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
Spherical aberration High-angle rays focused more strongly An ideal lens is parabolic, not spherical
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
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
Effect of f in Image Plane : 1 1 1 p q f Otherwise: Expand: 1 1 1 p q q f f 1 1 x 1 1 x x... f f q q f f q q f f
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
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 14.61 opt.77 18 3 34 14 34 min.61 3 3.91 optimal semi-angle of collection practical resolution
Electromagnetic lenses Current-carrying coils Enclosed in iron Bore Gap Pole Pieces Water-cooled Twin lens
Immersion Lens idealied objective lens immersion objective lens sample in pole piece
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
Lens field axial and radial components axial symmetry strongest field on axis, near pole piece radial component reverses from top to bottom
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
Model of lens field: B Model: Bell-shaped field ("Glockenfeld") B 1 a Determine radial component: B B, a 1 a
Model: Uniform B in lens B B uaua B B B a a
Focusing action Side View Radial field lateral velocity Axial field helical motion + Top View Helical motion radial force Parallel incident beam is focused.
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
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
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
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
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 a @ 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
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
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
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
Ray Diagram: Uniform B Lens f f f Rays follow straight-line paths outside of lens. f
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
Focal length Increasing field
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