Abstract:
A charged particle shaped beam column includes: a charged particle source; a gun lens configured to provide a charged particle beam approximately parallel to the optic axis of the column; an objective lens configured to form the charged particle shaped beam on the surface of a substrate, wherein the disk of least confusion of the objective lens does not coincide with the surface of the substrate; an optical element with 8N poles disposed radially symmetrically about the optic axis of the column, the optical element being positioned between the condenser lens and the objective lens, wherein N is an integer greater than or equal to 1; and a power supply configured to apply excitations to the 8N poles of the optical element to provide an octupole electromagnetic field. The octupole electromagnetic field is configured to induce azimuthally-varying third-order deflections to the beam trajectories passing through the 8N-pole optical element. By controlling the excitation of the 8N poles a shaped beam, such as a square beam, can be formed at the surface of the substrate. The 8N-pole element can be magnetic or electrostatic.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Applications Ser. Nos. 60/895,126 filed Mar. 15, 2007 and 60/921,733 filed Apr. 3, 2007. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of Use for the Invention 
     This invention relates to the field of charged particle optics, and in particular to systems for generation of high current density shaped electron beams. 
     2. Description of the Related Art 
     The use of electron beams to lithographically pattern semiconductor masks, reticles and wafers is an established technique. The different lithography strategies may be characterized by the following key parameters: beam positioning strategy; and beam shape control. 
     There are two main approaches to the positioning of electron beams for the exposure of resist during the lithographic process:
         (a) Raster Scanning, where the beam is moved on a regular two-dimensional lattice pattern. This method has the advantage that the scan electronics is typically simpler, but the disadvantage is that the beam may spend large amounts of time moving across areas not needing to be exposed. In addition, in order to accomplish very precise pattern edge placement, sophisticated gray-scale and/or multiple-pass scanning may be required.   (b) Vector Scanning where the beam is moved two-dimensionally directly to areas to be written. This method has the advantage of reduced time over areas not needing to be exposed, but the disadvantage of more complicated and expensive deflection electronics. Precise pattern edge placement is also easier, utilizing the beam placement capability on a 2D address grid much smaller than the beam size.
 
Each approach is advantageous in certain circumstances, the optimum choice depending on the critical dimensions of the pattern, pattern density (% of area to be written), and also on the profile of the beam current distribution.
       

     There are two well-known approaches to the shaping of the electron beam used to expose the resist on the substrate:
         (a) Gaussian beams are characterized by the highest current densities (typically &gt;2000 A/cm 2 ) since in these systems, an image of the electron source is focused onto the substrate surface, thereby taking full advantage of the high brightness of the source. A key disadvantage of Gaussian beams is their long tails of current, stretching far outside the central beam diameter—only 50% of the beam current at the substrate falls within the FWHM of a two-dimensional Gaussian distribution.   (b) Shaped Beams are formed by electron optical columns typically having several intermediate shaping apertures, combined with additional deflectors and lenses to form a focused image of the aperture(s) on the substrate surface. These systems typically have beam current densities orders-of-magnitude lower (e.g. 20-50 A/cm 2 ) than for the Gaussian beams. An advantage of these systems is the reduced current tails outside the desired beam shape, making patterning less susceptible to process fluctuations. Another advantage is that effectively a large number of pixels may be written simultaneously since the area of the variable shaped beam may be large in comparison to a single pixel—this has the effect of increasing the writing throughput since fewer “flashes” of the electron beam are required to write a pattern.       

     There is a need in the semiconductor industry to achieve the highest patterning throughputs, both for mask and reticle writing as well as potentially for the direct writing of wafers. Either of the two approaches to beam positioning can be combined with either of the two approaches to beam shaping, but none of these four combinations is capable of fully meeting the semiconductor industry&#39;s needs. Clearly there is a need for an electron lithography system having high throughput (at least several wafers/hour or less than an hour to write a reticle), combined with the ability to pattern very small CDs with edge placement accuracies &lt;CD/8, as well as the simplest possible electron optical design to ensure adequate system reliability, long mean-time-between-failures (MTBF) and short mean-time-to-repair (MTTR). 
     A third possible contribution to increasing throughput is to use multiple beams in parallel to lithographically pattern a single wafer. The challenges associated with using multiple beams include: scaling electron beam columns to fit multiple columns over a single wafer; stitching together the areas patterned by different columns; and the complexity and hardware costs associated with multiple columns. 
     In order to achieve high throughput, there is clearly a need to have a writing system with two or three of the following characteristics: 
     1) multiple beams writing in parallel on the same substrate; 
     2) a high beam current density in a shaped beam; 
     3) an efficient writing strategy such as vector scanning. 
     There is a need for a lithography system which makes best use of the above three characteristics. 
     SUMMARY OF THE INVENTION 
     The present invention provides an optical column for charged-particle direct-writing which generates a high current density charged particle beam, coupled with the ability to dynamically shape the beam into non-circular profiles at the substrate being written on. According to aspects of the invention, a first embodiment of the charged particle shaped beam column includes: a charged particle source; a gun lens configured to provide a charged particle beam approximately parallel to the optic axis of the column; an objective lens configured to form the charged particle shaped beam on the surface of a substrate, wherein the disk of least confusion of the objective lens does not coincide with the surface of the substrate; an optical element with 8N poles disposed radially symmetrically about the optic axis of the column, the optical element being positioned between the condenser lens and the objective lens, wherein N is an integer greater than or equal to 1; and a power supply configured to apply excitations to the 8N poles of the optical element to provide an octupole electromagnetic field. The octupole electromagnetic field induces azimuthally-varying third-order deflections to the beam trajectories passing through the 8N-pole optical element. These beam deflections, when combined with spherical aberration in the optical system and defocus in the objective lens, induce an azimuthally-varying effective spherical aberration which causes the beam profile to deviate from circularity. By controlling the excitation of the 8N poles, it is possible to generate a square beam at the substrate, or a partially-square beam with rounded corners. The 8N-pole element can be a magnetic 8N-pole element, where the excitation is a current, or an electrostatic 8N-pole element, where the excitation is a voltage. The charged particle beam may be an electron or ion beam. 
     The 8N pole optical element allows for a fully rotatable octupole field for N&gt;2. The larger the value of N, the more control there is over the quadrupole and octupole fields generated. However, large values of N result in greater complexity and cost. The invention is not limited to generating square beams at the surface of the substrate. Other shapes, such as rectangles may also be generated using the structure and method of the present invention. For example, with the addition of non-octupole excitations, rectangular or parallelogram-shaped beams are possible. 
     Further aspects of the first embodiment of the invention include a method of forming a charged particle shaped beam in a charged particle optical column. The method includes the steps of: forming a charged particle beam approximately parallel to the optic axis of the charged particle column; creating an octupole electromagnetic field to induce azimuthally dependent deflection of the charged particle beam, wherein the azimuthal angle is about the optic axis of the charged particle column, in a plane perpendicular to the optic axis; and forming a charged particle shaped beam on a substrate. 
     A second embodiment of the present invention enables more complete control of the beam profile at the substrate. According to aspects of the invention, a second embodiment of a charged particle shaped beam column includes: a charged particle source; a gun lens configured to provide a charged particle beam approximately parallel to the optic axis of the column; an objective lens configured to form the charged particle shaped beam on the surface of a substrate; and four non-circular symmetry optical elements, each comprising 8N poles, where N is greater than or equal to 1, and N may be different for each optical element. The first 8N-pole element is excited to generate a quadrupole electromagnetic field which induces a defocusing action on the beam in a first plane (see  FIG. 12A ), and a focusing action on the beam in a second plane perpendicular to the first plane (see  FIG. 12B ). Due to this defocusing/focusing action of the first 8N-pole element, the beam profile is a first line at the second 8N-pole element. The second 8N-pole element is excited to generate a combined quadrupole and octupole electromagnetic field which induces a focusing action on the beam in the first plane and no focusing action on the beam in the second plane. Combined with the focusing action at the second 8N-pole element, the octupole excitation applied to the second 8N-pole element induces a third-order beam deflection along a first axis (the first axis being contained within the first plane). 
     Due to this focusing action of the second 8N-pole element, the beam profile is a second line at the third 8N-pole element, where the second line is oriented 90° azimuthally with respect to the first line at the second 8N-pole element. The third 8N-pole element is excited to generate a combined quadrupole and octupole electromagnetic field which induces no focusing action on the beam in the first plane and a focusing action on the beam in the second plane. Combined with the focusing action at the third 8N-pole element, the octupole excitation induces a third-order beam deflection along a second axis (the second axis being contained within the second plane). 
     Due to this focusing action of the third 8N-pole element, the beam profile is circular at the fourth 8N-pole element. The fourth 8N-pole element is excited to generate a combined quadrupole and octupole electromagnetic field which induces a focusing action on the beam in the first plane and a defocusing action on the beam in the second plane. Combined with the focusing action at the fourth 8N-pole element, the octupole excitation induces an azimuthally-varying third-order beam deflection. 
     The combination of the third-order beam deflections at the second, third and fourth 8N-pole elements combines with the spherical aberration (which is azimuthally-symmetric) and defocus (also azimuthally-symmetric) to generate an azimuthally-varying beam deflection at the surface of the substrate to be written on. With proper control of the octupole excitations on the second, third and fourth 8N-poles, it is possible to generate either a square beam or a square beam with rounded corners at the surface of the substrate. 
     The advantage of the second embodiment over the first embodiment is the more complete control of the beam profile, including the beam shape and edge acuity (i.e., the rate of current drop at the edge of the beam, measured in A/cm 2  per nm of distance perpendicular to the beam edge.). The advantage of the first embodiment over the second embodiment is a simpler optical system, requiring the addition of only a single 8N-pole element. 
     Further aspects of the present invention include a high throughput charged particle direct write lithography system including the charged particle shaped beam columns described herein. The system includes: a charged particle optical assembly configured to (1) produce a multiplicity, N, of high current density charged particle non-circular shaped-beams focused on the surface of a substrate and (2) vector scan the charged particle shaped-beams across the surface of the substrate; wherein each of the multiplicity of high current density charged particle shaped-beams has a current density, I a , and an area A which satisfy the equations:
 
I a ≧1000 Ampères per square centimeter;
 
300≧N≧10;
 
A=p 2 ; and
 
120&gt;p&gt;10 nanometers; and
 
wherein said charged particle optical assembly includes N charged particle columns, each of the charged particle columns forming a charged particle beam, each of the charged particle columns including at least one optical element with 8N poles disposed radially symmetrically about the optic axis of the column, N being an integer greater than or equal to 1, each of the optical elements being configured to produce azimuthally dependent deflection of the corresponding charged particle beam, the azimuthal angle being about the optic axis of the corresponding charged particle column, in a plane perpendicular to the optic axis.
 
