MPhys Final Year Project Dissertation by Andrew Jackson

Transcription

1 Development of software for the omputation of the properties of eletrostati eletro-optial devies via both the diret ray traing and paraxial approximation tehniques. MPhys Final Year Projet Dissertation by Andrew Jakson Abstrat: This projet onerns the development of software designed to simulate eletro-optial devies; the potential being alulated via the finite element method, and the optial behaviour determined by paraxial and diret ray raing. The diret ray traing tehnique, previously only used for finite differene grids, has been developed by using new interpolation tehniques (to ope with the variability in mesh point distribution inherent in the finite element tehnique), and also by testing various different integration algorithms under these onditions. The new software has been found to be at as a useful tool for visualisation of lens behaviour, and has been found to give reasonable quantitative results for the predited lens properties in omparison to the paraxial approximation and to the results published in the literature for the standardised two-tube eletron lens. 1. Introdution Charged partile optis forms an important part of modern physial researh, and the simulation of suh devies forms an important part of their development. The omputational tehniques used in the software written to assist the design proess an be broken down into two parts. Firstly, the determination of the eletri and/or magneti fields inside the devie (via the boundary element [1,11], finite differene [1,3,4,5] or finite element [1,6,9] methods), and seondly the simulation of the trajetories of harged partiles through those fields (via the paraxial approximation [1,5,9] or by diret ray traing [1,3,4] ). While the existing software is apable of produing reliable and aurate data in most ases, it is generally diffiult to use and restritive in the kinds of behaviour it an simulate. The diffiulty for the user arises from the fat that the software usually requires a grid or mesh in order to find the eletri and/or magneti fields, and that this mesh must be ompletely defined by the user (as opposed to implementing any form of automati mesh generation). The restrition on the kind of behaviour to be simulated is a onsequene of the dependane of most of the software on the paraxial approximation, whih while aurate for partile trajetories near the optial axis, annot simulate the partile motion near the boundaries of the devie. This projet onerned the development of a user-friendly piee of software designed to allow the finite element solution of the eletri field of any rotationally symmetri eletrostati eletro-optial devie (assisted by semi-automati mesh generation), and the simulation of the eletron trajetories inside the 1

2 devie both under the paraxial approximation and by diret ray traing. Before this projet began a signifiant amount of ode had already been written, but while the eletri field solution and paraxial ray traing parts of the ode were thought to be working reasonably well, the ode designed to arry out the diret ray traing was working somewhat inaurately, leading to the predition of learly nonphysial behaviour. Thus, the aim of this projet was primarily to re-write the diret ray traing ode to give more realisti behaviour and more aurate results, and also to hek the eletri field and paraxial ray solutions produed by the original ode. In order to asertain the auray of the ode, results from the different tehniques an be ompared with the published results of others, for example the predited potential and lensing properties of a standard onfiguration two-tube eletron lens [3,5,11]. One the diret ray traing results have been ompared with those of the paraxial approximation and the theoretial and experimental results of others, then the validity of this implementation of the diret ray tehnique an be determined, it s weaknesses noted and possibilities for further development an be proposed. 2. Theory 2.1 Physial Priniples The software under development onerns only eletrostati devies, and so the field problem redues to the solution of the Laplae equation for the eletrostati potential: Ñ 2 V = 0 (1) This simplifiation, oupled with the rotational symmetry of the devies we wish to investigate, means that the software needs only to onsider the axial plane in order to ompletely desribe the behaviour of the system. In this way the problem beomes the determination the two-dimensional potential V(r,z), suh that: d 2 V(r,z) dr r Whih is the expanded form of the Laplae equation in ylindrial polar oordinates for the axial plane. The potential is thus defined by the boundary onditions of the system, in other words, by the harged plates whih make up the eletrostati eletro-optial devie being simulated. One the potential has been found, then the eletri field is defined by the gradient of the potential: E = - ÑV (3) Whih for this two-dimensional problem is equivalent to: E r =- dv dr Diret ray traing an now be arried out by alulating the trajetories of eletrons as they move through the devie, their motion being governed by the Lorentz fore: d (mv) = ee(r) dt dv(r,z) dr + d2 V(r,z) dz 2 = 0 E z =- dv dz (2) (4) (5) 2

