Linear Systems. : object : image. Describes the output of a linear system. input. output. Impulse response function

Size: px

Start display at page:

Download "Linear Systems. : object : image. Describes the output of a linear system. input. output. Impulse response function"

Timothy Walton
6 years ago
Views:

1 Linear Systems Describes the output of a linear system Gx FxHx- xdx FxHx F x G x x output input Impulse response function H x xhx- xdx x If the microscope is a linear system: : object : image G x S F x

2 Transfer Function H(x) is also called the transfer function of the system

3 Microscope as a Linear System F x H x Gx Convolution in direct space: G x F x H x Fourier Transforms: F u G u H u F x G x H x Convolution Theorem: Multiplication in reciprocal space G u F u H u

4 Contributions to H(u) H u A u E u e Au E u u i u : aperture function : envelope (damping) function : aberration (phase) function Most important for phase-contrast imaging: We need to find (u):

5 Path Length Correction due to Lens Ideal lens: p q f s p r q r sr r r r p q sr p q r 1 1 p q sr p q r s r pq s r f We expect the same net path length for all focused rays: sr s pq r s r f

$cn 1 1 1 v f cn refractive indices n sin n sin 1 1 This applies to any type of wave (not only light).$

6 Snell s Law The wave crests must be continuous across the interface: sin sin 1 1 c f sin sin 1 1 v f cn v f cn refractive indices n sin n sin 1 1 This applies to any type of wave (not only light).

Thin Optical Lens (I) air glass air Snell's law: Small angles: sin n sin 1 1 sin nsin 1 n 1 n 1 1 1 1 n n n r tan r f n1 f dt r

7 Thin Optical Lens (I) air glass air Snell's law: Small angles: sin n sin 1 1 sin nsin 1 n 1 n n n n r tan r f n1 f dt r tan dr n 1 f r r dr r trt T T t r r n1 f n1 f Thickness change: r r t r n1 f f p parabola focal length Optical focal Length: f f p n 1

8 Thin Optical Lens (II) off-axis rays have a greater distance to travel Wavelength: vacuum: c f c medium: nf n Phase difference: lens thinner at edges 1 1 r T tr n1tr s r r dt r dr n 1 f 1 s r n t r ds r dr f Optical path-length difference: r rdr r sr r f f

9 Uniform-Field Electron Lens off-axis rays have a greater distance to travel less momentum, longer wavelength

10 Path Length Correction due to Lens A path-length correction for each trajectory through the lens should be included. Generalize: Non-ideal lens: f r f f r d s r r dr f f r r r rf r s r dr f r f d s r r f r 1 dr f f r f f The derivative of the path-length difference determines the focal length.

11 Influence of Spherical Aberration From Chap. 6: r f rcs f r r r r sr C s dr f r f f 4 r 1 r Cs f 4 f normal focusing aberration

12 Spherical aberration (wave optics) When object is at focal point, if no aberration, all rays on image side (back) of lens will be parallel to lens With aberration, off-axis rays will be tend toward the optic axis Cr f 3 s 4 ds dr Cr ds dr dr f C f 3 s 4 4 r s 4 r 4 4 f s 3 s r r dr C r

13 Path-length difference: general case Assume object is on-axis, but not in focus p q f Axial Ray: s z q Lens: Off-axis Ray: s z r q r sr r r r z q sr z q s s s r r net 4 r 1 r s r C f 4 s f r r snet r Cs z q f 4 f 4

14 Combine defocus and sample height p zz f f f z q f pz q f f z f p q f p f z f q q q p f f p f zf p f z q f f M 1 1 f q only the difference matters

15 Optical path-length difference 1 r 1 r snet zf Cs f 4 f 4 r z 1 z 1 z s zf θ C θ f 4 s f 4 4 z z 1 f 1 f q 1 f q z f Net Effect: 1 1 s zf C 4 s 4 θ θ

16 Phase correction Phase shift: s 4 f z 1 C s Scattering angle: R u L d Frequency representation: 3 4 u f z u C u s

17 Image function for weak-phase object A weak phase object produces an image function: First term: * G x F x H x G x 1 i Vt x H x 1 H x u Hu uhu H u 1 G x iv x H x 1 t u The overall phase doesn't matter, so we might as well pick Hu u 1

18 H x Simple situation We might expect the ideal transfer function to be: x 1 * 1 G x i Vt x H x i Vt x Intensity: 1 t Ix Gx V x 1 This gives no contrast to first order in We will need a relative phase shift between the incident and scattered beams to see phase contrast

