Ptychographic reconstruction with wavefront initialization

X-ray ptychography is a cutting edge imaging technique providing ultra-high spatial resolutions. In ptychography, phase retrieval, i.e., the recovery of a complex valued signal from intensity-only measurements, is enabled by exploiting a redundancy of information contained in diffraction patterns measured with overlapping illuminations. For samples that are considerably larger than the probe we show that during the iteration the bulk information has to propagate from the sample edges to the center. This constitutes an inherent limitation of reconstruction speed for algorithms that use a flat initialization. Here, we experimentally demonstrate that a considerable improvement of computational speed can be achieved by utilizing a low resolution sample wavefront retrieved from measured diffraction patterns as initialization. In addition, we show that this approach avoids phase singularity artifacts due to strong phase gradients. Wavefront initialization is computationally fast and compatible with non-bulky samples. Therefore, the presented approach is readily adaptable with established ptychographic reconstruction algorithms implying a wide spread use.


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X-ray ptychography can be regarded as a combination of scanning X-ray transmission microscopy 24 (STXM) and coherent diffraction imaging (CDI). STXM utilizes a lateral scan of the sample 25 through a focused X-ray beam, which provides the transmission function of the specimen [1]. 26 However, spatial resolution is limited by the focus size. CDI, on the other hand, employs an 27 extended X-ray beam larger than the sample and exploits effective oversampling contained in 28 diffraction patterns with additional constrains during algorithmic retrieval [2]. While spatial 29 resolution provided by CDI is at least in principle wavelength limited, the availability of high 30 quality, large X-ray beams practically limit the sample size. In the following, we will re-iterate elements of the frame work for refractive ptychography [18] 70 for the convenience of the reader. In the X-ray regime the complex refractive index of a material 71 is commonly expressed as with , the refractive index decrement and , the absorption index. For samples that are sufficiently 73 thin to avoid internal diffraction, the complex wave field (r) at the object plane point r = ( , ) with , the modulus of the wave vector and , the direction along the optical axis. The goal of 76 refractive ptychography is to retrieve the refractive object function˜(r). In the experiment the 77 object function is illuminated by a focused X-ray beam with the complex wave field of the probe 78 (r − R ) at R the -th scan position. The resulting complex wave field Ψ(r) is given as and the observable intensity is Then the modeled amplitude |ˆ(q)| is replaced with the amplitude of the measurements √︃ˆ( q) 89 and the resulting wave field is propagated back to the object plane with F −1 , the inverse Fourier transform. Finally, the object wave field is updated by and the probe wave field by The update strength is tuned by the parameters and . Initialization of the iterative procedure 93 refers to the starting values for the object function˜(r) and the probe (r). Usually, a flat 94 initialization for the object function is chosen, i.e.
The complex wave field of the probe (r) is usually well characterized, which allows a realistic 96 initialization.

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For the experimental demonstration of our proposed approach we will reuse a previously with and both integers indicating the horizontal and vertical order of the moment, respectively.

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In the following, we will be only interested in the three moments up to the first order. The 131 transmission signal of the object corresponds to 00 in a straight forward way: with flat (q) a diffraction pattern taken in a region outside the object for the purpose of where terms involving derivatives of the transmission signals vanish, since the probe has a finite 144 support: ∫ r (r) (r) = 2 (r)/2| ∞ −∞ = 0. The first term in eq. (14) corresponds to the object's phase gradient, which is of interest here. The second term constitutes a contribution of 146 the probe to 10 and is non-zero in the combined case of a non-vanishing absorption signal of the 147 object (R ) ≠ 1 and a non-vanishing phase gradient of the probe (r − R ) ≠ 0. The latter is 148 typically the case if the object is located outside of the beam focus, the optics are aberrated, or if 149 a purposefully structured probe is used.

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Assuming that the second term in eq. (14) can be neglected, the differential phase signals of 151 the object in the area of illumination defined by the probe (r − R ) at the scan point R can be 152 estimated as and where correction terms account for the pixel position of the flat field beam in the detector.

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If the second term in eq. (14) cannot be neglected, the above equations have to be corrected.

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For this, the influence of the object's transmission and the probe's differential phase signal must 157 be determined. This can be done by either using the second term directly or -more conveniently

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-by calculating virtual diffraction patternsˆ(q) provided by the pure absorption signal of the 159 object according to The moments 10 and 01 of these virtual diffraction pattern correspond exactly to the second 161 term in eq. (14). Thus, the differential phase signals can be estimated in this case as and Wavefront retrieval 164 In order to retrieve the phase image Φ(R ) from the estimated differential phase images Φ (R ) 165 and Φ (R ) we use the non-iterative, boundary-artifact-free wavefront reconstruction presented 166 in [26]. This approach starts with constructing an antisymmetric extension of the inputs and Then the integrated image is retrieved by calculating where interpolation between the coordinate systems defined by r and R is used as necessary.  Fig. 4 (b) and (c). These artifacts are absent for ptychographic 183 reconstruction with wavefront initialization (insets in Fig. 5b).

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In order to demonstrate the versatility of the wavefront initialization for ptychographic 185 reconstructions, the proposed approach was further applied to a ptychographic data set which was   probe. This has potential benefits for combining ptychography with other X-ray techniques that 205 would suffer from structured probes, such as X-ray fluorescence.

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Using wavefront initialization (eq. 23) implies that the bulk phase information associated  shape with a full width half maximum of 7 pixels, which is markedly larger than the smallest 222 sample features. The observable diffraction patterns were calculated according to eq. (4) and 223 wavefront retrieval was performed as described above.

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Although the estimated wave field for the object used for initialization has insufficient resolution 225 to sample the Siemenstar (Fig. 8a), the reconstructed object's phase distribution (Fig. 8b) shows 226 that structures smaller than the probe are still reliably retrieved using wavefront initialization.

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This demonstrates that wavefront initialization is compatible even with challenging samples.

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However, the comparison of the cost function between flat and wavefront initialization (Fig. 8c) 229 illustrates only a negligible difference in terms of convergence speed.

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Conclusion 231 We have demonstrated that the speed of ptychographic reconstruction algorithms which use a flat 232 initialization for the sample's complex wave field is inherently limited for bulky samples. This is 233 due to the fact that during the iterations the bulk phase information has to propagate from the 234 edges of the sample to its center. By instead using wavefront initialization, the reconstruction 235 speed is considerably increased as the bulk phase information is already present. In addition, we 236 have shown that wavefront initialization can avoid phase singularity artifacts associated with 237 large phase gradients.

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The input data for constructing the wavefront initialization is readily accessible in most