction factors for the four neighboring, and potentially overlapping pixels. Here, it may be observed that the worse-case rounding error of two effective gray levels does not occur. Instead, the maximal error is halved to one effective gray level. 
     In accordance with many embodiments  FIGS. 27A to 35B , illustrate the advantageous behavior of dose correction factors during dithering by using a uniform dose correction for each aperture-line along the write direction (i.e., the scanning direction sd, see  FIG. 3 ).  FIG. 27A  shows a typical case of a current profile (more exactly, current density profile) across the image-field of size 80 μm×80 μm, each value representing the deviation α(X,Y) (quantified in percentage [%]) of the current of a single beamlet located at the position (X,Y) of a respective area, relative to the mean current across the image field. The values listed in  FIG. 27A  depict a typical distribution according to a realistic scenario, with a roughly parabolic overall characteristic and a deviation range of several percent. Corresponding normalized correction dose factors q (see above, Section ‘Dose inhomogeneity correction’) are given in  FIG. 28A , which are proportional to the multiplicative inverse of the doses from  FIG. 27A .  FIG. 29A  shows the corrected dose profile: applying the factors from  FIG. 28A  to the measured doses shown in  FIG. 27A  yields a completely flat dose profile of constant value 0.960. Note that the overall dose is now lower by a constant 4% across the whole image-field. This decrement, however, can easily regained by increasing the overall source current by the corresponding amount. 
     The data in  FIGS. 27A to 29A  relate to the current profile, which corresponds to a (fictitious) pattern profile of uniform gray level y=1. In the case of a realistic pattern profile, where each of the pixels will receive an individual gray level value y with 0≤y≤1 (nominal dose value), the correction dose factors q are used to correct these nominal dose values y so as to compensate for the non-ideal current profile ( FIG. 27A ). This is done by dividing each nominal dose value y by the value q of the respective beamlet ( FIG. 28A ), so as to obtain a corrected dose value y′=y/q. This correction is made for each beamlet (i.e., each pixel or image element on the target) of the entire image field. 
     In order to avoid that the unfavorable worst-case rounding occurs, one computes the averaged dose profile along the scanning direction sd, which in this example is the X-axis. Re-normalized, this yields the effective dose-values shown in  FIG. 27B , which can be corrected via the dose-correction factors given in  FIG. 28B . Overall, the mean dose-intensity is reduced to the common value 0.974 as shown in  FIG. 29B . This is another advantage of this aspect of the invention, namely that the minimum dose correction factor q is larger due to this averaging. This has the effect that more effective dose levels are available for writing a pattern, because the lower this number is, the more gray-levels are effectively needed for dose-inhomogeneity-correction. 
     While the worst-case rounding behavior (of two effective gray levels) described earlier only occurs by chance, it may appear systematically if an overlapping stripe strategy (=multipass) is applied as described in U.S. Pat. No. 9,053,906; which disclosure is incorporated herein by reference. Consider again a double-grid double-dithering strategy with ordered dithering discretization as illustrated in  FIG. 14 . In addition, a double-pass writing strategy as described previously is applied, according to which the pixels p 1 , p 4  will typically be written in one pass, whereas the pixels p 2 , p 3  are written in the other pass. Pixels written in the same pass tend to have similar dose factors (as they originate from the same y-position of the charged particle source), or have already been equalized according to previously introduced method. Due to the structure of the dithering matrix, the pixels p 1 , p 4  are more likely to be rounded up, whereas p 2 , p 3  tend to be rounded down, leading to an intricate correlation between the dose factors and the rounding behavior, which under certain circumstances can introduce systematic edge placement errors across the beam-field when exposing a regular pattern. 
     An example is given in  FIG. 24 ; the reference symbols correspond to those of  FIGS. 22 and 23  mutatis mutandis. As before, the beamlets are assigned individual dose correction factors q, however, in this case a correlation between the dose factors and the dithering matrix arises. The solution presented in  FIGS. 27 to 29 , in which a common dose correction factor q is assigned to each aperture line improves the situation, but not to a satisfactory amount, because pixels p 1 , p 4  and p 2 , p 3  may now be written by beamlets from different aperture rows. Consequently, only two of the four dose factors will be identical. In this case, the maximal rounding error is again four times as large as the ideal quantization error with 3.33% (two effective gray levels). In some constellations, the error can be additionally increased due to the fact that beamlets from one pass now have the tendency to get rounded down, whereas for the other pass it is more likely that the beamlet is rounded up. In the example given in  FIG. 24 , for instance, the high-dose-factor beamlets have the tendency to be rounded down and low-dose-factor beamlets the tendency to be rounded up, leading to observable systematic CD errors with respect to the Y-position of a stripe. 
     The solution to this problem is presented in accordance with many embodiments and further illustrated in  FIGS. 30A to 32B . First, an effective dose-profile is computed by averaging the dose profile α(X,Y) (quantified in percentage [%] of the current of a single beamlet located at a position (X,Y) relative to the mean current across the image field) according to the (multipass) stripe overlap (compare with  FIG. 8A  and  FIG. 8B ). This stripe overlap is determined by the offset Y 1  between overlapping stripes, Y 1 =y 0 / 2 .  FIG. 30A  shows the deviation α(X,Y) of the effective dose-profile from the mean value in percent. For this effective dose-profile, one then computes the corresponding correction factors, as illustrated in  FIG. 31A . Optionally, the factors are additionally equalized along the X-direction, analogously, as for the non-overlapping stripe-exposure mode; this leads to the effective dose correction factors as illustrated in  FIGS. 30B and 31B . In particular, using the common correction factors as depicted in  FIG. 31B  in the double-pass mode, where stripes from different passes overlap with 50% of their width (as also shown in  FIGS. 8A and 8B ), a maximal rounding error of 1 effective gray level is achieved, since the neighboring pixels from different passes now have the same dose correction factor q, as e.g. depicted in  FIG. 25 . 
     It is straightforward to generalize the procedure described above to other scenarios. For instance, the quad-pass variant is illustrated in  FIGS. 33A to 35B . Here, the offset Y 1 ′ between overlapping stripes is ¼ of the stripe width y 0 , and every second image-field segment is averaged in Y-direction, corresponding to four passes; in other respects, the  FIGS. 33A to 35B  correspond to those of  FIGS. 30A to 32B , respectively. Note again, that the minimum dose correction factor over the image-field becomes larger the more averaging is performed. 
     Although various embodiments are presented herein, it should be understood that any suitable embodiment may be utilized. 
     Doctrine of Equivalents 
     As can be inferred from the above discussion, the above-mentioned concepts can be implemented in a variety of arrangements in accordance with embodiments of the invention. For example, though the method for correcting the written pattern from a multibeam-writer is described in relation to the various components of the MBW, other arrangements may be contemplated within the scope of the current disclosure. 
     Accordingly, although the present invention has been described in certain specific aspects, many additional modifications and variations would be apparent to those skilled in the art. It is therefore to be understood that the present invention may be practiced otherwise than specifically described. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive.