## 2 Theory  
Previously, researchers adopted RL deconvolution to accommodate spatial partial coherence into CDI [5,6] and ptychographic [11] iterative algorithms, and their mathematical formulations have been described in details in Ref [5,6,11]. To further develop the mathematical concept of partial coherence that suits ptychography, a different version of representation of partial coherence of electromagnetic waves has been suggested [7,9], which adopts the theorem that transverse and longitudinal partially coherent X-rays can be generally formulated equivalently by introduction of superposition states of multiple modes in both spatial and wavelength (spectra) regimes. In the spectral regime, multi-modal approach is also valid. Insertion device generated synchrotron X-rays always possess a certain amount of longitudinal incoherence, especially when no X-ray monochromators are present, the resulting X-rays are to be composed of significant broadening of the energy spectra. It is considered of general interest to utilize an undulator harmonic from X-ray intensity spectra that is generated by an insertion device because coherent intensity diffraction intensities can be mathematically recovered with iterative algorithms by applying multi-modal ptychography. The original research on broadband X-ray illumination in coherent diffraction imaging reported by [PERSON] et al. [12], include a priori calculated electromagnetic wave spectrum from a particular undulator harmonic, obtained by simulations of broadband X-rays, produced by an insertion device to guide the iterative algorithms in the reconstruction of the acquired CDI data. This approach was extended to Ptychography, both in visible [8] and X-rays regimes [13], with demonstrated enhanced experimental efficiencies achieved by allowing larger flux illumination on sample. Figure 3 shows reconstructions of test sample with multiple probe modes using broad bandwidth pink beam X-rays, and the reconstructions improved significantly when introducing multiple probe modes into the iterative algorithms. Figure 4 displays Multi-modal ptychography can be extended to electron regime, and a study demonstrates the applicability of probe modal decomposition using electron ptychography [14]. Figure 4 illustrates experimental reconstructions of both modulus and phase components of object, showing much improved data inversion quality when 16 probe modes are incorporated when compared to results with only a single probe mode in the ptychographic iterative algorithms. It is particularly important to note that recent development of fly-scan ptychography [15,16] introduces continuous motions of sample movements, thus, would result in blurring of measured diffraction intensities. To reconstruct  
Figure 2: Two different sets of reconstructions from the same data set using different initial guesses. (**a**) The first set of raw reconstructions; (**b**) The second set of raw reconstructions. The scale bars indicate a length of 300 m. The color wheel applies to all the images. To aid in display, the curvature of the probe phases has been removed. Figure adapted from [10], Reprint permission granted, 2016 The Optical Society. fly-scan ptychography datasets, incorporation of partial coherence is required; multiple probe modes are artificially introduced into data processing to account for the continuous motions of samples. Figure 5 shows experimental reconstruction of several probe modes as a function of sample scanning speed in their reported study with a collimated visible-light laser source [15]. The advantages of fly-scan ptychography are two-fold. First, the overhead of sample stages could be minimized to reduce total measurement time, and secondly, the sample stage scanning motions are much more stable and reliable, comparing to the case of excessive acceleration and deceleration of sample stage motors during ptychographic data acquisition.