Abstract:
A scanning system uses a multi-beam brush having a widely separated beams, a modulator that controls intensity of pixels in scan beams, an optical system that minimizes scan line bow at the expense of non-uniform scanning beam velocity, and a timing generator that generates a pixel clock signal having a variable period that compensates for the non-uniformity of pixel velocity. The wide separation of scan beams permits the modulator to turn beams on or off with a direction of brightening or darkening in the cross-section of the beams being opposite to the scanning direction. A novel arrangement of the beams in the brush permits a uniform indexing step size to uniformly expose an image region. In one embodiment, the timing generator includes: a source of pixel period values; a select circuit to select pixel period values for pixels, and a counter that loads a first value from the pixel period value selected for a pixel, counts for a period of time indicated by the first value, and asserts a signal marking an end of the period. An additional delay after the signal from the counter can be shorter than the period of a clock signal to the counter but also controlled by the pixel period value.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This patent document is related to and incorporates by reference in its entirety, co-filed U.S. patent application Ser. No. 09/272,947, entitled “Multi-Beam Scanner Including A Dove Prism Array”, now U.S. Pat. No. 6,271,514. 
    
    
     BACKGROUND 
     1. Field of the Invention 
     This invention relates to printing systems and methods and particularly systems and methods using multiple scan beams that have wide lateral separations. 
     2. Description of Related Art 
     Printing systems including scanners are suitable for a variety of applications including printing text on paper, patterning photoresist during integrated circuit manufacture, and creating masks or reticles for projection-type photolithography systems. For integrated circuit applications, the printing systems typically require submicron precision. FIG. 1A illustrates the basic architecture of a precision printing systems  100  that employs scanning. System  100  includes: a light source  110  such as a laser; an acousto-optic modulator  120  that controls intensity of one or more input beams  135 ; pre-scan optics  130  that control the position, shape, and collimation of input beams  135 ; a scanning element  140  such as a polygon mirror that sweeps scan beams  145  along a scan direction; and post-scan optics  150  that focus scan beams  145  on an image plane  160 . Scanning of scan beams  145  forms scan lines that expose a pattern in an image area of plane  160 . Acousto-optic modulator  120  modulates the intensity of input beams  135  to select the pattern that scan beams  145  expose. 
     A conventional acousto-optic modulator includes a block of material such as fused silica through which input beams propagate. To turn on, turn off, or change the intensity of an input beam, a transducer generates an acoustic wave that crosses the path of the input beam in the block. The acoustic wave locally changes the optical properties of the block and deflects part of the input beam. Typically, a beam stop later in the optical train blocks the undeflected portion of the beam. 
     A concern for a precision scanner having a conventional acousto-optic modulator is the orientation of the scanning direction relative to propagation of the acoustic waves that modulate the input beams. If the propagation direction and the scanning direction are not collinear, the turning on and turning off of beams can reduce sharpness of edges or create undesired skew or directional bias in a pattern being illuminated. FIG. 1B illustrates an illuminated region  170  of a scan line formed when an acoustic wave deflects an input beam in a direction  178  that (after convolution through the system optics  130  and  150 ) is perpendicular to a scan direction  172 . Deflection direction  178  typically corresponds to the direction of propagation of the acoustic wave in the acousto-optic modulator. As acousto-optic modulator  120  turns on input beam  135 , a cross-section  174  of the beam expands in direction  178 . Accordingly, the initially illuminated part of region  170  is narrow and toward one edge until the input beam has a fully illuminated cross-section such as cross-section  175 . Similarly, when acousto-optic modulator  120  turns off input beam  135 , one edge of the input beam darkens first, and a shrinking cross-section  176  of the beam causes illuminated region  170  to recede toward the opposite edge. This reduces sharpness at the edges of illuminated regions formed by multiple scan lines, skews rectangular illuminated areas, and causes pattern lines at 45° to the scan direction to differ in thickness from pattern lines at 135° to the scan direction. However, to provide independent control of the beam intensities and a narrow scan brush, acoustic waves in an acousto-optic modulator generally propagate at an angle relative to the scan direction. 
