Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use o

Efficient encoding algorithms for computer-aideddesign of diffractive optical elements by the use of electron-beam fabricationJiao Fan,David Zaleta,Kristopher S.Urquhart,and Sing H.LeeOne of the general requirements of a computer-aided design system is the existence of efficient1in datasize and running time2algorithms that are generally reliable for the broadest range of designinstances.The restricted data formats of the electron-beam machines impose difficulties in developingalgorithms for the design of diffractive optical elements1DOE’s2and computer-generated holograms1CGH’s2.Issues that are related to the development of CGH algorithms for e-beam fabrication of DOE’sand CGH’s are discussed.We define the problems the CGH algorithms need to solve,then introducegeneral curve drawing algorithms for the e-beam data generation of diffractive optical components.Anefficient algorithm for general aspherical DOE’s is proposed.Actual design and fabrication examplesare also presented.1.IntroductionDiffractive optical elements1DOE’s2or computer-generated holograms1CGH’s2can be used in many industrial and military applications,including optical interconnections for next-generation highly parallel computing systems.The most general DOE that can be expressed in analytic form for such applications is the aspheric DOE.Typically an aspheric element is defined as an optical element whose phase function can be specified by a polynomial of some specified type.Thus aspheric elements can be designed in standard optical design programs,such as CODE V, that will optimize the coefficients of the polynomial to satisfy the designer’s specifications.Once these coef-ficients are created,the next step is to generate the e-beam fabrication data for the aspheric DOE’s. Depending on whether the element will be fabricated by a direct-write e-beam method or by e-beam masks,files that contain such information as e-beam shapes, dose levels,and mask numbers must be generated. The problem,however,is that e-beam shapes are trapezoids rather than curved shapes.Further-more,current e-beam pattern generators for vector-based machines place stringent limitations on the types of trapezoids that can be generated,as we discuss in Section2.For instance,only trapezoids with horizontal tops and bottoms and with afixed set of angles for the trapezoid’s sides are allowed. Therefore the diffractive fringe patterns must be approximated by large numbers of these e-beam shapes.There are two major issues in generating curved-fringe DOE’s:112whether the algorithm is general enough,and122whether the data generated are efficient1in terms of data size2and accurate enough.A general algorithm should be able to produce correct results for any defined input parameters.In the case of aspheric DOE’s,the input can be any polyno-mial phase function and have any aperture size, wavelength,and phase-error requirements.Math-ematically any continuous phase function defined on a rectangular or circular aperture can be approxi-mated by a polynomial.1This makes e-beam-fabri-cated aspheric elements attractive andflexible for many applications;however,at same time,this makes the task offinding a general algorithm for this purpose very difficult.Furthermore,as the number of trapezoids grows,the e-beam data size grows proportionally.Thus there is a trade-off in data generation between patternfidelity and data size or between optical noise and fabrication cost and time. Therefore there is a need for general algorithms that generate reduced data size for general aspheric DOE’s.The authors are with the Department of Electrical and Computer Engineering,University of California San Diego,9500Gilman Drive,La Jolla,California92093-0407.Received22August1994;revised manuscript received7Novem-ber1994.0003-6935@95@142522-12$06.00@0.r1995Optical Society of America.In this paper,we introduce the models,algorithms and a new encoding–fracturing method for solving these problems.Note that these algorithms and methods are practical and compatible with current computer and e-beam technologies.They have been integrated into a computer-aided design1CAD2tool for CGH and DOE fabrication,in which different types of e-beam format are supported.In Section2,the background information and basic e-beam machines and their associated limitations are discussed.Then, in Section3,we present the fundamental procedures for curve drawing and fracturing of phase contours based on e-beam machine limitations for rotationally symmetric DOE’s.In Section4,we present an aspheric tracing and fracturing algorithm,based on modified subdivision techniques,2that is stable,accu-rate,and data efficient.Examples,results,and dis-cussion are given in Section5.2.Background and Past ResearchIn this section,wefirst discuss the previous research conducted on generating e-beam data