     In further aspects of the invention the parameter space for the high throughput charged particle direct write lithography system may be varied. For example, I a ≧5000 Amperes per square centimeter; 100≧N ≧10; and 120&gt;p&gt;20 nanometers, where A=p 2 . 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
         FIG. 1A  shows a schematic side view of a column employing two lenses. 
         FIG. 1B  shows an isometric view of the two-lens column in  FIG. 1A . 
         FIG. 2  shows a side view of electron trajectories converging to a focused spot on a substrate surface in a two-lens column. 
         FIG. 3  shows a close-up side view of electron trajectories converging to a focused circle of roughly 72 nm diameter on a substrate surface in a two-lens column. 
         FIG. 4  shows a graph of the radii of the electron trajectories at the substrate surface against the radii of the trajectories at the objective lens in a two-lens column. 
         FIG. 5  shows a graph of beams transmitted to the substrate surface in a two-lens column. 
         FIG. 6  shows a schematic side view of a first embodiment of the present invention, showing a single combined quadrupole/octupole element  203  added to the two-lens column of  FIGS. 1A-B . 
         FIG. 7  shows a schematic view of an electrostatic 16-pole optical element as used in first and second embodiments of the present invention. 
         FIG. 8  shows a schematic view of an electrostatic 8-pole optical element as used in first and second embodiments of the present invention. 
         FIG. 9  shows the beam profile and force vectors induced by the quadrupole in a first embodiment of the present invention. 
         FIG. 10  shows a graph of the radii of the electron trajectories at the substrate surface against the radii of the trajectories at the objective lens in a first embodiment of the present invention. 
         FIG. 11  shows a graph of beams transmitted to the substrate surface in a first embodiment of the present invention. 
         FIG. 12A  shows a schematic side view in the +X+Y plane of a second embodiment of the present invention. 
         FIG. 12B  shows a schematic side view in the −X+Y plane of a second embodiment of the present invention. 
         FIG. 13A  shows a schematic isometric view of a second embodiment of the present invention, viewed in a direction approximately perpendicular to the +X+Y plane in  FIG. 12A . 
         FIG. 13B  shows a schematic isometric view of a second embodiment of the present invention in a direction 90° away from the viewing direction for  FIG. 12A  (the viewing direction is approximately perpendicular to the −X+Y plane in  FIG. 12B ). 
         FIG. 14  shows the beam profile and force vectors at Quadrupole #1 in a second embodiment of the present invention. 
         FIG. 15  shows the beam profile and force vectors at Quadrupole/Octupole #2 in a second embodiment of the present invention. 
         FIG. 16  shows the beam profile and force vectors at Quadrupole/Octupole #3 in a second embodiment of the present invention. 
         FIG. 17  shows the beam profile and force vectors at Quadrupole/Octupole #4 in a second embodiment of the present invention. 
         FIG. 18  shows a graph of the radii of the electron trajectories at the substrate surface against the radii of the trajectories at the objective lens in a second embodiment of the present invention with only Octupole #4 activated. 
         FIG. 19  shows a graph of beams transmitted to the substrate surface in a second embodiment of the present invention with only Octupole #4 activated. 
         FIG. 20  shows a graph of the radii of the electron trajectories at the substrate surface against the radii of the trajectories at the objective lens in a second embodiment of the present invention with Octupoles #2-4 activated. 
         FIG. 21  shows a graph of beams transmitted to the substrate surface in a second embodiment of the present invention with Octupoles #2-4 activated. 
         FIG. 22  shows a Venn diagram illustrating the interactions of the three contributions to the system throughput: 1) multiple beam column, 2) the high current density shaped beams, and 3) vector scanning. 
         FIG. 23  shows a schematic circuit diagram of drive electronics for the 16-pole element in  FIG. 7  for the first embodiment. 
         FIG. 24  shows a schematic circuit diagram of drive electronics for the 8-pole element in  FIG. 8  for the first embodiment. 
         FIG. 25  shows a schematic circuit diagram of drive electronics for the 16-pole element in  FIG. 7  for the second embodiment. 
         FIG. 26  shows a schematic circuit diagram of drive electronics for the 8-pole element in  FIG. 8  for the second embodiment. 
         FIG. 27  shows a schematic view of a magnetic 16-pole optical element as used in first and second embodiments of the present invention. 
         FIG. 28  shows a schematic view of a magnetic 8-pole optical element as used in first and second embodiments of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     The invention disclosed herein is a charged particle beam column comprising one or more quadrupole/octupole elements which deflect the charged particle beam going down the column. The beam deflections due to the quadrupole/octupole element(s) effectively create azimuthally-varying radial deflections to the beam trajectories which, when combined with spherical aberration and defocus in the objective lens, result in forming a high current-density shaped (i.e., non-circular) beam at the substrate surface. 
     The charged particle beam column of the invention can be either an electron beam or an ion beam column. The quadrupole/octupole optical elements can be electrostatic or magnetic elements. Many of the examples of the invention provided herein are examples of electron beam columns, with electrostatic quadrupole/octupole optical elements. However, the invention is equally applicable to ion beam columns and columns with magnetic quadrupole/octupole optical elements. 
     Two embodiments of the present invention are described in detail herein:
         1) Embodiment #1 which comprises a single additional quadrupole/octupole element (implemented using an 8N-pole optical element with combined quadrupole and octupole excitations), and   2) Embodiment #2 which comprises a quadrupole element followed by three quadrupole/octupole elements (wherein all four elements may be implemented using 8N-pole optical elements with combined quadrupole and octupole excitations).
 
The first embodiment is described in  FIGS. 6-11  and the second embodiment in  FIGS. 12A-21 . The relative advantages and disadvantages of the two embodiments are discussed in detail.
       

     Before describing the present invention, it is useful to first characterize the operation of a simple two-lens optical column in the absence of the present invention, as shown in  FIGS. 1A-5 . The present invention may be implemented in a one-lens column, but general industry practice (familiar to those skilled in the art) is to use at least two lenses in a charged particle optical column: 1) a gun (or “condenser”) lens in the electron gun to collect electrons emitted from the source (typically emitted into an expanding cone-shaped distribution) and focus their trajectories into a roughly parallel beam, which may converge to a crossover before the beam reaches the objective lens, and 2) an objective lens which focuses the electron beam generated by the gun onto a target surface. Such a two-lens column is shown in  FIGS. 1A-B , for the case where there is no intermediate beam crossover between the gun lens and objective lens. The first embodiment of the present invention is applicable to columns having no intermediate crossover, as well as to columns having a single intermediate crossover.  FIGS. 3-4  characterize the optical performance of the two-lens column shown in  FIGS. 1A-B . The particular settings of the gun and objective lenses which generate the trajectories in  FIGS. 3 and 5 , and the graph in  FIG. 4 , have been selected for their applicability to the first embodiment of the present invention. 
       FIG. 1A  shows a schematic side view of a column employing two lenses. Electrons  121  are emitted from electron source  125  in object plane  101 , which can be a thermionic source, a LaB 6  emitter, a cold field emitter, a Schottky emitter, or other type of electron source as is familiar to those skilled in the art. Gun lens  102  (with focal length  111 ) focuses electrons  121  into an approximately parallel electron beam  122  (with radius  114 ) which passes down the column a distance  112  before reaching the objective lens  103  (with radius  115 ). Objective lens  103  (with focal length  113 ) focuses electrons  122  into a converging beam  123  which intersects with the surface of substrate  104  at point  126 . Both lenses  102  and  103  are centered on the optical axis  127 . In  FIG. 1A , substrate  104  is at the paraxial focal plane of objective lens  103 . 
       FIG. 1B  shows an isometric view of the two-lens column in  FIG. 1A . The arrow  120  shows the direction of electron trajectories down the two-lens column. 
       FIG. 2  shows a side view of electron trajectories converging to a focused spot on a substrate surface in a two-lens column. At the left of the graph (position 0.0 along horizontal axis  131 ), the beam diameter is 300000 nm (i.e., 150 μm radius) on vertical axis  132 . At the resolution of this graph, electron trajectories  133  are seen to converge towards region  134  which is shown in greater detail in  FIG. 3 . The focal length of lens  103  (see  FIGS. 1A-B ) is 10.0060 mm. 
       FIG. 3  shows a close-up side view of electron trajectories converging to a focused spot on a substrate surface in a two-lens column at region  134  in  FIG. 2 . The substrate is shown as a dashed line  143  at 9.9997 mm from lens  103  (see  FIGS. 1A-B ), having a focal length of 10.0060 mm, thus the substrate  143  is (10.0060−9.9997) mm=6.3 μm above the paraxial focal plane. The left vertical axis  142  is at 9.9970 mm from lens  103 , which is (9.9997−9.9970) mm=2.7 μm above the substrate surface  143 , and the top horizontal axis  141  shows the position along the optical axis  127  from lens  103 . The central (on-axis) ray  144  passes from source  125 , through lenses  102  and  103 , and strikes the substrate  104  all on optical axis  127 . Ray  145  is the farthest off-axis ray at substrate  104 , but does not correspond to the farthest off-axis ray at lens  103  due to the combined effects of defocus and spherical aberration. The outer ray at lens  103  strikes substrate  104  at a radius of  146 , again due to the combined effects of defocus and spherical aberration. 
       FIG. 4  clarifies the effects of defocus combined with spherical aberration in a two-lens column. At the paraxial focal plane (10.0060 mm from lens  103 ), the beam displacement off-axis is due solely to spherical aberration:
 
δ X=−C   s   x ( x   2   +y   2 )= x -axis beam displacement at paraxial focal plane
 
δ Y=−C   s   y ( x   2   +y   2 )= y -axis beam displacement at paraxial focal plane
 
where x and y are the beam coordinates at lens  103 . Note that for electron lenses, C s  is always positive in the above formula, so δX and δY are always negative, thus spherical aberration causes the electron trajectories to cross optical axis  127  before reaching the paraxial image plane. Now if we move the substrate  104  above the paraxial image plane, we must add defocus terms to the equations for δX and δY:
 
δ X =(Δ f/f ) x−C   s   x ( x   2   +y   2 )
 
δ Y =(Δ f/f ) y−C   s   y ( x   2   +y   2 )
 
where f=the focal length  113  of lens  103 , and Δf=the amount of defocus (i.e., the distance above the paraxial focal plane where the substrate is positioned). Clearly, for small x and y, the linear terms dominate δX and δY, but as x and/or y is increased (corresponding to rays which are not paraxial at lens  103 ), eventually the cubic spherical aberration terms come to dominate δX and/or δY.
 
