3D Field Computation and Ray-tracing

Transcription

1 3D 3D Family 3D Field Computation and Ray-tracing 3D computes the properties of electrostatic and magnetic electron optical systems, using a fully 3D potential computation and direct electron ray-tracing around 3D electrode and polepiece structures. Equipotentials in 2D sections of the structure in any (x,y), (y,z) or (z,x) plane can also be plotted. In cases where the structure has a straight optical axis but a departure from rotational symmetry (e.g. an off-axis electrostatic secondary electron collector in a SEM column or a magnetic matrix lens), the axial field functions can be extracted and plotted. These axial field functions can then be used to compute the optical properties of the system, taking the 3D deflection and quadrupole fields into account. The spot shape at the image plane can then be plotted, including the beam shift, coma and astigmatism caused by the 3D fields. The finite difference method (FDM) is used to compute the potential distribution, wherein the potentials are obtained at points on a 3D rectangular grid. However, the surfaces of the electrodes do not have to conform to the grid lines and thus, 3D objects of varied shape can be analysed without the need for the user to define a complicated mesh to fit around the objects. The regular mesh topology afforded by the FDM and use of a point relaxation method to solve the equations enormously reduces the storage requirements compared with other methods (eg. finite element). The programs require one main data file to run, in which the FD mesh, the object structures and potentials (or permittivities), the boundary conditions and any required symmetry planes in x, y or z, are defined. The data file is in free format, uses key-words to define the data and allows the use of comment lines, thereby making the file easily readable and compact. The specification of the FD mesh is achieved by defining independently the x, y and z mesh line numbers and their positions; the use of the 3D Cartesian grid makes the mesh definition compact. The form of each electrode or dielectric structure involves three levels of hierarchy, namely: surfaces, objects and groups. Each object consists of a number of intersecting surfaces, and several objects at the same potential can be collected together in a group. The user can define each surface of the object separately, or can use key-words to define all the surfaces for the commonly encountered shapes listed below. Each object is usually composed of one solid object from Figure 5, plus one or more cutting planes or holes. Key-Word Arguments Purpose CYLINDER x 1, y 1, z 1, x 2, y 2, z 2, r 1, r 2 Defines a cylinder ELLIPCYL x 1, y 1, z 1, x 2, y 2, z 2, x 3, y 3, z 3, r 1, r 2 Defines an elliptical cylinder SPHERE x c, y c, z c, r 1, r 2 Defines a sphere BOX x c, y c, z c, x p, y p, z p, x q, y q, z q, s u, s v, s w Defines a box CONE x 1, y 1, z 1, x 2, y 2, z 2, α Defines a cone FRUSTUM x 1, y 1, z 1, x 2, y 2, z 2, r 1, r 2, r 3, r 4 Defines a frustum CYLHOLE x 1, y 1, z 1, x 2, y 2, z 2, r Defines a cylindrical hole ECYLHOLE x 1, y 1, z 1, x 2, y 2, z 2, x 3, y 3, z 3, r Defines an elliptical cylindrical hole SPHHOLE x c, y c, z c, r Defines a spherical hole CONHOLE x 1, y 1, z 1, x 2, y 2, z 2, α Defines a conical hole PLANE x p, y p, z p, x n, y n, z n Defines a plane cut GENERAL c, c 1, c 2, c 3, c 4, c 5, c 6, c 7, c 8, c 9 Defines a general quadric surface Figure 1: Command list for 3D object definition

