## 3 Simulation Result  
### The Performance of the Analytic Extrapolation in the Noise-Free Condition  
After the phase recovery by the BIFT algorithm, the optical field can be perfectly reconstructed. The recovered optical field in the Fourier plane can then be analytically extrapolated using the method described in Section 2.2. Given the range of the finite FOV and finite aperture, we calculated the oversampling ratio (OSR) for the optical field in the image plane and the Fourier plane, respectively. The OSR is defined as the practical sampling ratio relative to the Nyquist sampling ratio. To verify the performance of the analytic extrapolation super-resolution imaging, simulations were conducted. A United States Air Force (USAF) resolution target was simulated as the object function \(O(x_{0},y_{0})\) with the normalized absorbance 0 to 1 and the phase from \(-\pi/2\) to \(\pi/2\). Under the finite FOV and finite aperture constraints in the BIFT imaging system, the sampled amplitudes of the image plane and Fourier plane are shown in Figure 2.  
Firstly, we evaluated the effect of the extrapolation range ratio (ERR), defined as the ratio of the number of pixels of the extrapolated field to the number of pixels in the range of finite spatial frequency (\(ERR=M/N\)). To verify the final extrapolation error, an error function was calculated as follows:  
\[e_{rec}=\sqrt{\frac{\sum\limits_{n,m}\left|S(n,m)O(n,m)-f_{rec}(n,m)\right|^{2 }}{\left|S(n,m)O(n,m)\right|^{2}}}, \tag{13}\]  
where the \(f_{rec}(n,m)\) is the optical field in the image plane after the extrapolation and \(S(n,m)O(n,m)\) is the ground truth of the field under the constraints of finite FOV, which has a high resolution. For a comprehensive comparison, we also calculated the error function of the band-limited optical field as follows:  
\[e=\sqrt{\frac{\sum\limits_{n,m}\left|S(n,m)O(n,m)-f(n,m)\right|^{2}}{\left|S( n,m)O(n,m)\right|^{2}}}. \tag{14}\]  
We used Equations (13) and (14) to evaluate the relations between the ERR and the recovered \(e\), \(e_{rec}\) when the USAF target of Figure 2 was imaged. In the simulation, the OSR  
Figure 2: The simulated object and sampled amplitude in the BIFT system. (**a**) The simulated object (red circle stands for the FOV of the BIFT system). (**b**) The diffraction-limited image. (**c**) The finite spatial frequency (red circle stands for the cut-off frequency). of 8 in the image plane was used to guarantee the uniqueness for different extrapolation ranges, and the results were evaluated for different OSRs in the Fourier plane. As shown in Figure 3, with the extrapolation range becoming larger, \(e\), \(e_{rec}\) were both increasing. As the ERR of 2, the \(e_{rec}\) is 0.0157, while it increases to 0.2883 as the ERR increases to 4. This error increasing comes from the matrix theory that we described in Section 2.2. When \(M\leq 2N\), the Equation (12) will have the unique solution (if it is full rank). Consequently, the extrapolation error will be small. However, when \(M>2N\), the Equation (12) will have the multiple solutions, resulting in a significantly larger error. If we assume that an error of less than 0.1 is tolerable as a successful extrapolation, at this point, the extrapolation range is 2.5 times compared with the finite aperture range, which means that in the absence of noise, analytic extrapolation can achieve a 2.5-fold of SBP. This also implies that the resolution in the x and y directions will be improved by approximately \(\sqrt{2.5}\approx 1.58\) times. Apart from the impact of the extrapolation range on the accuracy of analytic extrapolation, the sampling rate in the Fourier plane also affects the reconstruction results. As shown in Figure 3, when the oversampling ratio in the Fourier plane is 1.25 and ERR = 2, the reconstruction error is 0.0488, which is higher than the reconstruction error at an OSR of 2.5. As the sampling rate further increases to 5, the reconstruction error becomes 0.00068.