     As shown in FIG. 1C, a separation  133  between beams  132 ,  134 ,  136 , and  138  inside acousto-optic modulator  120  must be sufficient for acoustic waves  122 ,  124 ,  126 , and  128  to independently modulate respective beams  132 ,  134 ,  136 , and  138 . Typically, separation  133  must be more than a beam diameter. To avoid the separation causing gaps between scan lines, a scanning direction  172  is selected so that beams  132 ,  134 ,  136 , and  138  overlap when viewed along the scan direction  172 . An advantage of overlapping beams is the narrow width  180  of the scan brush. Narrow brushes reduce scan line bow which is common for conventional f-θ scan lenses. (Scan line bow is the curvature of scan lines that are off the optical axis of a scan lens.) Also, scanning overlapping beams along scan direction  172  forms a band of scan lines without intervening gaps, which simplifies indexing of scan lines to cover the image area. As indicated above, disadvantages of the configuration of FIG. 1C are reduced sharpness at the edges in the image, skew of rectangular areas, and 45°/135° line thickness bias. 
     As shown in FIG. 1D, the scan direction  172  can alternatively be the same as or opposite to the direction of propagation of acoustic waves  122 ,  124 ,  126 , and  128  in acousto-optic modulator  120 . With this configuration, the separation  133  required for independent modulation of beams controls the separation between the scan lines. This creates a scan brush that is wider than the brush of FIG. 1C, and the wider scan brush increases scan line bow from a conventional f-θ scan lens, making the accuracy required for integrated circuit applications difficult to achieve. Other types of scan lenses can reduce scan line bow but generally cause scan beams to move with non-uniform velocity and therefore can distort the image. 
     Systems and methods are sought that use simultaneous scan beams for faster scanning but avoid scan line bow and image distortion and also avoid the skew, blurred edges, and directional bias associated with acousto-optic modulators having acoustic waves propagating at an angle to the scan direction. 
     SUMMARY 
     In accordance with the invention, a multi-beam scanner has a wide scan brush, a modulator that controls intensity of pixels in scan beams, an optical system that minimizes scan line bow at the expense of non-uniform scanning beam velocity, and a timing generator that generates a pixel clock signal having a variable period that compensates for the non-uniformity of pixel velocity. The wide separation of scan beams permits the modulator to turn beams on or off with a direction of brightening or darkening in the cross-section of the beams being opposite to the scanning direction. This allows the brightening direction to be opposite the scan direction to improve edge sharpness, avoid skew in rectangular regions, and avoid a directional bias in line thickness. 
     A novel arrangement of the beams in the brush permits a uniform indexing step size to uniformly expose an image region. In particular, a brush with b beams spaced a distance n apart uniformly covers the image region after repeated scanning and indexing by a distance m if the number of beams b and the distances n and m are such that the ratio of m to n is equal to the ratio of b to an integer q that has no common factors with b. In one embodiment, a diastemal brush has a top half including b beams uniformly spaced distance n apart and a bottom half including b beams uniformly spaced distance n apart. The distance between the top and bottom halves is 1.5*n. With this diastemal brush and a uniform indexing distance m, the top half forms uniformly spaced scan lines, and the bottom half forms scan lines midway between adjacent scan lines that the top half forms. Other embodiments of the scan bush include three or more sections of equal spaced beams separated by two or more diastema. 
     In one embodiment, the timing generator includes: a source of pixel period values and a counter. The counter loads a first part of a pixel period value selected for a pixel, counts for a period of time indicated by the first part, and asserts a signal marking an end of the period. An additional delay calculator circuit can delay the signal from the counter for a time shorter than the period of a clock signal to the counter. A second part of the pixel period value controls the delay. The combination of the times for the count and the delay forms the complete pixel period. After asserting a pulse for the pixel clock for a pixel period, the source supplies the next pixel period value which controls the count and delay for the next pixel period. 
