Depths of Field & Focus (I) First identify the location and size of the image of a flat (2-D) object by tracing a number of rays.

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Simon Harris
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1 Depths of Field & Focus (I) d First identify the location and sie of the image of a flat (-D) object by tracing a number of rays. d

2 Depths of Field & Focus (II) The image of a point on the object will be out of focus (broadened) in planes above and below the correct image plane. Find the depth of focus: Say the smallest image feature sie is The image is still in focus if D The resolution of the object is M D Recall that M 1 M depth of focus M The depth of focus indicates the range along the optic axis over which the image remains in focus.

3 Depths of Field & Focus (III) The images of points above and below the object plane will be broadened in the image plane. Find the depth of field: A point on the optic axis above of below the object will remain in focus if its rays in the image plane span a lateral distance in the less than the smallest image feature sie. D So, again which gives D D depth of field depth of focus M depth of field

4 Spherical aberration High-angle rays focused more strongly An ideal lens is parabolic, not spherical

5 Parabolic approximation of a sphere 1) Sphere: y R R x Small angle: y x R ) Parabola: y f or: y f x y x 4 f f R

6 Origin of spherical aberration Find C s for a sphere: R 1 cos R f f f f R 1 R R 1 cos 8 16 C f s R f 16 8 f f C s Cs r f r f r Cs f

7 Effect of f in Image Plane : p q f Otherwise: Expand: p q q f f 1 1 x 1 1 x x... f f q q f f q q f f

8 Effect of spherical aberration on resolution Ray misses crossover at Gaussian image plane C s : spherical aberration coefficient (typically 1-3 mm) f C s High mag: q q q M 1 p f f Image: tan Object: s f q q f M C q M q MC s M C s 3 3

9 Optimal Diffraction d.61 Spherical Aberration s Cs 3 Define: 3 C 14 s C s 14 Rewrite: Combine: d.61 s 3 net d s Minimie: d d net opt opt min optimal semi-angle of collection practical resolution

10 Electromagnetic lenses Current-carrying coils Enclosed in iron Bore Gap Pole Pieces Water-cooled Twin lens

11 Immersion Lens idealied objective lens immersion objective lens sample in pole piece

12 Magnetic fields and forces Magnetic fields are caused by electrical currents. Ampère's law: B ds I Moving charges in magnetic fields experience forces. Lorent force law: F q vb cross-product vb vbsin

13 Lens field axial and radial components axial symmetry strongest field on axis, near pole piece radial component reverses from top to bottom

14 Conditions on magnetic field No magnetic monopoles: 1 1 B B B B B //divergence in cylindrical coord s. B B B //Axial symmetry B B d d //Integrate B B Assume:, B B B B

15 Model of lens field: B Model: Bell-shaped field ("Glockenfeld") B 1 a Determine radial component: B B, a 1 a

16 Model: Uniform B in lens B B uaua B B B a a

17 Focusing action Side View Radial field lateral velocity Axial field helical motion + Top View Helical motion radial force Parallel incident beam is focused.

18 Force on Moving Electron FevB //Lorent force law r ρˆ ˆ //electron position vr ρˆ φˆ ˆ B B ˆ B ˆ ρ //velocity //magnetic field F ρˆ φˆ ˆ ρˆ φˆ ˆ qv v v e B B B B B F F ˆ F ˆ F ˆ ρ φ F F eb eb F eb eb

19 Equations of motion Fm r F ˆ F ˆ F ˆ ρ φ cylindrical cartesian ρˆ cos xˆ sin yˆ ˆ sin xˆ cosyˆ ˆ ˆ ρˆ ˆ ˆ //1 ˆ st derivatives ˆ ˆ ρˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ // nd derivatives Find force in cylindrical coord's.: r ρˆ ˆ r ρ ˆ ˆ ˆ r ρˆ φˆ ˆ r ρˆ φˆ ˆ 1 d ρˆ φˆ ˆ dt m m F eb d m F eb eb dt m F eb

20 B Solve (I): component d dt m e B eb B a a B B uaua d e B m aa e B u a u a dt d e B uaua dt eb L m a, a L u a u a L, a a, a Define: Assume: //Larmor Frequency //uniform field in lens //rotation in lens

21 B Solve (II): component B a a L a a m eb u a u a a a L dt uaua aa dt t L t d u u Notice: dt u u //integrate Define: v v L v, a v v v u a u a v, a a v, a Small reduction in velocity inside lens for off-axis rays

22 Solve (III): component m m eb L u a u a B B u a u a L L u a u a t cosl t Csin L t v ta //solution for t a a cos k a Csink a //Focusing! k v L v L v kk 3 L 3 v L L L k v 1 k 3 The lens exhibits spherical aberration. k

23 Find paraxial ray Assume: k k We have: cos k a tan k sink a Paraxial Ray: d d a cosk a Find trajectory exiting lens: ka sin ka a cos d d a k ka a ka sin cos //ray for >a

24 Find focal length Find back focal point: f f 1 1 a ka tan ka Find back principal plane: ka tan 1 a ka f f tan ka 1 1 a a a ka katan ka ka sin ka //Focal length

25 Find Ray II: Through Lens Center cos k a Csink a C tan ka sin sin k ka d d a k tan ka k 1 a tan Find back nodal plane: n ka tan ka n 1 a ka a Nodal and principal planes coincide. //ray for >a

26 Ray Diagram: Uniform B Lens f f f Rays follow straight-line paths outside of lens. f

27 Electron trajectory: : Uniform B Lens Eqn's of motion: Change to Cartesian coord's: d k ka d cos d k d 1 cos 1 cos 1 ysin sin x k a a k L v L 1 TL f T T Paraxial ray: Electrons move in circular orbit, but not about the lens center. v

28 Focal length Increasing field

29 Estimate Spherical Aberration Coefficient f f f k f k k f k kk //Expand in Taylor s series k v L f kk k 1 sin k a f 1 a f k k tan k a kk 1 k k 3 sin ka 1 //Assume strong excitation f kk 1 k f k kk f k f f f 1 k f f k k f f f C s 1 C s f v mv eb L

Depths of Field & Focus

Depths of Field & Focus Depth of Field: If the resolution is, then the entire object is in focus when: d D d D depth of field Depth of Focus: d D Image in focus if: d M D M M 1 M M D M M depth of focus