4.4.1 Coherence and ptychography
To study the influence of X-ray coherence on the resolution of ptychographic reconstruc-tion measurements of a strongly scattering test sample (FZP made out of Au with smallest structures of 30 nm) were performed. The FZP used as a focusing optics had diameter of 100µmand an outermost zone width of 70 nm, yielding a size of focus spot of 85.4 nm.
Different degrees of coherence were created by tuning the gap of the vertical slits from 10 to 50µmat X-ray energy of 1400 eV, in accordance with calculation presented in the Chapter 4.1.2. Since the spatial coherence of the beam at the slits position in horizontal direction is around 45µmthe opening of horizontal slits can be freely varied within this width in order to keep photon flux the same for different vertical gaps. Reconstructed phase images on figure 4.15 showed no loss in resolution for slits gaps 10, 20 and 30µm, since the spatial coherence is bigger than the focus spot. The images taken with vertical slits gap of 40µmsignificantly degraded, at 50µm- structures of the test target are not resolved. The observed degradation of the reconstructed images stay in a good agreement with estimated coherence values. However, even illumination spot with spatial coherence less than its size can provide usable ptychographic reconstruction as it was observed with the 40µmslits gap.
Figure 4.15: Phase reconstructions of resolution target (FZP) taken with different slit widths: a) 10×23µmb)20×16µmc)30×13µmd)40×12µme)50×11µm. The horizontal slit width varied in order to keep the total flux the same for different vertical slit opening.
4.4.2 Numerical aperture
The resolution of diffraction imaging methods is determined in the first place by the high-est scattering angle that can be measured by the CCD, that is expressed as a numerical aperture (NA) of the system. The property of the system to detect a diffraction pattern in accurate way relates to the sampling criteria. Since the diffraction pattern is recorded by the discrete pixels of CCD detector in reciprocal space we can estimate the sampling distance in the real space. Having feature size∆x, the CCD placed on distanceLfrom the sample and wavelengthλwe can calculate the speckle size in q-space as:
∆q= λL
∆x (4.8)
If pixel size of the CCD ispand amount of pixels in 1D direction isN the sampling size in the real space, or in other words, the pixel size of a reconstructed image is expressed as following:
∆x= λL
N p (4.9)
According to equation 4.9 a bigger the CCD chip area, higher energies or a shorter distance between sample and detector would result in a higher detection aperture and resolution.
Figure 4.16 shows the difference in numerical apertures with CCD placed at the distances Land2Lfrom the sample. The closer placement of the CCD results in higher q-space coverage and, potentially, in higher ptychography resolution.
Figure 4.16: Sketch of the different numerical aperture provided by two positions of the CCD rela-tive to the sample. In case of 8 cm distance at position 2 the CCD has twice smaller coverage of the reciprocal space in comparison with position 1.
At MAXYMUS microscope the conventional stages assembly allows minimal dis-tance between detector and sample of 8 cm. In this case the CCD size of12.7×12.7 mm2 at energy 1200 eV give the resultant pixel size of 6.5 nm. Removal of the piezo sample stage allows to reduce distance to 4 cm that gives twice smaller sampling distance in real space. However this configuration is not possible if experimental set up involves a magnet system which is placed in front the CCD restricting its movement closer than 8 cm. Fig-ure 4.17 shows the simulation of two ptychographic reconstructions taken by the systems with different numerical apertures. The a iffraction pattern a) captures smaller scattering angles in q-space cutting off high diffraction orders. The diffraction pattern b) have twice bigger NA, that for example could correspond to twice smaller distance between detector and sample. As a result we have reconstructions with different pixel sizes, figure 4.17 c) has pixel size of 19.4 nm that is twice bigger than pixel size of 9.7 nm in figure 4.17 d), that directly reflects on the resolution and contrast.
4.4.3 Flux and scattering power
Sufficient flux and photon count rate at high scattering angles are another important factors for image resolution. In this case a high number of photons per area results in better spatial resolution of ptychographic reconstruction. In order to evaluate coherent intensityIc at
Figure 4.17: Comparison of two ptychographic reconstructions of Siemens Star obtained by systems with different NA: a) is a diffraction pattern with twice reduces NA obtained from diffraction pattern b) by cutting off q values at high diffraction orders. Images c) with pixel size of 19.4 nm and d) with pixel size of 9.7 nm are corresponding ptychographic reconstructions.
the focus spot the following equation can be used [105]:
Ic= Fc
Aef f
∝Br×∆E
E ×N A2×T, (4.10)
whereFc - coherent flux obtained by the focusing optics,Aef f - the effective area of the diffraction limited focusAef f ∝ (λ/N A)2,Br- brilliance of X-ray source, ∆E/E -the energy bandwidth,N A- numerical aperture of FZP, andT is the efficiency of FZP.
Thus the highly efficient optics with big numerical aperture should be used for the larger fluence in diffraction signal. However ptychography needs comparatively big focus spot (around 100 nm) to ensure reasonably fast scanning that requires FZP with smaller NA.
In any case the intensity of a scattering signal strongly depends on the scattering power of a sample. For example, scattering from magnetic samples based on XMCD have
signifi-Figure 4.18: Averaged diffraction patterns from the ptychography scans of a) strongly scattering resolution target Siemens star made of Au at 800 eV, and b) weakly scattering magnetic domain structure at the Co-edge energy 780 eV. c) shows radial profiles of these diffraction patterns, where green curve corresponds to the resolution target, red curve - to the magnetic sample.
cantly lower intensities than from charge scattering samples. Figure 4.18 shows diffraction patterns averaged over two sets of ptychographic data, one of a resolution target (Siemens Star, Zeiss) made of Au about 150 nm thick at the energy of 800 eV, the other of a mag-netic domain sample scanned at Co edge (780 eV) with total thickness in the multilayer of 15 nm. Since the sample materials and structures are not the same it gives only general understanding of the difference between charge and magnetic scattering systems. Scatter-ing from the magnetic sample doesn’t show strong pronounced speckles, but a faint halo around zero order at the low q-space frequencies. The diffraction pattern from the reso-lution target has higher level of scattering as shown in the radial profile graph in figure 4.18 c). At higher energies diffraction speckles from the resolution pattern can be clearly
visible at the very edges of the CCD detector providing NA limited resolution.
Since the radiation damage of the magnetic multilayer systems can be neglected the scattering statistics is improved by increasing dwell time for every scanning step. The basic issue in experiments with long exposure times is the specimen contamination with carbon that occurs due to the interaction of X-rays and residual gases in vacuum chamber.
The carbon layer is built up on the surface of the scanned area that worsen the contrast of the resulting image. Also thermal drifts of the sample during long scans can be an issue causing resolution worsening.