## 2 Theory
Figure 3: Comparison between three reconstruction strategies: conventional reconstruction (**a**,**d**); reconstruction after background correction on diffraction data (**b**,**e**); and mixed-state reconstruction using 10 modes (**c**,**f**). For each frame, the image on the top displays the modulus of the retrieved probe and the image on the bottom is the phase of the reconstructed object. (**a**,**c**) (resp. (**d**,**f**)) Reconstructions from the scan at a distance of +0.35 mm (resp. +2.1 mm) from the focus of the illumination. The colour scale for all panels is the same as in (**a**). Figure adapted from Ref [13]. Reprint permission granted. 2014 American Institute of Physics. The number of sample modes are representatives of multiple sample states, sometimes they can be orthogonal to each other, in the same fashion to that of X-ray beam. Following the free-space propagation intensity equation derived from Equation (1) in the work of [PERSON] et al. [9],  
\[I(q)=\int\int B(\mathbf{r_{1}},\mathbf{r_{2}})S(\mathbf{r_{1}},\mathbf{r_{2}}) \exp[\frac{i\pi}{z\lambda}(\mathbf{r_{1}}^{2}-\mathbf{r_{2}}^{2})]\times\exp[ i(\mathbf{r_{1}}-\mathbf{r_{2}})\cdot q]d\mathbf{r_{1}}d\mathbf{r_{2}} \tag{1}\]  
where \(B(\mathbf{r_{1}},\mathbf{r_{2}})\) is the probe function, \(S(\mathbf{r_{1}},\mathbf{r_{2}})\) is the object function, \(\mathbf{r_{1}},\mathbf{r_{2}}\) represent two arbitrary points separated in the free-space propagation direction, and \(q\) represents the reciprocal-space unit vector, and \(\lambda\) is the wavelength of the illuminating X-rays. The mutual optical intensity (MOI) equations of both multiple X-ray beam \(J_{B}(\mathbf{r_{1}},\mathbf{r_{2}})\) and sample modes \(J_{S}(\mathbf{r_{1}},\mathbf{r_{2}})\) can be described as the following mathematical formulations, which are first proposed by [PERSON] and [PERSON] [7] :  
\[J_{B}(\mathbf{r_{1}},\mathbf{r_{2}})=\sum_{n=1}^{N}\eta_{n}B_{n}(\mathbf{r_{1} })B_{n}^{*}(\mathbf{r_{2}}) \tag{2}\]  
\[J_{S}(\mathbf{r_{1}},\mathbf{r_{2}})=\sum_{m=1}^{M}\eta_{m}S_{m}(\mathbf{r_{1 }})S_{m}^{*}(\mathbf{r_{2}}) \tag{3}\]  
where \(\eta_{n}\) and \(\eta_{m}\) represent the corresponding X-ray beam and sample modes weights individually according to specific experimental setup conditions and sample states respectively. \(n\) and \(m\) represent  
Figure 4: Specim reconstructions. (**a,b**) The phase and modulus of the specimen when 16 modes were engaged; (**c,d**) The phase and modulus reconstructed with a single mode. The main images are the magnification of the squared area by the red dashed line in the whole field of view inserted in the top corner. Figure adapted from Ref [14]. Reprint permission granted. 2016 American Physical Society. the number of modes in X-ray beam and sample respectively, and where \(*\) indicates complex conjugate operation, and \(N\) and \(M\) represent the total number of modes in X-ray beam and sample respectively. When the illumination demonstrates a statistically stationary mutual intensity function which depends only on the difference \(\mathbf{r_{1}}-\mathbf{r_{2}}\), the measured partially coherent intensity \(I_{pc}(\mathbf{q})\) adopts a particularly simple form:  
\[I_{pc}(\mathbf{q})=\mathcal{F}[J_{B}(\mathbf{r_{1}},\mathbf{r_{2}}).J_{S}( \mathbf{r_{1}},\mathbf{r_{2}})]=\mathcal{F}[\sum_{n=1}^{N}\sum_{m=1}^{M}\eta_ {n}B_{n}(\mathbf{r_{1}})B_{n}^{*}(\mathbf{r_{2}}).\eta_{m}S_{m}(\mathbf{r_{1} })S_{m}^{*}(\mathbf{r_{2}})] \tag{4}\]  
where \(\mathcal{F}\) denotes Fourier transform of the product of X-ray beam and sample states.