## 2 Method  
### The Principle of BIFT Phase Imaging System  
The scheme of the BIFT imaging system is depicted in Figure 1. By adding a Fourier lens into the conventional bright-field microscopy, the bandlimited image and its Fourierwere recorded simultaneously. The laser illuminates the object with a finite wide field of view \(S(x,y)\). The reflected or transmitted light from the object, containing the direct beam and the scattered field, is gathered by an objective, imaged through the tube lens, and then split into two beams by a beam splitter (BS). One of the beams is recorded in the imaging plane as the image-space intensity. The other beam is Fourier transformed by a lens (I) and then recorded at the focal plane of I as the Fourier-space intensity. BIFT microscopy adds a Fourier lens and an array detector to conventional microscopy. When the object with the complexed-valued response function \(O(x_{0},y_{0})\) was imaged, the object field was projected to the image plane \(f(x,y)\) and can be given by [26]  
\[f(x,y)=\iint S(x_{0},y_{0})O(x_{0},y_{0})h(x-x_{0},y-y_{0})dx_{0}dy_{0}. \tag{1}\]  
Then, the field was further Fourier transformed to the Fourier plane by a Fourier lens; the field \(U(fx,fy)\) at the Fourier plane can be given by  
\[U(fx,fy)=\iint f(x,y)\exp[-i2\pi(xfx+yfy)]dxdy, \tag{2}\]  
where \((x,y)\) and \((f_{x},f_{y})\) represent the two-dimensional spatial and spatial frequency coordinates, \(S_{1}\) is the finite size of illumination, \(h(x,y)\) is the point spread function (PSF) in image space as \(h(x,y)=\iint P(f^{\prime}_{x},f^{\prime}_{y})\exp[i2\pi(xf^{\prime}_{x}+yf^{ \prime}_{y})]d^{\prime}_{x}df^{\prime}_{y}\), where \(P(f^{\prime}_{x},f^{\prime}_{y})\) is the pupil function of the objective (numerical aperture NA, magnification \(M\)) with cut-off spatial angular frequency. \(f_{\rm c}=\) NA\(/\lambda\). Thus, the corresponding bandlimited image \(I(x,y)=f(x,y)f^{*}(x,y)\) and the Fourier intensities \(F(f_{x},f_{y})=U(f_{x},f_{y})U^{*}(f_{x},f_{y})\) were recorded, where the * donates the complex conjugate operation. With the recorded bandlimited image and its Fourier intensity \(I(x,y)\) and \(F(f_{x},f_{y})\), the optical fields \(U(x,y)\) in image space and \(F(f_{x},f_{y})\) in Fourier space can be recovered with the BIFT algorithm [25]. The BIFT algorithm has been modified based on the HIO algorithm for two intensities measurements [44] by applying the intrinsic band limit constraint of finite aperture to remove the non-trivial ambiguities and combining the recorded intensities constraint to  
Figure 1: Optical scheme of BIFT phase imaging method. remove the trivial ambiguities [26]. The specific modifications in the \(j\)-th iteration are: (1) The estimated \(f_{j}(x,y)\) is imposed to the recorded image intensity, which is given by  
\[f_{j}(x,y)=\sqrt{I(x,y)}\cdot\frac{f_{j}\left(x,y\right)}{\left|f_{j}(x,y) \right|}, \tag{3}\]  
where \(f_{j}(x,y)\) is the inverse Fourier transform of \(U(f_{x},f_{y})\) in the \(j\)-th iteration. (2) The feedback of the estimated \(U_{j+1}(f_{x},f_{y})\) is added in Fourier space to strengthen the band limit constraint that  
\[U_{j+1}\left(f_{x},f_{y}\right)=\left\{ \begin{array}{l}U_{j}\left(f_{x},f_{y }\right)+\beta\cdot\Delta U_{d}\left(f_{x},f_{y}\right),\left(f_{x},f_{y} \right)\in P(f_{x},f_{y})\\ U_{j}\left(f_{x},f_{y}\right)+\beta\cdot U_{j}^{\prime}\left(f_{x},f_{y} \right),\left(f_{x},f_{y}\right)\
ot\in P(f_{x},f_{y})\end{array} \right., \tag{4}\]  
where \(U_{j}(f_{x},f_{y})\) and \(U_{j}^{\prime}(f_{x},f_{y})\) are the optical field of the input and output in Fourier space in the \(j\)-th iteration, the \(P(f_{x},f_{y})\) is the amplified pupil function at the recording Fourier plane, and the \(\Delta U_{d}(f_{x},f_{y})\) is given by  
\[