## 3 Simulation Result
### The Performance of the Analytic Extrapolation in the Noise-Free Condition
Although this is lower than the error at an oversampling ratio of 2.5 (0.00157), the reduction in error is not significant. This is because satisfying the Shannon sampling theorem typically requires the signal to be sampled at more than twice its rate. When the oversampling ratio is less than 2, discrete sampling can cause spectral aliasing, leading to signal distortion, which prevents accurate representation of the analytic function. When the oversampling ratio is greater than 2, the Shannon interpolation theorem can accurately recover the analytic function, and thus, further increasing the sampling rate does not result in a significant improvement. To demonstrate the resolution improvement achieved by the analytic extrapolation method, we present the amplitude images of the diffraction-limited optical field, the analytically extrapolated field, and the ground truth shown in Figure 4. In Figure 4, Rows 1-3 correspond to ERR values of 2, 3, and 4, respectively. As shown in Figure 4, due to the cut-off effect of the finite aperture, the images in Figure 4(a1-c1) cannot display the details within the dashed line area. However, through analytic extrapolation, the structures within the dashed line area can be clearly resolved at different ERR values, as shown in Figure 4(a2-c2). As shown in Figure 4(a2,c2), at the ERR of 2, this area can be resolved and have the same value as the ground truth. At the ERR of 3, due to the range of the extension being larger, Figure 4(b2) has more detailed information in the rectangle area. In the red box, Figure 4(b2) clearly resolves the three lines in the resolution target, while Figure 4(a2)  
Figure 3: Illustration of the error \(e\) and \(e_{rec}\) with the function of EER at different OSRs of 1.27, 2.5, and 5 in the Fourier plane. only resolves two lines. As ERR increases further to 4, the three lines in the red box become blurred again. This is because, despite the increased extrapolation range, the reconstruction error also increases, leading to a reduction in resolution. In order to clearly see how the improved resolution related to ERR and error, Figure 5 plots the area of the dashed line in detail, and the area of the purple box stands for the highest resolution area of dash line area. In Figure 5(a1), the analytic extrapolation can perfectly recover the structure of the dash line area and successfully resolve the three peaks of the USAF target at the area of purple boxes. However, for the bandlimited BIFT imaging, only one peak in the purple box can be resolved by the imaging system. With the ERR increasing to 3, the recovery error becomes larger than the threshold of successful analytic extrapolation. Even though our method can still resolve every peak of the USAF targets, the three peaks can still be distinguished in the purple box. In the purple box of Figure 5(a3), our method can still resolve the three peaks of the target but have a small contrast compared with the ground truth. Similarly, the resolution improvement is also evident in the phase images in Figure 5(b1-b3), the phase image at ERR of diffraction-limited, analytic extrapolation, and ground truth are shown in Figure 5(b1-b3), respectively (the red box area corresponds to the dashed line region in Figure 4 at ERR of 2). The super-resolution phase can be perfectly recovered by analytic extrapolation comparing with ground truth and successfully resolved three peaks with the phase delay from \(-\pi\) to \(\pi\) compared with the traditional BIFT imaging method that can only distinguish one peak in Figure 5(b1). Figure 4: The super-resolution ability by analytic extrapolation in the BIFT method; (**a1–a3**) the diffraction-limited image, super-resolution image, and ground truth at ERR of 2; (**b1–b3**) the diffraction-limited image, super-resolution image, and ground truth at ERR of 3; (**c1–c3**) the diffraction-limited image, super-resolution image, and ground truth at ERR of 4.