     In alternative embodiments, the source of the pixel period values includes a set of registers, a register and a series of adders, or a look-up table. In one embodiment, the source of pixel period values includes a lookup table, a start index register, and a pixel counter, which initially loads from the start index counter and supplies an address to a lookup table. When a set of registers or a register and a series of adders provide the pixel period values, a multiplexer selects the pixel period value according to a select signal from a look-up table. The look-up table is indexed by pixel and selects an appropriate pixel period value for each pixel. The pixel counter increments the pixel index each time the timing generator marks a boundary of a pixel, and in response to the changed pixel index, the timing generator selects the next pixel period value. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1A shows a prior art printing system. 
     FIG. 1B illustrates the relation between the cross-section of a scan beam that is being turned on and then off and the shape of a resulting illuminated region. 
     FIGS. 1C and 1D show alternative orientations of the scan direction in an acousto-optic modulator. 
     FIG. 2 is a block diagram of a precision printing system in accordance with an embodiment of the invention. 
     FIG. 3 illustrates an interleaved scanning process in accordance with an embodiment of the invention. 
     FIGS. 4,  5 , and  6  are block diagrams of timing generators for a printing system such as illustrated in FIG.  2 . 
    
    
     Use of the same reference symbols in different figures indicates similar or identical items. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     In accordance with an aspect of the invention, a precision printing system employs a scanner and multiple scan beams arranged in a wide scan brush with separations between individual beams. An acousto-optic modulator or deflector in the printing system controls the intensities of individual scan beams using acoustic waves oriented along the scan direction. Accordingly, as the acousto-optic modulator turns a beam on or off, successive portions of a scan beam cross-section brighten or darken along the direction of propagation of the scan lines. The printing system uses a scan lens such as an f-sin θ scan lens that reduces scan line bow caused by the width of the scan brush, and a timing generator that generates a pixel clock signal with a variable period to compensate for variations in the velocity of scan beams. 
     FIG. 2 shows a precision printing system  200  that employs scanning in accordance with an embodiment of the invention. A pre-scan portion of system  200  includes a beam source  210 , an acousto-optic modulator (AOM)  220 , and pre-scan optics  230 . Beam source  210  forms multiple input beams  219  which are spaced along a line to form a brush. AOM  220  modulates the intensity of each input beam  219  independently and directs the modulated scan beams  229  to pre-scan optics  230 . In accordance with an aspect of the invention, acoustic waves in AOM  220  are oriented so that each beam  229  brightens in a direction perpendicular to the line of beams  229  as AOM  220  turns on the beam  229 . Pre-scan optics  230  direct the line of modulated input beams  229  onto a scanning element  240  so that the scanning direction resulting from movement of the scanning element is opposite the brightening direction of beams  229 . Pre-scan optics  230  optionally includes brush rotation optics, such as a K mirror or a dove prism, that rotates the line of the brush if necessary to align the brightening and scan directions. 
     Scanning element  240  directs multiple scan beams  249  into post-scan optics  250 . Scanning element  240  is preferably a rotating polygon mirror that during scanning rotates with a constant angular velocity. Alternatively, an oscillating mirror or a rotating holographic element could be employed. Post-scan optics  250  focuses the scan beams as the scan beams sweeps along scan lines on a surface of a workpiece. Post-scan optics  250  include a scan lens  252  and a reduction lens  258 . In an exemplary embodiment of system  200 , scan lens  252  is an f-sin θ lens, which reduces scan line bow for wide scan brushes. F-sin θ lens are known in the art. U.S. Pat. No. 5,018,807 to Shirota and U.S. Pat. No. 5,235,438 to Sasada describe examples of f-sin θ lenses and are hereby incorporated by reference in their entirety. Since lens  252  is an f-sin θ lens and scanning element  240  rotates with a uniform velocity, the scan beams which form scan lines on a workpiece move with non-uniform velocity in the image plane. A timing generator  226  as described further below provides a non-uniform pixel clock signal to synchronize modulation of the scan beams with the positions of the scan beams on the workpiece. 