       FIG. 4  shows a graph  153  of the radii  152  of the electron trajectories at the substrate  104  surface against the radii  151  of the trajectories at objective lens  103  in a two-lens column. The central ray strikes substrate  104  a position  154  (0 nm off-axis)—this corresponds to point  144  in  FIG. 3 . As the radius  151  at objective lens  103  is increased, defocus initially makes the radii at the substrate  104  increase from 0 mm at point  154  to point  155 —this is region  156 . For radii at objective lens  103  larger than point  155  (in region  159 ), spherical aberration starts to dominate and the radii  152  at the substrate  104  start to decrease, crossing the 0 nm axis at point  157  and ending up at point  158  which is on the opposite side of axis  127  (see  FIGS. 1A-B ) from point  155 . This is a common phenomenon familiar to those skilled in the art. The curve  153  is the same for any azimuthal (i.e., angle around the axis  127 ) initial position of the trajectory at lens  103  since the beam is circular (see  FIG. 5 ). Note that axis  152  includes both positive and negative numbers for the radius at the substrate  104 —in this case, a negative radius corresponds to a positive radius of the same magnitude, but rotated azimuthally by 180° around the optical axis  127 . 
       FIG. 5  shows a graph of the trajectories  163  along the X-axis  161  and Y-axis  162  at the substrate  104 . Since the beam-defining aperture (not shown) is round, the beam at the substrate  104  is also round. The distribution of current within the round beam is determined by the interaction of defocus and spherical aberration as illustrated in  FIG. 4 . Generally there is a concentration of current around the origin of the X-Y coordinate system at the substrate  104  as shown by the dark area at the center of  FIG. 5 . 
     First Embodiment 
       FIG. 6  shows a schematic side view of a first embodiment of the present invention. Electrons  218  are emitted from electron source  215  in object plane  201 , which can be a thermionic source, a LaB 6  emitter, a cold field emitter, a Schottky emitter, or other type of electron source as is familiar to those skilled in the art. The particular type of electron source is not part of the present invention. Gun lens  202  (with focal length  211 ) focuses electrons  218  into an approximately parallel electron beam  219  which passes down the column a distance  212  before reaching octupole  203 . Octupole  203  may be implemented in the column using an element with 8N poles, where N=1 (an octupole), 2 (a 16-pole), . . . as is familiar to those skilled in the art.  FIG. 7  shows a view of a 16-pole element (N=2). The excitation of octupole  203  is discussed in  FIG. 7 . Trajectories leaving octupole  203  pass a distance  213  down the column, reaching objective lens  204  (with focal length  214 ) which focuses electrons  220  into a converging beam  221  which intersects the surface of substrate  205  at location  216 . Both lenses  202  and  204  are centered on the optical axis  217 . 
     Implementations of Octupole Elements 
     There are a number of ways to physically implement an octupole element in an electron column. Two of these methods are illustrated in  FIGS. 7 and 8 . A pure octupole element (i.e., an element not also having dipole, quadrupole, hexapole, or other non-octupole excitations) is characterized by an electrostatic potential, V(x,y), with four-fold symmetry:
 
 V ( x,y )= A ( x   4 −6 x   2   y   2   +y   4 )+ B 4( x   3   y−xy   3 )
 
where A and B are constants, and x and y are the beam coordinates at the octupole element. Since the deflection of the electron trajectories passing through the octupole is proportional to the electric field, E(x,y)=−∇V(x,y), the beam deflections at the substrate, δX and δY, are:
 
δ X=K∂V ( x,y )/∂ x=KA (4 x   3 −12 xy   2 )+ KB (12 x   2   y −4 y   3 )
 
 δY=KV ( x,y )/ ∂y=KA (−12 x   2   y+ 4 y   3 )+ KB (4 x   3 −12 xy   2 )
 
Where K is a constant that depends on the beam energy passing through the octupole, the length and bore of the octupole poles, and the focal length of the objective lens. The constant A corresponds to an octupole oriented along the X- and Y-axes, while the constant B corresponds to an octupole oriented 22.5° relative to the X- and Y-axes. In the following discussion, B=0 for simplicity. For complete generality (i.e., arbitrary orientations of the shaped beam), both A and B would be non-zero.
 
       FIG. 7  shows a schematic view of an electrostatic 16-pole optical element that can be used for octupole  203  (see  FIG. 6 ) in a first embodiment of the present invention, and for elements  1203 - 1206  (see  FIGS. 12A-13B ) in a second embodiment of the present invention. The sixteen poles  233 - 248  are oriented relative to the X-axis  231  and Y-axis  232  as shown. Table I shows octupole excitation voltage polarities for poles  233 - 248  for this orientation—note that the voltage magnitudes are all the same, only the polarities differ between poles  233 - 248 . For the first embodiment, octupole  203  has no non-octupole excitations, thus the voltages on poles  233 - 248  will reflect the octupole voltages in Table I only. For the second embodiment, the octupole excitations are combined with quadrupole excitations, thus the voltages on poles  233 - 248  will be combinations of the octupole voltages shown in Table I with quadrupole voltages in Table V. 
       FIG. 8  shows a schematic view of an electrostatic 8-pole (octupole) optical element that can be used as an alternative to the 16-pole element described in  FIG. 7 . The eight poles  253 - 260  are oriented relative to the X-axis  251  and Y-axis  252  as shown. Table II shows octupole excitation voltage polarities for poles  253 - 260  for this orientation—note that the voltage magnitudes are all the same, only the polarities differ between poles  253 - 260 . For the first embodiment, octupole  203  has no non-octupole excitations, thus the voltages on poles  253 - 260  will reflect the octupole voltages in Table II only. For the second embodiment, the octupole excitations are combined with quadrupole excitations, thus the voltages on poles  253 - 260  will be combinations of the octupole voltages shown in Table II with quadrupole voltages in Table VI. 
     Table III shows a comparison of the relative advantages and disadvantages of the two octupole implementations shown in  FIGS. 7 and 8 . The key determinant between the two implementations would be whether all orientations of the beam shape are required for patterning the substrate. In general, usually only orientations along 0° and 45° are needed, so the simpler 8-pole implementation in  FIG. 8  would be preferred. If, however, all orientations are required, then it is necessary to use the more complex 16-pole implementation in  FIG. 7 . 
       FIG. 9  shows the beam profile and force vectors induced by quadrupole  203  in a first embodiment of the present invention, corresponding to the case where KA&lt;0 and B=0 in the formulas for δX and δY above. The beam profile is shown as a group of concentric circles  303 - 308  centered on the optical axis (X=Y=0). The X-axis  301  and the Y-axis  302  are shown in units of mm, with a maximum beam radius of 150 μm. The polarities of arrows  321 - 328  correspond to the 0° columns in Tables I and II. For a 45° orientation of the shaped beam, the directions of arrows  321 - 328  would be reversed as shown in the 45° columns in Tables I and II. 
     
       
         
               
             
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE I 
               
             
             
               
                   
               
               
                 Octupole excitation strengths and polarities for generating square beams 
               
               
                 in four orientations relative to the X- and Y-axes: 0°, 22.5°, 45°, and 
               
               
                 67.5°, with the 16-pole octupole implementation in FIG. 7 or FIG. 27. At 
               
               
                 these four angles, the excitation strengths on the sixteen poles 233-248 are 
               
               
                 the same. Orientations at other angles between 0° and 90° are possible 
               
               
                 if the excitation strengths are not the same, as is familiar to those skilled in 
               
               
                 the art. Angles ≧90° are equivalent to angles between 0° and 
               
               
                 90° since the excitation has four-fold symmetry. 
               
             
          
           
               
                   
                 Orientation of Square Beam 
                   
               
             
          
           
               
                 Pole # 
                 0 deg 
                 22.5 deg 
                 45 deg 
                 67.5 deg 
               
               
                   
               
               
                 233 
                 − 
                 − 
                 + 
                 + 
               
               
                 234 
                 + 
                 − 
                 − 
                 + 
               
               
                 235 
                 + 
                 + 
                 − 
                 − 
               
               
                 236 
                 − 
                 + 
                 + 
                 − 
               
               
                 237 
                 − 
                 − 
                 + 
                 + 
               
               
                 238 
                 + 
                 − 
                 − 
                 + 
               
               
                 239 
                 + 
                 + 
                 − 
                 − 
               
               
                 240 
                 − 
                 + 
                 + 
                 − 
               
               
                 241 
                 − 
                 − 
                 + 
                 + 
               
               
                 242 
                 + 
                 − 
                 − 
                 + 
               
               
                 243 
                 + 
                 + 
                 − 
                 − 
               
               
                 244 
                 − 
                 + 
                 + 
                 − 
               
               
                 245 
                 − 
                 − 
                 + 
                 + 
               
               
                 246 
                 + 
                 − 
                 − 
                 + 
               
               
                 247 
                 + 
                 + 
                 − 
                 − 
               
               
                 248 
                 − 
                 + 
                 + 
                 − 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE II 
               
             
             
               
                   
               
               
                 Octupole excitation strengths and polarities for generating a 
               
               
                 square beam in two orientations relative to the X- and Y-axes: 0° and 45° 
               
               
                 with the 8-pole implementation in FIG. 8 or FIG. 28. At these two angles, 
               
               
                 the excitation strengths on the eight poles 253-260 are the same. 
               