2 For the calculation of trajectories the initial positions, energies and emission angles of each charged particle are specified. The path of the particle (whose mass and charge can be specified) through the 3D fields is computed using a Runge-Kutta algorithm, and the data at the initial and final point on the ray are output, as is the position and velocity at each step of the ray-trace, if required. To compute the optical properties, the program reads the object and image planes, and the beam halfangle. The beam can be interactively focused at the selected image plane by adjusting any of the electrode potentials or the magnetic field either manually or automatically in an autofocus mode. The first-order optical properties and the normal and 3D aberration coefficients are printed out. The field computation parts of 3D are available separately, if required. These modules are used to compute the fields for 3D optical elements, such as multipole lenses and Wien filters, and the axial fields can be interfaced with many of our optical design packages, such as WIEN, MULTIPOLE and IMAGE. Also, the electrostatic and magnetic parts of the 3D package (EO-3D and MO-3D, respectively) are also available separately, if required. Sample input and output from the 3D package is shown below. TITLE Example Secondary Electron collector UNITS Millimetres MESH X-lines Y-lines Z-lines OBJECTS ELECTRODE lens FRUSTUM 2, 1, 5 1, 5 65 ELECTRODE lens FRUSTUM 35, 1, 5 7, 1 35 Lens polepiece Secondaries Tilted Sample Equipotentials Collector Ray and equipotential output GRID sample BOX , ,, ELECTRODE collector CYLINDER -65 5, -5 5, 1 DIELECTRIC light_pipe CYLINDER -5 5, -3 5, 1 BOUNDARIES N lens, N lens, chamber_base lens POTENTIALS GROUP lens -1. GROUP sample. GROUP collector 1. GROUP chamber_base -2. PERMITTIVITIES GROUP light_pipe 5. Finite difference mesh layout SYMMETRY S N N Example electrostatic data file

3 5.7e-5 electron beam T magnetic scalar equipotentials 1.19e MILLIMETRES 25. Axial deflection function Semi angle (mr) = 1. Beam energy (ev) = 5. Beam ripple (ev) = u m Dy Spherical Cs(mm) = Chromatic Cc(mm) = X-shift Dx(um) = Y-shift Dy(um) = Coma Cx(um) = coeffs Cy(um) = lens bores Astig. Ax(um) = coeffs Ay(um) = Dx 25. um Defocus Dzi(um) =. Spot for optic axis through side bore of matrix lens Rays and equipotentials in magnetic matrix lens Example Secondary Electron collector z-y plane, x = mm Conical Magnetic Lens with Side Hole z-y plane, x = mm lower z = -13 mm, upper z = 3 mm lower y = -7 mm, upper y = 7 mm Rays and equipotentials in combined field system Screen-shot of 3D software in action

4 Space Charge Upgrade for 3D Software 3D Family GO-3D is an upgrade to the 3D package that computes the properties of electrostatic electron optical systems, using a fully three-dimensional (3D) potential computation and direct electron ray-tracing around 3D electrode and dielectric structures, taking space-charge effects into account. The software also handles emission from flat cathodes. GO-3D takes magnetic fields into account in computing the electrostatic potentials and rays, if the corresponding 3D magnetic field has already been computed. The data format and running of the programs is similar to the 3D package. To compute the effects of volume space charge, the potential distribution, φ, should be a solution of Poisson s equation 2 = ρ φ ε At the outset of the solution, neither φ nor the space charge distribution, ρ, are known. Poisson s equation is solved by an iterative scheme, which involves successively computing the 3D electrostatic potentials, cathode emission current (if a flat cathode is present) and space charge density distribution, until a self-consistent solution is obtained. A flowchart showing how the program works when a cathode region is analysed is shown in Figure 51. If the cathode region is not analysed, the user can assign a specified current to each of the electron rays defined in the ray-trace initial conditions. Alternatively, GO-3D will interface with the SOURCE software and reads the output data from a 2D cathode region analysed using SOURCE and computes the space charge effects in the subsequent 3D electric and magnetic field regions. The space charge distribution is computed by tracing the currentcarrying rays through the system. As a ray carrying current I passes through each cell of the mesh (Figure 52), it deposits a charge Q = I. t, where t is the transit time of the ray through the cell. The space charge density in the cell is increased by ρ = Q vol where vol is the cell volume. An incremental charge density, δρ = ρ/8, is then assigned to each node bounding the cell. Figure 2: GO-3D Flowchart Flat cathode regions are defined using a relatively small number of key-words, specifying the cathode shape, temperature, work function, Richardson constant, etc. Some examples of the graphical output from the package are shown below. Figure 3: Assignment of space charge

5 cathode grid anode (b) (a) (c) Figure 4: Equipotentials and rays in triple in-line CRT gun. (a) Cross-section through the gun, along the optical axis. (b) Section normal to the optic axis through the grid, showing the circular apertures. (c) Section normal to the optic axis through the anode, cathode grid (a) Figure 5: Equipotentials and rays for Poisson solution of SCALPEL-type gun. (a) cross-section through the gun, along the optical axis. (b) cross-section through the grid, normal to the optical axis (b)