## 3 Numerical Results and Discussions
In spite of this, the large twist factor will degrade the beam quality and even totally blur the image; one may choose a tradeoff twist factor for practical use where the image quality and visibility are acceptable. Figure 4: (**a**) Image of the double slits; (**b**) Changes of coherence width with the parameter \(\chi_{T}\); (**c**) Dependence of CC on the parameter \(\chi_{T}\); (**d**) Dependence of \(V\) on the parameter \(\chi_{T}\). \[\sigma_{gz}=\sigma_{g0}\!\left(\frac{1}{\Delta(z)}+\frac{2\sigma_{g0}^{2}\pi^{ 2}k^{2}Tz}{3}\!\left(1+\frac{2}{\Delta(z)}\right)-\frac{2\sigma_{g0}^{2}\pi^{4 }k^{2}Tz^{2}4}{18\Delta(z)\sigma_{I0}^{2}}\right)^{-1/2}, \tag{31}\]0 (**a**) Image of the double slits; (**b**) Change of coherence width with the parameter \(\varepsilon\); (**c**) Variation of the quality of the image (CC) with the parameter \(\varepsilon\); (**d**) Variation of the visibility of image (\(V\)) with the parameter \(\varepsilon\). \[\sigma_{gz}=\sigma_{g0}\!\left(\frac{1}{\Delta(z)}+\frac{2\sigma_{g0}^{2}\pi^{ 2}k^{2}Tz}{3}\!\left(1+\frac{2}{\Delta(z)}\right)-\frac{2\sigma_{g0}^{2}\pi^{4 }k^{2}Tz^{2}4}{18\Delta(z)\sigma_{I0}^{2}}\right)^{-1/2}, \tag{31}\]1 (**a**) Image of the double slits; (**b**) Changes of coherence width with parameter \(\omega\); (**c**) Variation of the quality of the image (CC) with the parameter \(\omega\); (**d**) Variation of the visibility of image (\(V\)) with the parameter \(\omega\). Finally, we should emphasize that the results including the image of double slits, image quality, and visibility obtained in this article correspond to those from the numerical simulation with random mode decomposition when the number of random modes (measurements) \(N\) is sufficiently large or infinity. In fact, there are two methods to calculate the results for the ghost image in numerical calculation/simulation. One directly uses the CSD function of the partially coherent beam (random beam) [24; 31; 32], and another is based on random mode decomposition of the beam [30; 47; 48]. The CSD function shown in Equation (1) can be considered as the results for the ensemble average over the random fluctuations. In the latter method, the random beam is usually expressed as  
\[\sigma_{gz}=\sigma_{g0}\!\left(\frac{1}{\Delta(z)}+\frac{2\sigma_{g0}^{2}\pi^{ 2}k^{2}Tz}{3}\!\left(1+\frac{2}{\Delta(z)}\right)-\frac{2\sigma_{g0}^{2}\pi^{4 }k^{2}Tz^{2}4}{18\Delta(z)\sigma_{I0}^{2}}\right)^{-1/2}, \tag{31}\]2  
where \(E_{n}\) represents the one realization of the random field. Under this scenario, the partially coherent beam is regarded as the \(N\) realizations of the incoherent superposition of the random fields, and the derived quantities depend on the number of realizations (measurements). Only when the number of realization is sufficiently large (approaches to infinity in theory), Equations (1) and (37) may establish the relation  
\[\sigma_{gz}=\sigma_{g0}\!\left(\frac{1}{\Delta(z)}+\frac{2\sigma_{g0}^{2}\pi^{ 2}k^{2}Tz}{3}\!\left(1+\frac{2}{\Delta(z)}\right)-\frac{2\sigma_{g0}^{2}\pi^{4 }k^{2}Tz^{2}4}{18\Delta(z)\sigma_{I0}^{2}}\right)^{-1/2}, \tag{31}\]3  
Therefore, all the results derived in our article are the cases of the number of measurement \(N\) being large enough (approaching infinity).