               
                 Orientations at other angles are not possible with an 8-pole configuration. 
               
               
                 Angles ≧90° are equivalent to angles between 0° and 
               
               
                 90° since the excitation has four-fold symmetry. 
               
             
          
           
               
                   
                 Orientation 
                   
               
               
                   
                 of Square Beam 
               
             
          
           
               
                 Pole # 
                 0 deg 
                 45 deg 
               
               
                   
               
               
                 253 
                 − 
                 + 
               
               
                 254 
                 + 
                 − 
               
               
                 255 
                 − 
                 + 
               
               
                 256 
                 + 
                 − 
               
               
                 257 
                 − 
                 + 
               
               
                 258 
                 + 
                 − 
               
               
                 259 
                 − 
                 + 
               
               
                 260 
                 + 
                 − 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                 TABLE III 
               
             
             
               
                   
               
               
                 A comparison of the relative advantages and disadvantages of 
               
               
                 the two alternative implementations of an octupole element as shown in 
               
               
                 FIGS. 7 or 27 compared with FIGS. 8 or 28. 
               
             
          
           
               
                 Octupole 
                   
                   
               
               
                 Implementation 
                 Advantages 
                 Diasadvantages 
               
               
                   
               
               
                 16-pole 
                 capability for beam shape 
                 increased complexity, more 
               
               
                 (FIG. 7 &amp; 27) 
                 orientation at any angle 
                 wires, more electronics 
               
               
                 8-pole 
                 simpler - fewer wires and 
                 beam can only be oriented 
               
               
                 (FIG. 8 &amp; 28) 
                 less electronics 
                 at 0 and 45 deg 
               
               
                   
               
             
          
         
       
     
       FIGS. 7  or  27  compared with  FIGS. 8  or  28 . 
       FIG. 10  shows a graph of the radii  342  of the electron trajectories at the substrate surface  205  against the radii  341  of the trajectories at the objective lens  204  in a first embodiment of the present invention. Comparison of  FIG. 10  with  FIG. 4  for a two-lens circular beam column shows that now there are two curves,  343  and  344 , instead of only one (e.g., curve  153  in FIG.  4 )—this is because the beam is shaped into a square by azimuthal control of the total spherical aberration, as described below. In this example, the goal is to generate a square-shaped beam with 61 nm sides, where the sides of the square-shaped beam are aligned with the X-axis  361  or Y-axis  362  in  FIG. 11 . Thus the distance from the center of the beam to the side is 30.5 nm (short-dashed line  350 ) and the distance to the corners is √2 (30.5 nm)≈43.1 nm (long-dashed line  351 ). By use of octupole element  203 , the total deflection of the beam now has the combined effects of three terms: a) defocus, b) spherical aberration in lens  204  (equivalent to lens  103  in  FIGS. 1A-B ), and c) the deflection due to octupole  203  (assuming B=0):
 
δ X =(Δ f/f ) x−C   s   x ( x   2   +y   2 )+ KA (4 x   3 −12 xy   2 )
 
δ Y =(Δ f/f ) y−C   s   y ( x   2   +y   2 )+ KA (−12 x   2   y+ 4 y   3 )
 
The terms in these equations can be rearranged:
 
δ X =(Δ f/f ) x +(4 KA−C   s ) x   3 −(12 KA+C   s ) xy   2  
 
δ Y =(Δ f/f ) y −(12 KA+C   s ) x   2   y +(4 KA−C   s ) y   3  
 
If K A &lt;0, since C s &gt;0, then the on-axis terms (i.e., terms with x 3  and y 3 ) are increased, while the off-axis terms (i.e., terms with x y 2  and x 2  y) are decreased. Curve  352  is equivalent to curve  153  in  FIG. 4 , corresponding to azimuthally-uniform spherical aberration. Note that the end point  353  of curve  352  is midway between the endpoint  349  of curve  344  and the endpoint  347  of curve  343 —this shows the effects of adding the octupole beam deflection due to element  203 . By proper choice of defocus Δf, combined with the value of A, it is possible to bring the tangent point  348  of curve  344  to match the required radius  351  of the shaped beam corner at (43.1 nm). At the same time, the tangent point  346  of curve  343  is matching the required radius  350  of the shaped beam sides (at 30.5 nm). All three curves  343 ,  344 , and  352  start at point  345  on axis  342  (i.e., at 0 nm radius at the substrate and at 0.00 mm radius at the objective lens  204 ).
 
       FIG. 11  shows a graph of the trajectories  363  along the X-axis  361  and Y-axis  362  at the substrate  205  (see  FIG. 6 ) with the use of the first embodiment of the present invention to shape the beam into a square, instead of the round beam  163  shown in  FIG. 5 , which would result if the octupole element  203  were inactivated (i.e., if all the poles  233 - 248  in  FIG. 7 , or all of the poles  253 - 260  in  FIG. 8  were set to the same voltage). The sharpness of corners  365  can be controlled by adjusting the strength of octupole element  203  (i.e., adjusting the value of constant A). Curves  343  and  344  in  FIG. 10  show that there is substantial overlap of the trajectories  363  in FIG.  11 —this overlap can be seen from the fact that both curves  343  and  344  show two different radii at the objective lens  204  (axis  342 ) for the same radius at substrate  205  (axis  341 ) in many cases. This overlap corresponds to a “folding over” of the beam on itself, thus making the beam smaller for a given number of trajectories reaching the substrate  205 . Since the number of trajectories is proportional to the total beam current, this means that the beam current density at the substrate  205  is increased compared with the case of first-order imaging (the conventional method of beam-shaping) in which there is no folding over of the trajectories at the substrate. 
     If A is set=0, and B≠0, then a square rotated 45° to that shown in  FIG. 11  would result. Note that because endpoint  347  has a larger magnitude of radius than line  350 , a small number of trajectories  364  strike the substrate  205  outside the desired 61 nm square shape. This has only a minor effect since only a very small number of trajectories are in this group. The second embodiment of the present invention reduces or eliminates this effect. 
     Second Embodiment 
       FIGS. 12A-21  illustrate a second embodiment of the present invention. Table IV shows a comparison of the relative advantages and disadvantages of the first and second embodiments of the present invention. The second embodiment utilizes four elements  1203 - 1206 , as shown in  FIGS. 12A-B  between the gun lens  1202  and the objective lens  1207 . Elements  1203 - 1206  may be implemented using either 16-poles as in  FIG. 7 , or 8-poles as in  FIG. 8 . 
       FIG. 12A  shows a schematic side view of a second embodiment of the present invention in a plane containing two lines: a) a line midway between the +X-axis and +Y-axis, and b) the Z-axis=the optical axis−hereinafter this plane will be referred to as the (+ X+Y )− Z  plane.  FIG. 12B  shows a schematic side view of a second embodiment of the present invention in a plane containing two lines: a) a line between the −X-axis and +Y-axis, and b) the Z-axis=the optical axis−hereinafter this plane will be referred to as the (−X+Y)− Z  plane. Note that this plane is perpendicular to the (+ X+Y )− Z  plane of  FIG. 12A .  FIG. 13A  shows a schematic isometric view of a second embodiment of the present invention, viewed in a direction approximately perpendicular to the +X+Y plane in  FIG. 12A . 
       FIG. 13B  shows a schematic isometric view of a second embodiment of the present invention in a direction 90° away from the viewing direction for  FIG. 13A  (the viewing direction is approximately perpendicular to the −X+Y plane in  FIG. 12B ). The following discussion refers to all of  FIGS. 12A-13B . Electrons  1221  and  1231  are emitted from electron source  1241  in object plane  1201 , which can be a thermionic source, a LaB 6  emitter, a cold field emitter, a Schottky emitter, or other type of electron source as is familiar to those skilled in the art. The particular type of electron source is not part of the present invention. Gun lens  1202  (with focal length  1211 ) focuses electrons  1221  and  1231  into approximately parallel electron beams  1222  and  1232 , respectively, of diameter  1312  which pass down the column a distance  1212  to reach quadrupole #1  1203  at a diameter  1313 . In the (+X+Y)−Z plane ( FIG. 12A ), quadrupole #1  1203  is a diverging lens, while in the (−X+Y)−Z plane ( FIG. 12B ), quadrupole #1  1203  is a converging lens. The focal length of quadrupole #1  1203  is set equal to the distance  1213  between optical elements  1203  and  1204 . Thus, in the (−X+Y)−Z plane ( FIG. 12B ), beam  1233  is brought to a focus at the center of quadrupole/octupole #2  1204 . In the (+X+Y)−Z plane ( FIG. 12A ), beam  1223  is twice as far off-axis at quadrupole/octupole #2  1204  as at quadrupole #1  1203 . The effect of quadrupole #1  1203  on beams  1222  and  1232  is shown in  FIG. 14 . 
     Due to the focusing effects of quadrupole #1  1203 , the beam profile at quadrupole/octupole #2  1204  is a line  1314  (seen most clearly in  FIG. 13A ) that is twice as long as beam diameter  1313 . The effect of quadrupole/octupole #2  1204  on beams  1223  and  1233  is shown in  FIG. 15 . Because in the (−X+Y)−Z plane the beam is on-axis, there is no focusing effect due to quadrupole/octupole #2  1204 . In the (+X+Y)−Z plane, the beam  1223  is strongly focused towards optical axis  1240 , generating converging beam  1224 . In the (−X+Y)−Z plane, the beam  1234  diverges away from optical axis  1240 . 
     In the example shown here, the relationships between the spacings of elements  1203 - 1206  are as follows:
 
Spacing 1214=2 (spacing 1213)=2 (spacing 1215)
 
Spacing  1211  is the focal length of gun lens  1202 , while spacing  1217  is approximately the focal length of objective lens  1207 . As long as the beam is assumed parallel after lens  1202 , spacing  1212  is unimportant. As long as the beam is parallel after quadrupole/octupole #4  1206 , spacing  1216  is also unimportant. Midway between quadrupole/octupole #2  1204  and quadrupole/octupole #3  1205 , the beam is circular with a diameter  1318 .
 
