Paper presented at Tenth Anniversary International Conference Varna on Electron Beam Technologies, Varna, Bulgaria, 1-4 June 2012. Proceedings published in 'Elektrotechnica & Elektronica' Vol 47 No 5-6/2012
By The Union of Electronics, Electrical Engineering and Telecommunications/CEEC/, Bulgaria
The design of electron guns, lenses and deflection systems has advanced significantly since the introduction of computer modelling of electrostatic and electromagnetic systems. In particular, development of high power guns used for welding and melting, where space charge plays a significant role in determining the beam qualities, has depended upon accurate modelling.
Typically, the design process has involved taking a first estimate at the design, analysing this and then seeking to improve the output by adjusting the geometry or operating parameters. The accuracy and solution speed available today has allowed a new approach to electron system designs to be taken. Trending uses automatic model generation to look across a wide variety of operating characteristics, geometries and material properties. Rather than providing a single optimized design, hundreds of models are analysed to quantify the effect of multiple changes. This results in a design that is the best available within practical constraints, rather than one locally optimized. Also an understanding is gained of the problem space and identification of critical design features. Examples will be described where trending has been successfully applied to provide high integrity electron beam designs.
Introduction
The development of computer algorithms and the increased speed of personal computers have enabled new approaches to computer modelling to be taken. In particular this paper will look at new approaches to electron gun optics that have been enabled by these developments. To harness this potential for more varied and higher performing systems some rules are suggested, with justification for their reasoning.
Simple characterisation
Design of electron guns or lens or deflection systems can only be shown to be effective if characterisation techniques are developed which are:
Quantitative - allowing designs to be compared systematically
Capable of being validated - ie properties of the electron beam are characterised in a way that can be measured in real life
One of the key advantages of electron optics analysis software is that it allows electron trajectories and fields to be visualised in way that could never be directly observed on a gun design. This also carries with it the risk that the designer does not verify the model predictions with the observed performance, not least as the model predictions, unless properly assessed may not be directly observable.
So this leads to a primary rule I have imposed upon our own electron optic design: The end result of the model should be observable in the real world.
This rule should not be seen as restrictive but rather one of enabling design without falling into the pit falls of errors generated by modelling method or computing algorithm errors. Within this rule it is also reasonable to optimise design around controlling field patterns (e.g. in a deflection coil) or electron trajectories (e.g. as they escape from the cathode surface) that could not be directly observed in the real world, with the proviso that it is possible to calibrate the model through real world observations e.g. beam current, beam angle or spot size.
An example of this simple characterisation approach is to understand an electron gun through examination of the beam parameters over the working current range of the gun, and this approach involves the following steps
- Develop a means of quantifying the beam characteristics
- Develop macro code for adjusting the bias level in the model
- Develop macro code to allow the bias to be swept over a range
- Predetermine through estimate and trial the range of bias voltage required to sweep the beam current over the required range.
The quantification of beam characteristics in a model is typically carried out by some interrogation of the trajectories produced by the solution, at a position beyond the cathode where the electrons are travelling without acceleration (i.e. in a straight line at near-constant velocity). Some of this functionality is available within post processors although within this work the specific parameters of interest required additional macro software development.
The ideal electron beam is laminar (beamlets appear to originate from a single point source). Such a beam can be focused to a single point. Conversely a beam where the beamlets are not laminar will not be focused to a single point. Beams will not be perfectly laminar, but a measure of the merit of a gun design is to determine how close to this ideal has been achieved, and in quantifying this comparisons can be made between different gun designs.
The beam quality metric should be an inherent quality of the beam (i.e. can be measured to be the same at any z position once the beam is fully accelerated). From a file containing the data on the velocity (vrn, vzn ), position (xn, yn, zn ) and current (in ) of trajectories intersecting a plane after the anode, the apparent source of each trajectory is calculated (zsourcen), see equation (1).
Current weighted averages are used to compute the apparent source position (zimean ), see equation (2), the apparent source size (rimean ), see equation (5) and the beam divergence angle (Ang_divimean ). The calculation of variance in source position (zvar ), see equation (3) and percentage variation in proportion to the distance to the focal plane of the first lens (uivar ), see equation (4) have also been found to be useful beam characteristics that can be quantified.
The use of current weighted averages determines overall beam characteristics such as beam brightness, see equations (6) to (8). Depending on the electron-optical requirements these characteristics can be used to determine the performance of any potential modelled system.
Macro code is software written to automatically drive the modelling software and is available in various forms for most modelling packages.
Characterisation through multi-parameter sweep
It is a single step from the simple characterisation approach above to one where the 'working envelope' of a potential solution is fully explored and quantitatively characterised. For example, to fully assess the performance of a gun design the emerging beam current and brightness can be assessed for a range of cathode temperatures and bias (Wehnelt electrode) potentials.
This is executed as two nested loops in macro code so that for each cathode temperature the full bias potential range is explored and at each combination of cathode temperature and bias potential the emerging beam current and beam brightness are recorded.
