Depths of Field & Focus

Transcription

1 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 M depth of field

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

3 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

4 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

5 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

6 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

7 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

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

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

10 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

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

12 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

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

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

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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

23 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

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

25 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

26 Focal length Increasing field

27 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