     Due to the focusing effects of quadrupole/octupole #2  1204 , the beam profile at quadrupole/octupole #3  1205  is a line  1315  (seen most clearly in  FIG. 13B ) that is equal in length to line  1314 , but rotated 90° azimuthally. The effect of quadrupole/octupole #3  1205  on beams  1224  and  1234  is shown in  FIG. 16 . Because in the (+X+Y)−Z plane ( FIG. 12A ) the beam is on-axis, there is no focusing effect due to quadrupole/octupole #3  1205  and the beam  1225  diverges away from optical axis  1240 . In the (−X+Y)−Z plane ( FIG. 12B ), the beam  1234  is strongly focused towards optical axis  1240 , generating converging beam  1235 . 
     Due to the focusing effects of quadrupole/octupole #3  1205 , the beam profile at quadrupole/octupole #4  1206  is a circle  1316 . The effect of quadrupole/octupole #4  1206  on beams  1225  and  1235  is shown in  FIG. 17 . Because in both the (+X+Y)−Z and (−X+Y)−Z planes the beam is off-axis, there is a lens effect for all positions on the beam diameter  1316 . In the (+X+Y)−Z plane ( FIG. 12A ), beam  1225  is focused towards optical axis  1240 , generating parallel beam  1226 . In the (−X+Y)−Z plane ( FIG. 12B ), the beam  1235  is focused away from optical axis  1240 , generating parallel beam  1236 . The parallel beams  1226  and  1236  reach objective lens  1207  on circle  1317 , where all electrons are focused towards the substrate  1208  at point  1242 . 
     In the preceding discussion, only the first-order focusing effects of elements  1203 - 1206  have been discussed—these are the optical effects of the quadrupole excitations of elements  1203 - 1206 . In order to shape the beam, however, it is necessary to add octupole excitations to elements  1204 - 1206 , as will be described in  FIGS. 15-17 . 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE IV 
               
             
             
               
                   
               
               
                 Comparison of the advantages and disadvantages of the first and 
               
               
                 second embodiments of the present invention. 
               
             
          
           
               
                 Embodiment 
                 Advantages 
                 Diasadvantages 
               
               
                   
               
               
                 First 
                 simpler - no added 
                 some rays outside square, 
               
               
                   
                 components, only addition 
                 less ability to fine-tune 
               
               
                   
                 of octupole excitation to 
                 corner rounding 
               
               
                   
                 existing deflectors (8N- 
               
               
                   
                 poles) 
               
               
                 Second 
                 no rays outside square, 
                 more complex - need to 
               
               
                   
                 ability to fine-turn corner 
                 add some 8N-pole 
               
               
                   
                 rounding 
                 elements to column design, 
               
               
                   
                   
                 added wires and 
               
               
                   
                   
                 electronics 
               
               
                   
               
             
          
         
       
     
       FIG. 14  shows the beam profile and force vectors induced by Quadrupole #1  1203  in a second embodiment of the present invention. The beam profile is shown as a group of concentric circles  1403 - 1408  centered on the optical axis (X=Y=0). A pure quadrupole element (i.e., an element not also having dipole, hexapole, octupole, or other non-quadrupole excitations) is characterized by an electrostatic potential, V(x,y):
 
 V ( x,y )= C ( x   2   −y   2 )+ D 2 xy  
 
where C and D are constants, and x and y are the beam coordinates at the quadrupole element. Since the deflection of the electron trajectories passing through the quadrupole is proportional to the electric field, E(x,y)=−∇V(x,y), the beam deflections at the next element (e.g., at element  1204  due to deflection by element  1203 , etc.), δX o , and δY o , are:
 
δ X   o   =Q∂V ( x,y )/∂ x=QC 2 x+QD 2 y  
 
δ Y   o   =Q∂V ( x,y )∂ y=−QC 2 y+QD 2 x  
 
where Q is a constant that depends on the beam energy passing through the quadrupole and the length and bore of the quadrupole poles. The constant C corresponds to a quadrupole oriented along the X- and Y-axes, while the constant D corresponds to an quadrupole oriented 45° relative to the X- and Y-axes. In the following discussion, C=0, corresponding to the requirement to generate line foci  1314  and  1315  oriented 45° relative to the X- and Y-axes. For complete generality (i.e., arbitrary orientations of the shaped beam), both C and D would be non-zero. Use of quadrupoles to shape beams down an electron beam column is familiar to those skilled in the art.
 
     The beam profile at quadrupole #1  1203  is shown as a group of concentric circles  1403 - 1408  centered on the optical axis (X=Y=0). The X-axis  1401  and the Y-axis  1402  are shown in units of mm, with a maximum beam radius of 150 μm. The four double arrows  1410 - 1413  represent forces on the beam due to the quadrupole excitation as shown in the columns for 45° in Table V (for the 16-pole in  FIG. 7  or  FIG. 27 ) and Table VI (for the 8-pole in  FIG. 8  or  FIG. 28 ). 
       FIG. 15  shows the beam profile and force vectors induced by Quadrupole/Octupole #2  1204  in a second embodiment of the present invention. The X-axis  1501  and the Y-axis  1502  are shown in units of mm, with a maximum beam distance off-axis of 300 μm, or twice the 150 μm radius in  FIG. 14 , as described in  FIGS. 12A-13B , above. The two double arrows  1510  and  1511  show the first-order converging effects of the quadrupole excitation of quadrupole/octupole #2  1204 . The two single arrows  1521  and  1522  show the third-order diverging effects of the octupole excitation of quadrupole/octupole #2  1204 . Note that both the quadrupole and octupole effects act only along the (+X+Y)-axis where the beam has a non-zero radius since in all cases the beam deflection is a function of the beam radius. Quadrupole #1  1203  generates a line beam at quadrupole/octupole #2  1204  so that the octupole excitation of quadrupole/octupole #2  1204  can act on the beam only along the (+X+Y)−direction, thereby adjusting the sharpness of two diagonal corners of the final shaped beam at the substrate  1208 . 
     
       
         
               
             
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE V 
               
             
             
               
                   
               
               
                 Quadrupole excitation strengths and polarities for generating line 
               
               
                 beams in eight orientations relative to the X- and Y-axes: 0° to 157.5° 
               
               
                 in steps of 22.5°, with the 16-pole implementation in FIG. 7 or 27. 
               
               
                 Two people strength magnitudes are shown for these orientation 
               
               
                 angles: 1.000 and 0.414 = (√2 − 1). Orientations at other angles 
               
               
                 between 0° and 180° are possible with other pole strengths as is 
               
               
                 familiar to those skilled in the art. Angles ≧180° are 
               
               
                 equivalent to angles between 0° and 180° since the 
               
               
                 excitation has two-fold symmetry. 
               
             
          
           
               
                   
                 Orientation of Line Focus 
               
             
          
           
               
                   
                 0.0 
                 22.5 
                 45.0 
                 67.5 
                 90.0 
                 112.5 
                 135.0 
                 157.5 
               
               
                 Pole # 
                 deg 
                 deg 
                 deg 
                 deg 
                 deg 
                 deg 
                 deg 
                 deg 
               
               
                   
               
               
                 233 
                 + 
                 + 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
               
               
                 234 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
               
               
                 235 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
                 − 
                 − 
               
               
                 236 
                 − 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
                 − 
               
               
                 237 
                 − 
                 − 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
               
               
                 238 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
               
               
                 239 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
                 + 
                 + 
               
               
                 240 
                 + 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
                 + 
               
               
                 241 
                 + 
                 + 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
               
               
                 242 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
               
               
                 243 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
                 − 
                 − 
               
               
                 244 
                 − 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
                 − 
               
               
                 245 
                 − 
                 − 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
                 −a 
               
               
                 246 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
                 + 
                 + 
                 +a 
               
               
                 247 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
                 + 
                 + 
               
               
                 248 
                 + 
                 +a 
                 −a 
                 − 
                 − 
                 −a 
                 +a 
                 + 
               
               
                   
               
               
                 a = cos(2 * 33.75 deg)/cos (2 * 11.25 deg) = 0.414 
               
             
          
         
       
         
         
           
             Table V. Quadrupole excitation strengths and polarities for generating line beams in eight orientations relative to the X- and Y-axes: 0° to 157.5° in steps of 22.5°, with the 16-pole implementation in  FIG. 7  or  27 . Two pole strength magnitudes are shown for these orientation angles: 1.000 and 0.414=(√2−1). Orientations at other angles between 0° and 180° are possible with other pole strengths as is familiar to those skilled in the art. Angles ≧180°are equivalent to angles between 0° and 180° since the excitation has two-fold symmetry. 
           
         
       
    
     
       
         
               
             
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE VI 
               
             
             
               
                   
               
               
                 Quadrupole excitation strengths and polarities for generating line beams 
               
               
                 in four orientations relative to the X- and Y-axes: 0°, 45°, 90° and 
               
               
                 135° with the 8-pole implementation in FIG. 8 or 28. At these four 
               
               
                 angles, the excitation strengths on the eight poles 253-260 are the same. 
               
               
                 Orientations at other angles are between 0° and 180° are not possible 
               
               
                 with an 8-pole implementation. Angles &gt;180° are equivalent to angles 
               
               
                 between 0° and 180° since the excitation has two-fold symmetry. 
               