     Reduction lens  258  reduces the scan line size and separation and the resulting image size as required for the image to be formed on the workpiece. For the exemplary embodiment, the workpiece is a mask, a reticle, an unprocessed wafer, or a partially processed wafer that is coated with a layer of photoresist. A precision stage system  260  which is connected to an interferometer  262  and an alignment system  264  positions and moves the workpiece as required for alignment and indexing after each scan line. Alignment system  264  identifies the positions of alignment marks on the workpiece as viewed through reduction lens  258  and accordingly determines the position and orientation of the workpiece relative to the scan lines. Interferometer  262  monitors the movement of the workpiece for indexing. 
     Beam source  210  includes a light source  211 , beam-shaping elements  212  and  214 , a beam steering system  213 , a beam splitter  215 , and brush optics (or telescope)  216 . Light source  211  is preferably a laser that generates a coherent beam of suitable power and wavelength for the printing performed. In the exemplary embodiment of the invention, light source  211  is a laser that produces a monochromatic beam of deep ultraviolet light having power greater than about 0.1 W. For example, a “Sabre Fred” system available from Coherent, Inc. delivers a 0.5-W beam at 244 nm or 257 nm. This deep UV laser has a resonant cavity free of oxygen and moisture to prevent formation of ozone and degradation of the BBO doubling crystal. Such lasers with appropriate components in the rest of system  200  can achieve a minimum feature size of  360  nm with a uniformity of ±20 nm and a placement accuracy of less than 20 nm at exposure doses up to 200 mJ/cm 2  using multiple pass printing. The environment for the optical system and workpiece is a clean environment that is kept at a temperature controlled to ±0.05° C. High purity nitrogen is a suitable purge gas where required. 
     Beam preparation optics  212  and beam steering system  213  direct the beam from light source  211  to beam splitter  215  which splits the beam into multiple input beams  219  for AOM  220 . In the exemplary embodiment, beam splitter  215  and telescope  216  form thirty-two input beams  219  which are along a line with a 404.8-μm on center spacing. The center two beams are separated by an extra 202.4 μm for a total separation of 607.2 μm (one and one half times the normal spacing). The separations between input beams  219  are sufficient for AOM  220  to modulate the intensity of each beam independently. 
     AOM  220  is a block of material such as fused silica having a patterned layer of lithium niobate bonded to one surface. Electric signals applied to contacts lithographically defined in a conducting layer overlying the lithium niobate layer create multiple acoustic waves. Each acoustic wave propagates through the path of an associated input beam in the block, deflects the associated input beam, and controls the intensity reaching an aperture that selects the diffracted beam. A rasterizer  224  generates the signals that create the acoustic waves and as a result controls the intensity of modulated beams  229 . In particular, rasterizer  224  divides each scan line into pixels and generates the signals as required for each pixel to have a desired intensity. For timing, a facet detection system  242  detects the orientation of scanning element  240  to identify the beginning of scan lines, and a timing generator  226  generates pixel clock signals to identify the beginning of each pixel in a scan line. In a system where scan beams scan at a uniform rate, the pixel clock signal is a uniform periodic signal. In the exemplary embodiment, the pixel clock signals have periods that vary according to variations in the scan rates for associated scan lines. Timing generators suitable for non-uniform scan rates are described below. 
     To provide the maximum space for separate acoustic waves that control individual beam intensities, the acoustic waves propagate along a direction perpendicular to the line of input beams  229 . The direction of propagation of the acoustic waves is the same as the direction in which successive portions of a beam  229  become illuminated as AOM  220  turns on the beam. This direction is sometimes referred to herein as the brightening direction. In accordance with an aspect of the invention, the brightening direction for the scan beams at the image plane of system  200  is opposite the scan direction. This prevents the blur, skew, and line thickness bias described above, but also leaves a separation between the scan beam along the direction perpendicular to the scan direction. The scan beams thus form a “brush” for simultaneous illumination of multiple scan lines that are separated from each other. 
