Dr. Omer SiseAfyon Kocatepe University, TURKEYomersise@aku.edu.tr

Lecture 1. Introductory Guide to Electrostatic Lenses

Charged Particle Optics: Theory & Simulation

My Current Adress:Suleyman Demirel University, TURKEYomersise@sdu.edu.tr

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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 or

cylinders.• ability– calculate the focal and aberration properties of a range of

lenses– design a beam transport system

Why we need electrostatic lenses anyway?

Many physicists using electron or ion spectrometers, electron/ion gun and beam transport system needs some knowledge of electrostatic lenses.

This lecture is an introduction and I will do my best to let you in on the basics first and than we will discuss some of the applications of electrostatic lenses.

This is recommended to people who are studying or work with vacuum electronics and charged-particle beam technology: students, postgraduate students, engineers, research workers.

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What is Electrostatic Lenses?

Some types of specially shaped electric fields possess the property to focus electron beams passing through them. The fields are known as electrostatic lenses.

An electrostatic lens is a device that assists in the transport of charged particles.

Electrostatic lenses are used for • acceleration/deceleration of charged

particles, • confinement of beams in beam

transporting units,• focusing beams prior to their entrance

into the energy or mass analyzer.

• CPO design tools needed to understand electrostatic and magnetic systems.

• They have served to broaden the understanding of electron motion in vacuum throughout the instruments such as electron microscopes, cathode ray tubes, accelerators, and electron guns.

Simulation tools

Basic tools of the trade

Many other specialized programs, e.g for CPO design (e.g. ion traps, electron microscopes) not covered in this lecture series.

SIMION ion optics simulation program (computing electric and magnetic fields and ion trajectories)

http://simion.com/

Simulation of a round field emission cathode at the tip of a cone in CPO-3D

http://electronoptics.com/

An understanding of the principles and behavior of lens systems is essential and readily described by simulation of their fields and trajectories.

Recommended Reading

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 and 2,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 without optical lenses. Therefore, Charged Particle Optics requires knowledge of optical elements: electrostatic and magnetic lenses.

• Analogy between Light Optics and Charged Particle Optics is useful but limited.

• It is customary in charged particle optics discussions to make use of the same terminology and formulae. Terms have a one-to-one correspondence between the two fields.

Introductory remarks

(a)

(b)

Electrostatic Lenses

Optical Lenses

Charged particles

Light rays

V1

n1

V2

n2n1

Optical Analogy• Charged particles optics are very

close analogue of light optics, and one can understand most of the principles of a charged particle beam by thinking of the particles as ray of light.

• In CPO, we can employ two electrodes held at different potentials for focusing, where the gap between the cylinders works as a lens. In light optics, refraction is accomplished when a wavelength of light moves from air into glass.

In light optics:

In charged particle optics:

Snell Law

1221 /sin/sin nn=αα

( ) 21

1221 /sin/sin VV=αα

The path of the ray of the light refracts on crossing boundary between two media having refractive index n1 and n2 while the trajectory of the charged particle deviates on a boundary separating regions having potentials V1 and V2. The directions in two regions being related by Snell's law is determined by sinθ1/sinθ2 = n2/n1 in light optics, but in charged particle optics this equation is formed by sinθ 1/sinθ2 = (V2/V1)1/2.

Significant differences• In light optics there is one refractive surface when the ray passes to

another region; however, in particle optics, there are an infinite number of equipotential surfaces which deviate the beam of charged particles at different regions. Changes in the “refractive index” are gradual so rays are continuous curves rather than broken straight lines.

• The other difference is the effect of space charge, due to the mutual repulsion of the charged particles, on image formation. An excellent summary of this and other limitations of the analogy between 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 be formed arbitrarily due to the Laplace equation (Scherzer theorem).

Action of a Lens

Converging Lens

Diverging Lens

The equipotential lines in the plot indicate the intersection with the plane of the drawing of surfaces on which the electrostatic potential is a constant.

Lens Parameters

• For any of electrostatic lens it is possible to define focal points, principal planes, and focal lengths in the same manner as for light lenses and to determine with their aid linear or angular magnification for any object position. All the 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 the

object space, and behind the lens it is called the image space.

Focal and Principal Planes

• A paraxial trajectory entering the lens from the object space parallel to the optic axis is bent by the lens field;

• This trajectory (or its asymptote in the backward direction) in the image space cross the optic axis at the profile plane, called the focal plane of the lens.

• The asymptotes of the considered trajectory from the object space and from the image space intersect at some plane H2; this plane is called the principal plane of the lens.

• The distance f2=F2-H2 is called the focal length of the lens.

• Similarly, one can consider a paraxial particle trajectory entering the lens field in the backward direction from the image space parallel to the optic axis.

Electron Optical Properties• Electron optical properties of a lens are determined by

positions of the principal planes and one of focal lengths. • Because of that the set of focal and principal planes is

called cardinal elements of a lens. • When on both sides of a lens potential is constant and has

the same value it is called a unipotential (or einzel) lens. • If potentials are constant but have different values on the

two sides of the lens, it is known as an immersion lens.

Helmholtz-Lagrange Law• The linear magnification, M, relates the size of the image to

the size of the object. M is given simply by the ratio of the final to the initial beam diameter in the radial axis, r2/r1.

• Equally important in electron optics is to understand how the angular divergence of an electron beam will change during the image formation process. The so-called angular magnification is then given by Mα . It is interesting, and also a very important result, that if we multiply together the two equations for the angular and linear magnifications, the product is always equal to

This is the Abbe–Helmholtz sine approximation to the Helmholtz–Lagrange law described by

• we may evaluate the magnitude of the final image

• This expression shows that the cross section of the beam in the target plane (reducing r2), can be obtained by reducing the cathode size (r1), the potential in the near cathode region (V1) and the aperture angle α1 at the cathode side. However, the reduction of the numerator of this expression can hardly be accomplished in practice.

Prof. G. King, Lecture Notes

In CPO, apertures are used in electrostatic lenses to define the beam. A window aperture defines the radial size of the beam and a pupil aperture defines the angular extent of the beam. The lens produces an image of the window. As the beam has passed from potential V1 to V2, there has also been a change of energy. The angular extent of the beam is minimized by placing the pupil at the focal length of the lens. This produces a zero beam angle and hence 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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