An example for a triode gun with a directly heated filament is shown in Fig.1. This shows the peaking characteristic curves (beam current vs bias voltage) for a wide range of cathode temperature. Fig.1(a) shows the curves predicted by the models (each point on the graph is a distinct model solution), and Fig.1(b) shows the measured characteristics over a range of filament currents (each current relating to a cathode temperature - the form and absolute values of these curves are in good agreement.
A similar approach has been carried out to assess a variety of lens designs. In this case the optimum combination of lens pole piece gap (ie the length of the lens between the shroud end plates) and the bore of the lens pole pieces was sought to give the minimum beam aberration at a specific focal strength. The designs are electrically different as the coils will each carry a different value of ampere-turns in order to provide an equivalent focal strength lens. They are also geometrically different in order to have a shorter or longer pole piece gap, or a smaller or larger diameter bore. To address this problem macros were written to:
- Calculate the lens excitation in ampere-turns from an empirical formula,
- To generate alens geometry of a given pole piece gap and lens bore.
The modelling software was then used to resolve the magnetic field distribution for the coil and shroud assembly and then to plot the trajectory of a beam of electrons of fixed width in order to assess the aberration of the lens through measurement of the focal spot size. Although the thin lens formula provides an approximation of the lens aberration, in many cases, particularly for electron beam processing equipment, the lenses do not fall within the thin lens approximation. This investigation allowed the optimum lens geometry to be found for a specified beam diameter and required focal strength.
Random solution analysis
So far we have described looking at the results of analysis when particular electrical parameters or geometric dimensions are varied over a fixed range. Nested loops within macro code produce a model for each combination of parameters and the beam quality is analysed.
When there are many degrees of freedom it is less convenient to specify each range and produce a model for each combination of parameters. As the number of models required is the product of the number of settings for each parameter or dimension, the solution times can become prohibitive when many parameters and settings are being examined. An alternative approach was adopted where the parameters or dimensions are set to random values and solutions obtained across the problem space.
With this type of solution method, there are frequently combinations of parameters and dimensions that do not provide a valid model. Methods of error trapping were investigated, although it was found that the large number of constraints and potential interactions between parameters made trapping all potential errors difficult. It was found that the most effective way to trap errors was to use the modeller itself as a means of checking physical validity.
There are a number of disadvantages with this approach - not least that there is no built in tendency for convergence in finding optimum settings - it is simply a random investigation of the problem space. The development was useful in respect of laying a foundation for two potential alternative methods. One of those - design evolution - is described below. The other method, the use of a Design of Experiments approach to provide an understanding of the parameters, both individually and in combination, influence on the required characteristics is work that is still on going and will be described more fully in a later paper.
Design evolution
Evolution can be described as the process of continuous design optimisation through selection of the best, and then deriving descendents with some random variation from this standard. This process has a tendency to converge and has the following key features:
- Survival depends on superior performance relative to other 'current best' solutions,
- Survivor descendents are similar but varied.
Most of the required properties to set up a design evolution environment have already been developed in the previous modelling approaches. In order to satisfy the above criteria we must develop methods to measure the performance in a manner where a direct comparison can be carried out against alternative designs. This will enable us to determine which design survives. Also it is necessary to have a method of generating descendent designs from the survivor - each design slightly different to the others, but by no more than incremental steps to one or two of the degrees of freedom in the model. The error trapping method can also be applied and it essentially prevents the evolution method entering into a combination of parameters and dimensions that are not possible to model.
This process was applied to a 60kV electron gun diode design. The number of degrees of freedom can be chosen by the designer - for example for the gun electrode geometries each vertex can be moveable and more vertexes can be added as seen fit. Some parts of the geometry can be fixed if required - e.g. the cathode diameter. Other parts of the geometry were fixed to enable ready comparison of designs - i.e. the cathode z position.
When this methodology is applied we see a convergent process but one which can identify a number of solutions on successive runs. The process is also a flexible one allowing a number of design challenges in electron optics to be developed.
The design of electron optical systems has evolved over the past decades through a process of optimisation and development of best design practice. It is anticipated that with adoption of the techniques described in this paper characteristics will be quantitatively assessed and designs evolution will be accelerated to rates at least an order of magnitude greater.
Acknowledgements
The research leading to these results has received funding from the European Union's Seventh Framework Programme managed by REA - Research Executive Agency [FP7/2007-2013] under grant agreement number 286695 (for FastEBM), 286762 (for HiResEBM) and 286603 (for RingMan).
Colin N. Ribton CPhys, CEng, MInstP, MWeldI - TWI, Cambridge, UK
Colin Ribton graduated from The University of Nottingham in 1984 with a joint honours degree in Pure and Applied Physics. He joined TWI's Electron Beam group in 1985. In 2001 he left TWI to join a company developing novel antenna technologies, where he became Vice President of Application Engineering. He re-joined TWI in 2003 to lead the Electron Beam group.
His roles at TWI have involved him in the computer modelling of electron optics and high voltage components, the design of high voltage power supplies, the design and optimisation of radiation shielding, real-time control system architecture, and the design of digital and analogue electronics. In particular, this has been involved in the development of processes to manufacture major components in power generation, nuclear, aerospace and medical applications. He is presently Technology Consultant in the EB group where he is active in promoting EB technology for new applications.