     FIG. 3 illustrates an exemplary brush configuration, the relative positions of the scan beams during seven scans S 0  to S 6 , and the accumulated exposure for scans S 0  to S 6 . The exemplary brush configuration includes  32  beams B 0  to B 31 . In FIG. 3, the size of beams B 0  to B 31  and the spacings between beams B 0  to B 31  are indicated in arbitrary “grid units.” For example, each of scan beams B 0  to B 31  has a radius of about 2 grid units, and the center-to-center separation between adjacent beams is 6 grid units (except for the central beams B 15  and B 16  which are separated by 9 grid units.) The actual separations and sizes of beams change according to the optical properties of system  200 . At AOM  220 , the separations for beams B 0  to B 32  (i.e., beams  219 ) are 404.8 μm or 607.2 μm, but the demagnification between AOM  220  and the image plane of system  200  is about 1/400 so that the separations at the image plane are on the order of 1 μm. 
     Interleaved scanning of the scan brush of FIG. 3 covers the image area. With the exemplary brush, which has a 50% wider separation between central beams B 15  and B 16 , a constant displacement for indexing after each scan interleaves the scan beam for uniform coverage of an image area IA. For example, indexing in FIG. 3 displaces the scan brush by 32 grid units relative to the object being scanned. In system  200 , indexing occurs when precision stage  260  moves the object being scanned perpendicular to the scan line direction. The distance that stage  260  moves the object is equivalent to 32 grid units in the image plane. After scans S 0  to S 6 , an accumulated exposure ACC in image area IA includes scan lines with a uniform center-to-center separation of one grid unit. The one grid unit center-to-center spacing overlaps the scan lines for smoother imaging. Adding additional scans after scan S 6  will expand the area of uniform exposure IA. 
     The interlaced scanning illustrated in FIG. 3 can be generalized. Specifically, if a brush containing b uniformly spaced beams having centers separated by n units is repeatedly scanned with an increment of m units between each scan, uniform coverage (i.e., evenly spaced scan lines) will be achieved if the scan parameters satisfy Equation 1.                Equation                 1        :                                            m   n     =     b   q                                              
     In Equation 1, the parameter q is an integer having no common factors with the number of beams b. The interleaved scanning can also be employed with diastemal brushes such as illustrated in FIG.  3 . In particular, a beam having two uniformly spaced halves that independently satisfy Equation 1 and a diastema of 1.5 times m between the halves will write uniformly spaced scan lines with the top half forming scan lines exactly between the scan lines that the bottom half forms. For the scanning illustrated in FIG. 3, the number of beams b per half is 16. The number of units n between beams is 6. The number of scans NS required for uniform coverage is 6, and the offset m between scans is 32. As a result of the scan in the desired coverage area IA, scan lines are uniformly spaced one unit apart. The radius of the beams can then be selected to provide the desired coverage or overlap of the scan lines. 
     An alternative diastemal scan brush includes three or more sections of uniformly spaced beams where the separations between pairs of sections differ from the spacing of beams within a section. For example, a diastemal scan brush can include three sections containing beams uniformly space one unit apart and two diastema providing separations of one and a third units between sections can provide uniform scan coverage. Many other diastemal scan brushes having multiple diastema are possible. 
     Returning to FIG. 2, rasterizer  224  controls the intensity of the individual beams to form an image containing a rectangular array of uniformly sized pixels. To form a rectangular array of pixels, scan lens  252  in the preferred embodiment is an f-sin θ scan lens which forms straight scan lines even for scan beams that are significantly off axis when passing through scan lens  252 . With an f-sin θ scan lens, the scan position is not linearly related to the polygon angle, and timing generator  226  varies the timing between pixels slightly across each scan line to correct for the non-linearity of scan rate along the scan lines. Additionally, the f-sin θ scan lens causes different beams to have different scan positions depending on the offset of the beam from scan plane passing through the optical axis of scan lens  252 . In an exemplary embodiment, this amounts to a 13 nm lag between a central beam and an edge beam for a full half field defection of about 12 degrees. If desired, timing generator  226  could generate separate timing signals for separate beams to compensate for different lag for each beam. However, in the exemplary embodiment timing generator  226  generates a single pixel clock signal for all the beams. 
