## 1 Introduction  
Phase imaging, which measures the optical path caused by light passing through an object, provides an effective method to obtain the surface topography at the nanoscale and is employed in numerous scientific areas, including X-ray crystallography [1], astronomy [2], cell morphology [3, 4], holography [5], and so forth [6]. However, current detection devices based on the photoelectric effect are too slow to follow the oscillations of the field at optical frequencies of \(\sim\)10\({}^{15}\) Hz and, therefore, cannot directly record the phase information but only the amplitudes. Nevertheless, the phase information can still be recorded through phase contrast image [7] and interference fringes [8] with the interference to a known light field or through its own diffraction patterns [9]. However, since only the amplitude images can be captured, the recovered phases always have multiple solutions preventing the quantization of the phase [10]. Such multiple-phase solutions from single-shot measurement include non-trivial and trivial ambiguities [6]. One common approach to removing these ambiguities relies on prior knowledge of the samples. These object-reliant priors, such as the \"tight\" support [11], sparsity [12, 13, 14], and non-negativity [15, 16], are widely used in coherent diffraction imaging (CDI) [9] with single-shot measurement and achieved the successful recovery. However, such strong assumptions often conflict with the optical properties of samples; for example, the \"tight\" support is hard to obtain for transparent bio cells. Other methods to guarantee the uniqueness of phase recovery always require redundant measurements. Representative techniques include a phase-shifting method in holography [17, 18, 19, 20] and phase contrast microscopy [21], multi-image measurement at different propagation planes in in-line holography [22], multi-overlapped illuminations in ptychography [23] and Fourier ptychographic microscopy (FPM) [24]. Such multiple measurements inevitably increase the measuring time. Moreover, the required electrical controllable elements or mechanical scanning parts make them sensitive to small vibrations during the acquisition process, causing signal fluctuation. In previous work, we reported a single exposure phase imaging method based on phase retrieval that simultaneously recorded the bandlimited image and its Fourier intensity (BIFT). By applying the intrinsic constraints of finite aperture in a band-limited imaging system, uniqueness was guaranteed without prior knowledge of the object [25; 26]. With the unique phase solution and stable BIFT algorithm, the phase can be recovered to the global minimum [26]. However, in the band-limited imaging system of BIFT, the finite aperture prevented high spatial frequency entering the BIFT system, and the more detailed structures of the samples under the diffraction limit could not be resolved. In the band-limited imaging system, there always exists a tradeoff between the Field of View (FOV) and the bandwidth of spatial frequency [27]. Given a specific objective lens, the product of the field of view and the bandwidth of spatial frequency is a constant known as the space-bandwidth product (SBP). Without sacrificing the FOV to expand the bandwidth of spatial frequency, current methods commonly rely on the techniques of synthetic aperture technique, which includes synthetic holography [28] or Fourier ptychographic microscopy (FPM) [24]. The key idea of the synthetic aperture technique involves using multiple mechanical scanning [29; 30], angle-varied illumination [24; 30], or different structured patterns [31; 32] to shift an object's high-frequency information to a low-frequency region where it can be collected by the band-limited imaging system. With the multiple-shot images and recombined spatial frequency with the higher spatial frequency width, the SBP can be increased, and the super-resolution phase image can be recovered.