3 Where the magneti field omponent has been negleted. For this two dimensional problem, we find: d dt (mv r ) = ee r (r,z) d dt (mv z ) = ee z (r,z) (6) Thus the high auray integration of the Newtonian equations of motion of an eletron being ated on by the Lorentz fore, with a range of initial onditions (position and veloity), an be used to investigate the properties of the eletro-optial devie. It should be noted that this formulation of the problem is not omplete and does plae some restritions on the kinds of behaviour being simulated. Firstly, the behaviour of a beam of eletrons is alulated by the simulation of single eletron trajetories through the eletrostati field, and so this approah annot take the spae harge effet into aount. Fortunately, this effet is only signifiant for relatively high harge densities, and eletron beams of suffiient onentration only our in some eletron guns. Seondly, the equations of motion are lassial, and so relativisti effets are not taken into aount. However, these effets only beome notiable for beam energies greater than 1keV [8, pp 91], and so we only onsider devies whih funtion at energies below this level. 2.2 The Paraxial Approximation Muh of the work on eletron optial devies makes use of the onepts and terminology of ray optis, as these onepts provide a onise way of desribing the behaviour of a given devie. The basis of ray optis theory is the paraxial approximation, and added on to this is the theory of aberrations whih aounts for the deviations from first order (paraxial) behaviour. The paraxial approximation [1,9,7] assumes that the eletrons remain lose to the optial axis of the lens, and this allows the variation of the field perpendiular to the optial axis to be simplified. By using this simplifiation we find we only need to know the eletri field along the optial axis to arry out our alulations. However, when the eletrons are not moving lose to the axis, then the basi approximation begins to fail and aberrations start to form. Although there are various types of aberration, espeially for non-rotationally symmetri devies, for our purposes the most important is spherial aberration. Spherial aberration ours when a hange in the angle at whih the eletron beam enters the lens auses the foal point of the beam to move along the optial axis. Figure 1 ompares the idealised paraxial approximation behaviour with the behaviour of the more realisti ase of spherial aberration for a plane lens. From this diagram it an be seen that spherial aberration alters the effetive foal length of the lens, forming not a point fous, but a irle of least onfusion (or spot). Thus the paraxial approximation has to be modified in order that spherial aberration may be taken into aount. This 3

4 (a) objet lens plane image (b) objet lens plane image paraxial fous spot Figure 1: Foussing properties of a plane lens demonstrating (a) idealised paraxial behaviour and (b) spherial abberation, showing the position of the irle of least onfusion, or spot. behaviour is best desribed in terms of an aberration oeffiient suh that: Dr = MCs q 3 (7) Where M is the linear magnifiation, and Dr is the transverse displaement at the paraxial image plane of a ray whih leaves the point objet at an angle q to the axis [11]. The saling fator Cs is known as the third order aberration oeffiient. However, when arrying out diret ray traing, it is more onvenient to use the form: zf = zfp - Cs tan 2 a (8) Where zfp is the paraxial foal length, and where zf is the foal point and a is the angle of the alulated eletron trajetory as it rosses the optial axis [4]. The fifth order abberation term has been omitted as this relatively small effet is negligible whilst this piee of software is under development. It should be noted that the eletron optial lenses onsidered here generally behave more like thik lenses than thin lenses, and so the foal length and aberration oeffients will depend on whether the rays enter the lens from the left or right hand side of the lens. While spherial aberration is by far the most prominent, it should also be noted that two other aberrations an beome signifiant in some ases. The first is relativisti aberration, where the foal point moves as a onsequene of the relativisti mass of the partile, and the seond is hromati aberration, where the fat that any harged partile beam will have some variation in the veloities of eah of the partiles means that the foal point beomes blurred. Relativisti aberration is signifiant only for devies using relatively high potentials, and as noted above, we an ignore these ases and still produe useful results. Unfortunately, the same annot generally be said for hromati aberration, beause for the foussing properties of any devie using a small aperture hromati aberration will dominate over spherial aberration [12]. However, the effet does not signifiantly affet the foal properties of devies using weaker potentials, and so for the time being we ignore the hromati aberration in the diret ray alulation. 4

5 2.3 Examples of Eletro-Optial Devies The simplest eletro-optial devie is the two-tube eletrostati eletron lens, and numerous examples of the predited properties of this standardised form of this devie have been published in the past [3,5,11]. The devie is omposed of two ylindrial eletrodes of idential diameter, with a small gap between them (see figure 2). In order to ensure the devie has a standard definition, the parameter for the gap, g, and for the lengths of the two ylinders, L1 and L2, are all expressed in terms of the lens diameter D. The properties of the devie an then tabulated for a range of g/d, and for a range of potential differenes, V2/V1. g V V D 1 2 L Figure 2: Shemati representation of a simple two-tube eletrostati eletron lens. The work of Natali et al [3] provides an exellent example of the high auray omputation of the potential, and so an be used to hek the finite element ode, whereas the papers by Liu & Ximen [5] and Read et al [11] provide tabulated values for the foal length and spherial abberation oeffiients, and so an be used to hek the paraxial and diret ray traing. L Method & Development As mentioned in the introdution, a signifiant amount of ode had already been written when this projet began. This meant that muh time was spend simply getting used to the ode, understanding the way in whih devies are desribe and how to use and adapt the user interfae. Also, as the main aim of this projet was to improve the diret ray traing ode, only a basi outline of the finite element and paraxial approximation ray traing ode will be given here. However, the following breakdown of the various tehniques that have been inorporated into the ode will follow the struture of the program and also of it's development. 3.1 Method for the Finite Element Solution of the Eletrostati Potential When we wish to simulate a devie using this software, we must first define the physial spae whih the devie oupies so that it may be broken up into a finite element mesh, and then define the potentials that lie along the boundaries of that mesh. First we note that the symmetry of the problem means that given the rotational axis at r=0, we only need define the problem for r>0, as V(-r,z) must be equal to V(r,z). From this geometrial basis, the struture of the devie is broken down to a set of nodes, whih when entered into the program an be joined up to form a set of (usually quadrilateral) 5