for DOE and CGH applications,and then we discuss the limita-tions in e-beam shapes for standard e-beam ma-chines.A.Electron-beam Data ConsiderationsAn important aspect of a CGH design system is its capability of generating e-beam data in a format that is readily acceptable by common commercial e-beam systems that are supported.Examples of these for-mats are Perkin-Elmer’s MEBES and Cambridge Instru-ment’s EBMF.The basic characteristics of these e-beam systems have been discussed in Refs.3and4. Summarizing these characteristics,wefind that both e-beam systems accept only a limited number of shapes,which are generally trapezoids with horizon-tal upper and lower edges.MEBES uses a raster-scan technology,and it allows trapezoids whose sides can have arbitrary orientations.In contrast,EBMF uses a vector-scan technology,and it accepts only trapezoids with a limited number of orientations along their sides,i.e.,0°,690°,627°,645°,and663°.All patterns must be described in terms of the coordi-nates and the size of the trapezoids that make up the pattern.In this paper,we call the minimum addres-sable point an‘‘exel,’’and it can have a diameter of between0.125and0.0125µm,depending on the e-beam machine.Note that each e-beam shape is made up of many exels.MEBES shapes require be-tween8to14bytes to describe,whereas each EBMF shape takes8bytes to describe.Both e-beam sys-tems divide the area of a pattern into several sections. MEBES divides the pattern into segments and stripes. Each segment is32,76812152exels wide and each stripe may be256,512,or1024exels high.Simi-larly,EBMF divides the pattern into squarefields and subfields.The size of eachfield in each direction is 32,76812152exels,and that of each subfield is1024 exels,i.e.,eachfield consists of32332subfields. E-beam shapes are not allowed to cross the bound-aries betweenfields and subfields or between seg-ments and stripes.As subfield boundaries includefield boundaries,we use the term subfield to stand forbothfield and subfield in this paper.In the appen-dix,we outline some low-level functions that arecapable of achieving this subfield fracturing.Because MEBES machines have moreflexibility inthe e-beam angle requirements for mask production,any encoding algorithms for EBMF should also work for MEBES.However,one reason for using EBMF ma-chines for CGH and DOE fabrication is their capabili-ties of direct-write applications.By direct write,wemean both direct alignment techniques1typicallycalled direct-write in integrated-circuit fabrication2and analog resist techniques1called direct-write inDOE fabrication2.5Both of these methods have showngreat promise in reducing the number of fabricationsteps or in reducing the misalignment during fabrica-tion and are topics of current research.In both ofthese direct-write methods,it is necessary to assigneach shape in the pattern one of several differentdosage levels that is specified in the data format1in EBMF one of up to16different levels2.The MEBES format,unfortunately,does not provide the dosage information in the datafile.All the algorithms presented are applicable to both MEBES and EBMF with only slight modifications.B.Past ResearchUsing the computer to calculate the position and the width of fringes for an arbitrary wave front wasfirst proposed by Lee.6Since Arnold4,7first proposed the use of the e-beam writer in generating DOE’s,there have been many publications on the subject.8–11 There are two types of approaches that were proposed to take specification data,such as the coefficients of an aspheric DOE,and generate the desired e-beam data:fringe-tracing-based algorithms that trace fringe boundaries and pixel-based algorithms that draw small areas1pixels2based on the average wave-front value at the point.Pixel-based methods draw fringes that are based on the average phase-function value of each small region1pixel2.Lee’s method6and the mosaic-encoding method described by Arnold4are examples of this pixel-based approach.Because these types of algorithms rely totally on the phase function at each pixel,they are stable and general.However, because of the high-resolution requirement for CGH’s and DOE’s,the data size for this type of method is usually very high.Fringe-tracing methods,on the other hand,are based on tracing the contour curves that are produced when the phase function and a number of constant phase levels are intersected and the area between the appropriate contour curves is then fractured into valid e-beam shapes.This type of method has the advantage of being more data efficient because,if the fringe boundaries arefirst determined,larger e-beam shapes can be used to produce the fringes.Arnold’s staircase and polygon-encoding methods4are of this type.However,even through all polynomials are smooth and continuouslydifferentiable to any order at all points,the projec-tions of certain polynomials may not be smooth,and