## 3 Simulation Results  
To simulate sample multiple states for ptychographic reconstructions, we performed reconstructions using the first 5 orthogonal modes of canonical Lena test image given by Singular Value Decomposition (SVD) algorithm [17; 18]. Simulated data in which the Lena object \((L)\) under illumination was the real-part only canonical Lena test image of size 512 \(\times\) 512 pixels. The orthogonal modes in both probe and object are particular ranks of the singular values of the image matrices. Selecting more numbers of singular values produces higher numbers of ranks of the matrices. The orthogonal modes can represent real modes in materials systems, and they can be scientifically interesting. Electrical or magnetic field with mutually orthogonal modes could be present in waveguides. The first two orthogonal modes have the highest proportions of weights, and the weights become much less for orthogonal modes with increasing numbers of singular values. We used SVD to decompose a probe simulated by propagating illumination after a circular pinhole into 5 orthogonal modes, and the related reconstructions of the probe, which are displayed in Figure 6. The first 5 orthogonal modes of canonical Lena test image by SVD and the related reconstructions are depicted in Figure 7. Both probe and object orthogonal modes simulated by SVD are used to simulate multi-modal ptychographic diffraction intensities to test the feasibility of information retrieval by the reconstruction algorithms. Multi-modal diffraction intensities are simulated by performing summation over all the 25 individual coherent diffraction intensities that result from 5 orthogonal modes of both probe and object test pattern. The overlap ratio in our simulations was selected to be around 85%, and the total number of scan positions in the ptychographic datasets is 323, and the ptychographic scan was performed with a spiral geometry to intentionally break the possible translation symmetry. Only the first orthogonal mode of simulated probe is fed into the reconstruction algorithms, the remaining modes of the probe and all sample modes are unknown when starting the reconstructions. The reconstruction of simulated data was performed with 1000 iterations. The reconstructed images in Figures 6i and 7i are compared to Figures 6f and 7f respectively, where superposition of 5 orthogonal modes of both probe and sample are calculated to compare the reconstructed and simulated probe and sample. The central region of the superposition images of Figure 7a-e is selected for simulation analyses. Comparison of the superposition of the central region of the image (Figure 7f) to the reconstructed results (Figure 7i) on sample is performed. The reconstructed images in Figures 6i and 7i are compared to Figures 6f and 7f by using Real-space error metric [6] to be 0.0176 and 0.0154 respectively. The reconstructions of first 5 modes do not match the original 5 modes, because the reconstruction fails to recover separated modes. In real experiments, usually Poissonian or Gaussian noises can be present in the measured datasets. These noises can be effectively modelled and taken into account by using The maximum-likelihood (ML) formulation for ptychography [7; 19]. Figure 6: Five first fundamental modes (**a-e**) from SVD of a real-only probe propagated after a pinhole; (**f–h**): simulated sum of 5 modes; 10 modes and 20 modes; (**i**) reconstructed sum of 5 modes of probe. Images are plotted with a fire-style colormap with an arbitrary maximum value. The Figure 6i is reconstructed results of Figure 6f. The minima and maxima of colormap legend bar for Figure 6a–i are as follows respectively: [0, 1.1531]; [0, 0.2707]; [0, 0.3694]; [0, 0.3410]; [0, 0.3273]; [0.0381, 3.1813]; [0.0820, 3.1817]; [0.0916, 3.1818]; [0, 1.3419]. Figure 7: Five first fundamental modes (**a–e**) from SVD of a real-only Lena image; (**f–h**): simulated sum of 5 modes; 10 modes and 20 modes. (superposition of central part of mode Figure 7a–e); (**i**) reconstructed sum of 5 modes of object.