6 polygons overing the domain of the devie. Then, an automati mesh generation program is used to break eah polygon up into a set of triangular elements aording to user speified information on how fine the mesh should be, and on how the mesh within eah polygon should be graded (ie whether the mesh should be uniform of whether it should beome finer towards one side or orner). In order to solve this finite element problem, the potential must be speified along the boundaries of the mesh. These boundaries fall into one of two atagories, Neumann and Dirihlet. The optial axis (r=0) represents a Neumann boundary, a where the gradient of the potential aross the boundary is known (due to symmetry, the gradient must be zero). All the other boundaries are Dirihlet boundaries, where the value of the potential is known. A set of eletrodes, eah of a given potential, is entered by the user, along with any gaps (in whih the program assumes the potential to be varying linearly with z). This information, ombined with the left and right hand end potentials, ompletely defines the problem. The finite element solution then proeeds by the solution of a matrix problem (ie a set of linear simultaneous equations defined by the mesh and it's boundaries). In order to make the solution of the matrix problem more effiient, the mesh points are re-numbered so that the matrix beomes tri-diagonal in form. For more details on the nature of the finite element method, see Kikuhi [10]. One the potential has been found, the optial properties an be alulated by traing a set of eletrons thought the potential, from a point eletron soure with a user speified position, energy and angular range. This alulation was arried out by the subroutine TRAJEC, and the development of the program onsisted of alterations and modifiations to this part of the ode (supplied in appendix B). 3.2 Ray Traing Via The Paraxial Approximation This part of the ode reates a ubi spline of the axial potential for interpolation purposes (subroutine AXPOT), and then uses this potential V(z) to find for the paraxial trajetories by solving the differential equation desribing the first order properties of the devie, as derived from the theory of Hamiltonian optis (in AXPATHS). The exat form of this approah in not onsidered here (see refs. [1,4,5,9,12] ), and it is suffiient simply to note that the paraxial approximation plots the alulated trajetories on a plot of the lens, and reports the foal point position to the user, along with the third order spherial and first order hromati aberration oeffiients (PARAXABBERS, ABBERS) 3.3 Diret Ray Traing While the finite differene and paraxial approximation ode have undergone no development during this projet, the diret ray traing ode (ontrolled by the PATHS routine) has been altered many times, with many different algorithms being tested. The diret ray traing alulation breaks down into three main tasks. The aurate interpolation of the finite element mesh, the differentiation of the potential to 6

7 find the eletri field, and the integration of the equations of motion of the eletron. The following three setions addresses eah of these tasks by explaining the initial state of the ode, and then overing the other tehniques whih have been implemented Interpolation of the potential mesh: In order to solve the equations of motion for the eletron in this eletrostati potential, it is neessary to be able to alulate the potential at any point, V(z,r), in other words to find the value of the potential by interpolation inside eah mesh element. While this poses no problem in a finite differene alulation (where the mesh points follow a simple geometrial form, eg artesian, for whih high auray interpolation is well understood [1,3,4,5] ), the finite element method an allow any arrangement of mesh points, and so while it is more flexible in terms of the kinds of devie geometry it an aurately represent, it requires that the interpolation tehnique is equally as flexible. It should be noted that of all the published work that has been examined, none have used diret ray traing in ombination with the finite element approah, presumably beause of the diffiulties in reating an aurate interpolation algorithm Nearest Neighbour Averaged Linear Interpolation: This tehnique uses the routine FINDEL to find the referene number of the element at the urrent point, and the uses FINDNN to find the referene numbers of the three neighbouring elements. The geometrial entres of eah of the neighbouring elements an then be determined, and the value of the potential at the three elements entres alulated by linear interpolation of the values of the potential at the three nodes of eah element (using LINPOT). The way in whih this is done refers bak to the roots of the finite element method (by expressing the geometrial entre of eah element in terms of areal oordinates), and is equivalent to fitting a plane to eah of the elements nodes (in z,r,v spae) and then using this plane to find the value of V at the midpoint. The three sets of (z,r,v) oordinates for the ell entres are then expressed as another plane, and this an then be used to find the eletri field. The main problem with this method is that it uses a very simple linear approximation over a relatively wide physial range, and so an miss the finer details of the shape of the potential surfae Linear Interpolation: This approah is losely related to the nearest neighbour averaging algorithm above, but muh simplified. Instead of averaging the potential over the three nearest neighbour elements, it simply takes the linear interpolation of the three nodes of the urrently oupied element. This means that the potential being used is exatly that whih was alulated by the finite element method, and while it take no aount of the general form of the potential in the region surrounding the urrent element, and so is not a partiularly aurate approah, it does have the advantage of simpliity, and of forming a reliable 7