     FIG. 4 is a block diagram of a timing generator  400  suitable for use as timing generator  226  in the system of FIG.  2 . Timing generator  400  includes a start index register  430 , a pixel counter  440 , a lookup table  450 , a counter  460 , and a delay calculator  470 . Start index register  430  and pixel counter  440  provide to lookup table  450  a signal INDEX corresponding to the position of a scan beam at the start of the next pixel. Lookup table  450  then supplies to counter  460  and delay calculator  470  a pixel period value that controls the period between the start of one pixel (e.g., one rising edge of a signal PIXELCLK) and the start of the next pixel (e.g., the next rising edge of signal PIXELCLK). 
     Pixel clock period values in lookup table  450  differ from each other to correct for systematic scan direction non-linearity in the motion of scan beams as projected by the scan optics. In particular, the table contains pixel period values representing pixel periods T i  that approximately satisfy Equation 1.                Equation                 1        :                                            f                   sin              [     ω                   (       ∑     i   =   1     N                     T   i       )       ]       =     X   N                                              
     Where f is the focal length of the f-sin θ scan lens, ω is a constant angular frequency for the scanning, and X N  is the position of the pixel corresponding to index value N. Pixel period values are symmetric about the zero index which corresponds to an angle of zero degrees. 
     In the exemplary embodiment, lookup table  450  contains pixel period values for more pixels than are required for a scan line. This allows correction for position errors that precision stage  260  introduces in the scan direction. In particular, the angle of the scan beam at start of a scan line on a workpiece depends on the position of the workpiece, and the correct pixel periods are selected according to the angular positions of the pixels. After determining the error in stage position for a scan line, start index register  430  is loaded with the index corresponding to the pixel period value for the first pixel position in the exposed scan line. The index value from register  430  is used to initialize pixel counter  440  for the pixel at the start of the scan line. Pixel counter  440  generates a signal INDEX that indicates an address for lookup table  450  and selects the correct pixel period value for counter  460  and delay calculator  470 . 
     The pixel period value indicates a number of full periods of a clock signal CLK and a fraction of the period of signal CLK. In the exemplary embodiment, signal CLK has a period of 2 ns, and the pixel period values are 8-bit values including a 5-bit count of the full periods and a 3-bit value indicating the fraction. To generate the period for a pixel, counter  460  loads the pixel period value and then counts according to clock signal CLK until reaching a terminal count. Counter  460  then asserts a terminal count signal to delay calculator  470 . Delay calculator  470  delays asserting pixel clock signal PIXELCLK by a fraction of the period of signal CLK. The fraction is a combination of the fraction from the pixel period value and a fractional delay used for the last assertion of signal PIXELCLK. The result of the combination has a fractional part that is the fractional delay for the current pixel. When delay calculator  470  calculates a delay greater than one period of clock signal CLK, delay calculator  470  signals counter  460  with a signal WAIT that changes the terminal count or otherwise causes counter  460  to wait one count longer than its programmed value before asserting the terminal count signal. Delay calculator  470  asserts signal PIXELCLK after receiving the terminal count signal and waiting for a fraction of the period of signal CLK. In this manner, timing generator  400  generates pixel periods with finer resolution than one period of clock signal CLK. 
     In the embodiment of FIG. 4, delay calculator  470  includes registers  472  and  476  for current and previous fractions, an adder  474 , and a programmable delay  478 . At the start of a pixel (e.g., assertion of signal PIXELCLK) registers  472  and  476  respectively register a fraction from the pixel period value and a previously determined fraction from adder  474 . At the same time, counter  460  registers the number of full periods from the pixel period value and begins counting. To determine the fractional delay for a pixel period, adder  474  adds the factions from registers  472  and  476 . If the resulting sum is greater than one period of signal CLK, adder  474  asserts a carry bit as signal WAIT and to delay by one clock cycle assertion of the terminal count signal from counter  460 , and the fraction from adder  474  controls the amount of delay. In one specific embodiment, adder  474  has a 3-bit width so that the output signal adder  474  applies to programmable delay  478  selects one of eight delays. Since the programmable delay spans less than one 2-ns clock cycle, each delay increment is 2 ns/8 or about 0.250 ns. Alternatively, adder  474  could be of greater width than three bits to permit finer divisions of the programmable delay. 