the boundaries of these fringes from the projection may present a singular point3e.g.1x21y22213 x2y2y3504or loops.12Arnold did not discuss in detail how to solve this problem,and the solution given in Ref.11is not general enough to handle arbitrary cases.Finding the intersections of con-stant phase planes and the surface of an arbitrary polynomial function can be difficult because of the numerical nature of the problem.In fact,even if the task of approximating the fringes into valid e-beam angles is neglected,finding optimum ways of determin-ing surface-to-surface intersections is still an active research topic in computer graphics.13–16As we discussed above,EBMF machines have an advantage over MEBES machines in their direct-write capability.However,the shape limitations for EBMF machines present more difficulties in the fracturing of curved fringes.In fact,the staircase method that has been proposed for EBMF fracturing is less data efficient1,25times compared with MEBES polygon encoding2and can have optical noise introduced by the corners of the rectangles.Hawley and Gallagher17 and Newman et al.18have proposed an efficient way to solve this problem by combining Lee’s method and the Lempel–Ziv–Welch compression scheme.This ap-proach determines the binary value1expose or not expose2of each exel in the CGH and compresses the information directly into a modified e-beam machine format.It can achieve good accuracy and generate data efficiently for binary CGH masks.However,it requires hardware and software changes on the e-beam machine,and handling of the e-beam direct-write case was not considered.In this paper we introduce an approach that com-bines the pixel-based and the fringe-tracing methods for the e-beam fabrication of aspheric DOE’s.There are two major contributions from this paper:112a new fracturing method for DOE’s with a rotationally symmetric phase function is presented for curve fringe fracturing that tries to minimize the data size as well as the optical noise introduced from sharp corners of the rectangles when Arnold’s staircase encoding for EBMF e-beam machines is used,and122a general stable method is presented forfinding and fracturing aspheric DOE fringes while trying to keep the e-beam data size minimum.C.Electron-Beam Data-Generation HierarchyBecause of the strict constraints on e-beam shapes and theflexibility requirements of CGH designs,we need to generate the e-beam shapes in a hierarchical fashion.There are three hierarchical conversion lev-els from high-level specifications to actual e-beam shapes:Level112Converting high-level design param-eters into fringe information1usually defined by a set of points or an analytic equation2.Level122Approximating the fringes with trap-ezoids required by the e-beam system.Level132Fracturing the trapezoids with subfield boundaries so that thefinal trapezoids will be valid e-beam shapes.This hierarchy can imply different algorithms for different types of elements.For instance,DOE’s that are rotationally symmetric,which are commonly used in optical design,can be more efficiently frac-tured than general aspheric DOE’s.Although gen-eral aspheric DOE’s include rotationally symmetric DOE’s,the algorithm specifically designed for rotation-ally symmetric DOE’s can be much more efficient than a general aspheric algorithm.Therefore we separate our discussion of level112and level122for rotationally symmetric DOE’s and general aspheric DOE’s into Sections3and4,respectively.The basic functions for fulfilling level132in the hierarchy are given in Appendix A for interested readers.3.Fracturing of Rotationally Symmetric DOE’sIn this section we discuss how to fulfill level112and122 in the hierarchy for rotationally symmetric DOE’s by introducing a set of routines that are capable of fracturing their curved fringes into valid e-beam shapes.By rotationally symmetric,we mean DOE’s with phase functions that can be expressed as func-tions of the radius3i.e.,f1r24.Thus either spherical or certain types of aspheric phase functions can be fractured by the techniques described below.Note that these routines also serve as the foundational functions for the development of a general aspheric fracturing algorithm given in Section4.Because the fringe boundary of these types of DOE’s can be expressed in analytic form,the task for level112is to approximate the analytic curves with a set of points that trace the fringes with valid e-beam angles.The traces can then be converted to trapezoids with valid e-beam properties3level122in the conversion hierarchy4. These trapezoids are then further fractured into valid e-beam shapes by the use of a function called Fracture-Trapezoid3level132in the conversion hierarchy4,which is defined in Appendix A.Tofind a particular circular fringe boundary from a starting point P,we need to trace the points on the circle1within some phase-error