8 base on whih more omplex tehniques an be developed Loalised Nth Order Spline: This tehnique attempts to better desribe the potential surfae by fitting a non-linear spline it's form as opposed to using a linear approximation. The tehnique is related to the Chebyshev algorithm (see Press et al [2] ), exept that the spline is fitted loally to the point at whih we require to know the potential (as opposed to a set of splines over the whole domain), and that it must work in two dimensions. Eah spline has the form: y = an+1x n + anx n-1 + an-1x n a2x + a1 (9) And so learly an nth order spline required n+1 oeffiients (ai) in order for it to be properly defined, and so we need n+1 data points (xi,yi) in order to form the n+1 linear simultaneous equations required to find the set of oeffiients ai). This straightforward onept is ompliated by the two-dimensional nature of the problem in that in order to fit a nth order spline surfae to the potential we need enough sets of n+1 points to reate at least two splines. For this problem two sets have been used, one parallel to the z axis, and one parallel to the r axis (using these two perpendiular splines not only improves the auray of the interpolated potential surfae, but also assists the differentiation of the surfae to find the eletri field). This onept is illustrated in figure 3 below: Potential Þ Interpolated value In order for this approah the work, the ode must find the appropriate length sales to fit the splines over, and this must take into aount of the variation of element size in the positive and negative z and r diretions. This is ahieved by finding the relative positions of the entres of the nearest neighbour elements (using FINDNN and FINDELCENT), and then finding the overall z/r span of this set of spatial vetors in both the positive and negative diretions. Spatial oordinate Figure 3: Shemati example of averaging two perpendiular ubi splines for the purpose of two dimensional interpolation. These span lengths are then saled aording to the order of the approximation so that enough different elements will be overed to supply enough different information to make the approximation meaningful, although the best value for the saling parameter is by no means immediately apparent. The n+1 points are distributed over these z/r ranges, and then LINPOT is then used to find the potential at these 2(n+1) points (note that this means LINPOT now has to all FINDEL as the 2(n+1) point are not neessarily inside the urrent element). 8

9 This method also requires that no spline points lie outside the boundary of the finite element mesh, where the potential is undefined, and so the ode must also find the maximum spline in the positive and negative z and r diretions and make sure the interpolation splines fit within these maximum values (MinSplin). The only exeption to this is the ase of interpolation near the rotational axis, where the spline must be allowed to fall outside the mesh (ie into negative r values) by using the fat that V(-r,z) is equal to V(r,z). The omputational solution proeeds as follows. The subroutine MeshSpline uses FINDEL to find the element at the urrent point, then uses FINDNN to find it's nearest neighbours, and then uses FINDELCENT to alulate the entres of these ells. This data is then used to onstrut the harateristi ell dimensions in the positive and negative z and r diretions (Zmax, Zmin & Rmax, Rmin). These harateristi lengths are then modified suh that they over a suffiiently large spatial range, using the expressions: Zmin := nths (n+1) Zmin Zmax := nths (n+1) Zmax Rmin := nths (n+1) Rmin Rmax := nths (n+1) Rmax Where nths is the saling parameter mentioned above. One a set of spline lengths have been found, the ode alls MinSplin, whih returns the largest allowed values of Zmax, Zmin, Rmax and Rmin as defined by the boundary onditions for the urrent point. These two sets of values are then ompared, and the spline lengths are made to fit the boundaries of the devie where neessary. The proess ontinues by breaking the spline lengths down into n+1 evenly spaed point (z1 - zn+1 and r1 - rn+1), and using LINPOT to find the values of the potential at these points (ie Vi(zi,r) and Vi(z,ri)). These two sets of data are then passed to the routine nthapprox, whih uses a fully-pivoting Gauss- Jordan matrix routine (from Press et al [2] ) to solve for the spline oeffiients for eah set of data. One the oeffiients have been found, nthaeval an be used to evaluate the approximation to the potential for any point z,r in the region of the spline Loalised Nth Order Least-Squares Fit Polynomial: This fourth tehnique is losely related to the loalised spline alulation, and differs only on the fat that instead of using n+1 points to find an nth order spline to approximate the potential surfae, it uses 2n points and fits a least squares polynomial to potential data. As this tehnique uses more data from the mesh to form the same order of interpolation over the same spatial area, the form of the polynomial should more losely approximate the potential surfae. The routine used to perform this approximation is SVDFIT (from Press et al [2] ) whih replaes my own nthapprox routine, and has the advantage of 9

