Waveoptical properties and spatial resolution in pointprojection microscopy and holography
 Abstract number
 415
 Event
 Virtual Early Career European Microscopy Congress 2020
 Presentation Form
 Submitted Poster
 DOI
 10.22443/rms.emc2020.415
 Corresponding Email
 [email protected]
 Session
 PST.1  Phase Microscopy
 Authors
 Dr Faruk Krecinic (1), Dr Ralph Ernstorfer (1)
 Affiliations

1. Fritz Haber Institute of the Max Planck Society
 Keywords
electron waveoptics
field emission source
inline holography
pointprojection microscopy
 Abstract text
Pointprojection microscopy is a lowenergy (typically <200 eV) electron imaging technique that uses a sharp metallic tip as a point source of electrons to project a magnified image of a sample, without any additional electronoptical lens elements. The image magnification is simply given by the geometric ratio between the sampledetector and the tipsample distance, i.e. M=d_{td}/d_{sd}. The use of fieldemission sources that produce highly coherent electron beams has lead to the development of lowenergy electron holography (LEEH), where at sufficiently large magnification the projected image becomes an inline hologram that can be inverted to retrieve the realspace image of the object [13].
Although some early publications investigated the possibility of atomic scale spatial imaging with LEEH [4, 5], so far the best reported spatial resolution is 7 to 8 Å [6], where the lack of atomic resolution was mainly attributed to environmental mechanical vibrations. However, classical wave optical studies on the effect of (partial) coherence on the spatial resolution showed that increasing the size of a fully coherent source proportionally reduces the the numerical aperture and limits the spatial resolution [7, 8]. This conclusion is at odds with the observation by Cho et al [9] that showed an increase in the opening angle at which fringes are visible when the emitter is cooled, increasing the spatial coherence length. A shortcoming of the classical wave optical treatments that could explain this discrepancy is the omission of electron emission and propagation effects. Raytracing simulations are therefore necessary to estimate the effects of e.g. the initial electron momentum distribution [10], electronoptical aberrations, and the emission intensity distribution [11] on the spatial resolution. However, in this case it is challenging, especially when dealing with a partially coherent and spatially extended source, to relate the classical electron trajectories in a straightforward and consistent way to the electron wave function required to simulate holographic image formation.
Here we introduce a semiclassical model for a physically consistent treatment of wavemechanical and geometric electronoptical properties of fieldemission electron sources. We use it to investigate the spatial resolution limits in lowenergy electron holography (LEEH), where the aberration and coherence properties of the electron source are crucial and interrelated. In the semiclassical approximation the wave function can be expressed as
Ψ(r) = G(r) exp[iS(r)]
where G(r) is the amplitude function, and S(r) is the phase function [12, 13]. The quantum mechanical phase S for a single classical electron trajectory ending at point r is given by
S(r) = ∫^{t(r)} [2K(τ)]^{1/2} dτ
where K is the kinetic energy of the classical particle and the line integral is along the classical path, which is parametrized by the time τ with the upper limit of the integral given by the time of arrival t at r. The amplitude function G is determined from the initial electron momentum distribution, the spatial coherence length at the emitter, and the conservation of flux, i.e. the spread of the wave function during propagation. Summing the quantum mechanical phases and amplitudes for a complete set of classical electron trajectories arriving at point r we obtain the quantum mechanical wave function at that point, e.g. at the sample plane.
We numerically model, using the semiclassical approximation, inline hologram formation in LEEH with a concentric spherical electrode model illustrated schematically in Fig. 1. This model captures the essential electronoptical characteristics of the problem while remaining relatively simple and transparent. We investigate the influence of four factors: (i) the spatial coherence length, (ii) partially coherent emission from an extended surface, (iii) the geometry of the cathode, and (iv) electronoptical aberrations effects. For fully coherent emission from the source surface we show that the source surface curvature can effectively counteract the reduction in numerical aperture indicated by classical wavemechanical models, see Fig. 2(a). Nevertheless, the acceleration of electrons in the electrostatic field also clearly leads to aberrations in the phase of the wave function, see Fig. 2(b). The nonspherical wave front aberration is more pronounced for smaller coherent emitters, since coherent emitters with a spatial extent on the order of the cathode radius have initially a wave function that conforms more closely to the spherical source surface. Partially coherent emission is modeled by incoherently summing the emission from all fully coherent subpatches of the emitting surface, where the size of the subpatches is determined by the spatial coherence length at the source. Since wave front aberration is less pronounced for larger coherent emitters, the model shows that holographic interference fringe visibility of the partially coherent projection images is increasing proportionally to the spatial coherence length, see Fig. 3, which is indeed consistent with the experimental observations by Cho et al [9].
The semiclassical model introduced here provides a general framework to investigate and simulate partially coherent electron sources and evaluate the effect of the spatial coherence length, source geometry and electronoptical aberrations on the emitted electron beam.
Figure 1. Schematic illustration of the semiclassical concentric sphere model. The classical trajectories (blue/orange solid lines) originating from each point source emitter are normal to equiphase surfaces (green/red dashed lines) of the corresponding semiclassical wave function. The semiclassical electron wave function at any point on the object surface (R_{o}) is the sum of all incoming trajectories at that point.
Figure 2. Semiclassical simulation of an inline hologram of a perfect edge on the object surface projected by the concentric spherical electrode model with R_{s}=50 nm, R_{o}=1 μm and V_{s}=100 V. The wave function magnitude (a) and phase (b) are shown for varying coherent emitter sizes ranging from an atomic emitter, i.e. σ_{s}=2 Å, up to a spatially extended emitter of 80 Å.
Figure 3. Semiclassical simulation of the projection image from a partially coherent emitter for a range of coherence lengths of 2 to 80 Å. The electron emission from the spherical cathode surface is taken to be uniform over a spatial extent much larger than the coherence length.
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