19 Contrast transfer function Intensity: I x Gx 1iV 1 t x H x ivt x H x 1iVt x H x H x 1V Im t x H x I x 1V x T x t T x Im H x

20 T u I x V x T x 1 t T x CTF in reciprocal space I u 1 Vt x T x u Vt u T u Im H x * i H x H x i dx duh u e H u e e x u iu ux * iu ux e e u x x * u * iu x * iux iux i du H u dx H u dx i du H u u u H u u u T u i H u H u

21 Special Case: H(u) is an even function If then H u H u * * H u H u * T u i H u H u T u Hu * i H u H u Im

22 Using the CTF H u A u E u exp i u Assume: H u H u Im sin T u H u A u E u u 1 t uvt utu sin I u V x T x I u u V u A u E u u t

23 Ideal transfer function for phase object (I), u 1 u, u 1 b b H u i u 1, u 1 b e i, u 1 b H u 1 u u T u, u 1 ImH u, u 1 b b

24 Ideal transfer function for phase object (II) H u i u 1, u 1 b e i, u 1 b 1 sinc H x i x i x b ReHu ReHx b b ImHu ImHx Hu Hx causes relative phase change for u 1 b

25 Ideal CTF Assume Au Eu 1 Tu sinu, u u, u T u, u, u T(u)> positive phase contrast

26 Real CTF: (case 1: C s =) u f u f < : underfocusing f > : overfocusing

27 Real CTF: (case : f=) 1 s 3 4 u C u Contributes to negative contrast

28 Real CTF: (general) 1 s 3 4 u f u C u Underfocus gives a region with constant phase, positive contrast

29 d Stationary Phase: Scherzer Defocus u 3 3 du f ucs u d u du uu We need to pick what phase we want stationary: min sch s f f C u min 3 Tumin sin umin Tmax 3 u min 3 1 C s 3 4 u min sin u min 3 4 u min 3C s 3 14 f sch 4 3 C s 1 1.C s 1 This choice gives a relatively constant CTF.

30 Phase At Scherzer defocus C s. mm E 15 KeV

31 E=15 kev C s =. mm CTF at Scherzer defocus f sch 93.4 nm

32 Resolution at Scherzer defocus sin u sch u sch f sch u sch 1 C s 3 u sch 4 we already know: f sch 4 3 C s 1 u sch 3C s C s 3 d sch 1.66C s 3 14 u sch 14

33 Passbands We could make the phase stationary at higher u:

34 Damping due to temporal incoherence E u u c 4 exp chromatic aberration coefficient E I Cc E I E E I I : Variation in energy : Variation in objective-lens current

35 Beam convergence: one-lens condenser (I) underfocus overfocus A range of illumination angles may be incident on each sample point. The illumination semiangle s is not the same as the beam convergence angle.

36 Beam convergence: one-lens condenser (II) Semi-angle of illumination at a point on the specimen: r H s 1 z z r Illumination semi-angle in lens plane: r r p Max. illumination angle limited by aperture: s max s max s Ra H r z s max s r z max r, r s, otherwise Semi-angle of convergence: R R a a r q a q The limiting rays cross the axis at: q a 1 1 r 1 q R a p

37 Beam convergence: one-lens condenser (III) location of probe image: radius of probe image: convergence semi-angles: above below r parallel crossover max s CL excitation Large overfocus is usually better than underfocus. For underfocus, s is never less than r.

38 Beam convergence: two-lens condenser (I) Semi-angle of illumination at a point on the specimen: s r r z L H 1 1 q q 1 r r p 1 s max Ra HL L f H r z s max s, r z max s, otherwise r

39 Beam convergence: two-lens condenser (II) Two-lens condenser system: underfocus (CL off) CL slightly overfocused nearly parallel nearly optimal (CL highly overfocused) Allows small, parallel beam on sample. Parallel beam formed when probe image formed by CL is at front focal point of CM. Highly overfocused CL gives small illumination semi-angle on a sample point.

40 Beam convergence: two-lens condenser (III) location of probe images: radii of probe images: convergence semi-angles: crossover parallel above below CL excitation Beyond crossover, s decreases with increasing CL3 strength ("Brightness"). Parallel beam formed with CL3 overfocus to specific value; good for diffraction. Probe size increases with increasing CL3 strength below crossover.

