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# Lens lect 02

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Lecture 02. Focal and Zoom Lens Properties

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### Lens lect 02

1. 1. Dr. Omer SiseAfyon Kocatepe University, TURKEYomersise@aku.edu.trLecture 2. Focal and Zoom Lens PropertiesCharged Particle Optics:Theory & Simulation1My Current Adress:Suleyman Demirel University, TURKEYomersise@sdu.edu.tromersise.com
2. 2.  So far, we covered: the bits of optics we need to know (ray diagrams,optical elements, lens equation, magnification,focus) why we need electrostatic lenses & how we get it how they work, how electrons travel throughthem This presentation covers the focal and zoom-lens properties of electrostatic lenses I will also mention the (primary) aberrations.Introduction2
3. 3. We use conventional terminology, from light optics, to describe many similar featureshere!Photon Optics and ChargedParticle Optics are EquivalentPhotons or ions can be trapped ina resonatorR RLV VEk, qV>Ek/qV1 V2V1<V2Physical PrinciplePhotons or ions can be energyanalyzed usin a filter3
4. 4. αConvex LensOptic axisPointObjectPointImageExcludedRays• Fraction of rays from the object gathered by the lens is defined bythe semiangle, α• The lens forms a magnified or de-magnified image of an objectPhoton Optics:Imaging with a simple lens4
5. 5. All parallel rays (whether parallel to the optic axis or not) are brought to afocus in a plane at a position depending on their angle to the axisParallel RaysBrought to FocusConvexLensOptic axisObjectPlaneImagePlaneFocalPlaneImage formed after each lens isrotated by 180owith respect to theobjectPhoton Optics:Image Formation5
6. 6. Photon Optics:AberrationsComa AstigmatismField Curvature DistortionSpherical Chromatic6
7. 7. defocusingfocusingdefocusing0 V +V 0 VCharged Particle Optics:Electrostatic LensesLenses are used to focus the beam and adjust the current .Purpose: control the size and shape of the final beam spot. 7
8. 8. Electrostatic Lenseselectronelectronionion0 V + V 0 V0 V - V 0 V8
9. 9. Cardinal Points, Principal Planes,And etc.The representation of the four cardinal points, and the focal and mid-focallengths of the einzel lens is also shown. 9
10. 10. • Electrostatic lenses are fairly simple static fieldsystems, composed of a few electrodes held atpossibly different potentials.• They are widely used in electron and ion beaminstruments, both in sources and for extraction andimaging.• Electrodes of lenses are often made of coaxial• cylinders or• thin apertures(O. Sise, Master Thesis, July 2005)(N. Okumus, BSC Thesis, July 2007)Types of Electrostatic Lenses10
11. 11. Some Examples11
12. 12.  The two-electrodelens is often usedwhen the image andobject are required tobe in space ofdifferent potential. The second electrodeis placed at a higheror lower potential,thus providingacceleration ordeceleration of thebeam.Two-element lens12
13. 13. The electron-opticalproperties of a two-element lens can bepresented in a diagram,showing the imageposition corresponding toa given object distance,with the acceleration ratio(V2/V1>1) as a parameter(the corresponding datafor retarding lenses(V2/V1<1) can be obtainedfrom the same diagram).Magnification lines arealso indicated.Two-element lens:P-Q Diagrams13
14. 14.  In order to focus a beam without changing itsfinal energy, a lens with at least threeelectrodes is necessary.Three-element lens14
15. 15. In an acceleration ordeceleration lens, it isvery often desirable tobe able to keep theimage of a givenobject fixed when theacceleration /deceleration ratio andmagnification arechanged. A lensoperated in this way isusually referred to asa zoom lens.Three-element lens:Zoom Lenses15
16. 16. Calculation of zoom-lenscurves using simplexoptimization which is usedto identify locations of somerepresentative set ofminimums (dots) for athree-aperture lens withA/D = 1, P/D = 5 and Q/D =5. One can roughly see fromthe plots where the surfaceminimums are, but it doesnot precisely identify thelocations of the minimums.Therefore, an optimizationis necessary.Three-element lens:Zoom Lenses16
17. 17.  The paraxial approximation applies whenever the angleand distance between the system’s optical axis and theray of interest are small. This allows the use of the small angle approximations(sin(α) ≈ α, tan(α) ≈ α and cos(α) ≈ 1) when tracing thepath of the ray through an optical system (such as alens). However, when the electrons are not moving close tothe axis, then the basic approximation begins to fail andaberrations start to form. As in light optics, the optics of charged particles suffersfrom a number of image aberrations and distortions. Ofprimary concern are spherical and chromaticaberrations.Paraxial Approximation & Aberrations17
18. 18.  Spherical aberration is most important ascompared with other lens aberrations. It appearsin the cubic dependence of the angle oftrajectory refraction on its radial position.Aberrations:Spherical Aberration18
19. 19. y (µm)-4 -2 0 2 4z(µm)-4-2024y (µm)-4 -2 0 2 4z(µm)-4-2024Analogously, in chargedparticle optics, sphericalaberration refers to thevariation in the focalproperties of the lens withdistance (or angle) from theoptical axisAberrations:Spherical Aberration19
20. 20. 0246810Numberofparticles(x102)0246810r - radial exit displacement (mm)-4 -2 0 2 4012345-4 -2 0 2 4(a) (b)(c) (d)(e) (f)V2/V1=5.0V2/V1=4.0V2/V1=5.5V2/V1=4.5V2/V1=6.0V2/V1=5.0V2/V1=6.5V2/V1=5.5V2/V1=7.0V2/V1=6.0V2/V1=7.5V2/V1=6.5A/D=0.5A/D=0.5A/D=0.5A/D=0.5A/D=0.5A/D=0.51 1111 1Under focusUnder focus Under focusFocusOver focus Over focus20Aberrations
21. 21. The spherical aberration coefficients are obtained fromΔr = −MCsα03.Aberrations:Spherical AberrationCs is a fourth-order polynomial in 1/M:Cs(M) = Cs0 + Cs1/M + Cs2/M2+ Cs3/M3+ Cs4/M421
22. 22. The particles in a beam will vary some in velocity, soparticles with slightly different energies (δE) get focusedat different image planes, and the focal point becomesblurred.Always some spreadin initial electronenergies as leavecathodeW ~ 2 eVLaB6 ~ 1 eVFE ~ 0.2 to 0.5 eVMinimize by decreasein αAberrations:Chromatic Aberration22
23. 23. The chromatic aberration coefficient in the image plane isdescribed by δr = −MCcα0 δE/E0.The same reasoning can be applied to find the chromaticaberration coefficients Ci.Cc = chromatic aberrationcoefficientα = convergence angleDirectly related to focallengthMuch less significant at highE0Aberrations:Chromatic Aberration23
24. 24. Aberrations:Aberration Patterns24
25. 25.  It is not yet clear the extent to which type of lenses is moredesirable over the other.This largely depends on thespecific application and the overall length of the lenssystem employed. Nevertheless, computer optimization of electrostaticlenses is certainly important.This will save time andresources in design of future charged particle opticalsystems. To provide more flexibility, both in terms of energy rangeand for optimization with respect to optical properties,more independent lens elements have to be added. If more degrees of freedom are desirable, it is probablybetter to use some combination of such basic unitsseparated by field-free regions than to design a lens wheremore electrodes are closely spaced.Summary25