10 using single value deomposition as opposed to Gauss-Jordan elimination (whih an ope better with extreme ases suh as near singular matriies). In order to find out whih of the above methods of interpolation performs the best, the ode an be made to produe a interpolated version of the axial potential, and then this an be ompared with the potential on eah of the nodes along the axis. If the interpolation is working well, the form of the interpolated potential urve should be smooth, and should agree with the potential from the finite element alulation at eah node. In this way it should be possible to determine whih approah works the best, and in the ase of the latter two to find the optimum values for the order of the interpolation (n) and the saling parameter (nths). It should be noted here that during the development of the program, some of the most basi routines, whih existed before this projet began, were found to be working inorretly. Firstly, the routine to identify the element whose area inludes a given point r,z, FINDEL, was searhing the element array inorretly, and somtimes assumed a point to be outside the finite element domain when this was not the ase. Also, the routine designed to find the nearest neighbour elements (FINDNN) was failing, and returning invalid element referene numbers. Both these faults have bee orreted. Also, the optimum element numbering for the finite element solution need not orrespond the the spatial arrangement of the ells, but FINDEL works more quikly when the referene numbers of the ells are arranged so that they orrespond to the spatial order, and so the ode runs muh faster if the elements have been sorted aording to their position. Before the projet began, a very basi sorting was being used (taking tens of seonds to sort the mesh data), and during the odes development this routine has been replaed by QuikSort from Press et al [2], with the onsequene that the sorting proess is now at least one order of magnitude faster than before Calulation of the Eletri Field: Given the potential in it's interpolated form, we need to find the derivative of the potential at the urrent point, in other words the eletri field (Ez, Er), in order to integrate the equations of motion of the eletron. Two different shemes were ompared for this alulation Linear Surfae Gradient: This tehnique was applied to both the linear interpolation tehnique results, and is the original routine from the ode before this projet began. When alled, the routine GRAD takes the plane formed by the interpolation ode and returns the gradient of the plane in the z and r diretions as the values for the eletri field. The main drawbak of this approah is that as the interpolation uses the same single linear approximation for eah element, then no matter where the eletron lies in that element the 10

11 eletri field will be found to be the same. In other words, this tehnique annot take the general form of the potential into aount, and so it an be expeted to perform badly when it omes to finding the derivative of the potential Nth order polynomial derivative: In the ase of the two nth order polynomial interpolation routines, the derivative of the potential at the urrent point an be alulated by diret differentiation of equation (9) suh that: dy/dx = n an+1x n-1 + (n-1) anx n-2 + (n-2) an-1x n a3x + a2 (10) As the interpolation routine takes a broad physial range into aount, this approah should form a better approximation to the form of the potential and so form a muh more aurate representation of the eletri field than the linear gradient approah. The validity of these routines an be heked in a similar way to the interpolation. Instead of plotting V as a funtion of z, the two derivatives Ez and Er an be plotted as a funtion of z and r respetively for any arbitrary trajetory through the devie. Although there is no exat form of the eletri field with whih we an ompare these results, as there was with the interpolation hek, we an still use these plots to hek that the alulated eletri field is varying smoothly and onsistently. Also, if the results of the tests of the different interpolation are inonlusive as to whih is better, then the more hallenging differentiation hek an be used Integration of the equations of motion Whilst the details of the different integration routines vary, the general struture of the integration does not. Eah is designed to solve a set of N first-order differential equations, and so in order to solve the problem the equations of motion (6) are re-expressed to form a set of four oupled first order differential equations: and dvz = e Ez dz = vz (11) dt m dt dvr = e Er dr = vr (12) dt m dt The integration routine, when alled from PATHS, alulates the hange in veloity and position for one time step, h, and the DERIVS routine supplies the values of the eletri field and veloity at the urrent position (using one of the sets of interpolation and differentiation routines outlined previously). The new point of the trajetory is then stored and plotted on the sreen. 11