41 Damping due to beam convergence (I) Assuming a parallel beam: G u F u H u For a non-parallel source, we have to integrate over u: Gaussian spot profile: S u 1 k s e u k There are two distinct ways to combine the incidence angles: coherently and incoherently coherent: Intensity: incoherent: * I u du G uu G u u G u du G u u S u u I u du I u u S u u s

42 Damping due to beam convergence (II) Assume only effect is a phase shift : e i u G u F u Let's assume the different incidence angles combine coherently. u i u u k 1 s Gu dufuue e u k s If the input is a delta function, the output is the transfer function: H x S x F x xfu 1 H u u i u u k 1 s due e u k s

43 Damping due to spatial incoherence (II) Assume k s is small. Do the integral: u uuuu uucu u uu i u 1 icuu H u e due e u k s u k s i u 1 e ducosc uue u k s s i u ks i u e exp H u C u e E s u Spatial incoherence is represented by a damping function: u k E s u ks exp u

44 Combine incoherence effects We can combine temporal and spatial incoherence functions: E u E u E u c s

45 Uniform-Field Electron Lens (I) pmvqa //canonical momentum BAB ˆ B z ˆ ρ z //magnetic field in axially symmetric lens B B B za za B B uzauza z //uniform-field model 1 A A z A Az ˆ 1 A A ˆ A ˆ z z z A B u z a u z a ˆ //magnetic vector potential

46 Uniform-Field Electron Lens (II) v ρˆ ˆ z zˆ v kv sin k za cos kz a vzˆ z z L Assume: ˆ ρˆ za L cos k za kv sin z k za z v z pmvqamvea mk v sin kza m cos kza L mv zˆ z z eb coskza ˆ ˆ ρˆ eb L m v v z z L k v p mv ˆ ˆ z k sin k za ρz L z

47 re ir Uniform-Field Electron Lens (III) h pˆ i pˆ hrr r= p h f r = d n h p r r rri f 1 n r d h p r r r rri pmv ˆ ˆ z k sin k za ρz dr dρˆ dˆ dzzˆ //electron wave function //momentum operator //momentum proportional to gradient of phase // phase change //change in number of wave fronts //momentum in uniform-field lens pdr mv sin z k k za ddz //needed to find phase change

48 Electron Lenses (III) d k sin k z a dz p dr mv z k sin k za 1 dz //differential change z mv z n, z k sin k za 1 dz h za u k kv uv L z z v v 1u z z h mv z h mv z h z z 1u mv z n, z z k sin k za 1 dz z za //# of wave fronts along z

49 Uniform-Field Electron Lens (IV) kz z 1 n z k sin xka1dx, a ka xka x k z 1 xka sin xka k x ka 4 k z z k z n, z 1 1 sin ka 1 a a 4ka sin a kz xka i: radius at z a i: angle w.r.t. optic axis at z a tan i k i 1 tan i tan k i

50 Uniform-Field Electron Lens u 1 Expand to lowest order in u: z z u z n u, z 1 sin ka 1 sin a a 4ka a i i z z u z nu z 1 sin ka 1 a a 4ka a In this limit, the lens has no spherical aberration

51 Object function: Image of periodic specimen (I) Fx H u g F g A u e igx Transfer function (no attenuation): e i u Gu F g ug A u g Image function: lim K iux G x G u e du K uk e i u F u lim K i u iux Fg u g A u e e du K u K g Gx F Age e g g i g igx F g Fu F ug g F x g g e igx

52 Image of periodic specimen (II) FT of image function: lim L iux G u G x e dx L xl lim L i g igx iux Fg Age e e dx L xl lim L i g igx iux Fg Age e e dx x L g L lim L i g igux Fg Age e dx x L g L i g F Age ug g Gu G ug g g g g G F Ag g g Gx g G g e i g e igx

53 Three-beam case: Image of periodic specimen (III) Fu F u g F uf ug g g GuG ug G ug ug g g

54 Example Given: 1 sin F x ib gx Au 1, u g u, g u Write: B B F x F F F e 1 e e e igx igx igx igx g g ib ib G x B gx igx igx e 1 e 1 sin

Spatial Frequency and Transfer Function. columns of atoms, where the electrostatic potential is higher than in vacuum

Image Formation Spatial Frequency and Transfer Function consider thin TEM specimen columns of atoms, where the electrostatic potential is higher than in vacuum electrons accelerate when entering the specimen