Validity of the empty-beam correction in near-field imaging

Extended wavefronts are used for coherent full field imaging of objects based on solving the inverse Fresnel diffraction problem. To this end, the conventional data correction step is given by division of the recorded object image by the intensity pattern of the empty beam. This division of intensities in the detection plane is a rather crude approximation for the separation of the complex valued object and probing fields. Here we present a quantitative error estimate, along with its mathematical proof, and confirm the prediction with numerical simulations. Finally the problem is illustrated with experimental results.

Figure 1

Illustration for the standard correction scheme used to treat artifacts in propagation imaging related to aberrated illumination function. The projection image (projection hologram) recorded for a nematode c.elegans in a highly divergent beam shows artifacts from mirror focusing and the defocused holographic wavefront distortions of other spurious elements, such as dirt on vacuum windows. Note that only one part (near the end) of the worm is illuminated. The intensity pattern is shown for (a) the worm in the beam as well as (b) the empty beam, followed by (c) the divided image. Strong imperfections remain after this simple correction scheme, indicating that the product approximation in the detector plane is in this case not satisfactory. The square image size corresponds to $133\mu \text{m}$ in the sample plane.

Figure 2

Illustration of the empty-beam correction for a nearly perfect quasipoint beam illumination provided by a x-ray waveguide (left column), compared to the strongly aberrated case which is deliberately achieved by insertion of a wavefront modifier consisting of W stripes inserted between the waveguide and the object (Siemens star test pattern). Panels (a), (b), and (c) and (d), (e), and (f) show the empty beam, the raw hologram, and the divided image (corrected hologram), respectively. Panel (c) is close to the ideal hologram calculated for ideal spherical or parallel beam (in the two equivalent geometries, as discussed in the text). One pixel corresponds to 25.5 nm in the object plane.

Figure 3

Cone-beam and parallel-beam geometry.

Figure 4

Overview of the simulation setup and data generation. In panels (a) and (b) we choose as synthetic standard $P$ the image of a mandrill as phases and Dürer's Melencolia I as amplitudes. The phases are in the range of $[-0.4$ 0.4] rad and the amplitudes in [0.8 1.2]. The resolution is $512\times 512$ pixels. (c) Phases of sine grating test pattern. The phase shift is chosen as $-0.3$ rad, the amplitude is uniformly 1. The periodicity of this realization has $\Delta x_{\mathcal{O}}$ of 32 pixels. (d) Intensities of the simulated hologram of the empty beam $P$ for $F=0.0014$ (Fresnel number for one pixel). (e) Intensities of the simulated hologram of $PO$ for $F=0.0014$. (f) Empty-beam-corrected hologram of $\mathcal{O}$ [(e) divided by (d)]. For better visual comparison insets show a zoom into the region indicated by the red square. (g) Hologram of $\mathcal{O}$ simulated with a plane-wave probe. (h) $\delta$-error matrix as defined by Eq. (12). (i) Error matrix $|\text{(g)}-\text{(f)}|/\text{(g)}$.

Figure 5

Simulation results for Eq. (12). (a) The error ${\delta }_{\text{max}}$ (maximum of $\delta$ over all points) as a function of the normalized cutoff frequency $\xi /{\xi }_{\max }$ in the filter applied to the probe $P$, for different periodicities $\Delta x_{\mathcal{O}}$ (grid spacings). The alternative $x$ label on the top indicates the effective source size $a$ in units of pixels, corresponding to $\xi$. (b) Test of the inequality in Eq. (12). ${\delta }_{\text{max}}$ is plotted versus the minimum of the right-hand side (RHS) of Eq. (12). The purple solid curve is the plot of min(RHS) versus min(RHS). Same color coding as in panel (a). The simulation can be carried out in unitless pixel variables and Fresnel numbers, but parameters correspond to the following experimental parameters: $\lambda =0.07\mathrm{nm},\Delta x=25$ nm, $N=2048,z_1=6$ mm, $z_2=519$ mm, $z_{\text{eff}}=5.93$ mm, and ${\mathfrak{f}}_{\text{eff}}=2381$.