# Improvement of Fresnel Diffraction Convolution Algorithm
## 3 Experiments and Results  
To compare D-FFT, T-FFT, D-FFT (rect), and D-FFT (Tukey), [PERSON]'s double-hole diffraction issue was analyzed with the parameters listed in Table 1. The slit was used as  
Figure 1: (**A**) Tukey windows for different \(\beta\) values; (**B**) phase of \(\text{H}(\text{u},\text{v})\) before improvement; (**C**) phase of \(\text{H}(\text{u},\text{v})\) after improvement. the light source. The simulation results of the intensity profiles calculated with different algorithms are shown in Figure 2.  
As shown in Figure 2, the results when using the ideal low-pass filter window (Figure 2C) were much better than those before the improvement (Figure 2A). However, due to the ringing effect, there was still jumping at the edges of the picture and the peak of the diffraction fringes in Figure 2C. It is worth mentioning that the edges also sharply jumped when using the T-FFT algorithm, as shown in Figure 2B, which was inconsistent with the experimental observation. However, [PERSON]'s double-hole diffraction results when using D-FFT (Tukey) (Figure 2D) were in good agreement with the experimental results. There was neither sharp jumping at the edges nor absence of distortion at the peak of the diffraction fringes. According to the theory of [PERSON], the distribution of diffraction light field at a point on the observation surface is mainly derived from a limited-width (\(4\sqrt{\lambda\mathrm{d}}\)) area around the corresponding point on the diffraction surface [16]. That is to say, a spatial filter limiting is added to filter out some of the high-frequency components of the spectrum without increasing the sampling point. The amplitude of the D-FFT algorithm's transfer function is always equal to 1 while the amplitude of the high-frequency component of the T-FFT algorithm's transfer function is much lower than the amplitude of the low-frequency component [25]. That is why the T-FFT algorithm has better results than the D-FFT algorithm for \(\mathrm{d}\geq\mathrm{L_{o}}^{2}/(\lambda\mathrm{N})\). However, in practice, the amplitude and phase of the transfer function \(\mathrm{H}(\mathbf{u},\mathbf{v})\) used in the T-FFT algorithm cannot completely avoid the undersampling effect, so the errors of the T-FFT algorithm still cannot be ignored, especially when \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Variable** & **Value** \\ \hline Wavelength & \(\lambda\) & 0.589 \(\mathrm{\SIUnitSymbolMicro m}\) \\ Distances from object to light source & s & 100 \(\mathrm{mm}\) \\ Distances from object to observation & d & 500 \(\mathrm{mm}\) \\ Diameter of holes & D & 0.05 \(\mathrm{mm}\) \\ Distance of holes & a & 0.5 \(\mathrm{mm}\) \\ Illuminating slit width & b & \(s\)\(\lambda\)/2a \\ Number of sub-light sources & M & 41 \\ Size of hologram & \(\mathrm{L_{o}}\) & 10 \(\mathrm{mm}\) \\ Number of sampling points & N & 1024 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of [PERSON]’s double-hole diffraction. Figure 2: Intensity profiles through center of [PERSON]’s double–hole experimental fringes obtained by (**A**) D-FFT; (**B**) T-FFT; (**C**) D-FFT (rect); and (**D**) D-FFT (Tukey) (\(\beta=0.2\)). dealing with diffraction problems with lots of intricate structures, for example, microscopic digital holography. It should be noted that the more \(\beta\) increases, the more the high-frequency component of the transfer function is lost, and that not all values of \(\beta\) apply to this diffraction problem. In order to visualize the effect of different values \(\beta\) on the diffraction results of [PERSON]'s double-hole diffraction issue, we considered cases where \(\beta\) took other values, as shown in Figure 3.  
In Figures 2 and 3, it can be seen that when \(\beta\) is smaller than 0.2, the intensity profile of [PERSON]'s double-hole diffraction oscillates at the fringes, the ringing effect is not eliminated, and the fringes located at the edges of the picture are drowned in the noise.