specification2that satisfy the e-beam shape requirements when they are connected.Given a spherical phase function f1r25 f0,we know that the fringe boundaries are all of the form r25x21y25r i2,where5r i6is a set of constants determined for the appropriate phase transitions of the phase function.Now assume that e is the phase-error requirement that can be specified as a fraction of a wavelength:e5l@e.Because the DOE is rota-tionally symmetric,the magnitude of the gradient for a single fringe can be treated as a constant over the whole fringe1note that this assumption cannot be made for general aspheric DOE’s2.Then,for a given fringe with r j11and r j as the boundaries,the toler-ance,d,of the fringe tracing can be approximately given byd<21r j112r j2@e.13.12Without loss of generality in the following fringe-tracing description,we assume that the fringe is open and start from top to bottom,as one can always break a fringe up into a few fringes to satisfy this condition. The procedure for tracing the fringe boundary r j over the desired region R starts with point P within6d of the fringe boundary and follows an initial direction that is compatible with any e-beam angle restrictions. Typically P is chosen on some suitable position on the boundary of R1in this case we assume that the boundary of R is made up of lines with valid e-beam angles2.An intersection point can be determined by the solution of the linear equation of the e-beam direction with one of the error boundaries:r25 1r j1d22or r251r j2d221Fig.12.Then,depending on whether it intersects with the inner or the outer boundary,it searches for the next valid e-beam angle that will intersect the opposite error boundary and that is closest to the current direction.If the intersec-tion point cannot be found on the opposite error boundary,the current error boundary can be used again for the intersection.For example,e-beam line l1intersects r251r j1d22,and e-beam line l2has the closest angle to l1that intersects r251r j2d22.We then save this intersection point and then try to solve the next intersection in the same fashion;the process continues until the trace intersects the boundary of R. If the angle closest to the current direction is chosen, it is possible not only to maximize the e-beam shape’s sizes but also to help maintain smooth transitions between the e-beam shapes.This tracing algorithm can be applied to both EBMF and MEBES cases;of course,the implementation for the MEBES format has much more freedom in choosing the angles compared with that of the EBMF format.Therefore the MEBES format will generate less data.Formally,the e-beam tracing function can be described byDefinition:Given f,P,and e discussed above,and given the aperture of region R,define Trace-Ring 1f,P,e,R,ring2to be the function described above, which returns an array of points,ring.Using the above procedure,we canfind two arrays of points that define both boundaries r j11and r j1ring1 and ring22of any open fringe;we then define a function tofind trapezoids that satisfy any e-beam constraints in the following:Definition:Given two sets of points1making up the concentric polygons in Fig.22,ring1and ring2, which represent sampled boundary points along the two edges of a fringe,we define Find-Trapezoids 1ring1,ring2,trap2as a function that slices one trap-ezoid trap off of the fringe and also deletes this trapezoid area from the fringe for the next slice.The function returns this trapezoid.We can implement this function by slicing from each point horizontally until it reaches the opposite fringe boundary.Thus trapezoids withflat horizon-tal tops and bottoms will be created that have the desired e-beam angles found in the fringe-tracing section of the algorithm.Given the above tracing and fracturing functions,a generic binary or direct-write optics e-beam driver for rotational symmetric DOE’s can be defined as the following:Draw-Curve1f,e,R2102Choose a boundary function pair1f1,f22from function f defined on R such that f15f1n p@m, f25f11n112p@m,where n50,2,4,...,for a binary mask,n50,1,2,...,for a multiphase-level direct write,and m is the number of phase levels. 112Find the two starting points P1and P2ofthe Fig. 1.Fringe tracing by Trace-Ring function used in bothrotationally symmetric and general aspheric fracturing algorithms.Note that r1t,n1250°1the e-beam angle of l12.Fig.2.E-beam fracturing hierarchy for rotationally symmetricDOE’s.fringe boundary by intersecting f 1and f 2on the boundary of R .122Call Trace-Ring 1f ,P 1,e ,R ,ring12for the first set of points ring1.132Call Trace-Ring 1f 2,P 2,e ,R ,ring22for the second set of points ring2.142Call Find-Trapezoids 1ring1,ring2,trap 2to find a trapezoid trap .152Call Fracture-Trapezoid 1trap 2to write trap to the e-beam data file.162Repeat steps 142and 152until the fringe is covered by the trapezoids.172Repeat steps 102–162until all fringes have