0 of the same magnitude. If we don t use an OA and ignore any damping, the CTF is

Transcription

1 1 4. Image Simulation Influence of C Spherical aberration break the ymmetry that would otherwie exit between overfocu and underfocu. One reult i that the fringe in the FT of the CTF are generally farther apart for a given underfocu f 0 than for a correponding overfocu f 0 of the ame magnitude. If we don t ue an OA and ignore any damping, the CTF i Tu, f in u, f where u, f f u C u The fringe create zero in the FT of an amorphou ample, which ideally ha a continuum of patial frequencie. Finding C, f Apparently, we will have extrema and zero in the FT of the CTF given by: 1 u u n in 1 in u0 un The zero will occur when 1 n f u C u 3 4 Analyi One can experimentally determine C by analyzing FFT acquired at different amount of defocu. We can meaure the radii of the intenity minima in the FFT and plot their location a n v. u, which can be fit to a quadratic function. The trend i clearet if we aign negative value of n to the zero for underfocu. It i the light curvature in the curve that allow u to find C.

2 We find C 0.99 mm. We can then proceed to find 1 4 fch C 57 nm 3 and point reolution 3 C 14 ch nm Example: phae contrat A an imple imulation of phae contrat, we aume two cloely paced, circular phae object eparated by a ditance lightly greater then ch. Simulated image at different value of focu are hown below. We have aumed C.0 mm, which give 93.7 nm and 0.34 nm. f ch ch E 15 KeV and

3 3 While not entirely realitic, the point eem clear that the bet, poitive phae contrat i produced at a light underfocu near f. ch Multilice method Multilice i another ueful mean to imulate image from thin pecimen. A pecimen i conidered to contain many thin lice, perpendicular to the incident beam. It each lice i thin enough, it can be conidered a weak-phae object. We can then conider the lice to be an infiniteimally thin plane repreented by the projected potential. If we aume the plane lie at the top of the lice, then the ret of the lice i jut vacuum. So after tranmitting the incident beam through the plane, it mut be propagated through the remaining thickne of the lice. Thi i a little more complicated than it may eem at firt. Multilice propagation The algorithm i depicted below,. It proceed from top to bottom. Firt, the incident wave 0 xy, i multiplied by the pecimen function f1 x, y for the firt lice. Then, the product i propagated through the vacuum that comprie the lice thickne z1, uing the propagator px, y, z1, which we will derive hortly, howing that the propagation i expreed by a convolution. Thi give u the incident wave 1 xy, for the ubequent lice. Thi proce i repeatedly computationally through the entire foil thickne.

4 4 If every lice ha the ame projected potential, the pecimen function only need to be computed once, but thi i not a trict requirement. But one can imagine applying the method on a zone axi, where every lice repreent a crytallographic lattice plane of the pecimen. Propagator We ued Huygen principle to explain why diffraction occur from a narrow lit. After an incident wave i tranmitted through a thin plane (object) we hould conider contribution to a point on a ubequent plane (image) from the pherical wavelet emitted at every point on the exit urface of the object. Thee wavelet reaching our image point have different path length and make different angle q w.r.t the incident wave direction. Deriving the propagator The formalim here i baed on the Frenel-Kirchhoff integral.: ikr e 1co x, yz, zik x, y, z dxdy R ource

5 5 In TEM, we can fortunately make the mall-angle approximation when evaluating the obliquity factor : 1co 1 The radiu i given by R xx y y z Becaue the angle are mall, we aume lateral eparation are much le then the lice thickne. So the phae factor become xx yy ikz ikr ikz z z e e e The denominator i xx 1 y y 1 R z z z Combining thee, keep only lowet-order correction, we have ik x x y y ikr ikz z e e e R z The wave at the bottom of the lice i then ikz ik e xx yy z x, yz, z ik x, y, z e dxdy z x, y Notice that thi ha the form of a -D convolution of our tranmitted wave with omething. The propagator function We can write thi convolution a,, e ik z x yzz x, y, z pxx, y ydxdy x, y where we have defined the propagator ik ik x y z px, y e z Alo, there i no good reaon to keep the phae factor exp( ik z), ince the overall phae i arbitrary. So x, yz, z xyz,, p xy,, z Multilice algorithm Here i what we have o far. The wave function below lice n 1 i given by n1x, y pxy,, zn1fn1xy, n xy, We need to tranmit the wave through the projected potential for a lice, propagate through the vacuum that comprie the ret of the lice, then repeat. Symbolically

6 6 1 pz1f10 pzf 1 pzfpz1f10 Multilice in reciprocal pace Convolution are often eaier to perform by combining them with Fourier tranform. The convolution theorem tell u n1xy, = pxy,, zfn1xy, nxy, We need to know the -D FT of the propagator. A little thought how that px, y, z= pux, uy, z e Now iux uy z k 1 iu, = e x u y z n k xy f xy, xy, n1 n1 n We can multiply the incident wave by the pecimen function, take the FT, multiply that by the FT of the propagator, then perform an invere FT to find the incident wave on the next lice. Note that, to ue FFT, we will want to enure periodic boundary condition for our projected potential. Otherwie, artifact how up from edge effect. Alo note that even ingle atomic layer or ome material may not be pure phae object. For example, it i argued by ome that ingle Au atom are not pure phae object. So we would need to lice up atomic plane to perform accurate imulation of thee material uing multilice. Simulation: pherical particle, multilice Notice that the Bloch-wave method aumed an infinite crytal. So it i not practical to ue Bloch wave to imulate data from le ideal object, like nanoparticle. Below are hown ome multilice imulation uing a reaonably realitic model of a pherical, crytalline nanoparticle. One et of lattice plane i included to how their appearance in the image. Cro ection of thi object at very mall increment were ued to compute the projected potential for each lice. It wa aumed that E 00 KeV and 1 C.0 mm. A imple quadratic damping with damp.0 nm wa included to make the imulation a bit more realitic. The image at the Scherzer defocu (underfocu) how the bet poitive phae contrat with the fewet artifact. The in-focu image ha fairly low contrat, while the overfocu image ha negative phae contrat. The overfocu alo how delocalization of the lattice fringe. Thi i becaue perfect patial coherence wa aumed in the calculation.