12 For this proess to work at all, it is neessary to know what value of h would be reasonable for the integration, and this annot be set arbitrarily beause the potential of the system (and so the veloities involved) an differ greatly between devies. To get around this problem, PATHS looks at the potentials on the plates and uses this to estimate the speed of an eletron moving through the devie along the axis. This is then used as the basis for the estimation of the required timestep. The different integration algorithms that have been implemented are as follows: Adaptive Runge-Kutta This routine, taken from Press et al [2], takes the fourth order Runge-Kutta routine (see below), and modifies it's implementation suh that the time step, h, an be altered during the ourse of the integration. In this way many small steps an be used to tiptoe through the quikly varying parts of the potential, and a few muh larger steps employed to over the smoother areas, thus making the proess more time-effiient. However, this routine was suspeted not to be working orretly, and so was replaed with a somewhat simpler method Simple Euler Integration This routine is extreme easy to implement, and represents the simplest possible method of integration, where equations of the form: dx/dt = f(x) are solved using Dx = Dt f(x) expressed in the program as: xn+1 = xn + h f(x) While this routine is not partiularly aurate (error ~ h 2 ), and oasionally unstable, it has the advantage of being almost impossible to implement wrongly. In this way, the possible failure of the adaptive Runge-Kutta algorithm an be heked Runge-Kutta The Runge-Kutta algorithm is based of the priniple of taking set of Euler type steps, and then using the information obtained to fit a Taylor expansion up to some higher order. The routine RK4 (from Press et al [2] used here represents the form of this tehnique when four Euler steps are taken, leading to a Taylor approximation with an error of the order of h 5. While this tehnique represents a great inrease in auray over the Euler approah, a simple way to improve it is by using extrapolation. 12

13 Extrapolative Runge-Kutta This tehnique, taken from Hawkes & Kasper [1], inreases the auray of the Runge-Kutta routine by means of an extrapolation of the form: xn+1 = x (2) n /15 (x (2) n+1 - x (1) n+1) where x (1) n+1 represents the results of a single Runge-Kutta step over an interval h, and where x (2) n+1 represents the results obtained by using two steps of interval h /2. The main draw bak is that the Runge- Kutta routine needs to evaluate the eletri field four times, and so the extrapolating algorithm requires 16 evaluations per time step. This an lead to a rather slow integration proess when, as in this ase, the value of the eletri field requires quite a degree of omputation. Beause of this the struture of the ode was hanged so that instead of onstruting a new interpolation for the mesh every time the DERVIS routine is alled, the interpolation is only onstruted for every time step. This makes the assumption that the form of the potential does not hange signifiantly over the distane overed in the time interval h, and as this is required to be true in order for the integration to work aurately anyway, this assumption perfetly reasonable Hamming Preditor-Corretor This multistep algorithm (also from Hawkes & Kasper [1], and similar to that used by Natali et al [3] ) differs from the previous routines in that instead of using only the urrent point to predit the next point via Euler type steps, the new point is alulated by forming an appropriate linear ombination of the preeeding ones. Sine this an be done in different ways, it automatially provides an auray ontrol, Whereas all of the other routines tend to propagate and aumilate errors as the integration proeeds. The Hamming algorithm breaks down into two parts, the preditor (whih uses the previous four positions to predit the next one), and the orredtor (whih orrets the predited position by using information supplied by the DERIVS routine). The auray hek omes from omparing the predited and orreted results, suh that the time step is halved if the auray falls below some user defined level (errs), and doubled if the auray is signifiantly better than errs (two hundred times better in this partiular ase). While this hanging time step should mean that the ode works effiiently and auratly, the ode does have the drawbak that it is not self-starting, requiring four Runge-Kutta steps to be arried out before it an be used, and also for restarting the algorithm when the time step is altered (as the stored points orrespond to the old time step). From Natali et al [3], this drawbak should lead to an error of only 0.01%. Clearly, the extrapolative Runge-Kutta will perform better than the Euler and basi Runge-Kutta routines, and so the question that remains is whether the Adaptive Runge-Kutta, the Extrapolative Runge-Kutta or the Hamming Preditor orretor is the most aurate. This an be asertained firstly be simply heking that the eletron trajetories have a single, well defined foal point for rays near the 13

14 axis, and also by heking that at higher angles the foal point follows the expeted form due to spherial aberration. As well as this, we know that the potential and partile form a energetially losed system, and so if we add together the potential energy for the eletron at it's urrent position and it's kineti energy, then we an hek whether this figure is onstant, as it should be. This tehnique an also be used to investigate the quality of the interpolation and differentation algorithms. 3.4 Calulation of Lens Properties: As the eletron trajetories are alulated, part of the PATHS routine heks whether the last step has aused the eletron to ross the optial axis, in other words to see if the foal point has been alulated. Using this ondition the routine then stores the foal point (zf) and the slope of the ray (tan(a)) in an array, whih an then be used to alulate the paraxial fous (zfp) and the third-order spherial aberration oeffiient (Cs) using equation 8. This is ahieved by the FINELABBERS routine by using the SVDFIT least-square algorithm to finding the oeffiients of a polynomial of the form: y = a2x 2 + a1 Where x orresponds to the slope tan(a), y orresponds to the diret ray foal point zf, and a1 and a2 orrespond to the paraxial fous and the aberration oeffiient respetively. When the oeffiients have been found they are printed to the sreen, along with an estimate of the error in eah figure (alulated by SVDVAR, another routine from Press et al [2] ). Unlike the paraxial routine, the diret ray alulation only returns the value of the spherial aberration when it is atually affeting the foal length, and the quality of the fit depends on the number of trajetories that have been alulated. 14