             
          
           
               
                   
                 Orientation of Line Focus 
                   
               
             
          
           
               
                 Pole # 
                 0 deg 
                 45 deg 
                 90 deg 
                 135 deg 
               
               
                   
               
               
                 253 
                 + 
                 0 
                 − 
                 0 
               
               
                 254 
                 0 
                 + 
                 0 
                 − 
               
               
                 255 
                 − 
                 0 
                 + 
                 0 
               
               
                 256 
                 0 
                 − 
                 0 
                 + 
               
               
                 257 
                 + 
                 0 
                 − 
                 0 
               
               
                 258 
                 0 
                 + 
                 0 
                 − 
               
               
                 259 
                 − 
                 0 
                 + 
                 0 
               
               
                 260 
                 0 
                 − 
                 0 
                 + 
               
               
                   
               
             
          
         
       
         
         
           
             Table VI. Quadrupole excitation strengths and polarities for generating line beams in four orientations relative to the X- and Y-axes: 0°, 45°, 90° and 135° with the 8-pole implementation in  FIG. 8  or  28 . At these four angles, the excitation strengths on the eight poles  253 - 260  are the same. Orientations at other angles between 0° and 180° are not possible with an 8-pole implementation. Angels &gt;180° are equivalent to angles between 0° and 180° since the excitation has two-fold symmetry. 
           
         
       
    
       FIG. 16  shows the beam profile and force vectors induced by Quadrupole/Octupole #3  1205  in a second embodiment of the present invention. The X-axis  1601  and the Y-axis  1602  are shown in units of mm, with a maximum beam distance off-axis of 300 μm, or equal to the maximum beam distance off-axis in  FIG. 15 , as described in  FIGS. 12A-13B , above. The two double arrows  1610  and  1611  show the first-order converging effects of the quadrupole excitation of quadrupole/octupole #3  1205 . The two single arrows  1621  and  1622  show the third-order diverging effects of the octupole excitation of quadrupole/octupole #3  1205 . Note that both the quadrupole and octupole effects act only along the (−X+Y)-axis where the beam has a non-zero radius since in all cases the beam deflection is a function of the beam radius. Intuitively, quadrupole/octupole #2  1204  generates a line beam at quadrupole/octupole #3  1205  so that the octupole excitation of quadrupole/octupole #3  1205  can act on the beam only along the (−X+Y)-direction, thereby adjusting the sharpness of two diagonal corners of the final shaped beam at the substrate  1208  (the two corners not adjusted by quadrupole/octupole #2  1204 ). 
       FIG. 17  shows the beam profile and force vectors induced by Quadrupole/Octupole #4  1206  in a second embodiment of the present invention. The beam profile is shown as a group of concentric circles  1703 - 1708  centered on the optical axis (X=Y=0). The X-axis  1701  and the Y-axis  1702  are shown in units of mm, with a maximum beam radius of 150 μm, equal to the radius in  FIG. 14 , as described in  FIGS. 12A-13B , above. The four double arrows  1710 - 1713  show the first-order converging effects of the quadrupole excitation of quadrupole/octupole #4  1206 . The eight single arrows  1721 - 1728  show the third-order converging and diverging effects of the octupole excitation of quadrupole/octupole #4  1206 . Note that both the quadrupole and octupole effects act in all directions azimuthally since the beam has non-zero radius for all trajectories. Intuitively, quadrupole/octupole #3  1205  generates a circular beam at quadrupole/octupole #4  1206  so that the octupole excitation of quadrupole/octupole #3  1205  can act on the beam in all directions, in the same way that octupole  203  acts on the beam in the first embodiment. The combined effects of elements  1204 - 1206  is to shape the beam into a square, at the substrate  1208 , but with increased adjustability of corner sharpness compared with the first embodiment due to the additional octupole excitations in elements  1204 - 1205  (see comparison in Table IV). 
       FIG. 18  shows a graph of the radii  1802  of the electron trajectories at the substrate surface  1802  against the radii  1801  of the trajectories at the objective lens in a second embodiment of the present invention, in which only quadrupole/octupole #4  1206  has an octupole excitation—this example is only for illustration, and basically corresponds to operating in a mode similar to the first embodiment. In this case, elements  1204  and  1205  function only as quadrupoles. Comparison of  FIG. 18  with  FIG. 4  for a circular beam column shows that now there are two curves,  1803  and  1804 , instead of only one—this is because the beam is shaped into a square by azimuthal control of the spherical aberration, as described below. The goal is to generate a square-shaped beam with 66 nm sides. Thus the distance from the center of the beam to the side is 33 nm (short dashed line  1805 ) and the distance to the corners is √2 (33 nm)≈46.7 nm (long dashed line  1806 ). By use of quadrupole/octupole element  1206 , the total deflection of the beam now has the combined effects of three terms, defocus, spherical aberration in lens  1207  (equivalent to lens  103  in  FIGS. 1A-B ), and the deflection due to quadrupole/octupole  1206 , as was described above for the first embodiment. In  FIG. 18 , the octupole excitation of quadrupole/octupole  1206  has intentionally been set low to leave the corners of the beam in  FIG. 19  rounded. In many applications where sidewall coverage during deposition is an issue, it is preferable not to generate a beam with sharp corners, since the resulting etched square hole (typically a contact or via) would be difficult or impossible to completely fill with conductive material (such as tungsten, copper, aluminum, etc.). Because of the corner rounding in  FIG. 19 , the minimum  1807  of curve  1803  does not quite reach the desired 33 nm beam radius of  1805  for the sides of the square beam. Similarly, the minimum  1808  of curve  1804  does not quite reach the desired 46.7 nm radius of  1805  for the corners of the square beam. Note that axis  1802  includes both positive and negative numbers for the radius at the substrate  1208 —in this case, a negative radius corresponds to a positive radius of the same magnitude, but rotated azimuthally by 180° around the optical axis  1240 . 
       FIG. 19  shows a graph of the trajectories  1813  along the X-axis  1811  and Y-axis  1812  at the substrate  1208  with the use of the second embodiment of the present invention to shape the beam into a square with rounded corners  1814  in which only quadrupole/octupole #4  1206  has an octupole excitation. As described in  FIG. 18 , in many applications, some rounding of the corners of the beam may be advantageous to improve sidewall coverage during deposition into the contact or via. Curves  1807  and  1808  in  FIG. 18  show that there is substantial overlap of the trajectories  1813  in FIG.  19 —this overlap can be seen from the fact that both curves  1807  and  1808  show two different radii at the objective lens  1207  (axis  1802 ) for the same radius at substrate  1208  (axis  1801 ) in many cases. This overlap corresponds to a “folding over” of the beam on itself, thus making the beam smaller for a given number of trajectories reaching the substrate  1207 —this is the same phenomenon seen in  FIGS. 10-11 . Since the number of trajectories is proportional to the total beam current, this means that the beam current density at the substrate  1208  is increased compared with the case of first-order imaging (the conventional method of beam-shaping) in which there is no folding over of the trajectories at the substrate. 
       FIG. 20  shows a graph of the radii  1902  of the electron trajectories at the substrate  1208  against the radii  1901  of the trajectories at the objective lens in a second embodiment of the present invention, in which elements  1204 - 1206  all have octupole excitations as shown in  FIGS. 15-17 . Comparison of  FIG. 19  with  FIG. 4  for a circular beam column shows that now there are two curves,  1903  and  1904 , instead of only one—this is because the beam is shaped into a square by azimuthal control of the total spherical aberration, as described below. The goal is to generate a square-shaped beam with 66 nm sides. Thus the distance from the center of the beam to the side is 33 nm (short dashed line  1905 ) and the distance to the corners is √2 (33 nm)≈46.7 nm (long dashed line  1906 ). By use of quadrupole/octupole elements  1204 - 1206 , the total deflection of the beam now has the combined effects of three terms: defocus, spherical aberration in lens  1207  (equivalent to lens  103  in  FIGS. 1A-B ), and the deflection due to the octupole excitations in quadrupole/octupole elements  1204 - 1206 . In  FIG. 19 , the effects of elements  1204 - 1207  have combined to generate the square beam in  FIG. 21  which has no rays outside the desired square beam profile. The minimum  1907  of curve  1903  is tangent to the desired side radius 33 nm  1905 . The minimum  1908  of curve  1904  is tangent to the desired corner radius 46.7 nm  1906 . Note that axis  1902  includes both positive and negative numbers for the radius at the substrate  1208 —in this case, a negative radius corresponds to a positive radius of the same magnitude, but rotated azimuthally by 180° around the optical axis  1240 . 
       FIG. 21  shows a graph of the trajectories  1913  along the X-axis  1911  and Y-axis  1912  at the substrate  1208  with the use of the second embodiment of the present invention to shape the beam into a square with corners  1914  in which elements  1204 - 1206  all have octupole excitations as shown in  FIGS. 15-17 . The corners  1914  of the beam are now sharp, which may be useful for lithography applications with substantial blurring in the resist—in these cases, the resist profile must be as sharp as possible to achieve the best final etched shape in the substrate  1208 . Curves  1907  and  1908  in  FIG. 20  show that there is substantial overlap of the trajectories  1913  in FIG.  21 —this overlap can be seen from the fact that both curves  1907  and  1908  show two different radii at the objective lens  1207  (axis  1902 ) for the same radius at substrate  1208  (axis  1901 ) in many cases. This overlap corresponds to a “folding over” of the beam on itself, thus making the beam smaller for a given number of trajectories reaching the substrate  1207 —this is the same phenomenon seen in  FIGS. 10-11  and  FIGS. 18-19 . Since the number of trajectories is proportional to the total beam current, this means that the beam current density at the substrate  1208  is increased compared with the case of first-order imaging (the conventional method of beam-shaping) in which there is no folding over of the trajectories at the substrate. 
       FIG. 22  shows a Venn diagram illustrating the interactions of the three contributions to the system throughput:
         1) Multiple beam column assembly (circle  2201  enclosing areas  2204 ,  2210 ,  2211 , and  2213 )—a column assembly which can produce multiple electron beams is described in U.S. Pat. No. 6,943,351 B2, “Multiple Column Charged Particle Optics Assembly” issued Sep. 13, 2005, incorporated by reference herein. Clearly, increasing the number of beams which are simultaneously writing on a substrate will lead to a nearly-proportional increase in writing throughput. The multiple beam column technology described in the reference may be applied to the generation of both one- and two-dimensional arrays of beams, with inter-beam spacings in the range of 30 mm in X-Y, where X and Y are the coordinates in the plane of the substrate. Typical arrays of beams might comprise up to 10 beams in a line or 10×10 beams in a two-dimensional array. Area  2204  represents a system with a multiple beam column assembly using conventional low current density beam shaping and raster scanning.   2) High current density shaped beams (circle  2202  enclosing areas  2205 ,  2210 ,  2212 , and  2213 )—one method for achieving high current density shaped beams is the present invention. Another method for achieving high current density shaped beams is described in U.S. Patent Application Publication No. 2006/0145097 A1, “Optics for Generation of High Current Density Patterned Charged Particle Beams” filed Oct. 7, 2004, incorporated by reference herein. Both methods are capable of being implemented in the multiple beam column assembly described in the section above. The key requirement for this is the need for each column to fit within the small available X-Y footprint (typically, approximately 30 mm×30 mm) within the multiple beam column assembly. This requirement for a small column footprint generally precludes the use of complex columns with many lenses, apertures and deflectors, as are commonly used in the production of lower current density shaped beams as is familiar to those skilled in the art. The increase in throughput due to increased current density in the beam is almost proportional to the magnitude of the current density increase, assuming that blanking times between successive flashes are reasonably short compared to the flash (i.e., writing) times. In the beam shaping methods described above, current density increases of 25 to &gt;50 times over the conventional beam shaping approaches are possible. Area  2205  represents a system with a single column using a high current density shaped beam and raster scanning.   3) Vector scanning (circle  2203  enclosing areas  2206 ,  2211 ,  2212 , and  2213 )—the third contribution to throughput comes from the method of deflecting the beam around on the substrate. There are two widely-used scanning methods: 1) raster-scanning where the beam always traverses an X-Y pattern and is blanked on/off to write the pattern, and 2) vector scanning where the beam is moved directly from the position of a flash to the position of the next flash. The raster approach has the benefits of greater electronic simplicity at the expense of slower writing since the beam spends a lot of time over regions not to be written (where the beam is blanked). The vector scanning approach is more complex electronically, but has the substantial benefit of reducing writing times since the beam needs to be blanked a smaller percentage of the overall writing time. Depending on the pattern density, throughput increases due to vector scanning may range from 2× to 5× compared with raster scanning. Area  2206  represents a single column system using a low current density shaped beam and vector scanning (this is the prior art shaped beam approach).       

