Diffraction imaging: The limits of partial coherence

Coherent diffraction imaging (CDI) typically requires that the source should be highly coherent both laterally and longitudinally. In this paper, we demonstrate that lateral and longitudinal partial coherence can be successfully included in a CDI reconstruction algorithm simultaneously using experimental x-ray data. We study the interplay between lateral partial coherence and longitudinal partial coherence and their relative influence on CDI. We compare our results against the coherence criteria published by Spence et al. [Spence et al., Ultramicroscopy 101, 149 (2004)] and show that for iterative ab initio phase-recovery algorithms based on those typically used in CDI and in cases where the coherence properties are known, we are able to relax the minimal coherence requirements by a factor of 2 both laterally and longitudinally, potentially yielding significant reduction in exposure time.

Figure 1

Schematic of the experimental setup at 2IDB beamline of APS. The widths of the entrance and exit slits can be changed to change the coherence of the source. The monochromator can be removed to get a “pink” beam.

Figure 2

(a) The scanning electronic microscopy (SEM) image of the sample. (b) The backside of the sample. The scale bar is $2\mu \mathrm{m}$. The triangular feature is largely absent on the backside of the sample and so this feature has not been fully etched through the substrate.

Figure 3

Fitted lateral coherence length ${\sigma }_{s,x}$ (top) and longitudinal coherence length ${\sigma }_l$ (bottom) for different entrance/exit slit combinations from double slit interference pattern. The data for some exit slit widths are missing due to the saturation of the CCD when even the fastest shutter speed is used.

Figure 4

(a) The diffraction pattern of the fully coherent mode (b)–(d) are the diffractions in the same area bounded by the white box in (a) for slit settings as $S_l=5.78\mathrm{and}{S}_s=0.87$, $S_l=0.73\mathrm{and}{S}_s=0.55$ and $S_l=0.37\text{and}{S}_s=0.39$, respectively. The decrease in the contrast of the diffraction is readily seen. All data are shown in logarithmic scale.

Figure 5

The reconstruction of the amplitude of transmission function of the sample. The entrance slit width was set at 20 $\mu \mathrm{m}$ and exit slit was set at 5 $\mu \mathrm{m}$. (a) CDI reconstruction assuming perfect coherence. (b) Partially coherent CDI reconstruction. No obvious difference can be observed in the two reconstructions, meaning that the effect of partial coherence is negligible in this experimental setup. The scale bar is 2 $\mu \mathrm{m}$. The color scale is linear between an amplitude transmission of 0 (perfect absorption) and 1 (complete transmission).

Figure 6

Reconstructed amplitude of the transmission function using different entrance/exit slit combinations. (a) $S_l=5.78,S_s=0.87$; (b) $S_l=3.91,S_s=0.65$; (c) $S_l=0.73,S_s=0.55$, (d) $S_l=0.57,S_s=0.69$; (e) $S_l=0.37,S_s=0.39$; (f) $S_l=0.37,S_s=0.33$. The reconstructed results is acceptable until (d). In (e), the quality of the reconstruction degrades as the $S_s<0.5$. While in (f), the reconstruction fails. The scale bar is 2 $\mu \mathrm{m}$ and the color scale is same as for Fig. 5.

Figure 7

$R$ values of different setups. The $R$ values in Figs. 6a, 6b, 6c, 6d, 6e, 6f are labeled accordingly here. We choose $R<0.04$ as a criteria of a “good” reconstruction. By this criterion, (a)–(e) are acceptable, (f) is not acceptable, which is consistent with Fig. 6.

Figure 8

Reconstructed amplitude of the transmission function from simulated data with $S_s=0.83$. (a) The simulated sample (b) $S_l=1.67$, (c) $S_l=0.72$, and (d) $S_l=0.33$. The scale bar is 2 $\mu \mathrm{m}$.Color scale is as for Fig. 5 except for (a) where for clarity, a binary image is shown.