been converted to trapezoids.Creating e-beam data for an on-axis Fresnel lens can be easily achieved by the use of the function call:Draw-Curve 522p @l 31x 21y 21f 221@22f 4,e ,R i 6,where R i is the i th quarter of the two-dimensional plane for i 51,2,3,4.The element is divided into four quarters to ensure the no-loop condition of the driver.Figure 2shows the relationship of the three levels of conver-sion hierarchy and functions we have defined so far.Figure 3shows a quarter of a binary mask for a Fresnel lens generated by the Draw-Curve function with a phase error of l @100.4.Aspheric Diffractive Optical Element Electron-Beam Fracturing AlgorithmBecause of the generality implied in the aspheric case,the rotational symmetry required by Draw-Curve 1described in Section 32cannot be assumed.In this section,we formally describe the aspheric e-beam fracturing problem to be solved and present an algorithm capable of providing a general solution to the problem.To solve this problem,two difficult hurdles need to be overcome.One is to find thecontour lines of an arbitrary wave front 3level 1124.The other is to use optimum combinations of the valid e-beam shapes to approximate the fringes 3level 1224.Our approach is to divide the element into smaller subdivisions so that the task of tracing fringes can become more manageable and more stable or robust in the smaller areas.A.General Aspheric ElementsThe phase function for a general aspheric element can be given in the form of a polynomial asf 1x ,y 25o k 50M o j 50kCkj xk 2j y j,14.12where x and y are coordinates in the DOE plane,C kj are the aspheric coefficients,and M is the order of the polynomial.These coefficients depend on the de-sired function of the DOE,and they are usually obtained from an optical design program such as CODE V .19The aspheric algorithm uses these coefficients to locate the fringes in the DOE pattern.B.Problem FormulationBecause aspheric designs are much more general,they are more difficult for CAD programs to ensure high quality and efficiency.As an input to the algo-rithm,Equation 14.12is used to find the location of fringes and the polygon vertices approximating them.This can be done by finding the equiphase contours of f 1x ,y 2that define the edges of the fringes.The equiphase contours of f 1x ,y 2are defined by the locus of roots of the equation,f 1x ,y 25f 0,14.22where f 0is the constant phase ofinterest.Fig.3.Binary masks of a spherical element generated by the Draw-Curve function:1a 2EBMF ,1b 2MEBES .Formally,Let R be the aperture region of a DOE A:x0,x, x1and y0,y,y1.Let S55E0E is an e-beam shape6be the set of allowable e-beam shapes.Let f1x,y2be a polynomial phase function of A defined in region R,which represents the desired wave front of interest.Given the fractional fringe phase value,2p@m,where m is the number of phase levels and a phase-error bound e,our problem is to partition A into a set of fringes A i such thatA i551x,y201i21212p@m2,f1x,y2#i12p@m26;i.14.32 For each fringe A i,a number f i is assigned to indicate the quantized phase for this area.For ex-ample,when DOE’s are L phase level direct write,f i 512p@L21i mod L2for all i.The task of the algorithm is to approximate fringe A i efficiently with a subset S i8 of S such that the phase errors caused by undercov-ered or overcovered areas in the DOE plane are bounded by e.That is,for any i,we want tofind S i8 such that1i21212p@m22e,f1x a,y a2,i12p@m21e14.42 for all points1x a,y a2in S i8.Note that inequality14.42 indicates the valid region for each fringe in the DOE plane.The following is a more detailed discussion about how to trace a particular fringe to ensure that inequality14.42is satisfied.C.Error Estimation and Fringe TracingFor a general phase function f1x,y25f01e.g.,a polynomial2with a phase-error requirement e1usually a user-specified number such as l@1002,the curve tracing begins byfinding the error boundaries.This phase error is the local error between the desired phase surface and the approximated constant phase surface1Fig.42.However,the translation of this phase-error value to a contour plane tolerance around the desired fringe boundary is not necessarily a constant,as shown in Fig.4,where d1x1,y125d1fid25d1x2,y22.The phase-error requirement can be mathematically specified as0Df1x,y20<0≠f1x,y2≠x D x1≠f1x,y2≠y D y0#0≠f1x,y2≠x D x010≠f1x,y2≠y D y0#e.14.52 Therefore,if we letd1x,y25e2min31/≠f1x,y2≠x,1/≠f1x,y2≠y4,14.62then for each point within the range of d on the DOE plane,approximation14.52is satisfied.In other words,if D x and D y are less than or equal to d in Eq.14.62,then the phase-error criterion of approximation 14.52is met.Sometimes,to simplify the computation, we use the minimum value for the entire area of the element or the minimum e-beam pixel size,whichever is larger.However,Eq.14.62does provide a way to adjust the step size in the fringe tracing