     The assertion of signal PIXELCLK marks the start of a pixel and causes counter  460  and delay calculator  470  to register the current pixel period value and pixel counter  440  to increment and select the next pixel period value from lookup table  450 . A scan line is complete when the number of pixel clock periods generated is equal to the number of pixels in the scan line. The process then begins again for the next scan line with an update of the value in the start index register  430  according to the positioning of the workpiece for the next scan line. 
     FIGS. 5 and 6 are block diagrams of alternative embodiments of timing generator  226 . FIG. 5 shows a timing generator  500  which generates a pixel clock signal PIXELCLK that marks the beginnings of pixels in a scan line. For timing generator  500 , the period between the start of one pixel and the start of the next pixel is one of N different times where N is 2 or more. Generator  500  includes N digital storage elements  510  (e.g., N registers or ROM cells) which store pixel period values corresponding to the different periods. Storage elements  510  are coupled to the inputs of a multiplexer  530  which has a select terminal coupled to a lookup table  550 . Lookup table  550  contains i-bit selection values, where i is an integer such that 2 i  is equal to or greater than N. The selection values correspond with pixels, and each selection value identifies which of the N pixel period values corresponds to the period for a corresponding pixel. 
     As described above, pixel counter  440  is initialized according to a positioning error for the workpiece before the start of each scan line and  20 . increments a pixel count each time signal PIXELCLK is asserted. The pixel count from counter  440  provides an address signal for lookup table  550 . Lookup table  550  outputs to multiplexer  530  a select value corresponding to the pixel count. In response to the select value, multiplexer  530  selects one of the pixel period values from storage elements  510 , and applies one or more most significant bits of the selected period value to counter  460  and one or more of the least significant bits of the pixel period value to delay calculator  470 . When signal PIXELCLK is asserted to mark the start of a pixel, counter  460  loads a portion of the pixel period value from multiplexer  530  and begins incrementing the loaded value at a rate determined by clock signal CLK. When the count reaches the terminal count, counter  460  asserts the terminal count signal to delay calculator  470 . Delay calculator  470  asserts signal PIXELCLK after a delay that is a fraction of the period of clock signal CLK, the fraction being determined by the delay for the previous pixel and the least significant bits of the current pixel period value. 
     FIG. 6 is a block diagram of a timing generator  600  in which a storage element  610  and a set of adders  620  provide the period values to multiplexer  530 . In particular, storage element  610  stores a count representing the minimum period between the beginnings of consecutive pixels. Adders  620  add offsets to the minimum period to generate pixel period values associated with the various pixels. Multiplexer  530  selects a pixel period value from one of storage element  610  and adders  620  and applies the selected pixel period value to counter  460  and delay  470 . Timing generator  600  otherwise operates in the same manner as timing generator  500  of FIG.  5 . 
     Since each scan line scans at a slightly different rate from its neighbors, timing generator  226  can include multiple timing circuits such as generators  400 ,  500 , and  600 . Each such timing circuit provides a pixel clock signal for one or more scan lines. The exemplary embodiment of the invention uses one timing circuit and a single pixel clock signal for all of the scan lines being simultaneously formed. 
     Although the invention has been described with reference to particular embodiments, the description only provides examples of the applications of the invention and should not be taken as a limitation. For example, timing circuit  600  of FIG. 6 adders  620  add offsets to a minimum pixel period value to generate the range of pixel period values, but alternatively the range of pixel values can be generated from the minimum, the maximum, or an intermediate pixel period count using a variety of arithmetic or logic circuits. Various other adaptations and combinations of features of the embodiments disclosed are within the scope of the invention as defined by the following claims.