I will discuss t he influence of the geometric parameters on the focal points, magnification, and aberration c oefficient s.
the cathode emissivity cannot be as much as desired and, in order to obtain the beam current necessary for normal operation, one has to use a cathode with a comparatively large emissive surface. Reduction of the potential in the near cathode region is inexpedient for two reasons: (1) the lower the potential, the greater the influence of the initial velocities of the electrons, the higher the chromatic aberration of the lens; (2) the slower the electrons, the greater the effect of space charge leading to expansions of the beam due to the Coulomb repulsive forces.
Lens lect 01
Dr. Omer SiseAfyon Kocatepe University, TURKEYomersise@aku.edu.trLecture 1.Introductory Guide to Electrostatic LensesCharged Particle Optics:Theory & SimulationMy Current Adress:Suleyman Demirel University, TURKEYomersise@sdu.edu.tromersise.com
Goals and Objectives• knowledge– analogies between charged particle and light optics– control of electron and positron beams– principles of imaging in a lens– the optics of simple lens systems– paraxial approximation and aberrations– data on simple lenses consisting of multi-apertures orcylinders.• ability– calculate the focal and aberration properties of a range oflenses– design a beam transport system
Why we needelectrostaticlenses anyway?Many physicists using electron orion spectrometers, electron/iongun and beam transport systemneeds some knowledge ofelectrostatic lenses.This lecture is an introduction and I will do my best to letyou in on the basics first and than we will discuss some ofthe applications of electrostatic lenses.This is recommended to people who are studying or workwith vacuum electronics and charged-particle beamtechnology: students, postgraduate students, engineers,research workers.
4What is Electrostatic Lenses?Some types of specially shaped electric fields possessthe property to focus electron beams passing throughthem. The fields are known as electrostatic lenses.An electrostatic lens is a device that assists in thetransport of charged particles.
Electrostatic lenses are used for• acceleration/deceleration of chargedparticles,• confinement of beams in beamtransporting units,• focusing beams prior to their entranceinto the energy or mass analyzer.• CPO design tools needed to understandelectrostatic and magnetic systems.• They have served to broaden theunderstanding of electron motion invacuum throughout the instrumentssuch as electron microscopes, cathoderay tubes, accelerators, and electronguns.Simulation tools
Basic tools of the tradeMany other specialized programs, e.g for CPO design (e.g. ion traps, electronmicroscopes) not covered in this lecture series.SIMION ion optics simulation program(computing electric and magnetic fieldsand ion trajectories)http://simion.com/Simulation of a round field emissioncathode at the tip of a cone in CPO-3Dhttp://electronoptics.com/An understanding of the principles and behavior of lens systems is essentialand readily described by simulation of their fields and trajectories.
Other booksP. Grivet, Electron Optics, Pergamon Press, London, 1965.B. Paszkowski, Electron Optics, Iliffe, London, 1968.O. Klemperer, M.E. Barnett, Electron Optics, third ed., CambridgeUniniversity Press, Cambridge, 1971.E. Harting, F.H. Read, Electrostatic Lenses, Elsevier, Amsterdam,1976.H. Wollnik, Optics of Charged Particles, Academic Press, Orlando,1987M. Szilagyi, Electron and Ion Optics, Plenum, New York, 1988.P. W. Hawkes, E. Kasper, Principles of Electron Optics, vols. 1 and2,Academic Press, London, 1989.El-Kareh A B and El-Kareh J C J, Electron Beams, Lenses and Optics(London: Academic) 1970
• Light optics can hardly be discussed withoutoptical lenses. Therefore, Charged Particle Opticsrequires knowledge of optical elements:electrostatic and magnetic lenses.• Analogy between Light Optics and ChargedParticle Optics is useful but limited.• It is customary in charged particle opticsdiscussions to make use of the same terminologyand formulae. Terms have a one-to-onecorrespondence between the two fields.Introductory remarks
(a)(b)Electrostatic LensesOptical LensesChargedparticlesLightraysV1n1V2n2n1Optical Analogy• Charged particles optics are veryclose analogue of light optics, andone can understand most of theprinciples of a charged particlebeam by thinking of the particlesas ray of light.• In CPO, we can employ twoelectrodes held at differentpotentials for focusing, where thegap between the cylinders worksas a lens. In light optics,refraction is accomplished when awavelength of light moves fromair into glass.
