# Incoherent Digital Holography: A Review
## 2 Principle of OSH
### Point Spread Function and Pupil Engineering  
In the last subsection, the general OSH, for which the interference fringe is formed by a spherical wave and a plane wave, is considered. The point spread function (PSF) of OSH, according to Equation (3), is thus  
\[PSF_{OSH}=\text{circ}\Big{(}\frac{r}{a}\Big{)}\exp\left(\frac{-jkr^{2}}{2z}\right) \tag{6}\]  
In reconstruction, the point image is calculated by Fresnel diffraction, that is,  
\[\frac{jke^{-jkz}}{2\pi z}\exp\left(\frac{-jkr^{2}}{2z}\right)\text{F}\bigg{\{} PSF_{OSH}^{*}\exp\left(\frac{-jkr^{2}}{2z}\right)\bigg{\}}_{k_{x}=\frac{kr}{z} +\beta_{y}=\frac{kr}{z}}=\frac{jke^{-jkz}}{2\pi z}\exp\left(\frac{-jkr^{2}}{2z} \right)2\frac{J_{1}(NAkr)}{NAkr} \tag{7}\]  
where \(J_{1}(\cdot)\) denotes a Bessel function of the first kind of order 1. Therefore, the Rayleigh resolution of the reconstructed image is \(1.22\pi/kNA\), being the same as that in a conventional coherent imaging system. It should be noted that both the coherent mode of OSH and incoherent mode of OSH have the same PSF, regardless of the difference between spatial distributions of \(T(x,y;z)\) and \(|T(x,y;z)|^{2}\). This is different from conventional imaging systems or conventional DH.  
Figure 3: Reconstructed image at (**a**) \(z_{r}=90\,\text{mm}\), and (**b**) \(z_{r}=115\,\text{mm}\). The larger figures show the whole field of view, while the small figures are the zoom-in of selected regions (Reproduced with permission from [57], OSA publishing, 2015).  
Figure 2: Hologram of two coins recorded by optical scanning holography (OSH): (**a**) real part, and (**b**) imaginary part. (Reproduced with permission from [57], OSA publishing, 2015).  
In OSH, the PSF as well as the resolution of the reconstructed image can be modified by using different types of the scanning beam. This can be easily realized by modifying the pupils of the system, such as \(p(x,y)\) shown in Figure 1. For example, [PERSON] et al. have proposed to use two spherical waves with opposite curvatures to generate the heterodyne scanning beam [23]. In this way, the fringe density of the scanning beam is double that of conventional OSH, and thus the resolution of the reconstructed image is only half that of conventional OSH. Similarly, resolution enhancement along \(z\) or the preprocessing of holographic information can be realized by using a spiral phase plate as the pupil [58; 59]. In another example, an asymmetry pupil can limit the vertical bandwidth of OSH but retain full horizontal bandwidth [60]. In this way, scanned holograms can be recorded using a fine horizontal sampling rate but a coarse vertical sampling rate without aliasing error. This kind of vertical-bandwidth-limited hologram preserves only the horizontal parallax, but requires a scanning time as well as a data size only about 1/10 those of the full-parallax scanned hologram. Finally, by using a random-phase pupil [19; 61], an axicon pupil [62] or a configurable pupil [63], the axial resolution of the OSH is enhanced and thus makes sectional reconstruction possible. In another example, an optical Hilbert transform can be realized by using a quarter-plane pupil in OSH [64].