     Clearly to obtain the largest increases in writing throughputs, it is advantageous to combine two or all three of these contributions in one system. There are four possibilities:
         1) Multiple beam column assembly with high current density shaped beams using raster scanning (area  2210 )—the throughput advantage here is the product of the number of columns (10-100×) and the current density increase (25-50×)—giving an overall potential throughput increase of (250-5000×).   2) Multiple beam column assembly with low current density shaped beams and vector scanning (area  2211 )—the throughput advantage here is the product of the number of columns (10-100×) and the vector scanning throughput increase (2-5×)—giving an overall potential throughput increase of (20-500×).   3) Single beam column with a high current density shaped beam and vector scanning (area  2212 )—the throughput advantage here is the product of the current density increase (25-50×) and the vector scanning throughput increase (2-5×)—giving an overall potential throughput increase of (50-250×).   4) Multiple beam column assembly with a high current density shaped beam and vector scanning (area  2213 )—this represents the ultimate throughput improvement situation, since the advantage here is the product of the number of columns (10-100×), the current density increase (25-50×), and the vector scanning throughput increase (2-5×)—giving an overall potential throughput increase of (500-25000×).       

     Some examples of the parameters for combinations of multiple beam columns, high current density shaped beams and vector scanning to specify a high throughput lithography system of the invention are given below. 
     A first example is a system with a multiplicity, N, of columns, each with a high current density charged particle shaped-beam which has a current density, I a , and an area A, at the surface of the substrate, which satisfy the equations:
 
I a ≧1000 Ampères per square centimeter;
 
300≧N≧10;
 
A=p 2 ; and
 
120&gt;p&gt;10 nanometers.
 
     A second example is a system with a multiplicity, M, of columns, each with a high current density charged particle shaped-beam which has a current density, I b , and an area B, at the surface of the substrate, which satisfy the equations:
 
I b &gt;5000 Ampères per square centimeter;
 
100≧M≧10;
 
B=q 2 ; and
 
120&gt;q&gt;20 nanometers.
 
       FIG. 23  shows a schematic circuit diagram of drive electronics for the 16-pole element in  FIG. 7  used for element  203  in the first embodiment. Since element  203  only requires an octupole excitation, the voltages on poles  233 - 248  are driven by octupole driver  2302  (providing four signals: +Oct1, +Oct2, −Oct1, and −Oct2). Connections to the 16 poles  233 - 248  are as shown. The 4-fold symmetry inherent in an octupole excitation means that each of the four octupole signals is connected to four poles spaced 90° apart azimuthally around the optical axis. For example, signal +Oct1 connects to poles  233 ,  237 ,  241 , and  245 . Signals Oct1 and Oct2 are determined by the required rotation angle, θ, for the shaped beam. Table I illustrates some representative values for the voltages on poles  233 - 248  for four different orientations of a shaped beam. The general formulas for the voltage signals are:
 
 Oct 1 =A  cos [4θ+45°]
 
 Oct 2 =A  cos [4θ+135°]
 
where A&lt;0 is a particular voltage determined by the column optics design. Note that any rotation angle θ&gt;90° is equivalent to an angle between 0° and 90° due to the 4 θ term.
 
       FIG. 24  shows a schematic circuit diagram of drive electronics for the 8-pole element in  FIG. 8  used for element  203  in the first embodiment. Since element  203  only requires an octupole excitation, the voltages on poles  253 - 260  are driven by octupole driver  2402  (providing two signals: +Oct and −Oct). Connections to the 8 poles  253 - 260  are as shown. The 4-fold symmetry inherent in an octupole excitation means that each of the two octupole signals is connected to four poles spaced 90° apart azimuthally around the optical axis. For example, signal +Oct connects to poles  253 ,  255 ,  257 , and  259 . Signal Oct is determined by the required rotation angle, θ, for the octupole excitation of the 8-pole element, as is familiar to those skilled in the art. Table II illustrates some representative values for the voltages on poles  253 - 260  for two different orientations of a shaped beam. Since an 8-pole element can only generate two orientations of an octupole electrostatic field (θ=0° and 45°), the general formula for the voltage signal is: 
                   Oct   =     A   ⁢           ⁢     cos   ⁡     [     4   ⁢   θ     ]                     =     A   ⁢           ⁢     (       for   ⁢           ⁢   θ     =     0   ⁢   °       )     ⁢           ⁢   or                 =       -   A     ⁢           ⁢     (       for   ⁢           ⁢   θ     =     45   ⁢   °       )                   
where A&lt;0 is a particular voltage determined by the column optics design. Note that any rotation angle θ&gt;90° is equivalent to an angle between 0° and 90° due to the 4θ term.
 
       FIG. 25  shows a schematic circuit diagram of drive electronics for the 16-pole element in  FIG. 7  used for elements  1203 - 1206  in the second embodiment. Since elements  1204 - 1206  require both quadrupole and octupole excitations (element  1203  is a pure quadrupole), the voltages on poles  233 - 248  are driven by both quadrupole driver  2501  (providing eight signals: +Q1, +Q2, +Q3, +Q4, −Q1, −Q2, −Q3, and −Q4) and by octupole driver  2502  (providing four signals: +Oct1, +Oct2, −Oct1, and −Oct2). Connections to the 16 poles  233 - 248  are as shown. The two-fold symmetry inherent in a quadrupole excitation means that each of the eight quadrupole signals is connected to two poles spaced 180° apart azimuthally around the optical axis. For example, signal +Q1 connects to poles  233  and  241 . Signals Q1, . . . , Q4 are determined by the required rotation angle, θ, for the shaped beam. Table V illustrates some representative values for the quadrupole voltages on poles  233 - 248  for eight different orientations of a line focus. Note that the orientation angles for the line foci are different from the orientation angle for the shaped beam. For example, a shaped beam with a rotation angle θ would require the following line focus rotation angles (see  FIGS. 12A-13B  and Table V):
         Element  1203 : excitation has a θ+45° rotation—gives a line focus at θ+45° at element  1204     Element  1204 : excitation has a θ+135° rotation—gives a line focus at θ+135° at element  1205     Element  1205 : excitation has a θ+45° rotation—gives a round beam at element  1206     Element  1206 : excitation has a θ+135° rotation—gives a parallel round beam entering lens  1207         

     The 4-fold symmetry inherent in an octupole excitation means that each of the four octupole signals is connected to four poles spaced 90° apart azimuthally around the optical axis. For example, signal +Oct1 connects to poles  233 ,  237 ,  241 , and  245 . Signals Oct1 and Oct2 are determined by the required rotation angle, θ, for the octupole excitation of the 16-pole element, as is familiar to those skilled in the art. Table I illustrates some representative values for the voltages on poles  233 - 248  for four different orientations of a square beam. The general formulas for the voltage signals are:
 
 Oct 1 =A  cos[4θ+45°]
 
 Oct 2 =A  cos [4θ+135°]
 
where A&lt;0 is a particular voltage determined by the column optics design. Note that any rotation angle θ&gt;90° is equivalent to an angle between 0° and 90° due to the 4 θ term. Additive elements  2511 - 2518  combine the quadrupole and octupole voltages derived above. Additive elements  2511 - 2518  could be op-amp circuits if Q1-Q4 and Oct1-Oct2 are analog signals, or they could be digital circuitry if Q1-Q4 and Oct1-Oct2 are digital signals. In the latter case, additive elements  2511 - 2518  would also perform a digital-to-analog conversion to generate final (analog) drive voltages for poles  233 - 248 .
 