for different fringe sizes in the same element.Therefore,in the region of the DOE where the desired wave front is relativelyflat,the drawing accuracy requirement of the e-beam pattern can be relaxed so that the data size can be reduced.Note that it is possible that d could become less than a single exel.In this case,d will be clipped to the value of a single exel.Having determined the d,we now can redefine the Trace-Ring routine to handle general aspheric DOE’s. The contour trace method used here can also be explained by Fig.1.Similar to the rotational sym-metric case,if we are given one of the fringe boundary contours,f1x,y25f0,which needs to be approxi-mated by e-beam shapes,we set up two curves, f1x,y25f06e,to act as error boundaries for that contour.A small region R is assumed to ensure the stability.The algorithm starts from point P,which is within the error requirement3normally f1P25f04. From this point a tangent vector,t,and two normal vectors,n1and n21where n152n22,can be found that are tangent and normal,respectively,to the desired curve,f5f0.The problem is then reduced tofinding the intersection points of f1x,y25f06e and a one-dimensional function r1t,n12or r1t,n22. Here r is the straight line that approximates t,the tangent line of f2f0at point1x,y2.Depending on whether the value f1P2is greater than or less than f0, we choose an angle that is either larger1has a component along n12or smaller1has a component along n22than the tangent vector.The function r is e-beam format dependent and converts the angle to the closest allowable e-beam angle to thetangent Fig. 4.Relationship between the phase error and the trace tolerance in aspheric DOE’s.vector that intercepts the opposite error boundary.Note that if an intersection on the opposite error boundary cannot be found,a point that intersects the current boundary can be used.Unlike the Draw-Curve case,in which the intersection of a line and a circle can be solved easily,here a Newton–Raphson root-finding algorithm 20or other root-finding algo-rithms can be used to locate the intersection points.The tolerance d is used here to check and select the intersection points to ensure the pattern’s accuracy.D.Algorithm DescriptionTo solve the general aspherical data-generation prob-lem,we developed the algorithm described below.This algorithm is based on modifications of subdivi-sion methods found in computer science for surface-to-surface intersections.2These methods are known to be both reliable and accurate.The algorithm can be divided into the following steps 1Fig.52:Step 112Divide aspheric element A into N regions,R 1,R 2,..,R N 3Fig.51a 24.Step 122Divide each region R i into G i grids 3Fig.51b 24,where G i is dependent on the minimum feature requirement in that region R i .To ensure the accu-racy,there should be no more than one fringe bound-ary crossing a grid.Then Eq.14.62is calculated for each corner of the grid G i ,and the smallest value of the four is used to approximate the fringe-tracing tolerance requirement for that grid.Step 132Trace each fringe 1with Trace-Ring 2within each grid with valid e-beam angles 3Fig.51c 24.First,the roots on the boundary of G i are found.Then the angles that are the closest to the fringe section in the grid are found.The error of the trace is bounded by e or d .Step 142Form trapezoid shapes 1Find-Trapezoids 2based on these traced fringes inside a grid 3Fig.51d 24.Step 152Within each region R i ,we combine shapes so that the total number of shapes is reduced in that region 3Fig.51e 24and write the e-beam shape to the output file 1Fracture-Trapezoid 2.Note that two extreme cases of this algorithm will result in either pure fringe-tracing-or pixel-based methods.When N is set to 1and we ignore step 152,we have the case of pixel-based methods.In this case,the data size will be quite large 1see Section 52.When region R i is set equal to A i 1those regions covered by the fringes of interest 2for all i and we ignore step 122,we have fringe-tracing methods.The problem in this case is that it is difficult to divide the elements into these type of regions except for a few special cases,such as Fresnel lenses or other rotation-ally symmetric aspheric elements.In this paper we present results in which A is uniformly divided into equal rectangular regions 1R i ’s 2.Adaptively divid-ing A into regions of different sizes will be more effective in reducing the data size,but is a topic of futureresearch.Fig.6.Picture of a typical aspherical lens fabricated by use of the aspheric algorithm.There are eight phase levels.Direct-write method is used here.Phase-error requirement is l @50.parison of MEBES and EBMF data sizes for rotationally symmetric DOE’s:1a 2aperture size versus data sizes for both formats;1b 2relationship between EBMF and MEBES is close to a constant2.6.Fig.5.Algorithm for general aspheric element e-beam fabrica-tion.。