15 4. Results & Disussion 4.1 Validation Of The Finite Element Potential The axial potential of a standardised two-tube eletrostati eletron lens was ompared with the published results of Natali et al [3]. As the published data onsists of tabulated potential values (for one half of the eletron lens), the finite element mesh was defined suh that the points on the axis for the mesh math up with the tabulated values (Appendix A.1 shows the finite element mesh used). The tabulated values are presented below, along with a graph of the axial potential from both tables, with the present result plotted as points and the previous paper's potential plotted as a line (figure 4). Just by visual omparison the two results learly agree, and analysis of the tabulated data has shown that the differene between the two alulations was only 0.02%. On the basis of this evidene, the finite element potential alulation is seen to be working well. z/d Eletrostati Potential Present results Natali et al Data from Natali et al Finite Element Result Figure 4: Comparison of the present results with those of Natali et al [3]. 15

16 4.2 Analyis And Development Of The Diret Ray Traing Approah All the following results were taken for the same thin lens as that used by Liu & Ximen [5] (the form of the finite mesh used to desribe it is supplied in appendix A.2) Interpolation Of The Axial Potential From The Finite Element Mesh The first test arried out on the interpolation was to hek that the interpolated axial potential agreed with the values of the potential on the finite element nodes along the axis. Figure 5 ompares the four interpolation tehniques against the node potential. 1.2 Nearest Neighbour Linear Interpolation Nodal Potential 1.2 Linear Interpolation Nodal Potential Cubi Spline Interpolation Cubi Least-Squares Polynomial Interpolation Nodal Potential Nodal Potential Figure 5: Comparison of the axial potential for the four methods of interpolation with the nodal potential from the finite element alulation. Unfortunately, these results give no lear evidene of any one routine performing any better than any of the others, all of them agreeing reasonably with the finite element node potentials. This being the ase, the differential of the potential must be examined in order to determine whih between of the tehniques is the best. 16

17 4.2.2 Differentiation Of The Eletrostati Potential Figure 6 presents the z omponent of the eletri field for the two linear interpolation tehniques, and figures 7 and 8 ompare the same quantity alulated by the spline and least-squares approahes respetively. For these latter two, the left-hand plot orresponds to linear interpolation, the middle plot to quadrati and the right-hand plot to ubi interpolation. For eah of these a range of saling fators are ompared (nths = 1, 2, 3). 120 NN Linear 140 Linear element tehnique Figure 6: Plots of Ez as alulated by the nearest neighbour and single element linear interpolation nths = 1 nths = 2 nths = nths = 1 nths = 2 nths = nths = 1 nths = 2 nths = Figure 7: Comparison of the spline interpolation routines for linear, quadrati and ubi splines left to right. Eah diagram ompares the results from using a saling fator of 1, 2 and nths = 1 nths = 2 nths = nths = 1 nths = 2 nths = nths = 1 nths = 2 nths = Figure 8: Comparison of the least-squares polynomial interpolation routines for linear, quadrati and ubi splines left to right. Eah diagram ompares the results from using a saling fator of 1, 2 and 3. 17

18 Both of the linear interpolation tehniques an learly be seen to form a poor approximation to the eletri field, and loser inspetion of the results in figures 7 and 8 (by heking whih best onserves the total energy of the potential and the eletron) has shown that the least-squares ubi approximation forms the smoothst interpolation, whilst also being least dependant on the saling fator. This means that the tehnique should be able to ope better with the different variations in any finite element mesh it is applied to. The routine was refined by searhing for the value for the saling fator whih minimised the error in the total energy for a ray passing through the lens, and this investigation found that the best value of the saling fator for the ubi interpolation is nths = 1.75 (whih orrespond energy being onserved with 0.01% auray Integration Of The Equations Of Motion When the Adaptive Runge-Kutta routine was removed, and then replaed with the theoretially less aurate Euler routine, there was an immediate improvement in the quality of the results. This apparent paradox ourred beause while the Adaptive Runge-Kutta works well with most smoothly varying data, it appeared to be unable to ope with data that has a very small degree of variation. After disovering this, the Euler alrogithm was replae with the fourth order Runge-Kutta routine, and then with the extrapolative Runge-Kutta, with the auray of the results improving along the way. However, it was not immediately apparent whether the Hamming Preditor-Corretor or the extrapolative Runge-Kutta was giving the best results, and this setion deals with the differentiation between the quality of these two routines. The extrapolative Runge-Kutta routine was tested by alulating the error in the total energy for a single ray passing through the lens at a fixed angle (1 to the optial axis, starting on the axis at the lefthand side of the devie) for a range of values for the time step. The time step was altered by taking the estimated time step value and multiplying it by an onstant less than or equal to 1.0. In the ase of the Hamming preditor orretor routine, the same alulation was arried out using the best time step from the Runge-Kutta results as the initial time step and then setting the error parameter (errs) to a range of values to see how it affeted the results. The data from these tests is presented in the table overleaf. 18

