## 4 Conclusions  
In summary, we have proposed a method for achieving single-shot super-resolution phase imaging through analytic extrapolation. This approach utilizes the analyticity of the Fourier transform within the finite field of view. By leveraging the characteristic that partial information of an analytic function can be used to infer the complete information, our method can extend the limited spatial bandwidth of the BIFT system. This method does not require any prior information about the object and achieves super-resolution phase reconstruction with a single exposure. The mathematical theory and numerical simulations were carried out to investigate the relationship between the construction error of the light field after analytic extrapolation and three critical factors: extrapolation range, oversampling rate, and signal-to-noise ratio (SNR). Firstly, in the absence of noise, effective extrapolation requires the oversampling rate in the Fourier plane to be greater than 2. Secondly, the construction error is linearly increased with the increase of the extrapolation range. When the error is less than 1/10, the analytic extrapolation method can increase the space-bandwidth product (SBP) by about 2.5 times, which means the resolution will improve by approximately 1.58 times. Additionally, the SNR is the most crucial factor affecting the success of analytic extrapolation. The small eigenvalues can amplify noise during the extrapolation process, leading to the failure of extrapolation. To mitigate noise effects, a regularization method was employed. Using this method, we obtained a linear relationship between the extrapolation range and SNR. As SNR varies from 0 to 80 dB, the SBP can be extended by 1.3 to 1.8 times, corresponding to a resolution improvement of 1.15 to 1.34 times. Our BIFT phase imaging method is fundamentally different from previous single-shot methods of using coherent diffraction imaging [9]. The key constraints of our method are the finite aperture and finite FOV, which exist in common imaging systems such as microscopy, photography, remote sensing, astronomical telescopes, etc. Therefore, our proposed super-resolution method can have a wide range of applications. Our previous work showed that the BIFT dual-plane measurement can guarantee the uniqueness of phase retrieval, and the finite aperture provides a fixed, tight support for the stable recovery algorithm to converge to a global minimum, avoiding the need for prior knowledge of the object. Compared with the traditional interferential phase imaging methods, such as off-axis holography, phase contrast microscopy, and differential interference contrast microscopy, the non-interference nature of the BIFT method allows the imaging system to be more stable from environmental interference. In addition, for the spatially separating with conjugate inversion ambiguity to achieve the single-shot, the SBP of the off-axis holography is severely reduced compared with the in-line system of BIFT. Beyond this, in this paper, the SBP can be enhanced further by analytic extrapolation. Although the 1.58-fold resolution enhancement achieved by analytic extrapolation currently falls short compared to the 2-fold enhancement achieved by synthetic aperture techniques, we believe this method still holds significant potential. The key to improving analytic extrapolation capability is to enhance the distinguishability of the small eigenvalues of matrix \(A^{\prime}\). Currently, the overly small eigenvalues indicate that the high-order eigenmodes are severely attenuated. However, there still exist practical methods to allow the high-order eigenmodes to be detected with high SNR, which include the use of structured illumination, random modulation, speckle illumination and specific coding illumination. Additionally, to improve the SNR, partially coherent light illumination could be used to reduce speckle noise from coherent beams.