In light optics:In charged particle optics:Snell Law1221 /sin/sin nn=αα( ) 211221 /sin/sin VV=ααThe path of the ray of thelight refracts on crossingboundary between twomedia having refractiveindex n1 and n2 while thetrajectory of the chargedparticle deviates on aboundary separating regionshaving potentials V1 and V2.The directions in two regionsbeing related by Snells lawis determined by sinθ1/sinθ2= n2/n1 in light optics, but incharged particle optics thisequation is formed bysinθ /sinθ = (V /V )1/2.
Significant differences• In light optics there is one refractive surface when the ray passes toanother region; however, in particle optics, there are an infinitenumber of equipotential surfaces which deviate the beam ofcharged particles at different regions. Changes in the “refractiveindex” are gradual so rays are continuous curves rather than brokenstraight lines.• The other difference is the effect of space charge, due to themutual repulsion of the charged particles, on image formation. Anexcellent summary of this and other limitations of the analogybetween light and particle optics is provided in El-Kareh A B and El-Kareh J C J 1970.• In light optics, glass surfaces can be shaped to reduce aberrations,while in CPO aberrations cannot be avoided in round lenses.Because, spatial distribution of the electric potentials cannot beformed arbitrarily due to the Laplace equation (Scherzer theorem).
Action of a LensConverging LensDiverging LensTheequipotentiallines in the plotindicate theintersection withthe plane of thedrawing ofsurfaces onwhich theelectrostaticpotential is aconstant.
Lens Parameters• For any of electrostatic lens it is possible to define focalpoints, principal planes, and focal lengths in the samemanner as for light lenses and to determine with their aidlinear or angular magnification for any object position. Allthe ideal lens formulas apply to electrostatic lenses.• The field of the lens is restricted along the optic axis.• The field-free region in front of the lens is called theobject space, and behind the lens it is called the imagespace.
• A paraxial trajectory entering the lens from the objectspace parallel to the optic axis is bent by the lens field;• This trajectory (or its asymptote in the backwarddirection) in the image space cross the optic axis at theprofile plane, called the focal plane of the lens.• The asymptotes of the considered trajectory from theobject space and from the image space intersect atsome plane H2; this plane is called the principal plane ofthe lens.• The distance f2=F2-H2 is called the focal length of thelens.• Similarly, one can consider a paraxial particle trajectoryentering the lens field in the backward direction fromthe image space parallel to the optic axis.
Electron Optical Properties• Electron optical properties of a lens are determined bypositions of the principal planes and one of focal lengths.• Because of that the set of focal and principal planes iscalled cardinal elements of a lens.• When on both sides of a lens potential is constant and hasthe same value it is called a unipotential (or einzel) lens.• If potentials are constant but have different values on thetwo sides of the lens, it is known as an immersion lens.
Helmholtz-Lagrange Law• The linear magnification, M, relates the size of the image tothe size of the object. M is given simply by the ratio of thefinal to the initial beam diameter in the radial axis, r2/r1.• Equally important in electron optics is to understand how theangular divergence of an electron beam will change duringthe image formation process. The so-called angularmagnification is then given by Mα . It is interesting, and also avery important result, that if we multiply together the twoequations for the angular and linear magnifications, theproduct is always equal toThis is the Abbe–Helmholtz sine approximation to theHelmholtz–Lagrange law described by
• we may evaluate the magnitude of the final image• This expression shows that the cross section of the beamin the target plane (reducing r2), can be obtained byreducing the cathode size (r1), the potential in the nearcathode region (V1) and the aperture angle α1 at thecathode side. However, the reduction of the numerator ofthis expression can hardly be accomplished in practice.
Prof. G. King, Lecture NotesIn CPO, apertures are used in electrostatic lenses to define the beam. Awindow aperture defines the radial size of the beam and a pupil aperturedefines the angular extent of the beam. The lens produces an image of thewindow. As the beam has passed from potential V1 to V2, there has also been achange of energy. The angular extent of the beam is minimized by placing thepupil at the focal length of the lens. This produces a zero beam angle andhence the angular extent of the beam is solely defined by the pencil angle (θ).http://es1.ph.man.ac.uk/george-king/gcking.html
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