       FIG. 26  shows a schematic circuit diagram of drive electronics for the 8-pole element in  FIG. 8  used for elements  1203 - 1206  in the second embodiment. Since elements  1204 - 1206  require both quadrupole and octupole excitations (element  1203  is a pure quadrupole), the voltages on poles  233 - 248  are driven by both quadrupole driver  2601  (providing four signals: +Q1, +Q2, −Q1, and −Q2) and by octupole driver  2602  (providing two signals: +Oct and −Oct1). Connections to the eight poles  253 - 260  are as shown. The two-fold symmetry inherent in a quadrupole excitation means that each of the four quadrupole signals is connected to two poles spaced 180° apart azimuthally around the optical axis. For example, signal +Q1 connects to poles  253  and  257 . Signals Q1 and Q2 are determined by the required rotation angle, θ, for the shaped beam. Table VI illustrates some representative values for the quadrupole voltages on poles  253 - 260  for four different orientations of a line focus. Note that the orientation angles for the line foci are different from the orientation angle for the shaped beam. For example, a shaped beam with a rotation angle θ would require the following line focus rotation angles (see  FIGS. 12A-13B  and Table VI):
         Element  1203 : excitation has a θ+45° rotation—gives a line focus at θ+45° at element  1204     Element  1204 : excitation has a θ+135° rotation—gives a line focus at θ+135° at element  1205     Element  1205 : excitation has a θ+45° rotation—gives a round beam at element  1206     Element  1206 : excitation has a θ+135° rotation—gives a parallel round beam entering lens  1207 
 
The 4-fold symmetry inherent in an octupole excitation means that each of the two octupole signals is connected to four poles spaced 90° apart azimuthally around the optical axis. For example, signal +Oct connects to poles  253 ,  255 ,  257 , and  259 . Signal Oct is determined by the required rotation angle, θ, for the octupole excitation of the 8-pole element, as is familiar to those skilled in the art. Table II illustrates some representative values for the voltages on poles  253 - 260  for two different orientations of a square beam. Since an 8-pole element can only generate two orientations of an octupole electrostatic field (θ=0° and 45°), the general formula for the voltage signal is:
       

                   Oct   =     A   ⁢           ⁢     cos   ⁡     [     4   ⁢   θ     ]                     =     A   ⁢           ⁢     (       for   ⁢           ⁢   θ     =     0   ⁢   °       )     ⁢           ⁢   or                 =       -   A     ⁢           ⁢     (       for   ⁢           ⁢   θ     =     45   ⁢   °       )                   
where A&lt;0 is a particular voltage determined by the column optics design. Note that any rotation angle θ&gt;90° is equivalent to an angle between 0° and 90° due to the 4 θ term. Additive elements  2611 - 2618  combine the quadrupole and octupole voltages derived above. Additive elements  2611 - 2618  could be op-amp circuits if Q1, Q2 and Oct are analog signals, or they could be digital circuitry if Q1, Q2 and Oct are digital signals. In the latter case, additive elements  2611 - 2618  would also perform a digital-to-analog conversion to generate final (analog) drive voltages for poles  253 - 260 .
 
       FIG. 27  shows a schematic view of a magnetic 16-pole optical element that can be used for octupole  203  (see  FIG. 6 ) in a first embodiment of the present invention, and for elements  1203 - 1206  (see  FIGS. 12A-13B ) in a second embodiment of the present invention. The sixteen magnetic poles  2733 - 2748  are oriented relative to the X-axis  2731  and Y-axis  2732  as shown. The operation of magnetic 16-pole optical elements is essentially equivalent to the operation of electrostatic 16-pole optical elements, with the exception that the gaps in a magnetic 16-pole element are equivalent to the poles in an electrostatic 16-pole element. This can be seen from the fact that electrons are deflected perpendicularly to a magnetic field but are deflected parallel to an electric field. Each magnetic pole in  FIG. 27  is fabricated from a material with a high magnetic permeability and has a corresponding excitation coil, for example pole  2733  is excited by coil  2713  which surrounds pole  2733  next to flux return ring  2702 . Identical considerations apply to poles  2734 - 2748  with excitation coils  2714 - 2728 , respectively. The purpose of flux return ring  2702  is to connect together the magnetic flux generated by coils  2713 - 2728  to avoid excessive stray flux from adversely affecting the electron beam in parts of the column away from the 16-pole optical element. One polarity of current in an excitation coil (e.g.,  2713 ) will make the corresponding pole (pole  2733 ) a North pole, while the opposite current polarity will make the corresponding pole (pole  2733 ) a South pole, as is familiar to those skilled in the art of magnetic deflectors. It is also possible to avoid the use magnetic materials and fabricate the 16-pole optical element using only shaped coils. This approach has the advantage of avoiding hysteresis in the magnetic poles  2733 - 2748  and flux return ring  2702 , but with the disadvantage of requiring much higher excitation currents in coils  2713 - 2728 . 
       FIG. 28  shows a schematic view of a magnetic 8-pole (octupole) optical element that can be used as an alternative to the magnetic 16-pole element described in  FIG. 27 . The eight poles  2853 - 2860  are oriented relative to the X-axis  2851  and Y-axis  2852  as shown. As for  FIG. 27 , the operation of magnetic 8-pole optical elements is essentially equivalent to the operation of electrostatic 8-pole optical elements, with the exception that the gaps in a magnetic 8-pole element are equivalent to the Doles in an electrostatic 8-pole element. Each magnetic pole in  FIG. 28  is fabricated from a material with a high magnetic permeability and has a corresponding excitation coil, for example pole  2853  is excited by coil  2813  which surrounds pole  2853  next to flux return ring  2802 . Identical considerations apply to poles  2854 - 2860  with excitation coils  2814 - 2820 , respectively. The purpose of flux return ring  2802  is to connect together the magnetic flux generated by coils  2813 - 2820  to avoid excessive stray flux from adversely affecting the electron beam in parts of the column away from the 8-pole optical element. One polarity of current in an excitation coil (e.g., coil  2813 ) will make the corresponding pole (pole  2853 ) a North pole, while the opposite current polarity will make the corresponding pole (pole  2853 ) a South pole, as is familiar to those skilled in the art of magnetic deflectors. It is also possible to avoid the use magnetic materials and fabricate the 16-pole optical element using only shaped excitation coils. This approach has the advantage of avoiding hysteresis in the magnetic poles  2833 - 2740  and flux return ring  2802 , but with the disadvantage of requiring much higher excitation currents in coils  2813 - 2820 . 
     Table III shows a comparison of the relative advantages and disadvantages of the two octupole implementations shown in  FIGS. 27 and 28 . The key determinant between the two implementations would be whether all orientations of the beam shape are required for patterning the substrate. In general, usually only orientations along 0° and 45° are needed, so the simpler 8-pole implementation in  FIG. 28  would be preferred. If, however, all orientations are required, then it is necessary to use the more complex 16-pole implementation in  FIG. 27 . 
     The second embodiment is discussed herein with either four electrostatic 8N-pole optical elements or four magnetic 8N-pole optical elements. It is also possible to implement the second embodiment with a combination of 1-3 electrostatic 8N-pole optical elements and 1-3 magnetic 8N-pole optical elements, providing that there is a total of four 8N-pole optical elements. 
     Both the first and second embodiments may be implemented using combined electrostatic/magnetic 8N-pole optical elements, thereby enabling partial or complete correction for chromatic aberrations in the first- and third-order deflections—the use of combined electrostatic and magnetic optical elements for chromatic aberration correction is familiar to those skilled in the art. 
     The second embodiment may also be implemented using a configuration in which the first 8N-pole optical element has combined quadrupole/octupole excitations instead of, or in addition to, the combined quadrupole/octupole excitation on the fourth 8N-pole optical element. An advantage of this configuration is that two weaker octupole excitations (requiring may be used instead of the single, stronger, octupole excitation on the fourth 8N-pole optical element described above. A disadvantage of this configuration is that more complex electronics is required to drive the first 8N-pole optical element since it is required to generate both quadrupole and octupole fields, instead of only the quadrupole field described above. 
     
       
         
               
             
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE VII 
               
             
             
               
                   
               
               
                 Parameters assumed for the calculations modeling the first embodiment. 
               
             
          
           
               
                 Parameter 
                 Lenses 
                 Octupoles 
               
               
                   
               
             
          
           
               
                 Focal length lens 202 
                 arbitrary 
                   
                   
                   
               
               
                 Strength Octupole 203 
                   
                   
                 −0.000250 
                 1/mm{circumflex over ( )}3 
               
               
                 Focal Length Lens 204 
                 10.006 
                 mm 
               
               
                 Spherical Aberration Lens 204 
                 0.003 
                 1/mm{circumflex over ( )}3 
               
               
                 Distance Lens 204 to 
                 9.9997 
                 mm 
               
               
                 Substrate 205 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE VIII 
               
             
             
               
                   
               
               
                 Parameters assumed for the calculations modeling the second embodiment. 
               
             
          
           
               
                   
                 Quadrupoles/ 
                   
               
               
                 Parameter 
                 Lenses 
                 Octupoles 
               
               
                   
               
             
          
           
               
                 Focal length lens 1202 
                 arbitrary 
                   
                   
                   
               
               
                 Strength Quadrupole #1 1203 
                 0.06667 
                 1/mm 
                 0.000000 
                 1/mm{circumflex over ( )}3 
               
               
                 Strength Quadrupole/ 
                 −0.06667 
                 1/mm 
                 −0.000009 
                 1/mm{circumflex over ( )}3 
               
               
                 Octupole #2 1204 
               
               
                 Strength Quadrupole/ 
                 0.06667 
                 1/mm 
                 −0.000009 
                 1/mm{circumflex over ( )}3 
               
               
                 Octupole #3 1205 
               
               
                 Strength Quadrupole/ 
                 −0.06667 
                 1/mm 
                 −0.000180 
                 1/mm{circumflex over ( )}3 
               
               
                 Octupole #4 1206 
               
               
                 Focal Length Lens 1207 
                 10.006 
                 mm 
               
               
                 Spherical Aberration Lens 
                 0.003 
                 1/mm{circumflex over ( )}3 
               
               
                 1207 
               
               
                 Distance Lens 1207 to 
                 9.9997 
                 mm 
               
               
                 Substrate 1208