19 Runge-Kutta Hamming Preitor-Corretor Time Step Energy Error Auray Fator Energy Error e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-04 The result for the extrapolative Runge-Kutta routine is as would be expeted for any single step algorithm. While dereasing the time step initially dereases the error in the results, for very small time steps the aumulated errors begin to beome signifiant, and so the auray of the integration dereases. The balane between these two types of behaviour is therefore found to our at a time step fator value of 0.5, and so this value gives the best results for the Runge-Kutta integration. This figure was then used to find the initial time-step for the Hamming routine. The behaviour of the Hamming routine an be explained in terms of the auray fator (errs) as follows. For the larger values of the auray fator (errs = 1.0e-2-1.0e-5), the initial time step is suffiient to satisfy the error riterion, but as it is dereased furthur the routine has to hange the timestep in order to satisfy the same auray ondition. Unfortunately, when this happens the Hamming ode has to be restarted with the extrapolative Runge-Kutta, and this auses errors to form in the results (espeially if the time step is hanging rapidly, in whih ase the Runge-Kutta aumulates errors as before). However, in this region, the alulated error is approximately onstant for the different values of errs, whih implies that while the Hamming ode is less aurate than an extrapolative Runge-Kutta routine with an optimised time step, it is more onsistently aurate than the Runge-Kutta for a range of auray onditions. This means that the Hamming ode is more flexible and so more likely to work well for different problems, where the optimum time step may differ from the value found here. The only drawbak is that while the Runge-Kutta is always reasonable quik (~ 20-30s per ray), the Hamming ode slows down as the auray riterion starts to work, and so when a devie is being simulated it is best to start with errs ~ 1.0e-4 (to get a rough estimate of lens behaviour) and then hange this to errs ~ 1.0e-9 when more auray is required. Although the auray fator an be set higher than this, the ode starts to run very slow indeed (~ 2-3 minutes per ray) and there is very little inrease in auray (due to the Runge-Kutta/Hamming interfae). 19

20 4.3 Example Ray Diagrams For The Different Tehniques The overall ray diagram results from various tehniques are presented here for diret visual omparison. Figure 9 shows the ray diagram for a simple two-tube eletron lens as alulated by the paraxial approximation, and figures 10 and 11 ompare the results for the same lens as alulated by the diret ray traing ode before this projet began with the results the ode now produes Figure 9: Ray diagram for a simple two-tube eletron lens as alulated by the paraxial approximation Figure 10: Ray diagram for a simple two-tube eletron lens as alulated by the diret ray traing ode before this projet began Figure 11: Ray diagram for a simple two-tube eletron lens as alulated by the new diret ray trading algorithm. 20

21 The non-physial behaviour of the old ode has been eliminated from the diret ray traing ode to the point that the ray diagrams now agree with the paraxial ray diagrams for low angle rays. Moreover, the effet of spherial aberration is now lear, with the diret ray fous moving inward from the paraxial foal point as the angle inreases. The degree of agreement between the paraxial and diret ray traing will be examined further in the next setion. 4.4 Comparison With Published Data For The Two-Tube Eletron Lens The table below gives the results from a series of simulations of the standardised two-tube eletron lens (with g/d = 0.1) for a range of voltage ratios. The paraxial fous and third order spherial aberration oeffiients from the diret ray alulation are ompared with those from the paraxial approximation ode and the published results of Liu & Ximen [5]. The diret ray alulations were arried out using the Hamming PC ode (errs = 0.1e-6) and the mesh interpolation performed by the ubi least-squares approximation (nths = 1.75). f Cs V1/V2 Diret Paraxial Published Diret Paraxial Published Examining the progression in the foal point with inreasing voltage ratio, it appears that while the diret raytraing ode works well at lower voltage ratios, as that ratio inreases the auray of the ode dereases. This is beause the more rapidly hanging eletri field is more diffiult to desribe aurately, and so greater errors are made by the approximation whih are then arried through into the integration. The reason for the disrepany between the paraxial approximation and the published results is unknown, as the details of the paraxial algorithm have not been studied here. However, the paraxial approximation onsistently gives a reasonable approximation to the published results. The diret alulation of the third-order aberration oeffiient is somewhat more troublesome. This is mainly beause the diret tehnique requires a number of rays to be simulated where the effet of the aberration is fairly pronouned (due to the errors in the ray traing proess), and so when the aberration is small, it beomes more diffiult to get the simulation to atually display spherial aberration behaviour so that it an be alulated. The disrepany between the published results and the results of both the paraxial and diret ray traing ode is of unknown origin, but the fat that the paraxial ode and diret ode approximately agree (oupled with the onsistently reasonable behaviour of these methods for all the other results) implies that the published results may be inorret for this partiular lens definition. 21